United States Environmental Protection Agency Robert S. Kerr Environmental Research Laboratory Ada OK 74820 Center for Environmental Research Information Cincinnati OH 45268 Technology Transfer CERI-87-28 Seminar on Transport and Fate of Contaminants in the Subsurface Background Information DATE DUE EPA SEMINAR ON TRANSPORT CERI AND FATE OF 87-28 CONTAMINANTS IN THE 98110903 SUBSURFACE: BACKGROUND INFORMATION DEMCO 1N0I3 mi WATER i®£Y miiASY m NO 1 n ’c? u CONTENTS I. BACKGROUND INFORMATION ON GROUND-WATER MODELING Ground-Water Modeling: An Overview Ground-Water Modeling: Mathematical Models Ground-Water Modeling: Numerical Models Optimizing Pumping Strategies for Contaminant Studies and Remedial Actions II. QUALITY ASSURANCE IN GROUND-WATER MODELING Quality Assurance in Computer Simulations of Ground-Water Contamination Representation of Individual Wells in Two Dimensional Ground- Water Modeling Remedial Actions Under Variability of Hydraulic Conductivity A New Annotation Database for Ground-Water Models III. AVAILABILITY AND DOCUMENTATION OF MODELS Technology Transfer in Ground-Water Modeling: The Role of the International Ground-Water Modeling Center (IGWMC) Available Solute Transport Models for Ground Water and Soil Water Quality Management The Fundamentals of Geochemical Equilibrium Models with a Listing of Hydrochemical Models that are Documented and Available Price list of Publications and Services Available from IGWMC IV. U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP Report of Findings and Discussions of Selected Ground-Water Modeling Issues V. THE USE OF MODELS IN MANAGING GROUND-WATER PROTECTION PROGRAMS Digitized by the Internet Archive in 2020 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/seminarontranspoOOunse Ground-Water Modeling: An Overview by James W. Mercer and Charles R. Faust b ABSTRACT Ground-water modeling is a tool that can help analyze many ground-water problems. Models are useful for reconnaissance studies preceding field investigations, for interpretive studies following the field program, and for predictive studies to estimate future field behavior. In addition to these applications, models are useful for studying various types of flow behavior by examining hypothetical aquifer problems. Before attempting such studies, however, one must be familiar with ground-water modeling concepts, model usage, and modeling limitations. INTRODUCTION The use of aquifers is increasing as both a source of water supply and a medium for storing various hazardous wastes. As this usage expands, our knowledge of ground-water systems must also expand. Numerical ground-water modeling is a tool that can aid in studying ground-water problems and can help increase our understanding of ground-water systems. Numerical models have been extensively used for ground-water analysis since the mid-1960s, yet confusion and misunderstanding over their application still exists. As a result, some hydrologists have become disillusioned and have overreacted, concluding that models are worthless. At the other extreme are those who have been willing to accept any model results, regardless of whether or not they make hydrologic sense. 3 . • • This is the first in a series of papers on ground-water modeling. ^GeoTrans, Inc., P.O. Box 2550, Reston, Virginia 22090. Discussion open until September 1, 1980. The purposes of this series of papers are to introduce the basic concepts of ground-water modeling and to show how the various types of models can be effectively applied. We discuss ground-water flow modeling as well as transport modeling of both heat and hazardous wastes. Most of this material is available in the literature (e.g., van Poolen, et al., 1969; and Coats, 1969), but it is usually directed toward an audience assumed to be familiar with petroleum terminology or the mathematical terminology of numerical analysis. Our intent is to consolidate this material into a form that is meaningful to the experienced, though not mathematical, ground-water hydrologist. In so doing, we hope to eliminate some of the misunderstanding associated with ground-water modeling. MODELING APPROACHES Simulation of a ground-water system refers to the construction and operation of a model whose behavior assumes the appearance of the actual aquifer behavior. The model can be physical (for example, a laboratory sandpack), electrical analog, or mathematical. Other model divisions may be found in Karplus (1976) and Thomas (1973). A mathematical model is simply a set of equations which, subject to certain assumptions, describes the physical processes active in the aquifer. While the model itself obviously lacks the detailed reality of the ground-water system, the behavior of a valid model approximates that of the aquifer. Mathematical models may be deterministic, statistical, or some combination of the two. In this discussion, we restrict ourselves to deterministic Fig. 1. Logic diagram for developing a mathematical model. models, that is, those that define cause and effect relationships based on an understanding of the physical system. The procedure for developing a deterministic, mathematical model of any physical system can be generalized as shown in Figure 1. The first step is to understand the physical behavior of the system. Cause-effect relationships are determined and a conceptual model of how the system operates is formulated. For ground-water flow, these relationships are generally well known, and are expressed using concepts such as hydraulic gradient to indicate flow direction. For the movement of hazardous wastes, these relationships, especially those involving physical-chemical behavior, are only partially understood. The next step is to translate the physics into mathematical terms, that is, make appropriate simplifying assumptions and develop the governing equations. This constitutes the mathematical model. The mathematical model for ground-water flow consists of a partial differential equation together with appropriate boundary and initial conditions that express conservation of mass and that describe continuous variables (for example, hydraulic head) over the region of interest. In addition, it entails various phenomenological “laws” describing the rate processes active in the aquifer. An example is Darcy’s law for fluid flow through porous media; this is generally used to express conservation of momentum. Finally, various assumptions may be invoked such as those of one- or two-dimensional flow and artesian or water-table conditions. For solute (e.g., hazardous wastes) and heat transport, additional partial differential equations with appropriate boundary and initial conditions are required to express conservation of mass for the chemical species considered, and conservation of energy, respectively. Examples of corresponding phenomenological relationships are Fick’s law for chemical diffusion and Fourier’s law for heat conduction. Once the mathematical model is formulated, the next step is to obtain a solution using one of two general approaches. The ground-water flow equation can be simplified further, for example, assuming radial flow and infinite aquifer extent, to form a subset of the general equation that is amenable to analytical solution. The equations and solutions of this subset are referred to as analytical models. The familiar Theis type curve represents the solution of one such analytical model. Alternatively, for problems where the simplified analytical models no longer describe the physics of the situation, the partial differential equations can be approximated numerically, for example, with finite-difference techniques or with the finite-element method. In so doing, one replaces continuous variables with discrete variables that are defined at grid blocks (or nodes). Thus, the continuous differential equation, defining hydraulic head everywhere in an aquifer, is replaced by a finite number of algebraic equations that defines hydraulic head at specific points. This system of algebraic equations is generally solved using matrix techniques. This approach constitutes a numerical model, and generally, a computer program is written to solve the equations on a digital computer. Probably the most frequent application of ground-water models is that of history matching and prediction of site-specific aquifer behavior. Of the various types of models discussed, the numerical model offers the most general tool for simulating aquifer behavior. Physical models usually offer the most intuitive insight into aquifer behavior, but are limited in application (once constructed), and have the difficulty of scaling results to field level. Electric analog models can be applied to field problems, but are usually site-specific and expensive to construct. Deter¬ ministic mathematical models (both analytical and numerical) retain a good measure of physical insight while permitting a larger class of problems to be considered with the same model. Analytical methods, such as type curve analysis, are relatively easy to use. Numerical models, although more difficult to apply, are not limited by many of the simplifying assumptions necessary for the analytical methods. Finally, purely statistical methods are useful in classifying data and describing poorly understood systems, but generally offer little physical insight. Each type of model has both advantages and disadvantages. Consequently, no single approach should be considered superior to others for all applications. The selection of a particular approach should be based on the specific aquifer problem addressed. Whichever approach is taken, the final step in modeling a ground-water flow system is to translate the (mathematical) results back to their physical meanings. In addition, these results must be interpreted in terms of both their agreement with reality and their effectiveness in answering the hydrologic questions that motivated the model study. TYPES OF GROUND-WATER MODELS Four general types of ground-water models are listed in Figure 2. The problem of water supply is normally described by one equation, usually in terms of hydraulic head. The resulting model providing a solution for this equation is referred to as a ground-water flow model. If the problem involves water quality, then an additional equation (s) to the ground-water flow equation must be solved for concentration (s) of the chemical specie (s). Such a model is referred to as a solute transport model. Problems involving heat also require an equation in addition to the ground-water flow equation, similar to the solute transport equation, but now in terms of temperature. This model is referred to as a heat transport model. Finally, a deformation model combines a ground- water flow model with a set of equations that describe aquifer deformation. Ground-water flow models have been most extensively used for such problems as regional aquifer studies, ground-water basin analysis and near-well performance. More recently, solute transport models have been used to aid in understanding and predicting the effects of problems Applications WATER SUPPLY SEA WATER INTRUSION GEOTHERMAL LAND SUBSIDENCE REGIONAL AQUIFER LAND FILLS THERMAL STORAGE ANALYSES NEAR WELL WASTE INJECTION HEAT PUMP PERFORMANCE GROUND-WATER/ RADIOACTIVE THERMAL POLLUTION SURFACE WATER WASTE STORAGE INTERACTIONS DEWATERING • HOLDING PONDS OPERATIONS GROUND WATER POLLUTION Fig. 2. Types of ground-water models and typical applications. involving hazardous wastes. Some of the applica¬ tions include: sea-water intrusion, underground storage of radioactive wastes, movement of leachate from sanitary landfills, ground-water contamination from holding ponds, and waste injection through deep wells. Heat transport models have been applied to problems concerning geothermal energy, heat storage in aquifers, and thermal problems associated with high-level radioactive waste storage. Deformation models have been used to examine field problems where fluid withdrawal has decreased pressures and caused consolidation. This compaction of sediments results in subsidence at the land surface. This classification of ground-water models is by no means complete. All of the above models can be further subdivided into those describing porous media and those describing fractured media. Ground-water models can be combined with statistical techniques in an effort to characterize uncertainty in model parameters. These models can also be used to estimate aquifer parameters. In addition, there are other models that deal with multifluid flow (e.g., oil and water) and multiphase flow (e.g., unsaturated zone problems). Some resource management models combine flow models and linear programs, which are used to optimize certain decision parameters, like pumping rates. Other models combine some or all of the models in Figure 2. For example, a thermal loading problem may require that a heat transport model be combined with a deformation model. The type of model used will obviously depend on the application. For further information on the various models and their availability, the interested reader is referred to Bachmat, et al. (1978), and Appel and Bredehoeft (1976). A numerical model is most appropriate for general problems involving aquifers having irregular boundaries, heterogeneities, or highly variable pumping and recharge rates. The remaining sections are therefore generally concerned with numerical ground-water models. We discuss the use of the numerical modeling approach with respect to the various types of ground-water models, giving the most emphasis to ground-water flow models and the least emphasis to deformation models. MODEL USE Because the number of ground-water models available today is large, when beginning a study the first question that may come to mind is, “Which one should I use?” Actually, the first question one asks should be, “Do I need a numerical model study for this problem?” The answers to both of these questions can be determined by first considering the following: (1) What are the study objectives? (2) How much is known about the aquifer system; that is, what data are available? and (3) Does the study include plans to obtain additional data? The study objectives may be such that a numerical model is unnecessary. Or, if necessary, objectives may require only a very simple model. Additionally, lack of data may not justify a sophisticated model; however, if a field study is in its initial stages, the ideal approach is to integrate the data collection and analysis with a model study. Once it is decided that a model is necessary, the one used will, in part, depend on the study objectives. For example, if one is interested in the drawdowns near a well, then a regional model, where these local effects are lost due to the large spacing between nodes, should not be used. Instead, perhaps a radial flow model with small grid spacing would be sufficient. The application of a ground-water model to an aquifer involves several areas of effort. These are shown in Figure 3 and include: data collection, data preparation for the model, history matching, and predictive simulation. These tasks should not be considered separate steps of a chronological Fig. 3. Diagram showing model use. procedure; rather, they should be considered as a feedback approach. It is best to use the model not only as a predictive tool, but also as an aid in conceptualizing the aquifer behavior. For example, a model used in the early stages of a field study can help in determining which and how much data should be collected. Data preparation for the ground-water model first involves determining the boundaries of the region to be modeled. The boundaries may be physical (impermeable or no flow, recharge or specified flux, and constant head) or merely convenient (small subregion of a large aquifer). Once the boundaries of the aquifer are determined it is necessary to discretize the region, that is, subdivide it into a grid. Depending on the numerical procedure used, the grid may have rectangular or irregular polygonal subdivisions. Figure 4 shows typical two-dimensional gridding for both the finite-difference and finite-element methods. Once the grid is designed, it is necessary to specify aquifer parameters and initial data for the grid. For descriptive purposes, the following discussion refers to the finite-difference method, utilizing a rectangular grid. Required program input data include aquifer properties for each grid block, such as storage coefficients and transmissivities (see Table 1). For solute transport (that is, programs used for tracking hazardous wastes) and heat transport, additional data are required, such as hydrodynamic dispersion properties and thermal conductivity, respectively. Computed results generally consist of hydraulic heads at each of the grid blocks throughout the aquifer. These spatial distributions of hydraulic head are determined at each of a sequence of time levels covering the period of interest. For transport problems, computed results might also include concentrations and temperatures at each of the grid blocks. Initial estimates of aquifer parameters constitute the first step in a trial-and-error procedure known as history matching. The matching procedure (often referred to as model calibration) is used to refine initial estimates of aquifer properties and to determine boundaries (that is, the areal and vertical extent of the aquifer) and the flow conditions at the boundaries (boundary conditions); aquifer tests generally provide the initial estimates for storage coefficients and transmissivities. For certain ground-water problems, steady-state (or equilibrium) heads must also be determined and used as initial or beginning conditions. Simulated wells in the aquifer grid system are then allowed to pump at the observed rates, and computed (simulated) drawdowns are compared with observed drawdowns. Assuming that the model is correct, comparison between these two indicates the accuracy of the initial estimates of input data. It may be necessary to modify some of the input data until all observed and calculated data compare sufficiently well. In the past this has been done by trial and error; more Fig. 4a. Map view of aquifer showing well field and boundaries. Fig. 4b. Finite-difference grid for aquifer study, where Ax is the spacing in the x-direction. Ay is the spacing in the y- direction and b is the aquifer thickness. Fig. 4c. Finite-element configuration for aquifer study where b is the aquifer thickness. Table 1. Data Requirements to be Considered for a Predictive Model (after Moore, 1979) I. Physical Framework A. Ground-Water Flow 1. Hydrogeologic map showing areal extent, bounda¬ ries, and boundary conditions of all aquifers. 2. Topographic map showing surface-water bodies. 3. Water-table, bedrock-configuration, and saturated¬ thickness maps. 4. Transmissivity map showing aquifer and boundaries. 5. Transmissivity and specific storage map of confining bed. 6. Map showing variation in storage coefficient of aquifer. 7. Relation of saturated thickness to transmissivity. 8. Relation of stream and aquifer (hydraulic connection). B. Solute Transport (in addition to above) 9 . Estimates of the parameters that comprise hydro- dynamic dispersion. 10. Effective porosity distribution. 11. Background information on natural concentration distribution (water quality) in aquifer. 12. Estimates of fluid density variations and relation¬ ship of density to concentration. 13. Hydraulic head distributions (used to determine ground-water velocities). 14. Boundary conditions for concentrations. C. Heat Transport (in addition to above) 15. Estimates of thermal conductivities and specific heats of rock and water. 16. Background information on natural temperature distribution in aquifer, including heat flow measurements. 17. Estimates of fluid density variations and relation¬ ships of density and viscosity to temperature. 18. Boundary conditions for temperature. II. Stresses on System A. Ground-Water Flow 1. Type and extent of recharge areas (irrigated areas, recharge basins, recharge wells, etc.). 2. Surface-water diversions. 3. Ground-water pumpage (distributed in time and space). 4. Stream flow (distributed in time and space). 5. Precipitation. B. Solute Transport (in addition to above) 6. Areal and temporal distribution of water quality in aquifer. 7. Stream flow quality (distribution in time and space). 8. Sources and strengths of pollution. C. Heat Transport (in addition to above) 9 . Areal and temporal distribution of temperature in aquifer. 10. Strengths of heat sources. III. Other Factors A. Ground-Water Flow and Transport 1. Economic information of water supply. 2. Legal and administrative rules. 3. Environmental factors. 4. Planned changes in water and land use. recently the amount of work in matching has been reduced somewhat by using parameter estimation methods that modify initial estimates of input data in a more objective fashion. No hard and fast rules exist to indicate when a satisfactory match is obtained. The number of “runs” required to produce a satisfactory match depends on the objectives of the analysis, the complexity of the flow system and length of observed history, as well as the patience of the hydrologist. Once completed, the model can be used to predict the future behavior of the aquifer. Of course, confidence in any predictive results must be based on (1) a thorough understanding of model limitations, (2) the accuracy of the match with observed historical behavior, and (3) knowledge of data reliability and aquifer characteristics. The main purpose of prediction is estimation of aquifer performance under a variety of pumping schemes. While the aquifer can be developed only once at considerable expense, a model can be “pumped” or run many times at low expense over a short period of time. Observation of model performance under differing development schemes then aids in selecting an optimum set of operating conditions for utilizing the ground- water resource. More specifically, ground-water modeling allows estimates of: (a) recharge (both natural and induced) due to leakage from confining beds, (b) effects of boundaries and boundary conditions, (c) the effects of well locations and spacing, and (d) the effects of various withdrawal (or injection) rates. Other purposes for prediction include estimating the rates of movement of hazardous wastes from sanitary landfills and other containment areas. Models are used to predict the encroachment rate of salt water in coastal regions due to fresh¬ water withdrawal. They are also used to help determine what, if any, remedial action is best to take in a contamination situation. Finally, heat transport models are used to help predict the behavior of geothermal reservoirs and aquifers used for thermal storage. In addition to these site-specific applications, models are also used to examine general problems. Hypothetical (but typical) aquifer problems may be designed to study various types of flow behavior, such as ground-water/surface-water interactions or flow around a deep radioactive waste repository. The feasibility of certain proposed mechanisms for observed behavior can be tested. Parameters may be changed to learn what effect they may have on the over-all process. This is sometimes referred to as a sensitivity analysis, since results from these runs will indicate what parameters the computed hydraulic heads are most sensitive to. Sensitivity analysis is also useful for site-specific applications to indicate what additional data need to be determined and areas where additional data are needed. MODEL MISUSE There are a variety of ways to misuse models (Prickett, 1979). Three common and related ones are: overkill, inappropriate prediction, and misinterpretation. The temptation to apply the most sophisticated computational tool to a problem is difficult to resist. A question that often arises is, under what circumstances should simulation be three-dimensional as opposed to two- or even one¬ dimensional. Inclusion of flow in the third (nearly vertical) direction is often recommended only if aquifer thickness is “large” in relation to areal extent or if pronounced heterogeneity exists in the vertical direction (for example, high stratifica¬ tion). Another type of overkill is that grid sizes are used which are finer (smaller) than necessary considering available information about aquifer properties, resulting in additional work and expense. In some applications, complex models are used too early in the study. For example, for a hazardous waste problem, one generally should not begin with a solute transport model. Rather, the first step is to be sure the ground-water hydrology (velocity in particular) can be characterized satisfactorily, and therefore one begins by modeling ground-water flow alone. Once this is done to satisfaction, then solute transport can be included. One must assess the complexities of the problem, the amount of data that is available, and the objectives of the analysis, and then determine the best approach for the particular situation. A general rule might be to start with the simplest model and a coarse aquifer description and refine the model and data until the desired estimation of aquifer performance is obtained. One must always be aware that the history- match portion of the simulation occurred under a given set of field conditions, and that these conditions are subject to change during the prediction portion. For example, during the history- match portion, the aquifer may be confined, but be on the verge of becoming desaturated. Using a confined model for prediction will give erroneous results since the saturated thickness and storage coefficient will be incorrect. Because ground- water models deal with the subsurface, there are always unknown factors that could affect results. In general, one should not predict more than about twice the period used for matching, and only then under similar pumping schemes. Perhaps the worst possible misuse of a model is blind faith in model results. Calculations that contradict normal hydrologic intuition almost always are the result of some data entry mistake, a “bug” in the computer program, or misapplication of the model to a problem for which it was not designed. Proper application of a ground-water model requires an understanding of the specific aquifer. Without this conceptual understanding the whole exercise may become a meaningless waste of time and money. LIMITATIONS AND SOURCES OF ERROR IN MODELING In order to avoid model misuse, it is important to know and understand the limitations and possible sources of error in numerical models. All numerical models are based on a set of simplifying assump¬ tions, which limit their use for certain problems. To avoid applying an otherwise valid model to an inappropriate field situation, it is not only important to understand the field behavior but also to under¬ stand all of the assumptions that form the basis of the model. An areal (two-dimensional) model, for example, should be applied with care to a three- dimensional problem involving a series of aquifers, hydrologically connected by confining beds, since the model results may not be indicative of the field’s behavior. Errors of this type are considered conceptual errors. In addition to these limitations, there are several potential sources of error in the numerical model results. First, replacement of the model differential equations by a set of algebraic equations introduces truncation error; that is, the exact solution of the algebraic equations differs somewhat from the solution of the original differential equations. Second, the exact solution of the algebraic equations is not obtained due to round-off error, as a result of the finite accuracy of computer calculations. Finally, and perhaps most importantly, aquifer description data (for example, transmissivities, storage coefficients, and the distribution of heads within the aquifer) are seldom known accurately or completely, thus producing data error. The level of truncation error in computed results may be estimated by repeating runs or portions of runs with smaller space and/or time increments. Significant sensitivity of computed results to changes in these increment sizes indicates a significant level of truncation error and the corresponding need for smaller spatial and/or time increments. Compared to the other error sources, round-off error is generally negligible. Error caused by erroneous aquifer description data is difficult to assess since the true aquifer description is never known. An adage used to describe the error associated with this data is, “Garbage in, garbage out.” A combination of core analysis, aquifer tests, and geological studies often give valuable insight into the nature of transmissivity, storage coefficients, and aquifer geometry. However, much of this information may be very local in extent and should be regarded carefully when used in a model of a large area. As discussed, the final parameters that characterize the aquifer are usually determined by obtaining the best agreement between calculated and observed aquifer behavior during some historical period. SUMMARY Numerical ground-water models are an important tool for the ground-water hydrologist. They can be used to simulate the behavior of complex aquifers including the effects of irregular boundaries, heterogeneity, and different processes such as ground-water flow, solute transport and heat transport. The use of numerical models involves data collection, data preparation, history matching, and prediction. The process of constructing a model for an aquifer study forces one to develop a conceptual understanding of how the aquifer behaves. Models, therefore, can be used in all phases of the aquifer study including conceptualization and data collection, as well as prediction. To be most effective, the hydrologist must have a thorough understanding of the specific aquifer studied, must be familiar with alternative modeling techniques, and must realize the limita¬ tions and sources of error in models. Upon meeting these criteria, a successful model study will not only improve one’s understanding of the particular hydrologic system, but should also provide appropriate prediction and analysis of the problem under study. ACKNOWLEDGMENTS This effort was supported by the Holcomb Research Institute, Butler University, which is, in part, supported by EPA grant number R-803713. The authors also wish to acknowledge the National Water Well Association’s contribution for drafting, editing and publication. Finally, many of the ideas presented in this series of papers were formulated during the authors’ participation in training courses for the U.S. Geological Survey. REFERENCES Appel, C. A., and J. D. Bredehoeft. 1976. Status of ground- water modeling in the U.S. Geological Survey. U.S. Geological Survey Circular 737. 9 pp. Bachmat, Y., B. Andrews, D. Holtz, and S. Sebastian. 1978. Utilization of numerical groundwater models for water resource management. U.S. Environ. Prot. Agency Report EPA-600/8-78-012. 178 pp. Coats, K. H. 1969. Use and misuse of reservoir simulation models. J. Pet. Tech. (Nov.), pp. 1391-1 398. Karplus, W. J. 1976. The future of mathematical models of water resource systems. In System simulation in water resources (G. C. Vansteenkiste, ed.). North-Holland Publishing Co. pp. 11-18. Moore, J. E. 1979. Contributional ground-water modeling to planning. Journ. of Hydrology, v. 43 (Oct.), pp. 121-128. Prickett, T. A. 1979. Ground-water computer models — state of the art. Ground Water, v. 17, no. 2, pp. 167-173. Thomas, R. G. 1973. Groundwater models. Irrigation and Drainage Paper 21, Food and Agriculture Organization of the United Nations, Rome. 192 pp. van Poollen, H. K., H. C. Bixel, and J. R. Jargon. 1969. Reservoir modeling — 1: What it is, what it does. Oil and Gas Jour. (July), pp. 158-160. ♦ * * * James W. Mercer attended Florida State University and University of Illinois, where he received bis Ph.D. in Geology in 1973. Prior to graduation, he worked for Exxon Oil Corporation and the Desert Research Institute, University of Nevada. Most recently, he worked for the U.S. Geological Survey, and is currently President of GeoTrans, Inc. Dr. Mercer has taught ground-water modeling at the U.S.G.S. and at George Washington University. He has also coauthored several articles on modeling transport problems in ground water. Charles R. Faust received his B.S. and Ph.D. in Geology from Pennsylvania State University in 1967 and 1976. Until recently, he was a hydrologist with the Water Resources Division, U.S. Geological Survey. His interests there involved thermal pollution in rivers, geothermal reservoir simulation, and fluid flow in fractured rocks. Presently he is a principal with GeoTrans, Inc., where he is interested in quantitative evaluation of hazardous waste and radionuclide migration in fractured rocks. Ground-Water Modeling: Mathematical Models' by James W. Mercer and Charles R. Faust b ABSTRACT Ground-water modeling begins with a conceptual understanding of the physical problem. The next step in modeling is translating the physical system into mathe¬ matical terms. In general, the final results are the familiar ground-water flow equation and transport equations. These equations, however, are often simplified, using site-specific assumptions, to form a variety of equation subsets. An understanding of these equations and their associated boundary and initial conditions is necessary before a modeling problem can be formulated. INTRODUCTION The variety and complexity of mathematical models (usually partial differential equations) used in ground-water applications have increased dramatically during the last twenty years. This increase has been made possible by significant advances in digital computers, but has also been enhanced by environmental and energy concerns. This proliferation of mathematical models is often bewildering to the hydrologist who is trying to keep up with the research literature. Although the number of model types is large, only a few basic processes are considered. This is somewhat a This is the second in a series of papers on ground- water modeling. t>GeoTrans, Inc., P.O. Box 2550, Reston, Virginia 22090. Discussion open until November 1, 1980. ironic since the large number of “different” models is the result of various simplifying assumptions used to reduce a general set of equations to some solvable form. Fortunately, if one keeps in mind the fundamental processes being simulated, then different or simplified forms of equations are less confusing. The purposes of this paper are: (1) to review the basic processes of interest in ground-water applications, (2) to show how mathematical models are developed, and (3) to discuss some of the more commonly used equations. In order to apply mathematical models effectively, it is necessary to develop an intuitive knowledge of both the physical process and the corresponding mathematical model. Unfortunately the mathe¬ matics are often overwhelming. With this in mind, a review of the mathematical aspects is given in the appendices, whereas the main part of this paper emphasizes the physical processes. The appendices include a review of derivatives, alternative notation, coordinate systems, viewpoints, and a simplified derivation of the ground-water flow equation. BASIC PROCESSES The major processes that may be considered part of many ground-water problems are fluid flow, solute transport, heat transport and deformation. Even though only four general processes have been identified, the number of different models that can be conceived is very large. The reason for this is that many of the model variations are application 212 Vol. 18, No. 3-GROUND WATER-May-June 1980 dependent. For example, flow in fractured media behaves differently from flow in porous media, and is, hence, described by different equations. In addition to application-dependent variations, for convenience, equations describing the same process are often posed in terms of different dependent variables. Hydraulic head is usually used as the dependent variable (also called unknown or state variable) in the ground-water flow equation, but drawdown or fluid pressure is sometimes used. Table 1 provides a summary of the major processes, dependent variables, and application-dependent variations that are commonly encountered in ground-water applications. Rather than discuss each of the possible models that may be useful, attention in this report will focus on a few mathematical models that deal with single-phase flow in porous media. The basic processes that are considered include ground-water flow, solute transport, and heat transport. Ground-water flow is a process that can be, and usually is, modeled without considera¬ tion of heat or solute transport. Modeled by itself, it has important applications in ground-water supply and design of engineering structures, such as dams, mines, and excavations, that interact with the ground-water system. Both solute and heat transport require either simultaneous solution with or results (e.g., velocities) from a ground-water flow model. This is because the movement (transport) of solutes or heat is controlled partially by the ground-water movement. Solute transport models are used for a wide variety of ground-water quality problems, such as point source pollution (e.g., waste disposal wells), spread source pollution (e.g., landfills) or sea-water intrusion. Heat transport models are applied to problems involving thermal storage in aquifers, geothermal reservoir engineer¬ ing, and usage of ground-water heat pumps. MATHEMATICAL MODEL DEVELOPMENT The development of a mathematical model begins with a conceptual understanding of the physical system. Once these concepts are formulated they can be translated into a mathe¬ matical framework resulting in equations that describe the process. A variety of analytical and numerical techniques can be applied to solve the equations, resulting in practical tools (often referred to as models) such as type curves or finite-difference and finite-element computer programs. In this section we look at the procedure for translating ground-water concepts into mathe- Table 1. Major Processes with Dependent Variables and Application Variations Major Process Dependent Variable Application Dependent Variations Fluid Flow fluid pressure, Porous Media hydraulic head, Doubly Porous Media hydraulic potential Discrete Fractured Media or drawdown Single-phase Fluid Two-phase Fluid Heat Transport temperature, Same as for Flow plus enthalpy or Convection internal energy Conduction Radiation Solute Transport concentration Same as for Flow plus Convection Dispersion Chemical Source Terms Nuclide Decay & Daughter Products Adsorption & Desorption Precipitation & Dissolution Complexing Deformation strain or Elastic Media strain rate Plastic Media Visco — Elastic Media Visco — Plastic Media Discrete Fractured Media 213 matical terms, assuming that the reader already has a good conceptual understanding of ground-water systems and associated processes. Rigorous development of specific equations used in ground- water applications may be found in textbooks (e.g., Bear, 1972). The emphasis here is on (1) a review of basic considerations and assumptions, (2) the procedure for obtaining the final equations, and (3) a discussion of the equations in physical terms. As part of this discussion, we also include boundary and initial conditions as they relate to the solution of the equations. A summary of basic mathematical concepts and notation is given in Appendix 1. General Equations The derivations of equations used in ground- water applications are based on the conservation principles dealing with mass, momentum, and energy. These principles require that the net quantity (mass, momentum, or energy) leaving or entering a specified volume of aquifer during a given time interval be equal to the change in the amount of that quantity stored in the volume. The derivation of the particular conservation equation involves representing the balance in terms of mathematical expressions. Once the balance equation is developed in mathematical terms, it is necessary to specify additional relationships among variables so that the equations can be solved. These include thermodynamic (e.g., the effect of fluid pressure on density) and constitutive (e.g., the effect of fluid pressure on porosity) relationships. The result of the derivation is usually a set of general partial differential equations in the three-dimen¬ sional Cartesian coordinate system. As an example of the procedure, a simplified derivation of the ground-water flow equation is given in Appendix 2. General equations for ground-water flow, solute transport and heat transport are discussed in the remainder of this section. It should be noted that these equations were derived based on certain assumptions, and even more general (and complex) equations could be presented. The general equations that we discuss, however, are the ones most commonly used. Ground-Water Flow An equation describing unsteady ground-water flow in three dimensions can be written (see Appendix 2 for reference) as - = - ah V • K- Vh + R = S s —, (1) Fig. 1. Diagram of the major components of the ground- water flow equation. in which h is the hydraulic head (L); K is the hydraulic conductivity tensor (L/t); R is a general source or sink term (l/t), that is, volume of water injected per unit volume of aquifer per unit time; S s is the specific storage (1/L); and 7 is a differential operator (1/L). For general problems, R, S s and R can vary from point to point within the aquifer (that is, are functions of x, y and z). Equation (1) is known as a diffusion-type equation and is derived by combining the mass conservation (water balance) and momentum conservation (Darcy’s Law) equations, as shown in Figure 1 and outlined in Appendix 2. In a more explicit form, equation (1) can be written as, d ah v a , ah a- ah ah — (Kxx T~ ) + T~ (Kyy — ) + — (K zz —) + R - S s — 3x 3x 3y 11 3y 3z 3z 3t ( 2 ) In developing this equation it was assumed that the principal components of the hydraulic conductivity tensor are colinear with the Cartesian coordinate system (that is, the directions of the anisotropy line up with the coordinate system). To get an intuitive feel for what equation (2) expresses, consider a small control volume of the aquifer. The first three terms in (2) represent the difference in the rate of water flowing into and out of the control volume. R represents the rate of water gained or lost from some source or sink within or along the boundary of the volume. The right-hand side represents the change in the 214 amount of water stored in the control volume expressed as a rate. When it is necessary to consider the effect of other processes (such as variable-density solute transport or heat transport) on ground-water flow, a more general form of equation (1) in terms of pressure is used (Bredehoeft and Pinder, 1973): V • pk (Vp + pg Vz) + R = Hop) at (3) In this equation, p is water density (M/L 3 ); g is the gravitational constant (L/t 2 );p is dynamic viscosity (M/Lt); k is the intrinsic permeability tensor (L 2 ); and 0 is porosity (dimensionless). In general, p, p, R, and 0 can depend on the fluid pressure (M/Lt), p, as well as concentration of dissolved materials and fluid temperature. Pressure and hydraulic head are related by (Hubbert, 1940) h = z + / Po P dp gP where z is the elevation above some datum and p Q is some reference pressure (usually atmospheric). Solute Transport In general, a complete physical-chemical description of moving ground water would include the movement of the fluid and all species of material dissolved in the fluid, and chemical reactions among the various species. The difficulties encountered in solving the set of equations describing this interaction (for real problems) have forced hydrologists to consider simplified subsets of the general problem. The ground-water flow equation describes the rate of propagation of a pressure or head change in an aquifer. In order to describe the transport of dissolved chemical species in ground water, the transport equation and the ground-water flow equation must be solved simultaneously. The ground-water flow equation may be used to determine the velocity field of ground- water flow. To determine the movement of the material, an additional equation is necessary. This equation is also obtained by taking a mass balance of the material as shown in Figure 2 and may be written as (see, for example, Reddell and Sunada, 1970) - = - - 3 (0C) V • 0D • VC - V • qC + RC* = —— , (4) M 3t where C is the material concentration (M/L 3 ); C* is the concentration of the source/sink term (M/L 3 ); and D is the dispersion tensor (L 2 /t). The solute transport equation (also known as the convection- diffusion equation) contains (from left to right) a dispersion term, a convection term, a source term (which can include chemical reactions), and a rate change in concentration term. Although equation (4) mathematically describes solute transport, determining the parameters used in this equation usually proves to be quite difficult. These parameters include: (1) velocity, (2) source term properties, and (3) dispersion properties. Velocities are generally determined using Darcy’s equation (see Appendix 2). Therefore, porosity, hydraulic conductivity, as well as the hydraulic head distribution, must be known. Although not shown in equation (4), an additional source term can be incorporated that includes the effects of the various chemical processes and reactions operating in the ground-water system. These include precipitation and solution, co-precipitation, oxidation and reduction, adsorption and desorption, ion exchange, complexation, nuclear decay, ion filtration and gas generation. Not only are many of these processes poorly understood, but in pollution problems, often the type and strength of the source is inadequately described. For example, most landfills have no detailed records describing the materials buried in them. A common attempt to quantify some aspects of the source term is through the use of distribution coefficients (IQ). Fig. 2. Diagram of the major components of the solute transport equation. 4 In general, a high Kd indicates a strong tendency for sorption and therefore retards movement of material. Back and Cherry (1976) discuss this and other problems associated with incorporating geochemistry into transport models. Dispersion refers to the spreading and mixing caused in part by molecular diffusion and microscopic variation in velocities within individual pores (Anderson, 1979). For many field problems, molecular diffusion (described by Fick’s law) is small compared to mechanical mixing. The form of the dispersion tensor is complex and is discussed by Scheidegger (1961). A simplified representation that illustrates the basic components may be written as, D = f(v, d,, d 2 ) + f(D . dn where q n is some function that describes the boundary flow rate given the head at the boundary (hb)- 218 of the dependent variable specified everywhere inside the boundary. For example, in a confined aquifer for which the equations are linear, there is no need to impose the natural flow system since the computed drawdown can be superimposed on the natural flow system. In this case, the initial condition is drawdown (the dependent variable) equal to zero everywhere. Physical Interpretation Just as the physical ground-water system is idealized as continuous in deriving the differential equations, it is also expedient to idealize the conditions on the boundaries of the system in order that they too can be given mathematical expression. The boundary conditions of ground- water systems in nature are of several kinds, perhaps the most common being those describing the conditions at a well. Since the porous media stops at the well face, the aquifer not only has a boundary around its perimeter, but the outline or each well is also considered a boundary to the aquifer. The boundary conditions at wells are treated as constant or variable specified flux, or constant head, depending on which best describes the actual physical conditions. If the well is discharging or recharging at a given concentration or temperature, then these may also be specified, if the transport equations are being considered. Impermeable or nearly impermeable boundaries are formed by underlying or overlying beds of rock, by contiguous rock masses (as along a fault or along the wall of a buried rock valley), or by dikes or similar structures (Jacob, 1950). Permeable boundaries are formed by the bottom of rivers, canals, lakes, and other bodies of surface water. These permeable boundaries may be treated as surfaces of equal head (specified), if the body of surface water is large in volume, so that its level is uniform and independent of changes in ground-water flow. The uniform head on a boundary of this type may, however, change with time due to seasonal variation in the surface-water level. Other bodies of surface water, such as streams, may form boundaries with nonuniform distributions of head which may be either constant or variable with time. A small stream, for example, might be affected by a nearby withdrawal of ground water if that withdrawal occurred at a rate of the same order of magnitude as the flow in the stream. Then the boundary condition would not be independent of the ground-water flow; that is, it would be a head-dependent flux. The body of surface water may be a holding pond containing hazardous wastes. In solving the solute transport equation, the boundary condition at the holding pond must be specified. In theory, this may be simply accomplished by specifying some concentration at the bottom of the pond which may vary with time as the waste in the pond is changed. In practice, however, historical records rarely contain the necessary data to accurately define this boundary condition. This condition is perhaps the most critical portion of the model study since it describes the source of pollution. Often, it can only be estimated, and depends on the judgment of the hydrologist performing the study. VARIATIONS OF THE GENERAL EQUATIONS Ground-Water Flow As discussed, there are many subsets to the general equations. We present several of the more common ones used in modeling, and, by way of example, only present the modifications to the ground-water flow equation. Confined , Artesian Flow Equation (2) may be integrated over the thickness of an aquifer (see Pinder and Gray, 1977) to give, a ah v a 7 (T X x “ ) + ~ (Tyy 3y 77 3x 3x ah — ) + W = S ay ah at ’ (7) which is an areal ground-water flow equation where T is transmissivity; S is the storage coefficient; and W is a source/sink term. This is the equation most commonly used to examine ground-water flow in confined aquifers. Because all the terms in (7) are linear, it is a linear partial differential equation. This is important to note in terms of obtaining a solution. For more discussion on this equation, the interested reader is referred to Pinder and Bredehoeft (1968). In order to obtain equation (7), many simplifying assumptions were invoked to idealize the ground-water flow system to make a mathe¬ matical treatment possible. The major assumptions include: (1) porous media, (2) Darcy’s law, (3) slightly-compressible fluid, (4) small vertical variation in properties and head, (5) single aquifer with areal, confined flow, (6) linear elastic aquifer vertical compressibility, and (7) principal compo¬ nents of the transmissivity tensor are aligned with the coordinate axes. The assumption of a porous medium is not very restrictive; even fractured media over a large area generally behave as a porous media. Darcy’s law has been found to be valid for most field applications. The slightly-compressible fluid assumption implies that the density of water remains almost constant. For problems involving temperature variations or large differences in dissolved solids, the assumption of constant density may not be valid. For such problems, the ground- water flow equation may be reformulated in terms of pressure. If vertical variations in properties and head occur, the three-dimensional equation (1) must be used to describe the flow system. If the system is under water-table conditions, then the equation must be modified as discussed later. Finally, the assumption of linear vertical compressibility is valid for most field applications, except perhaps for problems where consolidation and subsidence are major factors. Leaky Artesian Conditions The source term in (7), W, can include well discharge, induced infiltration from streams, steady and/or transient leakage from confining beds, recharge from precipitation and evapotranspiration. These source terms are discussed in Prickett and Lonnquist (1971) and in Trescott, et al. (1976); as an example, steady-state leakage is discussed. It is assumed that the leakage through the confining bed into the aquifer is vertical and proportional to the difference in the head between the aquifer and the head in an overlying or under¬ lying source aquifer. It is also assumed that the head in the source aquifer is constant with time, that storage in the confining bed is neglected, and that the aquifer head does not fall below the bottom of the confining bed. Under these condi¬ tions, equation (7) may be written as, d 3h 3 — (T xx — ) + — (T 3x 3x 3y 3h W'-£ dtp dp ~T~~r ~T • (1.4) dt dp dt Mathematical Notation Perhaps one of the most confusing aspects of the mathematical description of ground-water flow is the wide variety of mathematical notations used. Basically, these notations amount to shorthand descriptions of the equations. Before discussing the notations, we first discuss some of the common nomenclature. Table 1.1 shows the definitions of a scalar, vector and tensor quantity. As may be seen, these quantities are used in every aspect of ground water, each having a different physical meaning, and therefore a different mathematical treatment. For example, consider head in an aquifer. Starting at a given point we wish to determine the rate of change in head with distance as we move away from the starting point. Even though head is a scalar, this rate of change will be different, depending on the direction we choose; conse¬ quently it is called a directional derivative. In rectangular coordinates, this derivative is expressed with the aid of the del operator (vector) as _ah T ah -ah , x V h = grad h = l— + j — + k— , (1.5) ax 3y 3z where i, j and k are vectors of unit length in the positive x-, y- and z-directions. This quantity is known as the gradient of head. This new quantity, Vh, is a vector, and may be used to determine the velocity vector, v. Thus, at each point of space, there is a vector, v, and the magnitude and direction of v may vary from point to point. If we apply the del operator to a vector quantity, it forms the dot product or divergence of V: 3vx 3vy 3vz V • v = div v =- +- + - . (1.6) 3x 3y 3z The divergence of Vh is called the Laplacian, which is found in the ground-water flow equation. There are several different ways of writing the Laplacian and these are summarized in Table 1.2. All of these various notations are found in the ground-water literature and all are equivalent. 223 Table 1.2. Laplacian Notation a 2 h 3 2 h a 2 h - 4- - -f — Definition dx 2 dy 2 3z 2 div grad h Vector notation V 2 h del notation a ah 3xj 3xj Summation notation, repeated index in a term implies summation 3 3h N 3 3h r ( r )= s i 3xj dxj i=l dxj dxj where N is number of dimensions —1, 2, or 3. Coordinate Systems The two most commonly used coordinate systems in ground-water flow problems are rectangular (Cartesian) and cylindrical. Regional problems are described using equations in terms of a rectangular system. In this system, the lines or axes are perpendicular to each other, and are generally designated x, y and z, as shown in Figure 1.2. Cylindrical coordinate systems are used in examining well problems. Cylindrical coordinates Fig. 1.2. Rectangular and cylindrical coordinate systems showing volume elements and transformation. (r, 0, z) are also shown in Figure 1.2, as well as a volume element and the equations for coordinate transformation. As may be seen, z is the vertical axis, r is the radius, and 0 is the angle in the horizontal plane measured from the x-coordinate. Note that cylindrical coordinates are simply polar coordinates in the x-y plane with z for the third variable. As an example of a coordinate transformation, consider the Laplacian in two dimensions V 2 h = d*h d*h ax 2 + ay 2 (1.7) Transformations of differential expressions from one coordinate system into another are frequently required in applications. Following an example given in Kreyszig (1968), we shall denote partial derivatives by subscripts and h(x,y,t), as a function of r, 0, t, by the same letter h. By applying the chain rule, we obtain, ah 3x h x = h r r x + hfl0 x . By differentiating this again with respect to x we first have hxx “ (hr r x)x + (h00 x )x - (h r )x r x + h r r xx + (h0) x 0 x + htf0 XX • (1-8) Now, by applying the chain rule again, we find (h r )x = hrrfx + h r 00 x and (h o)x = h0 r r x + h00 0 x • To determine the partial derivatives r x and 0 X , we have to differentiate r = V x 2 + y 2 and 0 = arc tan finding r _ x _ x e 1 / y y x r’ x 1 + (y/x) 2 x 3 ' r 2 Differentiating these two formulas again, we obtain r-xr x 1 x 2 y 2 2 2xy r XX = 2 = ~ ~ “ = ~ = -y(~ - ) r X - 4 • r 2 r r r r r Substituting these expressions into (1.8), using h r 0 = h e r , gives x 2 xy y 2 , y 2 , „ x y, ^xx = ~ hn- - 2 — hf0 + — h QQ -i 3 hr + 2 — h^ . r 2 r 3 r r r . (1.9) In a similar fashion we obtain 224 y 2 xy x 2 hyy = -pr hpr + 2 ~ hp0 + p" h Qd ( 1 . 10 ) By adding these two expressions we see that the Laplacian in polar coordinates is 3 2 h 1 dh 1 d 2 h dr 2 r 3r r 2 30 2 ( 1 . 11 ) Viewpoints Assuming we can identify or in some way understand the concept of ground-water flow and related processes of interest, the development of a mathematical model may be approached from (A. Klute, written comm., 1970): (1) the molecular, (2) the microscopic, or (3) the macroscopic viewpoint. A discussion of these viewpoints is important since confusion often arises between hydrologists over which viewpoint they are considering. An example of this is a modeler discussing chemical source terms at the macroscopic level with a geochemist who is describing the same chemical source terms at the molecular level. In the molecular point of view, theories and explanations of the mechanisms of flow are given in terms of the behavior of the water molecules. For this approach, statistical mechanical concepts might be used. Chemical reaction mechanisms are generally described using this viewpoint. At an intermediate level, the microscopic , a theory of flow may be developed treating water in the pores as a continuum and applying the principles of continuum mechanics, especially fluid mechanics, to work out the detailed behavior of the water within the pores. An example of this approach is using the Navier-Stokes equation for the flow of a viscous fluid to work out the detailed water velocity pattern within the pores. However, in natural environments, the identification of pore geometry and other conditions necessary for the solution of the equations is virtually impossible, and thus solutions are generally available only for rather simple pore space geometry, such as flow in straight capillary tubes, or between parallel plates. This approach may be used to describe flow in fractures assuming simple geometry. From a practical viewpoint, one generally must continue the theoretical development to the macroscopic level in order to create a useful tool. We cannot observe the behavior of individual molecules, nor can we observe the velocity and fluid pressure distributions that one might, in principle, calculate in the microscopic approach. All measurements in the field are generally made at the macroscopic level; therefore, to be useful, any theory of water movement must be developed to the point of describing flow on the macroscopic level. In the macroscopic approach, all variables are assumed continuous functions of space and time, with derivatives of as high an order as needed. The permeable medium is treated as a superposition of two continuous phases, solid and liquid. Velocities, hydraulic heads and other necessary variables are treated as point functions. That is, they are defined at every point in the region of interest. The way these variables are defined is important since it involves choosing a volume element of the permeable medium that is representative of the medium. For example, consider the bulk density, or mass of solids per unit bulk volume of permeable medium, at a point surrounded by some volume element. From Figure 1.3, we see that if we choose the volume element smaller than V,, then we may either have a volume of all solid or a volume of all liquid, causing the value for bulk density to vary. Alternatively, if we choose a volume element larger than V 2 , the value of bulk density again varies due to heterogeneous effects (e.g., major changes in rock type). In ■5 > a £ 2r ■8 molecular and microscopic effects I I I inhomogeneous porous medium y effects Fig. 1.3. Variation in bulk density as a function of the average volume, with inserts. practices the volume element (which in some literature is referred to as an REV or Representative Elementary Volume) is taken large enough to contain a representative assortment of pores, but at the same time small enough so that the values of macroscopic variables are approximately constant within the element. In our figure, this volume lies between V, and V 2 . In the following development, all equations and coefficients are considered at the macroscopic level. APPENDIX 2. SIMPLIFIED DERIVATION OF THE GROUND-WATER FLOW EQUATION A rigorous development of the ground-water flow equation (or equation of motion) generally begins with a mass and momentum balance. Momentum Balance For the momentum balance, Darcy’s equation, which is based on empirical evidence, is used and from Figure 2.1, may be written as, hi h 2 (2.1) Q = KA L . 1 jC O (2.2) OT A =q=K L ’ X Fig. 2.2. Volume element of a porous material showing flow across all faces. ah 9x ~ K xx — , 3x ah q y = -K yy — , (2.3) ah q z ~ ~Kzz — , 3z where Q is the rate of volume flow (vol/time); K is a proportionality constant (length/time); q is specific discharge (Darcy velocity) or discharge per unit area; and all other variables are shown in Figure 2.1. Using the definition of a derivative and choosing smaller and smaller values of L, allows us to write Darcy’s equation in differential form: Fig. 2.1. Apparatus to demonstrate Darcy's equation. where we have generalized the equation to allow the Darcy velocity to change with direction (see Domenico, 1972 for details). In equation (2.3), h is hydraulic head, q x , q y , q z are the components of Darcy velocity, and K xx , K yy , K zz are the principal components of the hydraulic conductivity tensor. Generally, the hydraulic conductivity tensor has nine components, describing its effects in all directions. We have reduced it to three components by assuming that it is symmetric and that the principal components are oriented with the x-, y-, z-directions, respectively. Mass Balance The mass balance is determined by considering changes in the mass of a small volume element of porous material over a small time interval (At). Figure 2.2 shows such a volume element with a source/sink term and flow across all faces. In words, the mass balance equation for our problem is, (mass leaving - mass entering) + (final mass - initial mass) = 0 .(2.4) Using the quantities shown in Figure 2.2, equation (2.4) may be rewritten to give, | [(Qp)x+Ax + (Qp)y+Ay + (Qp)z+Az1 - [(Qp)x + (Qp)y + (Qp)z + R AxAyAz] [ At + [(0p)t+At “ (0P)tl AxAyAz = 0 , (2.5) in which p is density (mass/volume); R is the volumetric injection rate/unit volume (1/time); and 0 is porosity (dimensionless). Dividing (2.5) by AxAyAz and rearranging terms gives [(Qp)x+Ax — (Qp)xl l(Qp)y+Ay — (Qp)yl AxAyAz AxAyAz [(Qp)z+Az ~ (Qp)zl D _ [(0p)t+At _ (0p)t 1 — -+ K-. \Z.o) AxAyAz At The Darcy velocity is the flow volume across the element face divided by the area of the face, or Qx _ Qy _ Qz -; Qy — ; q z — AyAz 7 AxAz AxAy (2.7) Using equations (2.7), and choosing smaller and smaller values of Ax, Ay, Az and At, and using the definition of a derivative, equation (2.6) becomes a(pq x ) _ 3(pq y ) _ d(pq z ) + _ 3(0 p) 3x by dz at ( 2 . 8 ) Final Equations For a slightly compressible fluid such as water, density changes are small. Under this condition, the spatial derivatives of density on the left side of the equation are generally negligible; whereas, the time derivative on the right side may be related to hydraulic head (see, for example, Davis and DeWiest, 1966). The resulting equation is dq x dqy 3x 3y (2.9) where S s is the specific storage. Substitution of equations (2.3) into (2.9) gives 3 3h 3 y 3h v 3 , 3h v 3h ~ (Kxx — ) + — (Kyy — ) + — (K zz — ) + R - S s — , 3x 3x 3y 77 3y 3z 3z 3t ( 2 . 10 ) which is the unsteady or transient, three-dimensional ground-water flow equation. It is sometimes called the diffusion equation, and equations of the same form occur in the theories of unsteady flow of heat and electricity. In mathematics, it is classified as a parabolic partial differential equation. Equation (2.10) states that the flow components in the x-, y-, and z-directions plus the source/sink term must balance the change in storage. Equation (2.10) is often written using a type of mathematical shorthand as V* K- Vh + R = S s — , (2.11) 3t where V is the differential operator, the single bar indicates a vector quantity, and the double bar indicates a second-order tensor quantity. For many problems, the velocity distribution, and hence the hydraulic head distribution, does not change with time; that is, the problem is steady - state. Many regional ground-water flow systems can be represented as a steady-state boundary value problem. For steady flow, 3h/31 = 0, so equation (2.10) becomes 3 , 3h 3 y 3h 3 , 3h v ~ (K X x r~ ) + r~ (Kyy —) + —- (K zz —) - 0 , (2.12) 3x 3x 3y 77 3y 3z 3z where for convenience the source/sink term has been dropped. In mathematics, the steady-state equation is classified as an elliptic partial differential equation. For a homogeneous medium, equation (2.12) reduces to 3 2 h 3 2 h 3 2 h (2.13) K xx I + KyV ^ + Rzz ; = 0 , 3x 2 yy 3y 2 z 3z 2 and for an isotropic medium, K xx = K yy and II * N N II 7 3 2 h 3 2 h 3 2 h v (2.14) K ( —- + —- + —- ) = 0 , 3x 2 3y 2 3z 2 or dividing (2.14) by hydraulic conductivity 3*h 3 2 h 3 2 h 3x 2 3y 2 3z 2 (2.15) which is Laplace’s equation. Note that if we had used the transient equation we would havg. obtained an S s /K term, which is called th^ry5faLlic diffusivity. Ground-Water Modeling: Numerical Models by Charles R. Faust and James W. Mercer b ABSTRACT Partial differential equations may be used to describe a large number of problems in ground-water hydrology. Without a solution, however, these equations are of little value. Only a simplified subset of the general equations can be solved by analytical means, and these often describe idealized situations that are limited in application. Numerical solution of these equations using high speed digital computers offers a logical alternative. INTRODUCTION Numerical models provide the most general tool for the quantitative analysis of ground-water applications. They are not subject to many of the restrictive assumptions required for familiar analytical solutions (e.g., Theis’ solution for radial flow to a pumping well in an infinite, confined aquifer). In spite of the flexibility of numerical models, their mathematical basis is acually less sophisticated than that of the analytical methods. Unfortunately, to the would-be-model user numerical methods seem complex. This perception results from two primary causes; the first is that the number of alternative methods appears to be very large. Actually, the number of basic alternative methods is few; only the number of minor variations is large. Each of these variations contributes to the second cause, unfamiliar terminology, by introducing new names and jargon. In this paper we introduce numerical methods commonly used a This is the third in a series of papers on ground-water modeling. bGeoTrans, Inc., P.O. Box 2550, Reston, Virginia 22090. Discussion open until January 1, 1981. in ground-water problems, while emphasizing basic ideas and only necessary terminology. For easy reference, common terminology is listed in Appendix 2. To develop a numerical model of a physical system (in our case, an aquifer), it is first necessary to understand how that system behaves. This understanding takes the form of laws and concepts (e.g., Darcy’s law and the concept of storage). These concepts and laws are then translated into mathematical expressions, usually partial differential equations, with boundary and initial conditions (the subject of the second paper in this series). Numerical methods provide a means for solving these equations in their most general form. Numerical solution normally involves approximating continuous (defined at every point) partial differential equations with a set of discrete equations in time and space. Thus, the region and time period of interest are divided in some fashion, resulting in an equation or set of equations for each subregion and time step. These discrete equations are combined to form a system of algebraic equations that must be solved for each time step. Finite-difference and finite-element methods are the major numerical techniques used in ground-water applications. The^ important components and steps of model development for the two alternative methods are shown in Figure 1. In the remainder of this paper we discuss these general methods in more detail showing how they lead to a matrix equation (system of algebraic equations). Various matrix solution techniques are reviewed, including direct and iterative methods. This is followed by a discussion of general consider¬ ations in applying models (e.g., initial and boundary conditions) and special techniques such as nonlinear techniques. Finally, the pros and cons of the various numerical techniques are presented. Fig. 1. Generalized model development by finite-difference and finite-element methods. FINITE-DIFFERENCE METHODS (FDM) One numerical approach that has been applied successfully to the ground-water flow equation involves finite-difference approximations. When using FDM to solve a partial differential equation, a grid is first established throughout the region of interest. For two-dimensional, areal problems, we overlay a grid system on a map view of the aquifer. There are two common types of grids: mesh- centered and block-centered. These are shown in Figure 2. Associated with the grids are node points that represent the position at which the solution of the unknown values (head, for example) is obtained. In the mesh-centered grid the nodes are located on the intersection of grid lines, whereas in the block-centered grid the nodes are centered between grid lines. The choice of the type of grid to use depends largely on the boundary conditions. The mesh-centered grid is convenient for problems where values of head are specified on the boundary, whereas the block-centered grid has an advantage in problems where the flux is specified across the boundary. From a practical point of view, the differences in the two types of grids are minor. Note that the grids shown in Figure 2 are rectangular and regular-, that is, the spacing in the x-direction, Ax, and y-direction, Ay, are constant. Often, irregular, rectangular grids with variable Ax and Ay are used to describe the aquifer, using smaller spacing for areas where more detail is required (as near a well field). Such a grid is shown in Figure 3. In addition, grids need not be rectangular. A. i ♦ X NODE POINT Si • • • • • • • • • • • • • • • MESH CENTERED BLOCK CENTERED Fig. 2. Grids showing mesh-centered nodes and block- centered nodes. B. Pa _ • • H, » - '1, t B E t D. „ ( 1 i- • -1 Axj Fig. 3. Four by five block-centered grid used in difference equation development (A) and typical connection (B) for node (i,j). 396 The integrated finite-difference method (IFDM) utilizes an arbitrary grid. Though less commonly applied than the conventional finite- difference approach, the IFDM has been used to study ground-water problems since the early 1960’s, when it was known by a different name. Tyson and Weber (1964) introduced the method to the ground-water field, but referred to it as a polygonal model technique. The method was further discussed by Cooley (1971) and Thomas (1973), and was finally given the current name by Narasimhan and Witherspoon (1976). For the IFDM, the region of interest must also be divided into smaller areas. Thomas (1973) refers to these as nodal areas, since they each have a node point which is used for mathematical purpses to connect each area with its neighbor. Further, as with other finite-difference methods, it is assumed that all recharge or withdrawal to and from the nodal area occurs at the node point and that water levels in the entire nodal area are the same as at the node point. For this reason, the polygon geometry as well as rectangular grids should be kept to a reasonable size to maintain accuracy. A typical node point, adjacent nodes, and the polygonal zone associated with it are shown in Figure 4. As may be seen, the triangles formed by connecting the node points have no interior angle greater than 90° since the polygon sides are perpendicular bisectors of the lines connecting the intersects. When rectangular nodes are used, the interconnects form rectangles. Thomas (1973) points out that construction errors of more than 5 to 8 degrees from the 90° limit will result in significant computational errors which cannot be easily identified in the results. Difference approximations may be developed using truncated Taylor series (that is, only the first few terms in the series are included). Taylor series are used frequently in ground-water hydrology for many purposes. For example, if water density is Fig. 4. Polygon geometry (after Thoma*, 1973). a weakly dependent function of temperature, it may be expanded about some reference density using a Taylor series. Usually the series is truncated after the first derivative (i.e., truncated in first- order terms), resulting in a linear relationship between density and temperature. For numerical methods, truncated Taylor series may be used to approximate the derivatives in partial differential equations. Alternatively, a more intuitive approach can be used to obtain the final equations by considering the fluxes into and out of a finite- difference block. This second approach is essentially that used to derive the equations for the IFDM. Prickett and Lonnquist (1971) develop finite- difference equations using the second approach and a mesh-centered grid; whereas Freeze and Cherry (1979) present a development based on the first approach and a block-centered grid. A similar approach to Freeze and Cherry’s is presented in Appendix 1. Regardless of the approach, the final result is an algebraic equation for each node in the grid system. For a rectangular grid the form of a typical equation is B i,j hj-i.j + D i,j B i,j-1 + E i,j E i,j + F i,j F i,j + 1 n + H i,j h i+l,j ( 1 ) The notation in equation (1) refers to the nodal locations shown in Figure 3 where h is the head at the designated node; the explicit definitions of the coefficients Bjj, Djj, Ejj, Fjj and Hjj are not given here, but may be found, for example, in Freeze and Cherry (1979). The main reason for presenting equation (1) is to demonstrate the form of the algebraic equation. This equation is for an arbitrary node (i,j) and as may be seen, it has contributions from four adjacent nodes: north (i,j+l), east (i+l,j), south (i,j — 1), and west (i— 1 ,j). These are evaluated at the new time level (n) and are related to some known quantity, Q-j 1 , which is computed from information at the old time level (n-1). Writing an equation similar to (1) for each node results in N equations with N unknown head values to be determined, where N is the total number of nodes. This may be formulated in matrix form and solved using matrix methods. These are discussed in a later section. A partial history of the use of this approach in solving the ground-water flow equation may be found in Pinder and Bredehoeft (1968) and Remson et al. (1971). Details of this approach applied to petroleum problems similar to ground-water problems may be found in Crichlow (1977) and Peaceman (1977). FINITE-ELEMENT METHODS (FEM) There are two fundamental problems in calculus: (1) examining the area under a curve, i.e., integration; and (2) examining the tangent of a curve at a point, i.e., differentiation. Both of these concepts were fairly well understood by the seventeenth century. For example, Archimedes demonstrated an understanding of integration by deriving his approximation for n. However, it was not until 1667 that Isaac Barrow (1630-1677), the teacher of Newton, discovered that integration and differentiation are essentially inverse to one another, which is the fundamental theorem of calculus (Allendoerfer and Oakely, 1959). Whereas FDM approximates differential equations by a differential approach, FEM approximates differential equations by an integral approach. Based on the fundamental theorem, one would expect the two methods to be related and to converge to the same solution, but perhaps from different directions. The FEM actually refers to the numerical method whereby a region is divided into subregions called elements, whose shapes are determined by a set of points called nodes (see Figure 5). Note that flexibility of elements enables consideration of regions with complex geometry; for example, a water-table aquifer with a meandering river can be outlined with elements fairly accurately. For transient problems, the time domain may also be approximated using finite elements. In general, however, most studies use finite-difference approximations for the time derivatives. Triangles were used in the grid shown in Figure 5 ; however, several other element shapes may be used. For one-dimensional problems, the elements are lines; for two dimensions, the elements may be either triangles or quadrilaterals; and for three dimensions, they are tetrahedrons or prisms. The first step in applying the FEM, as shown in Figure 1, is to derive an integral representation of the partial differential equation. This may be accomplished by several methods; two of the more popular ones include: (1) the method of weighted residuals and (2) the variational method. In the method of weighted residuals (see for example, Finlayson, 1972), one works directly with the differential equation and boundary conditions, whereas in the variational method (see for example, Zienkiewicz, 1971), one uses a functional (a function of a function) related to the differential equation and boundary conditions. The mathematics of both of these approaches is fairly straightforward, but not intuitive. The next step is to approximate the dependent variables (head, concentration or temperature) in terms of interpolation functions. The interpolation functions are called basis functions, and are chosen to satisfy certain mathematical requirements and for ease of computation. Although any system of independent functions can be chosen as the basis function, piecewise-continuous polynomial sets are often preferred because they are both easily integrated and differentiated. Since the element is usually small, the interpolation function can be sufficiently approximated by a low-order polynomial, for example, linear, quadratic, or cubic. As an example, consider a linear basis function for a triangular element. This basis function describes a plane surface including the values of the dependent variable (head) at the node points in the element. This is illustrated in Figure 6. For additional information on basis functions, see Desai and Abel (1972) or Zienkiewicz (1971). Once the basis functions are specified and the grid designed, the integral relationship must be expressed for each element as a function of the coordinates of all node points of the element. Next the values of the integrals are calculated for each element. The values for all elements are combined, including boundary conditions, to yield a system of first-order linear differential equations in time. As previously mentioned, this is approxi¬ mated using finite-difference techniques to produce a set of algebraic equations. As with finite-difference equations, matrix methods are required for solution. Aquifer Boundary Fig. 5. Finite-element configuration showing typical node and element. Node Element h Fig. 6. Surface described by linear basis function for a triangular element. For additional information on the FEM as used in ground water, see Verruijt (1970), Remson et al. (1971) and Pinder and Gray (1977). MATRIX SOLUTION TECHNIQUES As we have seen, each numerical approximation leads to an algebraic equation for each node point. These are combined to form a matrix equation, that is, a set of N equations with N unknown, where N is the number of nodes. The general form of these equations, written in matrix form is Ah = d , (2) where A is a matrix containing coefficients related to grid spacing and to aquifer properties, such as transmissivity; h is a vector containing the dependent variables to be determined, for example, head values at each node; and d is a vector containing all known information, for example, specified pumpage and boundary condition information. In general, a matrix equation may be solved numerically by one of two basic ways: (1) direct and (2) iterative. Some solutions may involve a combination of the two. In direct methods a sequence of operations is performed only once, providing a solution that is exact, except for machine round-off error. Iterative methods attempt solution by a process of successive approximation. They involve making an initial guess at the matrix solution, then improving this guess by some iterative process until an error criterion is attained. Therefore, in these techniques, one must be concerned with convergence, and the rate of convergence. Although solving the matrix equation is a mathematical problem, the hydrologist must be aware of some of its important aspects, since generally the matrix solution is the most expensive part of the computer costs. In very general terms, iterative techniques are more efficient than direct solution techniques for matrix equations that contain more than 1,000 unknowns. The relative merits of direct and iterative methods are shown in Table 1. It should also be pointed out that for some problems, the matrix A does not have to be regenerated eaclytime step. For a direct method, this means that A is decomposed only once, and a subsequent time step requires only back substitu¬ tion. Since back substitution is much less expensive than decomposition (elimination), this improves the efficiency of direct methods considerably. Direct Methods Direct methods can be further subdivided into (1) solution by determinants, (2) solution by successive elimination of the unknowns, and (3) solution by matrix inversion. According to Narasimhan and Witherspoon (1977), perhaps the most widely used direct approach for transient problems is that of successive elimination and back substitution, which includes the Gaussian elimination method (Scarborough, 1966) and the Cholesky decomposition method (Weaver, 1967). As shown in Table 1, direct methods have two main disadvantages. The first problem deals with storage requirements and computation time for large problems. The matrix in equation (2) is sparse (contains many zero values) and in order to minimize computational effort, several techniques have been proposed. Various schemes of numbering the nodes have been studied; an efficient one for finite-difference nodes is alternating direction (D4) ordering (Price and Coats, 1974). Other methods have been attempted with the finite-element method. However, for both finite-difference and finite-element methods storage requirements may still prove to be unavoidably large for three- dimensional problems. The second problem with direct methods deals with round-off errors. Because many arithmetic operations are performed, round-off errors can accumulate for certain types of matrices. Iterative Methods Iterative schemes avoid the need for storing large matrices, which make them attractive for solving problems with many unknowns. Numerous schemes have been developed; a few of the more commonly used ones include successive over¬ relaxation methods (Varga, 1962), alternating direction implicit procedure (Douglas and Rachford, 1956), iterative alternating direction 399 Table 1. Advantages and Disadvantages of Direct and Iterative Methods Advantages Disadvantages Sequence of operations performed only once. No initial estimates required. No iteration parameters required. No tolerance required. DIRECT METHODS May be inefficient in terms of storage and computation time for large problems. Can have round-off errors. Efficient in terms of storage and computation time for large problems. ITERATIVE METHODS Requires initial estimates. Requires iteration parameters. Requires tolerance. Matrix must be well conditioned. implicit procedure (Wachpress and Habetler, 1960), and the strongly implicit procedure (Stone, 1968). Since iterative methods start with an initial estimate for the solution, the efficiency of the method is dependent on this initial guess. This makes the iterative approach less desirable for solving steady-state problems (Narasimhan and Witherspoon, 1977). To speed up the iterative process, relaxation and acceleration factors are used. Unfortunately, the definition of best values for these factors is often problem dependent. In addition, iterative approaches require that an error tolerance be specified to stop the iterative process. This, too, may be problem dependent. According to Narasimhan and Witherspoon (1977), perhaps the greatest limitation of the iterative schemes is the requirement that the matrix be well conditioned. An ill-conditioned matrix can drastically affect the rate of convergence or even prevent convergence. An example of an ill-conditioned matrix is one in which the main diagonal terms are much smaller than other terms in the matrix. Such matrices can result from finite- element applications. SPECIAL TECHNIQUES For certain types of applications serious numerical difficulties are encountered. Sometimes these difficulties manifest themselves as unrealistic or inaccurate results. At other times, there may be no results at all—the numerical solution will “blow up.” The first type of difficulty is associated with convective-dominated transport problems, whereas the second type of difficulty is associated with nonlinear problems. Special techniques have been developed to deal with these problems, and some of these are discussed below. Method of Characteristics (MOC) The motivation for the method of character¬ istics (also known as particle-in-cell method) lies in the numerical difficulties that occur when solving a convection-dominated transport problem. With conventional finite-element and finite- difference approaches we have a choice between solutions that show numerical dispersion or • numerical oscillation. Numerical dispersion yields answers that are smeared out. Oscillatory solutions are not smeared out, but lead to physically unaesthetic consequences such as large negative concentrations for solute transport problems. To illustrate the two types of problems, consider the physical situation of one-dimensional convection between a line of injection wells and a line of pumping wells in a confined aquifer. If the concentration of the injection fluid is held at some constant value different from that of the aquifer and if there is no physical dispersion or chemical processes active, the concentration will move out as a sharp front (see Figure 7). Also shown in Figure 7 are typical results that demonstrate numerical dispersion and numerical oscillation. Usually numerical oscillation is associated with the finite-element method and numerical dispersion Fig. 7. Typical numerical solutions for transport problem with convection only. 400 is associated with the finite-difference method. However, depending upon the approximation used for the convective term, both methods can demonstrate either type of behavior. The method of characteristics was devised to alleviate these numerical difficulties. The MOC has been used in ground-water applications to solve the solute transport equation (see for example, Reddell and Sunada, 1970; or Pinder and Cooper, 1970). Although not a require¬ ment, it is usually used in conjunction with a finite-difference method; that is, finite-difference approximations are used for the flow equation and the finite-difference grid is utilized in solving the transport equation. The approach is not to solve the transport equation directly, but rather to solve an equivalent system of ordinary differential equations. The ordinary differential equations are obtained by rewriting the transport equation using the fluid particles as the point of reference. That is, instead of observing how the concentration changes with time at a fixed position in space, we observe changing concentration as we move with the fluid. Therefore, we need to know the velocity distribution. In two dimensions, the end result is three equations for x-velocity, y-velocity, and concentration, the solutions of which are called the characteristic curves, hence the name, method of characteristics. This is accomplished numerically by intro¬ ducing a set of moving points (or reference particles) that can be traced within the stationary coordinates of a finite-difference grid (Konikow and Bredehoeft, 1978). Points are placed in each finite-difference block and then allowed to move a distance propor¬ tional to the length of the time increment and the velocity at that point (see Figure 8). The moving points effectively simulate convective transport because the concentration at each node varies as different points having different concentrations enter and leave the area of that block. Once the convective effect is determined, the remaining parts of the transport equation are solved using finite- difference approximations and matrix methods. A more complete discussion of this method is presented by Garder, Peaceman, and Pozzi (1964); Pinder and Cooper (1970); Reddell and Sunada (1970); Bredehoeft and Pinder (1973); and Konikow and Bredehoeft (1978). In addition, applications of this method to field problems are presented by Bredehoeft and Pinder (1973); Konikow and Bredehoeft (1974); Robertson (1974); Robson (1974); and Konikow (1977). Nonlinear Techniques A differential equation is nonlinear if it includes products of dependent variables and/or derivatives of dependent variables. In unconfined flow problems, the transmissivity is a function of the saturated thickness and consequently, a function of head. The product of transmissivity (as a function of head) and the second derivative of head results in a nonlinear flow equation. Some unconfined problems, however, may be treated as linear if the water level changes are small compared to the total saturated thickness of the aquifer. Most confined ground-water systems are linear. For transport problems, the equations are nonlinear when changes in pressure, concentration, and temperature cause changes in density, viscosity, or porosity. With heat transport, this is generally the case; however, many solute problems may be approximated as linear systems because concentrations are too low to affect the flow field. In general, linear problems are easier to solve than nonlinear ones, and superposition may be used to add (or subtract) solutions. The principle of superposition simply states that any linear combina¬ tion of solutions to a linear equation is itself a solution. In terms of ground-water flow, this means that a drawdown solution can be subtracted from the natural head distribution to give the computed head distribution. This is why many analytical solutions are expressed in terms of drawdown. For nonlinear problems, equation parameters, such as density, are changing with time as pressure, concentration, or temperature change with time. This means that the density at the beginning of a time step will be different from the density at the X finite-difference node • reference particle o new location ^ flow line V / o / ? t // / / / oX // p / / / / Fig. 8. Finite-difference grid showing reference particles (after Konikow and Bredehoeft, 1978). 401 end of the time step. If this change is not treated properly, the numerical solution may have mass balance errors. In terms of numerical methods, the nonlinear differential equations result in the matrix equation (2) having a coefficient matrix that is a function of the unknown values. In general this requires some type of iterative procedure. The easiest approach is to make an initial estimate of the unknowns, calculate the coefficients and solve the matrix equation for the unknowns. The solution now becomes the new estimate and the procedure is repeated until the computed values of the unknowns do not change (or change only slightly). Unfortu¬ nately, this simple technique does not always work; that is, it fails to converge to a solution. This lack of convergence is serious for problems with severe nonlinearities; problems in which small changes in unknowns result in large changes in coefficients. Severe nonlinearities occur in unsaturated flow problems and in very thin unconfined aquifers. For these problems better nonlinear techniques have been developed. Two of the more general ones are quasilinearization and extrapolation (see Peaceman, 1977, for details of the various methods). NUMERICAL CONSIDERATIONS In applying numerical methods, we are concerned with three general characteristics of the solution procedure: (1) accuracy, (2) efficiency, and (3) stability. Accuracy deals with how well the discretized solution approximates the solution to the continuous problem it represents. Efficiency is a measure of how much computational work and computer resources are required to obtain a solution. Stability addresses the question of whether or not a solution is possible at all. These definitions are, of course, oversimplifications, but for practical purposes, sufficient. The important point to note is that the reason we have so many variations of numerical techniques is that each one was developed to improve at least one of these characteristics. From the study of numerical analysis, we obtain information on accuracy, efficiency, and stability. For example, by comparing finite- difference or finite-element approximations with Taylor series approximations, their order of accuracy can be determined. Similar <7 priori analysis into a particular method’s stability or efficiency can be made. From such analysis we may learn that the FEM offers more accurate approximations than the FDM or that quasilineari¬ zation is a more stable nonlinear technique than extrapolation. Even though numerical analysis provides measures of the important characteristic of alternative numerical methods, it does not tell us which one is optimum for a particular application. The reason for this is that field situations are much more complex than the ones assumed in any numerical analysis. For theoretical analysis it is often assumed that the properties and grid are uniform, whereas in applications this is rarely the case. Even when the theoretical analysis is more general, it still falls short of the field complexity we typically simulate. Consequently, the quantitative results of numerical analysis only serve as qualitative guides. The information from numerical analysis must be augmented with some numerical experi¬ mentation or practical experience in order to match numerical techniques to particular applications. PRACTICAL CONSIDERATIONS Design of Grids One of the critical steps in applying a ground- water model is designing the grid. Intuitively we would expect that the finer the grid the more accurate the solution. Numerical analysis confirms this intuition; therefore, we should use fine grids where we want accurate solutions, and we can use coarse grids where details are not important. No matter whether the FDM, IFDM, or FEM is used, some general guidelines should be followed (Trescott et al., 1976): (1) Locate “well” nodes near the physical location of the pumping well or center of the well field. (2) Locate boundaries accurately. For distant boundaries the grid may be expanded, but avoid large spacings next to small ones. (3) Nodes should be placed closer together in areas where there are large spatial changes in transmissivity or hydraulic head. (4) Align axes of grid with the major directions of anisotropy (that is, orient grid with major trends). Each of the numerical approaches has its own considerations as well. With the conventional FDM, for situations where the aquifer boundary is curved and not square or rectangular, the node points will generally not coincide with the boundary. This results in errors in approximating heads near the boundary. For aquifers, however, the subsurface 402 location of the boundaries is seldom known accurately enough to be concerned about these errors. Boundary approximation is not a serious problem with the IFDM or FEM. With these methods, care must be taken to avoid certain node and element shapes. The requirements of node shapes for the IFDM have already been mentioned. The topic is more involved for finite-element methods, because of the large variety of element types. A good feature of the FEM is the flexibility to use higher-order elements (for example, cubic or parabolic) in areas where accurate solutions are desired. For details of the practical design of finite- element grids, see Pinder and Gray (1977). Initial Conditions In many ground-water flow simulations, the important results are not the computed heads but the changes in head (drawdown) caused by a stress such as pumping wells. For this objective in a confined aquifer, for which the equations are linear, there is no need to impose the natural flow system as the initial condition, since the computed drawdowns can be superimposed on the natural flow system. Therefore, the initial conditions are simply zero drawdown everywhere. For nonlinear problems (e.g. water-table conditions), a head distribution must be specified as initial conditions. In this case, boundary and initial conditions must be compatible. To achieve this, the transient simulation should start from an equilibrium or steady-state position. To start from steady-state conditions in which flow is occurring, a model can be used to compute the initial head by leaving out the new stresses or changes in stresses (e.g., wells) and setting all storage coefficients to zero. Choice of Time Step An appropriate choice of time steps is of considerable practical interest in efficiently solving a given problem (Narasimhan and Witherspoon, 1977). One scheme is to progressively increase the size of the time step with advancing time. This scheme is particularly useful for variable pumping rates, where the time step is reduced at the beginning of each pumping period and allowed to increase (see Figure 9). Such progressive adjustments to the time step may be either arbitrary or determined by specific system behavior. While these adjustments may be relatively easy to make in linear problems with smooth boundary conditions, mass balance errors may result in strongly nonlinear problems with time-dependent boundary conditions. Fig. 9. Example of idealization of variable pumping rates, showing how time step is allowed to increase over each pumping period (after Prickett and Lonnquist, 1971). Treating Heterogeneities Field problems in ground water almost invariably demonstrate spatial variation in material properties. In the FEM, integration is performed over each element for which the material properties are specified. If material properties do not vary within each element, then heterogeneity is accounted for when element integrations are combined. If a material property does vary spatially within an element, Pinder et al. (1973) describe a way of accounting for the variability using functional coefficients. For FDM and IFDM, mean values of material properties are assigned to the node points and are assumed constant over the block or polygon. The fluxes between nodes, however, are evaluated at the interfaces between blocks or polygons. Therefore, appropriate mean values of permeability or transmissivity are used at the interfaces where the fluxes are evaluated. In many finite-difference models, this is referred to as interblock trans¬ missivity. There are-several ways to evaluate these terms (see, for example, Appel, 1976), however, one common way is to use the harmonic mean. Use of the harmonic mean (1) insures continuity across block boundaries at steady state even if a variable grid is used, and (2) makes the appropriate coefficients zero at no-flow boundaries (Trescott etal., 1976). For nonlinear problems, the interblock transmissivity terms may contain a parameter that is a function of head (for example, saturated thickness in a water-table aquifer). For this case, the upstream value of the saturated thickness may be used. The upstream node is located by 403 comparing the heads at adjacent nodes, and the interblock terms are evaluated using the thickness at the node having the greater head. Upstream weighting yields a lower-order approximation of the spatial derivatives but generally exhibits a more stable solution. Boundary Conditions Rigorous treatment of boundary conditions for FDM may be found in references such as von Rosenberg (1969); only a few helpful hints are given here. For a constant head boundary, the value of head at the node is known and need not be solved. This may be accomplished by not solving the finite-difference equation at the node in question, but another approach is to solve the trivial equation where hfj is the prescribed head and n is the new time level. A trivial equation can be created from the usual finite-difference equation by setting the storage coefficient equal to infinity (that is, a very large number, e.g., 10 7 ) and setting hpj 1 equal to h*j on the right-hand side (known side). For an impermeable boundary, that is, no flow (constant flux of zero), the interblock trans¬ missivities along the boundaries are set to zero in the FDM. That is, using equation (1), for i = 1, B = 0; for i = NC, H = 0; for j = 1, D = 0; and for j = NR, F = 0, where NC is the number of columns and NR is the number of rows. This eliminates the need for an extra boundary of blocks used in some models, e.g., Trescott et al. (1976). For the FEM, the boundary flux is incorporated in a surface integral; when the flux is zero, this term vanishes. Finally, for constant flux, not equal to zero, source terms may be used as approximations and included in the right-hand side of the matrix equation. Note that where it is impractical to include one or more physical boundaries (e.g., an alluvial valley that may be extremely long), the grid can be expanded to an artificial boundary. The artificial boundary should be located far enough from the project area so that it will have negligible effect on the area of interest during the simulation period, but can be much closer than the physical boundary. In this case the boundary condition is arbitrary (e.g., impermeable conditions), but the influence of the artificial boundary should be checked by comparing the results of two simulation runs using different artificial boundary conditions. Quality Control The steps in developing a numerical model consist of different levels of error elimination. The first step is to compile the program to remove FORTRAN errors. Next, the numerical solution is compared with analytical solutions to remove logic errors in solving the equation. Numerical solutions are compared with laboratory and field observations to remove logic errors in equations describing the physics. Finally, it is good programming practice to include mass and energy (if needed) balances as checks that the model is working properly. PROS AND CONS OF VARIOUS METHODS Many of the pros and cons associated with the various methods have already been discussed and some are listed in Table 2. These are summarized below according to the following categories: (1) ease in understanding the theoretical basis, (2) ease in programming, (3) ease in designing grid and preparing data input, (4) handling complex geometries, (5) approximation accuracy, (6) handling tensors, (7) handling low dispersion, and (8) solution efficiency. The FDM is based on truncated Taylor series, which are similar to the definition of the derivatives that they approximate. The theory is therefore straightforward and relatively easy to understand. According to Pinder and Frind (1972), “The theoretical development of the Galerkin method of approximation is possibly more abstract than finite difference theory.” As for the MOC, Konikow and Bredehoeft (1978) state, “Although it is difficult to present a rigorous mathematical proof for this numerical scheme, it has been successfully applied to a variety of field problems.” The IFDM again uses difference approximations and is therefore fairly straightforward. Table 2. Brief Summary of Important Advantages and Disadvantages of FDM and FEM (as They Are Commonly Used) Advantages Disadvantages FINITE-DIFFERENCE METHOD Intuitive basis. Low accuracy for some Easy data input. problems. Efficient matrix techniques. Regular grids. Program changes easy. FINITE-ELEMENT METHOD Flexible geometry. Mathematical basis is High accuracy easily included. advanced. Evaluates cross-product terms Difficult data input. better. Difficult programming. 404 Finite-difference approximations are relatively easy to program; the resulting code is usually short in length (see for example, Prickett and Lonnquist, 1971). Again quoting Pinder and Frind on the FEM, the development of an efficient computer code for the Galerkin procedure is a formidable task. Accord¬ ing to Pinder (1973), “Although it was simple in concept, the method of characteristics proved to be tedious to program for the computer and was not suitable for several situations commonly encountered in the field.” Because of the additional geometry considerations, the IFDM is perhaps slightly more difficult to program than the FDM. Designing a finite-difference grid is not a difficult task if the few rules outlined previously are followed. It simply consists of intersecting perpendicular lines. As for data input, only spacings in the respective directions are required. Designing a finite-element grid requires considerably more time and effort than that required by the FDM. According to Pinder (1973), the design of the finite-element mesh is probably the most critical step in applying the Galerkin, finite-element approach to field problems. Furthermore, as Pinder and Frind (1972) point out, experience has shown that errors in the input of nodal locations in the Galerkin model can lead to problems that are difficult to detect; this problem does not arise in the finite-difference model because the entire grid is specified by the spacing between rows and columns. Since the MOC uses a finite-difference grid, it requires the same amount of effort as the FDM. In the IFDM, care must be taken to design the mesh so that the lines joining nodal points coincide with the normals to the interfaces between the points (Narasimhan and Witherspoon, 1976). This restriction, as well as the requirement for providing geometrical parameters as input data, may require added effort in the design of networks for complex problems (Narasimhan and Witherspoon, 1976). The FDM grid consists of rectangles and, therefore, approximating a complex curved boundary is somewhat difficult. For many ground- water applications, the subsurface boundaries are not that well known and this difficulty is not a problem. These same comments apply to the MOC. On the other hand, both the FEM and the IFDM can be used to analyze systems with complex geometry. In fact, for ground-water flow modeling, Pinder and Frind (1972) state that in the final analysis the primary advantage of the Galerkin approach to digital modeling of aquifer systems is its flexibility in application. A common misconception is that the finite- element method is inherently more accurate than the finite-difference method. By using parabolic or cubic basis functions, high-order (higher- accuracy) solutions can be obtained with the finite-element method. High-order approximations can also be derived for finite-difference methods, but the procedure is somewhat cumbersome. Consequently, although it is not necessary, finite-difference models commonly use low-order approximations, whereas finite-element models often have convenient options for higher- order approximations. Perhaps another way of saying this is that a carefully designed model using finite elements may provide the same accuracy as a finite-difference model that uses many more nodes (Pinder and Frind, 1972). As we will see, however, this has little correlation with the relative costs of the two methods. In general, the finite-element method handles tensor parameters that include cross-product terms (e.g., for conductivity, K xy ) better than the finite-difference method. These terms are not well treated with the conventional finite-difference approximation, because diagonal linkings between adjacent nodes are not considered (see Appendix 1). Related to this are grid-orientation effects (i.e., different solutions depending on the orientation of the grid). Both of these problems can be eliminated by including (at some extra expense) the diagonal linkings in the finite-difference approximation. The finite-element method does not require any modifications because diagonal linkings are naturally included. Over the past several years, many researchers have attempted to solve transport problems using the FDM, and for various reasons felt that the results were inadequate. One of the difficult, but perhaps not commonly encountered, problems in flow through porous media involves sharp fronts. A sharp front refers to a large change in a dependent variable (e.g., concentration) over a small distance. The most common complaint about low-order, FDM and IFDM applied to sharp front problems is that the computed front is “smeared out.” As previously discussed, the process by which the front becomes smeared is generally referred to as numerical dispersion or diffusion (Lantz, 1971). An application of FDM applied to both the solute and heat transport equations, as well as a summary on numerical diffusion may be found in INTERCOMP (1976). In general, the FDM requires very small spacing to obtain reasonable results for these problems. 405 Sharp fronts are encountered in transport problems if hydrodynamic dispersion is negligible. This behavior is particularly amenable to solution by the MOC. The MOC has minimal numerical diffusion and is not constrained by small spacing. Other attempts to more accurately solve the transport equations generally make use of the FEM. In general, for linear problems the FEM can track sharp fronts fairly accurately, which reduces considerably the numerical diffusion problem. The FEM, however, is not problem-free and for nonlinear problems can demonstrate numerical oscillation that becomes unstable (Mercer and Faust, 1976). For a further discussion on this topic see Anderson (1979), and Pinder and Gray (1977). The final topic of comparison is solution efficiency. This is a difficult topic to assess since each computer code incorporates various techniques to improve efficiency; only very general concepts are discussed. Generating the matrix equation for the FDM is fairly straightforward and results in a matrix having properties that allow efficient solution (in terms of computer storage and time) by several different means. The FEM, on the other hand, requires integration over each element before the final matrix equation can be formulated. This integration process can be very time-consuming. Further, once the matrix is generated, it generally requires more storage and computer time to solve than that generated by the FDM. The I FDM also requires integration and results in a matrix that generally requires more storage than a FDM matrix. An attempt to reduce the time required for solution of an IFDM matrix equation is to use an explicit-implicit procedure (Narasimhan and Witherspoon, 1976). As for the MOC, tracking points and iterating between the convective and dispersive parts of the transport equation can be fairly expensive. SUMMARY The numerical techniques commonly used in ground-water applications are variations of two general methods—the finite-difference method (including the integrated finite-difference method) and the finite-element method. Occasionally, specialized techniques such as the method of characteristics are also used. All of these methods approximate the continuous partial differential equations with discrete equations, requiring matrix solution. There are two basic ways to solve matrix equations numerically: (1) direct and (2) iterative. In the direct methods, a sequence of operations is performed only once, and the results obtained are an approximation to the true results. The iterative methods attempt solution by a process of successive approximation. No particular combination of numerical technique and matrix solution procedure is best for all applications. For most ground-water flow problems, the FDM is probably adequate. For sharp-front problems the MOC or FEM will probably give better results. For deformation problems, the FEM is better because of its treatment of tensorial parameters. For any given class of problems the choice of the best approach depends on the processes being modeled, the accuracy desired, and the effort that can be expanded on obtaining a solution. Oftentimes, however, the hydrologist simply uses a technique that he is familiar with, and a computer code that is well documented. ACKNOWLEDGMENTS This effort was supported by the Holcomb Research Institute, Butler University, which is, in part, supported by EPA grant R-803713. The authors also wish to acknowledge the National Water Well Association’s contribution for drafting, editing and publication. Finally, many of the ideas presented in this series of papers were formulated during the authors’ participation in training courses for the U.S. Geological Survey. REFERENCES Allendoerfer, C. B., and C. O. Oakley. 1959. Fundamentals of Freshman Mathematics. McGraw-Hill, New York. 475 pp. Anderson, M. P. 1979. Using models to simulate the move¬ ment of contaminants through ground-water flow systems. Critical Reviews in Environmental Control, v. 9, issue 2, pp. 97-156. Appel, C. A. 1976. A note on computing finite difference interblock transmissivities. Water Resour. Res. v. 12, no. 3, pp. 561-563. Bredehoeft, J. D., and G. F. Pinder. 1973. Mass transport in flowing groundwater. Water Resour. Res. v. 9, no. 1, pp. 194-210. Cooley, R. L. 1971. A finite difference method for variably saturated porous media: Application to a single pumping well. Water Resour. Res. v. 7, no. 6, pp. 1607-1625. Crichlow, H. B. 1977. Modern Reservoir Engineering—A Simulation Approach. Prentice Hall, Englewood Cliffs, N.J. 345 pp. Desai, C. S., and J. F. Abel. 1972. Introduction to the Finite Element Method. Van Nostrand Reinhold, New York. 477 pp. 406 Douglas, J., Jr., and H. H. Rachford, Jr. 1956. On the numerical solution of heat conduction problems in two and three space variables. Trans. Am. Math. Soc. v. 82, pp. 421-439. Finalyson, B. A. 1972. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, 412 pp. Freeze, R. A., and J. A. Cherry. 1979. Groundwater. Prentice-Hall, Englewood Cliffs, N.J. 604 pp. Garder, A. O., D. W. Peaceman, and A. L. Pozzi, Jr. 1964. Numerical calculation of multidimensional miscible displacement by the method of characteristics. Soc. Petr. Eng. J. v. 4, no. 1, pp. 26-36. INTERCOMP. 1976. A model for calculating effects of liquid waste disposal in deep saline aquifers. U.S. Geol. Survey Water-Resources Inv. 76-61. Konikow, L. F. 1977. Modeling chloride movement in the alluvial aquifer at the Rocky Mountain Arsenal, Colorado. U.S. Geol. Survey Water-Supply Paper 2044. 43 pp. Konikow, L. F., and J. D. Bredehoeft. 1974. Modeling flow and chemical quality changes in an irrigated stream-aquifer system. Water Resour. Res. v. 10, no. 3, pp. 546-562. Konikow, L. F., and J. D. Bredehoeft. 1978. Computer model of two-dimensional solute transport and dispersion in ground water. U.S. Geol. Survey Techniques of Water-Resources Investigations. Book 7, Chap. C2, 90 pp. Lantz, R. B. 1971. Quantitative evaluation of numerical diffusion (truncation error). Soc. Pet. Eng. J. (Sept.), pp. 315-320. Mercer, J. W., and C. R. Faust. 1976. The application of finite-element techniques to immiscible flow in porous media. Paper presented at the International Conference on Finite-Elements in Water Resources, Princeton University, July 12-16. Narasimhan, T. N., and P. A. Witherspoon. 1976. An integrated finite difference method for analyzing fluid flow in porous media. Water Resour. Res. v. 12, no. 1, pp. 57-64. Narasimhan, T. N., and P. A. Witherspoon. 1977. Recent developments in modeling groundwater systems. Presented at the IBM Seminar on Regional Ground- water Hydrology and Modeling, Venice, Italy, May 25-26, 1976. 35 pp. Peaceman, D. W. 1977. Fundamentals of Numerical Reservoir Simulation. Elserview, New York. 176 pp. Pinder, G. F. 1973. A Galerkin-finite element simulation of ground-water contamination on Long Island, New York. Water Resour. Res. v. 9, no. 6, pp. 1657-1669. Pinder, G. F., and J. D. Bredehoeft. 1968. Application of the digital computer for aquifer evaluation. Water Resour. Res. v. 4, no. 5, pp. 1069-1093. Pinder, G. F., and H. H. Cooper, Jr. 1970. A numerical technique for calculating the transient position of the saltwater front. Water Resour. Res. v. 6, no. 3, pp. 875-882. Pinder, G. F., and E. O. Frind. 1972. Application of Galerkin’s procedure to aquifer analysis. Water Resour. Res. v. 8, no. 1, pp. 108-120. Pinder, G. F., E. O. Frind, and I. S. Papadopulos. 1973. Functional coefficients in the analysis of groundwater flow. Water Resour. Res. v. 9, no. 1, pp. 222-226. Pinder, G. F.; and W. G. Gray. 1977. Finite Element Simulation in Surface and Subsurface Hydrology. Academic Press, New York. 295 pp. Price, H. S., and K. H. Coats. 1974. Direct methods in reservoir simulation. Soc. Petrol. Eng. j. v. 14, no. 3, pp. 295-308. Prickett, T. A., and C. G. Lonnquist. 1971. Selected digital computer techniques for groundwater resource evaluation. Illinois State Water Survey Bull. 55, Urbana. 62 pp. Reddell, D. L., and D. K. Sunada. 1970. Numerical simulation of dispersion in groundwater aquifers. Colorado State Univ. Hydrology Paper 41. 79 pp. Remson, I., G. M. Hornberger, and F. J. Molz. 1971. Numerical Methods in Subsurface Hydrology. Wiley- Interscience, New York. 389 pp. Robertson, J. B. 1974. Digital modeling of radioactive and chemical waste transport in the Snake River Plain aquifer at the National Reactor Testing Station, Idaho. U.S. Geol. Survey Open-File Report, IDO-22054. 41 pp. Robson, S. G. 1974. Feasibility of digital water-quality modeling illustrated by application at Barstow, California. U.S. Geol. Survey Water-Resources Investigations 46-73. 66 pp. Scarborough, J. B. 1966. Numerical Mathematical Analysis. Johns Hopkins Press, Baltimore, MD. Stone, H. K. 1968. Iterative solution of implicit approximations of multidimensional partial differ¬ ential equations. Soc. Indust. Appl. Math., Jour. Numer. Anal. v. 5, no. 3, pp. 530-558. Thomas, R. G. 1973. Groundwater models. Irrigation and Drainage Paper 21, Food and Agriculture Organization of the United Nations, Rome. 192 pp. Trescott, P. C., G. F. Pinder, and S. P. Larson. 1976. Finite-difference model for aquifer simulation in two dimensions with results of numerical experiments. U.S. Geol. Survey Techniques of Water Resources Investigations. Book 7, Chap. Cl, 116 pp. Tyson, H. N., and E. M. Weber. 1964. Groundwater management for the nation’s future—Computer simulation of groundwater basins. J. Hydraul. Div. Amer. Soc. Civil Eng. v. 90, HY4, pp. 59-77. Varga, R. S. 1962. Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, N.J. 322 pp. Verruijt, A. 1970. Theory of Groundwater Flow. Gordon and Breach Science Publishers, New York. 190 pp. von Rosenberg, D. V. 1969. Methods for the Numerical Solution of Partial Differential Equations. Elsevier, New York. 128 pp. Wachpress, E. L., and G. J. Habetler. 1960. An alternating- direction-implicit iteration technique. J. Soc. Ind. Appl. Math. v. 8, pp. 403-424. Weaver, W., Jr. 1967. Computer Programs for Structural Analysis. D. Van Nostrand, Princeton, N.J. 300 pp. Zienkiewicz, O. C. 1971. The Finite Element Method in Engineering Science. McGraw-Hill, London. 521 pp. * * * * Charles R. Faust received his B.S. and Ph.D. in Geology from Pennsylvania State University in 1967 and 1976. Until recently, he was a Hydrologist with the Water Resources Division, U.S. Geological Survey. His interests there involved thermal pollution in rivers, geothermal 407 reservoir simulation, and fluid flow in fractured rocks. Presently be is a principal with GeoTrans, Inc., where he is interested in quantitative evaluation of hazardous waste and radionuclide migration in fractured rocks. James W. Mercer attended Florida State University and University of Illinois, where he received his Ph.D. in Geology in 1973. Prior to graduation, he worked for Exxon Oil Corporation and the Desert Research Institute, University of Nevada. Most recently, he worked for the U.S. Geological Survey, and is currently President of GeoTrans, Inc. Dr. Mercer has taught ground-water modeling at the U.S.G.S. and at George Washington University. He has also coauthored several articles on modeling transport problems in ground water. APPENDIX 1. DEVELOPMENT OF FINITE-DIFFERENCE EQUATION FOR TRANSIENT FLOW IN A CONFINED HOMOGENEOUS, ISOTROPIC AQUIFER When using FDM to solve the ground-water flow equation, as shown in Figure 1.1, a network of grid points is first established throughout the region of interest. As may be seen, the aquifer is divided into rectangular blocks having a thickness equal to that of the aquifer, b. Each block has hydrogeologic properties associated with it and a node at its center at which the hydraulic head is defined for the entire block. Also note from Figure 1.1 that some of the blocks may be the site of wells. The next step is to take a water balance over a typical block (see Figure 1.2). To do this we consider one of the interior blocks and the four blocks connected to it. In this case, the center block is labeled 1; and Q 2 i, for example, represents the volumetric flux from block 2 to block 1; Ax and Ay are the spacings in the x- and y-directions, K Fig. 1.1. Finite-difference grid, showing typical node connections (after Freeze and Cherry, 1979). Fig. 1.2. Water balance over a typical finite-difference block (after Freeze and Cherry, 1979). respectively. The equation of continuity for transient, saturated flow states that the net rate of flow into any block must equal the time rate of change of storage within that block. With reference to Figure 1.2, the water balance for block 1 may be written as 9hi Q 21 + Qai + Qai + Qsi = Ax, Ay, S, — , (1.1) where S, is the storage coefficient for block 1. In the general case, the source/sink term, W, should also be included in equation (1.1); it was omitted in this development for convenience. The third step in developing a finite-difference approximation to the ground-water flow equation is to evaluate the volumetric fluxes using Darcy’s equation. For example, the flux Q 2t may be evaluated as Q 21 - Axj T 21 ( — ) 2 i , (1-2) dy where T 2 , is a representative value for the trans¬ missivity between nodes 1 and 2. Similar expressions may be written for Q 31 , Q 41 and Q 51 . Using Figure 1.3, the hydraulic head derivative may be evaluated as h “ h Q 2 i « Axj T 2j - , (1.3) Ay which is a cord slope approximation. For simplifica¬ tion, we assume the aquifer is homogeneous and isotropic so that T 2 , = T 3I = T 41 = T s , = T and S, = S 2 = S 3 = S 4 = S 5 = S. In addition, we assume a uniform grid spacing, so that Ax = Ay. Substitution for the volumetric fluxes in (1.1) now gives 408 h 2 + h 3 + h 4 + h 5 - 4h, = ^Ax 2 3hj "dt (1.4) This leaves the time derivative on the right side to evaluate. Using an expression similar to that in equation (1.3) gives 3h, h? - hf 1 —— _ . _ ——— ^ 3t At (1.5) where n is the new time level and At is the time step. Substitution for the time derivative in equation (1.4) and using general notation, a finite-difference equation for node (i,j) may be written as n + h n n i,j-l T a i+l,j + ^i-l,j + ^i,j + l n n S Ax 2 n -4h"i =-(h": 1,J T At 1,J C) ( 1 . 6 ) An equation similar to (1.6) is obtained for each node in our grid; however, boundary nodes may require special consideration. In summary, applica¬ tion of the FDM to a ground-water flow problem involves: (1) divide aquifer into grid blocks with nodes; (2) node spacing is determined by Ax and Ay; (3) take time steps, At; (4) obtain an algebraic equation for each node, and (5) solve matrix equation. APPENDIX 2. NUMERICAL MODEL TERMINOLOGY Numerical models are often referred to on the basis of minor characteristics. For example, a finite- difference model that uses the ADI method for matrix solution may be called the “ADI model.” Because of the large number of minor variations, the terminology is confusing. A partial list of terminology (including those terms most commonly used) is presented in outline form. Many of the Derivative Midway Between Node 2 and Node 1 . . a h _ h ? - h, ~ Ay Fig. 1.3. Approximation of hydraulic head derivative. terms are defined in the text; others are self- explanatory. For those terms that are not defined, the interested reader is referred to standard texts on numerical methods, such as those listed in the references. I. Finite-difference methods may be classified by: A. Type of Grid 1. Block-centered 2. Mesh-centered 3. Irregular shape (integrated finite-difference) B. Type of approximation 1. Explicit in time 2. Implicit in time 3. Central difference in space 4. Upstream weighting in space C. Type of matrix equation solution 1. Direct methods a. Gauss elimination b. Cholesky method c. Sparse matrix methods d. Special ordering techniques (e.g. D4) 2. Iterative methods a. Alternating direction implicit (ADI) b. Strongly implicit procedure (SIP) c. Line successive over-relaxation (LSOR) d. Point successive over-relaxation (PSOR) II. Finite-element methods may be classified by: A. Type of element 1. Triangles, 2-D 2. Quadrilateral, 2-D 3. Tetrahedron, 3-D 4. Prism, 3-D B. Type of approximating (basis) function 1. Linear i 2. Quadratic 3. Cubic 4. Hermite C. Type of method used to obtain integral equation 1. Variational 2. Weighted residual a. Galerkin b. Collocation D. Type of time approximation E. Type of matrix equation solution III. Special numerical techniques include: A. Method of characteristics B. Nonlinear techniques 1. Newton-Raphson method (fully implicit) 2. Semi-implicit method 3. Quasi-linearization 4. Picard iteration 5. Extrapolation Optimizing Pumping Strategies for Contaminant Studies and Remedial Actions by Joseph F. Keely Abstract One of the more common techniques for controlling the migration of contaminant plumes is the use of pumping wells to produce desired changes in local flow rates and hydraulic gradients. When seeking to optimize an array of pumping well locations and dis¬ charge rates, it is important to consider the effects that non-ideal aquifer conditions, well construction and demographic constraints produce. Heterogeneous and anisotropic aquifer conditions seriously compli¬ cate siting and discharge rate requirements for pump¬ ing wells because of the distorted cones of depression that result from withdrawing water in such settings. Proper screen selection, gravel pack emplacement and well development are crucial factors affecting the operational characteristics and economics of pumping wells; these factors are generally recognized, though often undervalued. The impacts that well depth and diameter, and screen length and position have on the effectiveness of pumping efforts are also often under¬ valued, with detrimental consequences. Perhaps the most difficult problems to overcome in designing pumping schemes, however, are posed by demographic constraints. Denial of property access, vandalism and the unpredictability of nearby water supply and irri¬ gation pumpage tend to wreak havoc with the best of pumping strategies. Introduction Safe storage and disposal of hazardous wastes have become major social issues because of the discovery that many sites lack proper precautions for the pre¬ vention of soil and water contamination. Ground water contamination has received the major share of socie¬ ty’s attention to these issues, primarily because the route of human exposure by this pathway is direct. In practical terms, this means that the level of cleanup of the damage done by contamination incidents is often dictated by social concerns (e.g. health risk). Plume stabilization by interception and control with perime¬ ter wells, injection and recovery loops, and other pumping schemes may be chosen as the “remedial action” appropriate for a particular plume. The affected plume maybe held in place and treated, it maybe held and allowed to move on after alternate public supplies have been located for downgradient water systems, it may be held in place to allow biodegradation of particular constituents, or it may be held until better treatment procedures can be devised. Factors Alfecting Pumping Strategies Hydrodynamic control and recovery strategies vary considerably in their efficiencies. Besides the obvious need to choose the well locations and flow rates care¬ fully, a number of other considerations demand atten¬ tion (Figure 1). Non-ideal aquifer conditions are a reality for virtually all real-life situations; heterogeneity is the rule rather than the exception. Three-dimen¬ sional anisotropy, as expresed by the vertical vs. hori¬ zontal hydraulic conductivity ratio, is a near certainty for most strata. Less visibly pronounced, yet almost as prevalent, is an expressed anisotropy in the horizontal plane of many strata. These commonplace non-ideal aquifer conditions complicate our perception of where a given plume can go (Fetter 1981) under both natural flow conditions and remedial action pumping schemes. The preferential flow paths that are created by buried lake beds, glacial outwash gravels, streambeds, coastal deposits and the like cannot be delineated without expensive and time-consuming field tests. Likewise, it is nearly impossible to accurately predict the magni¬ tudes of distortion in the cones of depression created by wells pumping from heterogeneous, anisotropic aquifers. Variations in the properties of the fluid in an aquifer, particularly the solution density, also can significantly affect the behavior of contaminant plumes (Jorgenson et al. 1982). Immiscible plumes with lower density than that of the native ground water will float at the surface of the saturated zone, traveling along the same general gradient, but traveling at a different rate than the underlying ground water. Immiscible plumes with greater density than that of the native ground water will sink through the ground water, losing small but significant amounts of low solubility constituents as they move. Miscible plumes of any density, by definition, mix intimately with native ground water. The duration of time required to achieve a specific dilution by this mixing changes markedly, however, and is generally inversely related to the den¬ sity. For most situations, the greater the density, the shorter the mixing period. The exception to this gen¬ eral rule would be the case of a large volume of highly dense, miscible fluid penetrating a shallow aquifer quickly enough to reach bedrock in a relatively undis¬ turbed form. Considerations Non-Ideal Aquiler Conditions: • Heterogeneity • Anisotropism • Variable density Well Construction Effects: • Partial penetration • Partial screening • Incomplete development Anthropogenic Influences: • Property access • Vandalism • Unknown pumpage/injection Other Factors: • Physiochemical attenuation • Biological transformations • Operational failures Figure 1. General considerations for optimizing pump¬ ing strategies These complexities work against us if we are ignor¬ ant of them. Working up an appropriate recovery sys¬ tem for a contaminant plume can be compared to designing an oil production system. What you get out depends directly on what you put in—up to a point. Where that break-even point comes is hard to say, given unknowns like the source strength and timing, and immeasurables like the dollar value of additional cancer victims. What is abundantly clear, however, is that there is a substantial minimum for serious play. One does not blithely draw up plans to pump and treat a plume until considerable manpower and funds are expended to obtain information on the natural flow direction, gradient and velocity. The question is usually one of how much to spend to reach some desired level of detail; the level of detail is set by social concerns. This seems to be logical application of technology for social need, but the logic may be shortsighted. If social concerns (based on preliminary evaluation of a contaminant incident) are minimal, there is no guaran¬ tee that such social concerns are appropriate. Addi¬ tional studies, which could delineate preferential flow paths and quantify factors affecting contaminant behavior, might well generate findings that would justify considerably greater or lesser social concern. Quite often data from preliminary investigations are limited to samples from shallow on-site wells, which may fail to signify the potential impact of dense plumes or seasonally-occurring leachate plumes that have moved off-site. Additionally, the preliminary investiga¬ tion wells are not normally installed to a sufficient depth for appreciation of the local stratigraphic and lithologic characteristics of the aquifer. In addition to a better understanding of where contaminants might go with the natural flow, a second powerful argument to avoid “penny-wise and pound- foolish” investigations concerns the need to provide the best Information possible for targeting well loca¬ tions and pumping strengths in remedial actions. The occurrence of specific heterogeneities can be used to advantage by locating wells near low permeability clay units to generate greater drawdown for a given pump¬ ing rate. Likewise, knowledge of the direction of the principal horizontal axis i n anisotropic strata can help to maximize the arrangement of the “troughs of depression” for wells to be located in such settings; knowledge of the magnitude of vertical anisotropy can help determine the amount of water pumped from strata containing contaminants vs. the amount of “clean" water from the other strata open to the well. This latter factor, vertical anisotropy, leads to examination of some of the more controllable items to be considered in optimizing pumping strategies—well construction effects. For example, the impact that par¬ tial penetration of a fully screened pumping well can have on the estimate of potential for contamination of a water supply or on the effectiveness of a remedial action scheme is tremendous (Saines 1981). The effect is to cause exaggerated drawdowns near the well. The magnitude of the effect is inversely dependent on the degree of penetration of the well into the aquifer, being greatest for slight penetration. Naturally, partial screen¬ ing of a fully penetrating well results in the same effect-greater drawdown for a given pumping rate as compared with a fully screened, fully penetrating well. Again, knowledge of these factors can be used to enhance a pumping scheme that is, for example, de¬ signed to maintain hydrodynamic control of a plume at the lowest possible level of pumpage. Lack of appre¬ ciation of these well construction effects can result in poor estimates of potential contaminant impacts on supply wells and in poorly designed remedial action schemes. Another effect worthy of mention is that generated by well development practices. If a well is properly developed, the drawdown measurable inside the well will agree with the level projected by close observation wells. More often, however, a well is not perfectly efficient because the well development pro¬ cedures were not adequate to remove drilling fluid fines and locally disturbed aquifer material resulting from the drilling process. These materials lower the permea¬ bility of the gravel pack and formation immediately adjacent to the well. The greater the degree of well inefficiency caused by lack of proper development the greater the amount of non-productive drawdown inside the well; this means that the well may never be able to pump at design capacity without risk of running dry, and it means increased operating expense due to the additional pump lift required. What it may also portend, for seriously Inefficient wells, is that certain strata penetrated by the well may be effectively sealed by drilling mud or by natural clays that were smeared over the borehole face by the actions of the drilling operation. Such “sealed off’ strata may cany the bulk of the contaminants, resulting in poor recovery of the plume. Some of the most significant, though less control¬ lable, factors that should be considered when optimiz- 64 Summer 1984 ing pumping strategies concern direct anthropogenic influences: denial of property access, vandalism and unknown pumpage all tend to wreak havoc with the best laid plans. Bedient et al. (1984) describe efforts to delineate a plume of contaminants migrating under a residential area from an abandoned wood creosoting plant in Conroe. Texas: “Several wells exist in the general flow direc¬ tion, but not directly downgradient from the waste pit locations. Access was not granted for installing monitoring wells... Approximately 50 percent of the chloride plume has been defined since the monitoring well network is incomplete at this time... Completion of the monitoring well network is needed to capture the center of the contaminant plume. This will require more wells downgradient on land that has not previously been accessible for investigation." The granting of property access during investiga¬ tions of ground water contamination incidents in populated areas is no trivial matter. One typically finds it necessary to contact not only homeowners and land¬ lords for private property access, but also to negotiate with company engineers, vice presidents and attorneys for commercial property access. It is quite normal for such negotiations to be involved and protracted as city councils, educational boards, corporate headquarters and other bureaucratic entities are asked to concur in signing access agreements containing provisions deemed necessary to ensure against incurred liability and potential damage. The role played by unknown pumping and/or injection wells operating near a remedial action pump¬ ing system is subtle but far-reaching. Such unknown stresses can significantly distort the flow field and render remedial actions ineffective. Projections on plume movement made during an investigation of a ground water contamination incident would also be in error if unknown wells are causing distortion in the flow field: both the direction and the speed of the plume could be dramatically altered. The reason for the subtlety of the effects of many such wells is that their cyclic, seasonal or on-demand pumping sched¬ ules allow them to be detected only by continuous recording of water level changes at numerous points around the zone of interest. Since aquifer responses at a given observation point are somewhat non-unique, merely detecting extraneous sources of drawdown does not automatically result in identification of the sources. There are a few other important factors to consider that also affect pumping strategies. The physiochemi- cal properties of the contaminant itself can result in a need to pump several pore volumes from each unit volume of aquifer to be decontaminated. Sorption, ion exchange and speciation changes can result in re¬ tarded movement of contaminants relative to the average velocity of the water with which they are initially associated. Biotransformation of contami¬ nants may result in reaction products (daughter products) that are of greater or lesser toxicity, mobility and persistence—in other words, uncertain contami¬ nant behavior. Unlike the aquifer properties of storage coefficient, saturated thickness and hydraulic conduc¬ tivity, which can be readily determined, the current state-of-the-science with regard to determining the potentials for physiochemical attenuation and bio- transformation is not up to the level of routinely providing reliable answers on a site-specific basis. Finally, an obvious but often overlooked considera¬ tion involved in optimizing pumping strategies is the need to develop adequate contingencies for operational failures. This means some intentional overdesign for reserve capacity, total redundancy of key wells and electronic controls, backup power systems and so on. It also means bonding or insurance against unfore¬ seen catastrophies so that as little downtime is expended as possible. It may also mean that an escrow account or trust fund must be established to provide the necessary capital for replacement of bumed-out or inadequate pumps, deepening or abandonment of existing wells, or drilling of additional wells. Capture Zones vs. Zones of Pressure Influences Keely and Tsang (1983) introduced the term “cap¬ ture zone” to describe that portion of the aquifer affected by pumping which actually yields water to the well. They have shown that the capture zone is gener¬ ally much smaller than the zone of pressure influence because a balance is achieved, under steady-state conditions, between the pull of water back toward the well from its downgradient side and the tendency of the natural flow system to move on further downgra¬ dient. Figure 2 is a series of four idealized illustrations that present conceptualizations of how the size of the capture zone changes, relative to the zone of pressure influence/cone of depression, as the local gradient is increased. In Figure 2A the well is pumping from a stagnant aquifer, indicated by the flat pre-pumping surface, overlaid on the theoretical cone of depression that would occur during pumpage. For stagnant aquifer conditions, the capture zone is everywhere identical to the zone of pressure influence and flow is radial into the well. As the successive diagrams indi¬ cate, however, non-stagnant aquifer conditions lead to smaller capture zones (Figures 2B to 2D). The slopes of the pre-pumping surfaces are overlaid on the theoretical drawdown cones in each frame of Figure 2 to emphasize the interaction of the natural flow system with the pumping stress to yield a capture zone smaller than the zone of pressure influence. There is no intention to show the net surface resulting from pumpage by subtracting theoretical drawdown values from pre-pumping water elevations. These sketches do have the cosmetic drawback of showing crossing water level lines/curves, but the point is to illustrate the individual components of the net surface (cross- sectional view) and how they interact to yield a capture zone (three-dimensional view). The flow lines generated by pumping a well from an idealized aquifer (homogeneous, isotropic, constant density, etc.) under different natural flow conditions are shown in Figure 3. In Figure 3 a well pumping l,000m 3 /day froma 10m thick aquifer having a poros¬ ity of 0.10 and a hydraulic conductivity of lOOm/day has uniform radial flow under stagnant aquifer condi¬ tions (e.g. natural flow velocity equal to zero). When a mild hydraulic gradient (0.0001) is imposed on the same system (Figure 3B), the resulting natural flow velocity (0.1 m/day) is insufficient to significantly affect the flow lines, and uniform radial flow is nearly main¬ tained. With a more moderate hydraulic gradient (0.001), the resulting natural flow velocity (l.Om/day) is sufficient to sweep away many of the flow lines and the capture zone is clearly evident (Figure 3C). Where a steep gradient (0.01) is present, the capture zone diminishes to a small fraction of the zone of pressure VADOSE ZONE PRE-PUMPING SURFACE CONE OF DEPRESSION SATURATED ZONE CROSS-SECTIONAL CONCEPTUALIZATION VADOSE ZONE PRE-PUMPING SURFACE CONE OF DEPRESSION SATURATED ZONE £ * e V nat ). Using this relationship, it is apparent that the maximum width (W max ) of the capture zone of a well is directly and linearly related to its flow rate (Q), and is inversely related to the natural flow velocity (V nat ). For the example discussed here regarding a single well pumping l,000m 3 /day. the maximum width of the capture zone is 1,000m when the natural flow velocity is l.Om/day, and is 100m when the natural flow velocity is lOm/day. Each of the five wells in the second example discussed pumps at a flow rate equal to one-fifth the flowrate of the well in the first example (200m 3 /day), and each, therefore, has a capture zone the maximum width of which is one-fifth that of the single well (200m). Hence, by comparing Figure 3 with Figure 4, it is seen that the way in which the total pumpage is distributed does directly affect the distri¬ bution of the capture zone(s), but does not affect the magnitude or total area of the capture zone(s). Also to be seen in Figures 3 and 4 is that increasing the natural flow velocity estimate can have a dramatic impact on the effectiveness of the pumping strategy. Given the order-of-magnitude uncertainty so often associated with hydraulic conductivity estimates, it is not surprising that many seemingly acceptable reme¬ dial action schemes are doomed to fail miserably. A more complicated example provides further illus¬ tration of these points. Assume that we have the same 500 . 0. -500. - 1 -n- 1 i |-r- 1 - r - 1 - 500. i i r —i-1-1——i- r— —i- 1 - V*l: 0.0 m/d V*l: 0.1 m/d ' K . " - - * * - 0. * * r*• * * * ‘ * • *i 1_J_1_j_1_i_1_L_1_ -500. -*—*-—*-1—i-Li—i-1 i ■ -500. 0 . 500. -500. 0 . 500. Figure 5A Stagnant aquifer conditions Figure 5B. Mild hydraulic gradient (0.0001) 500. r-r-.-r I V«l: 1.0 m/d L I Figure 5C. Moderate hydraulic gradient (0.001) Figure 5D. Steep hydraulic gradient (0.01) Figure 5. Flow line plots tor a circle of eight wells, each discharging 125m 3 /day from an aquifer with 10m saturated thickness, 100m/day hydraulic conductivity and 0.10 porosity NOTE: Scale is in meters and natural flow proceeds from lower left-hand comer to upper right-hand comer of each plot at the velocity indicated. aquifer conditions and total pumpage limitation (l,000m 3 /day) as the preceding examples. We will distribute the pumpage uniformly by pumping each of eight wells at 125m 3 /day. The eight wells are evenly spaced around a circle of200m radius. We are trying to hold a plume within the circle. With stagnant aquifer conditions to low natural flow velocities, the plume appears to be stable; no flow lines pass through the circle (Figures 5A and 5B). At moderate to high natural flow velocities, however, the situation is quite different; flow lines readily pass through the circle, indicating that the plume stabilization attempt has failed (Fig¬ ures 5C and 5D). A pump and treat scenario can be examined by modifying the example shown in Figure 5 to change the operation of the eight wells from pumping to injecting and by adding a major pumping well in the center of the circle. The single pumping well will with¬ draw l,000m 3 /day from the plume. The withdrawn water will be treated and re-injected into the eight injections wells at 125m 3 /day each. At zero to low flow velocities, the injected water flows radially toward the central pumping well, forming a closed loop for recov¬ ery and treatment of the plume (Figures 6A and 6B). At moderate to high natural flow velocities, the recovery loop is broken and an increasing amount of the injected water and the plume are swept away by the regional flow (Figures 6C and 6D). It must be empha¬ sized that the cones of impression or depression of the wells overlap significantly for all of the multiwell exam¬ ples discussed so far. Despite those overlaps, the net surface resulting from the natural gradient and the 500 r 1 r "" r 1 V • 1: 0.0 m/d i .1.t-r-i-r* 500. » T •!! V*l: 0.1 m/d i r i~i- 0. - 0. - # * * X X X — 500. __i_L._i_i_1._i__i. -500. 0. Figure 6A. Stagnant aquiler conditions 500. ' r - ’■ T-.r ,| ,'i—.-.-i .V• I:.-1.0 m/d J 500. -500 _i— i i i -500. 0. Figure 6B. Mild hydraulic gradient (0.0001) i i i i 500 500. r V»l; 10 m/d T 250. x 0. - 0. X .-.x / x x x -250. - x X X -500. i.. j. . l- j -500. 0 i. .i . i 500 Figure 6C. Moderate hydraulic gradient (0.001) -500. I_L. -500. -250. Figure 6D. Steep hydraulic gradient (0.01) _I_l_.L. - -1 - J 0. 250. 500. Figure 6. Flow line plots lor a single well discharging l.OOOmYday, encircled by eight wells injecting 125m 3 / day Into an aquiler with 10m saturated thickness, lOOm/day hydraulic conductivity and 0.10 porosity NOTE: Scale Is In meters and natural flow proceeds from lower lelt-hand comer to upper right-hand comer o1 each plot at the velocity indicated. water level changes due to pumpage and/or injection is shaped such that the streamlines are truly as pre¬ sented here. For further discussion of capture zones and velocity distribution plots, see Keely and Tsang (1983). The detailed theoretical development and source code listings for the models that were used to generate the stream line plots shown here are given in Javandel et al. (1984). A Little More Detail It was quite clear in each of the preceding examples that the pumping strategy began to fail as the natural flow velocities became appreciable. The tendency to fail is generally becoming evident at a natural flow velocity of l.Om/day and is beyond question at a natural flow velocity of lOm/day. Figure 7 shows that failure of each design is certain at 5.0m/day as well; the point at which the flow lines break through must be at much lower natural flow velocities. In Figure 8 the natural flow velocity has been reduced to 0.5 and 0.4m/day for the last two examples only. Breakthrough of the streamlines (failure of the pumping strategy) occurs somewhere between the 0.4 and 0.5m/day natural flow velocities. Similar compar¬ isons for the first two examples are not presented because flow line breakthrough does not apply to the first example (a single production well) and the flow line did not indicate breakthrough at l.Om/day for the second example (a line of five wells). The presence of an unknown well is being studied in Figure 9. A major pumping well (1.000m 3 /day) has been arbitrarily located downgradient of the same line 0. r v y - r ,V«I:5.Q Wtf ’ T , T J "TT T 7 —y ~r 500. V*l: SlO.m/d r —1 r " .** .** . r • .* t * ■ • * . * ■ .1 ■ '.X • ’ . •' .* * • . * 4 v* .* .•* .•* ' X. . r" X ; 0. X . ■ . •• / .. ’. 'X /' /' /'.. - •' ■i *-•" i •' i- .i . f • l •1 ‘ 1. .1 -•* -500. ■■•■■.. 'i . i .1 ■' ■ 1 ■' I'' •• 1-’’ • •' i -• 0. 500 -500. 0. 500. -500. -500 Figure 7 A. Single well discharging l.OOOmVday 500. -~i—:—r——r-r Vrl: 5.0 m/d . i-r-r—-r Figure 7B. Line of five wells, each discharging 200m 3 /day 500 . [s y y 1 7 l 7y:-W V*h 5.0 m/d ■ • • A 250. - L' I r 0. . x x . x x. 0 X X X X ■■ j X . -250. - h I f -500. I l-J—L^L J:^1.l : :_J -5oo. J-: -500. 0 500. -500. -250 0. 250. ,:1 500. Figure 7C. Circle of eight wells, each discharging Figure 7D. Single well discharging l.OOOmVday, encircled 125m 3 /daY by eight wells injecting 125m 3 /day each. Figure 7. Comparison of pumping ar rays in an aquifer with 10m saturated thickness, lOOm/day hydraulic conductivity, 0.10 porosity and 0.005 hydraulic gradient NOTE: Scale is in meters and natural flow proceeds from lower left-hand comer to upper right-hand comer of each plot at the velocity indicated. of five wells discussed in the second example. Naturally, under stagnant aquifer conditions, the unknown well creates a hydraulic divide by distorting the flow field, but it does not cause breakthrough of the flow line from across the line of five wells (Figure 9A). With a natural flow velocity of 0.5m/day, however, flow lines do begin to break through the line of five pumping wells (Figure 9B). Substantial failure of the pumping scheme occurs at l.Om/day natural flow velocity (Fig¬ ure 9C). Contrast the onset of breakthrough due to unknown pumpage (Figure 9B) with the same situa¬ tion in the absence of the unknown pumpage (Fig¬ ure 9D). The impact of the unknown well is staggering, not only because flow line breakthroughs are occur¬ ring, but the collective size of the capture zones of the five pumping wells is being substantially reduced. Another illustration of the impact of an unknown well on the effectiveness of a pumping scheme is shown in Figure 10, which is the same example as discussed earlier (Figure 6) for a closed-loop aquifer rehabilita¬ tion system. Under stagnant aquifer conditions, the unknown well diverts flow away from two of the injec¬ tion wells (Figure 10A). At l.Om/day natural flow velocity, the unknown well diverts flow from five of the eight injection wells (Figure 1 OB). It also allows flow to break away from the well Held entirely, as indicated by the streamline leaving the uppermost injection well and heading downgradient in Figure 10B. The re¬ gional flow lines were omitted from Figure 10 and some of the diagrams in previous figures because inclusion of those flow lines would create confusion due to the excessive number of plotted points. 500. T 1 | •• r - r - i V •!: 0.4 1- f m / d 500. i .?..■• r- ,V#I: 0.5 m/d -500. . -500. i ■ J .• i 0. • 1 l i i -500. 1 — i .i t J 500. -500. 0. i •) 500. Figure 8A. Circle of eight wells, each discharging 125m 3 /day with 0.0004 hydraulic gradient Figure 8B. Circle of eight wells, each discharging 125mVdaY, with 0.0005 hydraulic gradient 500. Vel: 0.4 m/d 500. t V • I: T [ 0.5 m/d " 1" " r “r 250. - * x . x 0. — x x-" x x x X -250. - 250. 0 . -250. -500. _l_ . I -500. -250. 0. 250. 500. -500._l 1 i I i I ■ i -500. -250. 0. 250. 500. • Figure 8C. Single well discharging 1 .OOOmVday, encircled by eight wells injecting 125m 3 /day each, with 0.0004 hydraulic gradient Figure 8D. Single well discharging 1 .OOOmVday. encircled by eight wells injecting 125m 3 /day each, with 0.0005 hydraulic gradient Figure 8. Detailed views of the onset ol flow line breakthroughs lor two plume control strategies in an aquiler with 10m saturated thickness, lOOm/day hydraulic conductivity and 0.10 porosity NOTE: Scale Is In meters and natural flow proceeds from lower left-hand comer to upper right-hand comer ol each plot at the velocity Indicated. Conclusions Heterogeneity, anisotropy, partial penetration and so on distort drawdown patterns and associated velocity distributions. If known, such influences can be used to enhance recovery efficiencies for remedial actions. If unknown, such influences may cause recovery effi¬ ciencies to be substantially lowered. Similarly, predic¬ tions of plume migration in non-ideal aquifers under non-pumping/natural flow conditions will be strength¬ ened by specific knowledge regarding the occurrences, extent and magnitude of the non-ideal condition(s). Such predictions may be seriously in error if non-ideal conditions are not evaluated properly. Denial of property access, loss by vandalism and unpredictable operation of nearby wells are also major sources of uncertainty in predicting contaminant migration and in designing remedied actions. Though commonly perceived to be less of an impact on opti¬ mizing pumping strategies than non-ideal aquifer conditions, these factors may indeed be the most uncontrollable and the most detrimental to opera¬ tional success. Other factors that have major impacts are the physiochemical attenuation and biotransfor¬ mation potentials of the individual contaminant; it is not yet economically feasible to conduct adequately detailed studies of these potentials on a routine site- specific basis. Finally, a factor often overlooked that greatly impacts optimization efforts is the risk of 500 . [ 500. 250. - * .••• * o. - ■ ""■■■■ * ' ■250. - -500._i ! i I -500. -250. 0. 250. 500. Figure 9A. Stagnant aquiler conditions 500. r ...i- '---1 i !*V• 1: 0.5 m/d -500. -250. 0. 250. 500. -500. 0 . 500. Figure 9C. Hydraulic gradient of 0.001 Figure 9D. Hydraulic gradient of 0.0005—without the unknown well Figure 9. Influence of an unknown well discharging 1 ,000m 3 /day on flow line breakthroughs for a line of five wells discharging 200mVday each from an aquifer with 10m saturated thickness, lOOm/day hydraulic conductivity and 0.10 porosity NOTE: Scale Is In meters and natural flow proceeds from lower left-hand comer to upper right-hand comer of each plot at the velocity Indicated. mechanical and electrical operational failure; adequate contingency plans must provide certain minimal levels of excess/reserve capacity and redundancy of key sys¬ tem components. The capture zones of wells do not equal their asso¬ ciated zones of pressure influence (cones of depres¬ sion), except for stagnant aquifer conditions. Velocity distribution plots must be constructed to define potentials of contaminant migration. In particular, plotting the streamlines for various scenarios involv¬ ing pumping and/or injection wells subject to a spe¬ cific natural flow velocity can greatly assist the ground water professional in selection of an optimal pumping strategy. Acknowledgments Thanks go to Rosemary Keely and Christine Doughty for (computer) drafting the illustrations. Thanks also go to Renae Daniels for typing this manuscript. Disclaimer Although this article was produced by an employee of the United States Environmental Protection Agency, it has not been subjected to Agency review and therefore does not necessarily reflect the views of the Agency; no official endorsement should be Inferred. 500 . 250. 0 . -250. -500. -,-,-<— V«l: 0.0 m/d -1- —.-r —i— 500. ! 1 -T i V • 1: 1.0 m/d 1 i— 1 — — 250. — X - x:. X ** . 0. — "X.;. X •' ••x - - * X — * 250. — X ' ■_1_l_ _1_ . i i_ 500. i L i J_l— :X -500. -250. 0. 250. 500 Figure 10A. Stagnant aquiler conditions -500. -250. Figure 10B. Hydraulic gradient ol 0.001 _l_i. ..... 0. 250. 500 Figure 10. Influence ol an unknown well discharging 1,000mVday on flowline breakthroughs lor a single well discharging 1 .OOOmVday that is encircled by eight wells injecting 125m 3 /day into an aquiler with 10m saturated thickness, lOOm/day hydraulic conductivity and 0.10 porosity. NOTE: Scale is in meters and natural flow proceeds from lower left-hand comer to upper right-hand comer ol each plot at the velocity indicated. References Bedient, P.B..A.C. Rodgers, T.C. Bouvette, M.B.Tomson and T.H. Wang. 1984. Ground water quality at a creosote waste site. Ground Water, v. 22, no. 3, pp. 318-329. Fetter. C.W. 1981. Determination of the direction of ground water flow. Ground Water Monitoring Review, v. 1, no. 3. pp. 28-31. Javandel. I., C. Doughty and C.F. Tsang. 1984. Ground water transport: handbook of mathematical models. American Geophysical Union, Water Resources Monograph 10. Jorgensen, D.G., T. Gogel and D.C. Signor. 1982. Determination of flow in aquifers containing vari¬ able-density water. Ground Water Monitoring Re¬ view, v. 2, no. 2. pp. 40-45. Keely, J.F. and C.F. Tsang. 1983. Velocity plots and capture zones of pumping centers for ground water investigations. Ground Water, v. 21, no. 6, pp. 701-714. Saines, M. 1981. Errors in interpretation of ground water level data. Ground Water Monitoring Review, v. 1, no. 1, pp. 561-61. Biographical Sketch Joseph F. Keely (Robert S. Kerr Environmental Research Laboratory, U.S. EPA, P.O. Box 1198, Ada, OK 74820) received his B.S. in professional chemistry and M.S. in hydrology from the Uni¬ versity of Idaho (Moscow). He is employed as a hydrologist at EPA’s Robert S. Kerr Environ¬ mental Research Laboratory, where his efforts are directed toward geohydraulic and hydro¬ geochemical investigations of ground water con¬ tamination incidents, coordination of ground water modeling research and instructional assistance. He sits on the Policy Board of the International Ground Water Modeling Center and serves as expert witness to the U.S. Depart¬ ment of Justicefor Superfund cases. r V DATA QUALITY REQUIREMENTS ■ Data collected before or after model selection? ■ Spatial variability controls ■ Temporal variability controls ■ Sensitivity analyses ■ Comparability IGWMC GROUNDWATER MODELING REPRINT Quality Assurance in Coaster Simulations of Groundwater Contamination Paul K.M. van der Heijde International Ground Water Modeling Center reprint Bnviroimental Software, 1987, Vol 2, No. 1 GWMI 87-08 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 Groundwater Contamination: P.K.M. van der Hei/de Quality Assurance in Computer Simulations of Groundwater Contamination Paul K.M. van der Heijde International Ground Water Modelling Center, Holcomb Research Institute, Butler University, Indianapolis, IN 46208, USA ABSTRACT In the development of policies and regulations for groundwater protection, in permitting, and In planning monitoring and remedial actions, the role of mathematical models is growing rapidly. Because water-resource management decisions should be based on technically and scientifically sound methods, quality assurance (QA) needs to be applied to groundwater modeling, both In model development and field studies, and should also play an Important part In model selection. Important aspects of QA In groundwater model development are peer review, and verification and validation of the computer code and Its underlying theoretical principles. This paper discusses the role of review and testing as part of an overall QA approach, and addresses QA In model selection and field application. Key Words: groundwater, mathematical models, quality assurance, model validation, pollution, model selection INTRODUCTION The science of groundwater flow and contaminant transport Is not yet an exact field of knowledge. Although the physical processes Involved obey known mathematical and physical principles, exact aquifer and contaminant characteristics are hard to obtain and often make even plume definition a difficult task. However, where these characteristics have been reasonably estab¬ lished, groundwater models may provide a viable. If not the only, method to predict contaminant transport, to locate areas of potential environmental risk, and to assess possible remediation/corrective actions |lj. Mathematical models are used to help organize the essential details of complex groundwater management problems so that reliable solutions are obtained. Applications Include a wide range of technical, economic, and sociopolitical aspects of groundwater supply and protection (2, 3, 4, 5). A groundwater protection policy based on monitoring Is by Its very nature always reactive, not preventive; however, model-based policies and regulations can be both preventive and reactive. Because adequate on-site moni¬ toring Is not always feasible due to costs, available manpower, or site accessibility, models can provide a viable and effective alternative. An optimal approach to the management of groundwater resources Includes the Integrated use of modeling and monitoring strategies. The role of groundwater-flow and contaminant-trans- port models In the development of policies and regula¬ tions, In permitting, and In planning of monitoring and remedial action, Is continuing to grow. Some of the prin¬ cipal areas where mathematical models can now be used to assist In the management of groundwater protection pro¬ grams are (6 ]: • development of regulations and policies • planning and design of corrective actions and waste storage facilities • problem conceptualization and analysis • development of guidance documents • design and evaluation of monitoring and data collection strategies • enforcement Specifically, groundwater modeling plays or can play a role In: • determining or evaluating the need for regulation of specific waste disposal, agricultural, and Industrial practices • analyzing policy Impacts such as evaluating the consequences of setting regulatory standards and banning rules, and of delisting actions • assessing exposure, hazard, damage, and health risks • evaluating reliability, technical feasibility and effectiveness, cost, operation and maintenance, and other aspects of waste-disposal facility designs and of alternative remedial actions • providing guidance In siting of new facilities and In permit issuance and petitioning • detecting pollutant sources • developing aquifer or well-head protection zones • assessing liabilities such as post-closure liability for disposal sites These activities can be broadly categorized as either site-specific or generic modeling efforts, and these cate¬ gories can be further subdivided into point-source or non¬ point-source problems. The success of these modeling efforts depends on the accuracy and efficiency with which the natural processes controlling the behavior of ground- water, and the chemical and biological species It trans¬ ports, are simulated. The accuracy and efficiency of the simulations. In turn, depend heavily on the applicability of the assumptions and simplifications adopted in the model(s), on the availability of reliable data, and on subjective judgments made by the modeler and management. If litigation Is Involved, the model code itself and Its theoretical foundation may become contested. There- 0266-9836/87/010019-07 $2 00 * 1987 Computational Mechanics Publications Paper received on December 8, 1986 Referee: Dr. Paolo Zannetti ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 19 Groundwater Contamination: P.K.M. van der Heijde fore, adequate guidelines should be developed for selec¬ tion of simulation codes to be used under such circum¬ stances. Such guidelines should cover code review, vali¬ dation, and documentation and should be widely accepted. It Is of the highest importance that water resource management decisions be based on the use of technically and scientifically sound data collection. Information processing, and Interpretation methods. Quality Assurance (QA) provides the mechanisms to ensure that decisions are based on the best available data and analyses. This paper discusses QA guidelines applicable to groundwater modeling and the role of QA In the model selection process. QUALITY ASSURANCE IN GROUNDWATER MODELING Quality assurance In groundwater modeling Is the procedural and operational framework put In place by the organization managing the modeling study, to assure tech¬ nically and scientifically adequate execution of all pro¬ ject tasks Included In the study, and to assure that all modelIng-based analysis Is verifiable and defensible |7|. QA in groundwater modeling should be applied to both model development and model application and should be an Inte¬ gral part of all projects. The two major elements of quality assurance are quality control (QC) and quality assessment. Quality control refers to the procedures that ensure the quality of the final product. These procedures Include the use of appropriate methodology, adequate vali¬ dation, and proper usage. To monitor the quality control procedures and to evaluate the quality of the products of field studies, quality assessment is applied. It consists of two ele¬ ments: auditing and technical review. Audits are pro¬ cedures designed to assess the degree of compliance with QA requirements, conmensurate with the level of QA pre¬ scribed for the project. Compliance Is measured In terms of traceability of records, accountability (approvals from responsible staff), and fulfillment of commitments In the QA plan. Technical review consists of Independent evalua¬ tion of the technical and scientific basis of a project and the usefulness of Its results. QA Is the responsibility of both the project team (quality control and Internal evaluation) and the con¬ tracting or supervising organization (quality assessment). QA should not drive or manage the direction of a project nor Is QA Intended to be an after-the-fact filing of technical data. Various phases of quality assessment exist for both model development and application. First, review and testing Is performed by the author, and sometimes by other employees not involved in the project, or by Invited experts from outside the organization. Also to be con¬ sidered Is the quality assessment by the organization for which the project has been carried out. Again, three levels can be distinguished: project or product review or testing by the project officer or project monitor, by technical experts within the funding or controlling organization, and by an external peer review group. Decisions by natural resources and environmental managers rest on the quality of environmental data and data analysis; therefore, program managers in regulatory agencies should be responsible for: (1) specifying the quality of the data required from environmentally related measurements and for the level of problem-solving data analysis; and (2) providing sufficient resources to assure an adequate level of QA. QA procedures should be contained in a QA plan to be developed for each modeling study. The plan lists the measures required to achieve prescribed quality objec¬ tives. Major elements of such a QA plan are: (1) formu¬ lation of QA objectives and required quality level In terms of validity, uncertainty, accuracy, completeness, and comparability; (2) development of operational proce¬ dures and standards for performing adequate modeling studies; (3) establishing a paper trail for QA activities In order to document that standards of quality have been maintained; and (4) Internal and external auditing and review procedures. The QA plan should also specify Individual responsibilities for achieving these goals. Model Development Ideally, QA should be applied to all codes currently In use and yet-to-be-developed codes. Relevant QA proce¬ dures include such aspects as the verification of the mathematical framework, field validation, benchmarking, and code comparison. A detailed discussion of model testing and review Is described In the second part of this paper. QA for code development and maintenance should Include complete record-keeping of the model development, of modifications made In the code, and of the code-valida¬ tion process. The paper trail for QA In model development consists of reports and files on the development of the model. The reports should Include a description of: • assumptions • parameter values and sources • boundary and Initial conditions • nature of grid and grid design justification • changes and verification of changes made In code • actual input used • output of model runs and Interpretation • validation (or at least calibration) of model In addition, depending on the level of QA required, the following files may be retained (in hard-copy and, at higher levels. In digital form): • version of source code used • verification Input and output • validation Input and output • application Input and output If any modifications are made to the model coding for a specific problem, the code should be tested again; all QA procedures for model development should again be applied. Including accurate record keeping and reporting. All new Input and output files should be saved for inspec¬ tion and possible reuse. Model Application QA In model application should address all facets of the model application process: • correct and clear formulation of problems to be solved • project description and objectives • modeling approach to the project • Is modeling the best available approach and If so, Is the selected model appropriate and cost- effective? • conceptualization of system and processes. Including hydrogeologic framework, boundary conditions, stresses, and controls • explicit description of assumptions and slmplifIcatlons • data acquisition and Interpretation • model selection, or justification for choosing to develop a new model • model preparation (parameter selection, data entry or reformatting, gridding) • the validity of the parameter values used in the model application • protocols for parameter estimation and model call- 20 ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 Groundwater Contamination: P.K.M van der Heijde bration to provide guidance, especially for sensi¬ tive parameters • level of information In computer output (variables and parameters displayed; formats; layout) • identification of calibration goals and evaluation of how well they have been met • sensitivity analysis • postsimulation analysis (including verification of reasonability of results, interpretation of results, uncertainty analysis, and the use of manual or automatic data processing techniques, as for contouring) • establishment of appropriate performance targets (e.g., 6-foot head error should be compared with a 20-foot head gradient or drawdown, not with the 250-foot aquifer thickness!); these targets should recognize the limits of the data • presentation and documentation of results • evaluation of how closely the modeling results answer the questions raised by management A major problem in model use is model credibility. In the selection process special attention should be given to ensure the use of qualified models that have undergone adequate review and testing. As is the case with model development QA, all data files, source codes, and executable versions of computer software used in the modeling study should be retained for auditing or postproject re-use. MODEL REVIEW AMD TESTING Before a groundwater model is used as a planning and decision-making tool, its credentials must be established, Independently of Its developers, through systematic testing and evaluation of the model's characteristics. Code testing Is generally considered to encompass verifi¬ cation and validation of the model 18]. To evaluate groundwater models in a systematic and consistent manner, the International Ground Water Modeling Center (IGWMC) has developed a model review, verification, and validation procedure 15). Generally, the review process is qualitative In nature, while code testing results can be evaluated by quantitative performance standards. Model Review A complete review procedure comprises examination of model concepts, governing equations, and algorithms chosen, as well as evaluation of documentation and general ease-of-use, and examination of the computer coding |5, 9). If the model has been verified or validated by the author, the review procedure should include evaluation of this process. To facilitate thorough review of the model, detailed documentation of the model and its developmental history Is required. In addition, to ensure Independent evalua¬ tion of the performed verification and validation, the computer code should be available or at least accessible for implementation on the reviewer's computer facilities, together with a file containing the original test data used In the code's verification and validation. Review should be performed by experienced modelers knowledgeable in theoretical aspects of groundwater modeling. Because review is rather subjective In nature, selection of the reviewers is a sensitive and critical process. Model Examination Model examination determines whether anything funda¬ mental was omitted in the Initial conceptualization of the model. Such a procedure determines whether the concepts of a model adequately represent the nature of the system under study, and identifies the processes and actions pertinent to the model's intended use. The examination also determines whether the equations representing the various processes are valid within the range of the model's applicability, whether these equations conform mathematically to the intended range of the model's use, and whether the selected solution approach is the most appropriate. Finally, model examination determines the appropriateness of the selected initial and boundary conditions and establishes the applicability range of the model. For complex models, detailed examination of the implemented algorithms is required to determine whether appropriate numerical schemes, In the form of a computer code, have been adopted to represent the model (101. This step should disclose any inherent numerical problems such as non-uniqueness of the numerical solution, inadequate definition of numerical parameters, Incorrect or nonopti- mal values used for these parameters, numerical disper¬ sion, numerical instability such as oscillations or diver¬ gent solution, and problems regarding conservation of mass. In addition, the specific rules for proper appli¬ cation of the model should be analyzed from the perspec¬ tive of Its Intended use. These rules Include data assignment according to node-centered or block-centered grid structure for finite-difference methods; size and shape of elements In integrated finite-difference and finite-element methods; grid size variations; treatment of singularities such as wells; approach to vertical aver¬ aging In two-dimensional horizontal models or layered three-dimensional models; inclusion of partial solutions In analytical element, methods; and treatment of boundary conditions. Consideration Is also given to the ease with which the mathematical equations, the solution procedures, and the final results can be physically Interpreted. Evaluation of Model Documentation Model documentation Is evaluated through visual Inspection, comparison with existing documentation standards and guidelines, and through* its use as a guide In preparing for and performing verification and validation runs. Good documentation Includes a complete treatment of the equations on which the model Is based, of the under¬ lying assumptions, of the boundary conditions that can be Incorporated In the model, of the method used to solve the equations, and of the limiting conditions resulting from the chosen method. The documentation must also Include a user's manual containing Instructions for operating the code and preparing data files, example problems complete with Input and output, programmer's instructions, computer operator's instructions, and a report of the initial code verification. Evaluating Ease of Use The data files provided by the model developer are used to evaluate the operation of the code and the user's guide through a test-run process. In this stage special attention is given to the rules and restrictions (“tricks," e.g., to overcome restrictions in applic¬ ability) necessary to operate the code, and to the code's ease-of-use aspects 111]. Computer Code Inspection Part of the model review process Is the inspection of the computer code. In this inspection attention is given ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 21 Groundwater Contamination: P.K.M. van der Heijde to the manner In which modern programming principles have been applied with respect to code structure, optimal use of the programming language, and Internal documentation. This step helps reveal undetected programming or logic errors, hard to detect in verification runs. MODEL VERIFICATION The objective of the the verification process Is twofold: (I) to check the accuracy of the computational algorithms used to solve the governing equations, and (2) to assure that the computer code is fully operational. To check the code for correct coding of theoretical principles and for major programming errors ("bugs"), the code is run using problems for which an analytical solu¬ tion exists. This stage Is also used to evaluate the sensitivity of the code to grid design, to various domi¬ nant processes, and to a wide selection of parameter values |9, 12, 13. 141. Although testing numerical computer codes by com¬ paring results for simplified situations with those of analytical models does not guarantee a fully debugged code, a wel 1-selected set of problems ensures that the code's main program and most of Its subroutines. Including all of the frequently called ones, are being used In the testing. In the three-level test procedure developed by the International Ground Water Modeling Center (IGWMC), this type of testing Is referred to as level I (15). Hypothetical problems are ^used to test special features that cannot be handled by simple close-form solutions, as In testing Irregular boundary conditions and certain heterogeneous and anisotropic aquifer properties; this is the IGWMC level II testing. For both level I and level II testing, sensitivity analysis Is applied to further evaluate code characteris¬ tics. MODEL VALIDATION Model validation or field validation is defined as the comparison of model results with numerical data inde¬ pendently derived from laboratory experiments or obser¬ vations of the environment 1101. Complete model valida¬ tion requires testing over the full range of conditions for which the model Is designed. Model development Is an evolutionary process responding to new research results, developments In technology, and changes in user require¬ ments. Model review and validation needs to follow this dynamic process and should be applied each time the model is modified. The objective of model validation is to determine how well the model's theoretical foundation describes the actual system behavior In terms of the "degree of correla¬ tion" between model calculations and actual measured data for the cause-and-effect responses of the system. Obviously, a comparison with field data is required. Such a comparison may take either of two forms. One form, calibration. Is sometimes considered the weaker form of validation Insofar as It tests the ability of the code (and the model) to fit the field data, with adjustments of the physical parameters (131. Some researchers prefer to classify calibration as a form of verification rather than a form of validation. The other form of validation Is that of prediction. This Is a test of the model's ability to fit the field data with no adjustments of the physical parameters. In principle, this Is the correct approach to validation. However, unavailability and inaccuracy of field data often prevent such a rigid approach-. Typically, a part of the field data is designated as calibration data, and a cali¬ brated site-model Is obtained through reasonable adjust¬ ment of parameter values. Another part of the field data Is designated as validation data; the calibrated site model is used in a predictive mode to simulate similar data for comparison. The quality of such a test is there¬ fore determined by the extent to which the site model is "stressed beyond" the calibration data on which It is based (131. In the IGWMC testing procedure, this approach Is referred to as level III testing. For many types of groundwater models, a complete set of test problems and adequate data sets for the described testing procedure is not yet available. Therefore, testing of such models Is generally limited to extended verification, using existing analytical solutions, and to code IntercomparIson. Whether a model is valid for a particular application can be assessed by performance criteria, sometimes called validation or acceptance criteria. If various uses In planning and decision making are foreseen, different performance criteria might be defined. The user should then carefully check the validity of the model for the Intended use. Three levels of validity can be distinguished 110): (1) Statistical Validity: using statistical measures to check agreement between two differ¬ ent distributions, the calculated one and the measured one; validity Is established by using an appropriate performance or validity criterion. (2) Devlative Validity: if not enough data are available for statistical validation, a deviation coefficient D can be established, e.g., D - I(x-y)/x|100% where x = predicted value and y = measured value. The deviation coefficient might be expressed as a summation of relative devia¬ tions. If ED Is a deviative validity criterion supplied by subjective judgment, a model can considered to be valid if D < ED. (3) Qualitative Validity: using a qualitative scale for validity levels representing subjective judgment: e.g., excellent, good, fair, poor, unacceptable. Qualitative validity Is often established through visual inspection of graphic representations of calculated and measured data (161. The aforementioned tests apply to single variables and determine local-or-single variable validity; if more than one variable is present in the model, the model should also be checked for global validity and for validity consistency (10). For a model with several variables to be globally valid, all the calculated outputs should pass validity tests. Validity consistency refers to the variation of validity among calculations having different Input or comparison data sets. A model mlght.be judged valid under one data set but not under another, even within the range of conditions for which the model has been designed or Is supposedly applicable. Validity consistency can be evaluated periodically when models have seen repeated use. Often, the data used for field validation are not collected directly from the field but are processed In an earlier study. Therefore, they are subject to Inaccur¬ acies, loss of Information, Interpretive bias, loss of 22 ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 Groundwater Contamination: P.K.M. van der Heijde precision, and transmission and processing errors, resulting In a general degradation of the data. As noted earlier, for many types of groundwater models no field data sets are available to execute a complete validation. One approach sometimes taken Is that of code Intercomparison, where a newly developed model Is compared with existing models designed to solve the same type of problems as the new model. If the simulation results from the new code do not deviate significantly from those obtained with the existing code, a relative or comparative validity Is established. It Is obvious that as soon as adequate data sets become available, all the Involved models should be validated with those data. Further development of databases for field validation of solute transport models Is necessary. This Is also the case for many other types of groundwater * models. These research databases should represent a wide variety of hydrogeological situations and should reflect the various types of flow, transport, and deformation mechanisms present In the field. The databases should also contain extensive Information on hydrogeological, soil, geochemi¬ cal, and climatological characteristics. With the devel¬ opment of such databases and the adoption of standard model-testing and validation procedures, the reliability of models used in field applications can be Improved considerably. Validation Scenarios Often, various approaches to field validation of a model are viable. Therefore, the validation process should start with defining validation scenarios. Planning and conducting field validation should Include the following steps 1171: (1) Define data needs for validation and select an available data set or arrange for a site to study. (2) Assess the data quality in terms of accuracy (measurement errors), precision, and completeness. (3) Define model performance or acceptance criteria. (4) Develop strategy for sensitivity analysis. (5) Perform validation runs and compare model performance with established acceptance criteria. Sensitivity Analysis An Important characteristic of a model Is its sensitivity to variations or uncertainty In Input para¬ meters. Sensitivity analysis defines quantitatively or semlquantltatlvely the dependence of a selected model per¬ formance assessment measure (or an Intermediate variable) on a specific parameter or set of parameters 118). Model sensitivity can be expressed as the relative rate of change of selected output caused by a unit change In the Input. If the change In the Input causes a large change In the output, the model Is sensitive to that Input. Sensitivity analysis Is used to Identify those parameters most Influential In determining the accuracy and precision of model predictions. This Information Is of importance to the user, as he must establish required accuracy and precision in the model application as a function of data quantity and quality 117J. In this context the use of a sensitivity Index as described by Hoffman and Gardner |19] Is of Interest. It should be noted that If models are coupled, as In multimedia transport of contaminants, the propagation of errors and the Increase In uncertainty through the subsequent simulations must be analyzed as part of the sensitivity analysis. MODEL SELECTION Using models to analyze alternative solutions to groundwater problems requires a number of steps, each of which should be taken conscientiously and reviewed care¬ fully. After the decision to use an existing model has been made, the selection process Is Initiated. As model credibility Is a major problem In model use, special attention should be given In the selection process to ensure the use of qualified models that have undergone adequate review and testing. Selecting an appropriate model Is crucial to the success of a modeling project. Model selection Is the process of matching a detailed description of the modeling needs with well-defined, quality-assured characteristics of existing models, while taking Into account the objectives of the study and the limitations In the personnel and material resources of the modeling team. In selecting an appropriate model, both the model requirements and the characteristics of existing models must be carefully analyzed. Major elements in evaluating modeling needs are: (1) formulation of the management problems to be solved and the level of analysis sought; (2) description of the system under study; and (3) analysis of the constraints in human and material resources available for the study. Model selection is partly quantitative and partly qualitative. Many subjec¬ tive decisions must be made, often because there are Insufficient data In the selection stage of the project to establish the Importance of certain characteristics of the system to be modeled. Definition of modeling needs Is based on the manage¬ ment problem at hand, questions asked by planners and decision makers, and on the understanding of the physical system, Including the pertinent processes, boundary condi¬ tions, and system stresses. The major criteria In selec¬ ting a model are: (1) that the model Is suited for the Intended use; (2) that the model Is thoroughly tested and validated for the intended use; and (3) that the model code and documentation are complete and user-friendly. Regardless of whether problem-solving performance standards are set, management-oriented criteria need to be developed for evaluating and accepting models. Such a set of scientific criteria should include: • trade-offs between costs of running a model and accuracy • profile of model user and definition of required user-friendliness • accessibility In terms of effort, cost, and restrictions • acceptable temporal and spatial scale and level of aggregation If different problems must be solved, more than one model might be needed or a model might be used In more than one capacity. In such cases, the model requirements for each of the problems posed have to be clearly defined at the outset of the selection process. To a certain extent this Is also true for modeling the same system in different stages of the project. Growing understanding of the system and data availability might lead to a need for a succession of models of increasing complexity. In such cases, flexibility of the model or model package might become an Important selection criterion. It should be realized that a perfect match rarely exists between desired characteristics and those of available models. Many of the selection criteria are subjective or weakly justified. If a match is hard to obtain, reassessment of these criteria and their relative weight in the selection process Is necessary. Hence, model selection Is very much an Iterative process. In standardizing model selection, three major approaches are employed in characterizing the validation of numerical models. In one, the model is tested ENVIRONMENTAL SOFTWARE. 1987, Vol 2, No. 1 23 Groundwater Contamination: P.K.M. van der Heijde according to established procedures; when accepted, the model Is prescribed In federal or state regulations for use In cases covered by those regulations. This approach does not leave much flexibility for Incorporating the advances of recent research and technological development. The second approach Includes the establishment of a list of groundwater simulation codes as "standard" codes for various generic and site-specific management purposes. To be listed, a code should pass a widely accepted review and test procedure such as that described In a previous part of this paper. This approach Is suggested In a recent evaluation of the role of modeling In the U.S. Environ¬ mental Protection Agency |6|. It should be noted that establishing “standard" models will not prevent discussion of the appropriateness of a selected model for analysis of a specific problem nor of Its proper use In a particular decision-making process. In considering these two approaches, questions have been raised such as 16): • Are there legal liabilities for setting up certain models as acceptable? (For Instance, If an enforcement agency certifies a model for use, can that agency no longer criticize an Industry's use of that model?) • Does certification squelch the development of new, better models? • What balance should there be between using the newer, faster models and using mature models already subjected to peer review? A third approach Is to prescribe a revlew-and-test methodology In regulations of enforcement agencies, and require the model development team to show that the model code satisfies the requirements. This approach leaves room to update the codes as long as each version Is ade¬ quately reviewed and tested. An example Is the quality assurance program for models and computer codes of the U.S. Nuclear Regulatory Conwlsslon |20|. In any case, a general framework of nondlscrlmlnatory criteria should be established [6|. These criteria should Include: • publication and peer review of the conceptual and mathematical frame-work • full documentation and visibility of the assumptions • testing of the code according to prescribed methods; this should Include verification (checking the accuracy of the computational algorithms used to solve the governing equations), and validation (checking the ability of the theoretical foundation of the code to describe the actual system behavior) • trade secrets (unique algorithms that are not described) should not be permitted If they might affect the outcome of the simulations; proprietary codes are already protected by the copyright law Finally, as model selection Is very closely related to system concep-tualIzatlon and problem solving, "expert systems" Integrating system conceptualization and model selection on a problem-oriented basis promise to be valuable tools. Further information on groundwater model selection Is presented In (21, 22, 23, 24|. SUMMARY During the 1970s a rapidly Increasing awareness of the threat posed to groundwater resources by human-induced chemical and biological pollution has accelerated the development of sophisticated simulation models. These models are based on mathematical descriptions of the physical, chemical, and biological processes that take place In a complex hydrogeological environment. The extensive need for these models In assessing current and potential water quality problems has resulted In two groups of modelers: (1) model developers who are research- oriented and who generally apply models only for* verIfica- tlon and validation purposes, and (2) model users who apply models routinely to actual generic or site-specific groundwater problems. The economic consequences of model predictions and the potential liabilities Incurred by their use have brought quality guarantees and code credibility to the forefront as major Issues In ground- water modeling. Hence, quality assurance (QA) needs to be defined for both model development and model application. There Is a significant difference between these two: the first Is designed to result In a reliable code, and the second to Interpret correctly the simulation results. Both require stringent QA procedures to be established and enforced. As model credibility has become a major con¬ cern, model selection should focus on those codes that have undergone adequate review and testing. To further Increase the applicability of the models, good documenta¬ tion and user-friendliness of the computer coding Involved should receive proper attention. ACKNOWLEDGEMENT The research described In this publication has been funded In part by the U.S. Environmental Protection Agency through Cooperative Agreement #CR-812603 with the Holcomb Research Institute. It has not been subjected to the Agency peer and policy review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be Inferred. REFERENCES (1| van der Heijde, P.K.M. 1986. Modeling contaminant transport In groundwater: approaches, current status, and needs for further research and development. In Groundwater Contamination, proceedings conf., Santa Barbara, CA, November 11-16, 1984. |21 Mercer, J.W., and C.R. Faust. 1981. cround-water Modeling. National Water Well Association, 60 pp. (31 U.S. Office of Technology Assessment. 1982. Use of Models for Water Resources Management, Planning, and Policy. OTA Reoort 0-159, August, Congress of the United States, 233 pp. 14) Javandel, I., C. Doughty, and C.F. Tsang. 1984. Croundwater Transport: Handbook of Mathematical Models. Water Resources Monograph Series 10, American Geophysical Union, Washington, DC. [51 van der Heijde, P.K.M., Y. Bachmat, J. Bredehoeft, B. Andrews, D. Holtz, and S. Sebastian. 1985. Croundwater Management: The Use of numerical Models, 2nd ed. Water Resources Monograph 5. Washington, DC: American Geophysical Union. 180 pp. (6] van der Heijde, P.K.M., and R.A. Park. 1986. U.S. EPA Groundwater Modeling Policy Study Group, Report of Findings and Discussion of Selected Groundwater Modeling Issues. EPA/CERI, Cincinnati, OH (In press). 171 Taylor, J.K. 1985. What Is quality assurance? in J.K. Taylor and T.W. Stanley (eds.). Quality Assur¬ ance for Bnvironmental Measurements. ASTM Special Technical Publication 867, Am. Soc. for Testing and Materials, Philadelphia, PA, pp. 5-11. [8| Adrlon, W.R., M.A. Branstad, and J.C. Chernlasky. 1982. Validation, verification, and testing of computer software. ACM Computing Surveys, Vol.14(2), pp. 159-192. 24 ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 Groundwater Contamination. P.K.M. van der Heijde 19 j Huyakorn, P.S., A.G. Kretschek, R.W. Broome, J.W. Mercer, and B. H. Lester. 1984. "Testing and vali¬ dation of models for simulating solute transport In groundwater: development, evaluation, and comparison of benchmark techniques." GWM1 84-13, International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN, 420 pp. [10] ASTM. 1984. Standard practices for evaluating environmental fate models of chemicals. Annual book of ASTM standards, E 978-84, Am. Soc. for Testing and Materials, Philadelphia, PA. [11] van der Heijde, P.K.M. 1984. Availability and applicability of numerical models for groundwater resources management. In Practical Applications of Ground Water Models, proceedings NWWA/IGWMC COnf., Columbus, OH, August 15—17, 1984. [12] Sykes, J.F., S.B. Pahwa, D.S. Ward, and R.B. Lantz. 1983. The validation of SWENT, a geosphere transport model. In Scientific Computing, R. Stepleman et al. (eds.). New York: IMACS/North- Holland Publishing Company. |13[ Ward, D.S., M. Reeves, and L.E. Duda. 1984. Verification and field comparison of the Sandla Waste-Isolation Flow and Transport Model (SWIFT). NUREG/ CR-3316, U.S. Nuclear Regulatory Commission, Washington, DC. 1141 Gupta, S.K., C.R. Cole, F.W. Bond, and A.M. Monti. 1984. Finite-element three-dimensional ground-water (FE3DGW) flow model: Formulation, computer source listings, ’and user's manual. 0NWI-548, Battelle Memorial Inst., Columbus, OH. [15) van der Heijde, P.K.M., P.S. Huyakorn, and J.W. Mercer. 1985. Testing and validation of groundwater models. In Practical Applications of Groundwater Modeling, proceedings NWWA/IGWMC COnf., Columbus, OH, August 19-20, 1985. [16) Cleveland, W.S., and R. McGill. 1985. Graphical perception and graphical methods for analyzing scientific data, science 229:828-833. [17) Hern, S.C., S.M. Melancon, and J.E. Pollard. 1985. Generic steps In the field validation of vadose zone fate and transport models. in Hern, S.C. , and S.M. Melancon (eds.), Vadose Zone Modeling of Organic Pollutants. Chelsea, MI: Lewis Publishers, Inc., pp. 61-80. |18] Intera Environmental Consultants, Inc. 1983. A proposed approach to uncertainty analysis. 0NWI- 488, Battelle Memorial Inst., Columbus, OH, 68 pp. |19[ Hoffman,* F.D., and R.H. Gardner. 1983. in J.E. Till and H.R. Meyer (eds.). Radiological Assessment . NUREG/CR-3332, 0RNL-5968, U.S. Nuclear Regulatory Commission, Washington, DC. [20) Silling, S.A. 1983. Final technical position on documentation of computer codes for high-level waste management. NUREG/CR-0856, U.S. Nuclear Regulatory Commission, Washington, DC 11, pp. |21) Rao, P.S.C., R.E. Jessup, and A.C. Hornsby. 1981. Simulation of nitrogen in agro-ecosystems: criteria for model Selection and use. In MiLrogen Cycling in ecosystems of Latin America and the Caribbean. Proceeding Internat. Workshop, Cali, Colombia, March 16-21, 1981, pp. 1-16. [ 22 ] Kincaid, C.T., J.R. Morrey, and J.E. Rogers. 1984. Geohydrological models for solute migration; Vol. 1: Process description and computer code for selection. EA 3417.1, Electric Power Research Inst., Palo Alto, CA. (23) Boutwell, S.H., S.M. Brown, B.R. Roberts, and D.F. Atwood. 1985. Modeling remedial actions at uncon¬ trolled hazardous waste sites. EPA/540/2-85/001, U.S. Environmental Protection Agency, 0SWER/0RD, Washington, DC. |24| Siimnons, C.S., and C.R. Cole. 1985. Guidelines for selecting codes for groundwater transport modeling of low-level waste burial sites. Volume 1—Guideline approach. PNL-4980 Vol. 1. Battelle Pacific NW Laboratory, Richland, WA. ENVIRONMENTAL SOFTWARE, 1987, Vol 2, No. 1 25 IGWC Reprint REPRESENTATION OF INDIVIDUAL WELLS IN TWO DIMENSIONAL GROUND WATER MODELING Milovan S. Beljin International Ground Water Modeling Center Holcomb Research Institute, Butler University Indianapolis, Indiana 46208 presented at The NWWA/IGWMC Conference “Solving Ground-water Problems with Models” February 10-12, 1987 Denver, Colorado GWMI - 87-06 February 17, 1987 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 REPRESENTATION OF INDIVIDUAL WELLS IN TWO-DIMENSIONAL GROUND WATER MODELING Milovan S. Beljin International Ground Water Modeling Center Holcomb Research Institute, Butler University Indianapolis, Indiana 46208 Abstract Well-field simulation in ground water modeling requires grid blocks with dimensions that are much larger than the well diameter. The computed hydraulic head in the well block (i.e., in the block containing a well) represents an average head for the block and is not the head in a particular well. The accurate computation of hydraulic head in a well is needed for both flow and transport modeling. The effect of an individual well is normally represented in ground water modeling by an imposed discharge/recharge rate on the well block. The usual assumption is that the flow near the well reaches equilibrium rapidly and may therefore be treated as steady-state flow. Analytical methods for computing the nodal correction use the equivalent well block radius, which is defined as the radius at which the steady-state hydraulic head in the aquifer is equal to the numerically calculated head of the well block. Numerical methods to improve modeling of near-well zones are based on the localized mesh refinement. The objectives of this paper are to investigate the effects of several possible well situations on the nodal correction. These include a well not positioned at the center of the well block, a rectangular well block with different aspect ratios, an anisotropic medium, and more than one well within the block. Introduction In regional ground water modeling, the dimensions of a well are too small to be resolved by the computational grid. The effects of the well are included in the modeling by an imposed discharge/recharge rate on the block containing the well (the well block). The computed hydraulic head in the well block represents an average head for the block and is not the head in a particular well; but the accurate computation of hydraulic head in a well is 2 important for both flow and transport modeling. In flow modeling, the long¬ term predictions of drawdown in a well field are crucial to its design and determination of its lifetime. Simulation of solute transport in ground water systems requires the accurate determination of the velocity field, which is derived from the computed hydraulic heads. Advection and disper¬ sion, two major mechanisms of solute transport, are both functions of the velocity. Because of the radial nature of the flow near a well and the greatest velocity in the vicinity of the well, solute transport near the well must be modeled with special care. Despite the importance of the determination of the velocity near a well, relatively little research has been published on this topic in hydrological literature (Prickett 1967, 1971; Trescott et al. 1976; Prichett and Garg 1980; Bennett et al. 1982); most of the research related to this topic comes from reservoir modeling in the petroleum industry (e.g., van Poolen et al. 1970; Williamson and Chap- pelear 1981; Peaceman 1978, 1983; Abou-Kassem and Aziz 1985). The objectives of this paper are to identify the problems in repre¬ senting individual wells in ground water modeling and to point out the importance of the near-well zone in modeling. The effects of several pos¬ sible well situations are discussed. These include a well positioned at the center of a square or a rectangular well block, the case of an anisotropic medium, a well not positioned at the center of the block, and more than one well within the block. Concept of Equivalent Well Block Radius With horizontal ground water flow, mass conservation around a fully penetrating well can be expressed in polar coordinates by Q = o t/w dh 2irrKb — dr ( 1 ) where Q is the discharge rate [L 3 /T ]; r is the radius from well axis [L]; K is the hydraulic conductivity [L/T]; b is the thickness of the aquifer [L]; and h is the hydraulic head [L]. By integrating between r = r^ and some distance r, Equation (1) becomes Q = 2tt Kb h-h w 2,n(r/r ) w (2) where r is the radius of the well [L]; h* is the head at the r distance [L]; and h* is the head at the r w distance [L]. 3 Fig. I.a i.j i+l.i i + 2,j Fig. I.b Fig. I.c 4 4 2 in (4x/r e ) * h i 1 ,i I _I AyT AX Fig. 1. Flow from node (i + l,j) to node (i»j)* (a) sectional view, (b) equivalent radial flow; and (c) finite-difference representation. 4 From Equation (2), known as the Thiem equation, it follows that h will increase indefinitely as r increases, so that Equation (2) is valid only in the close proximity of a well as h approaches the original piezometric head ho. The Thiem equation is often used to extrapolate from the average head for the well block at radius r e to the head at the well radius r w (Prickett 1971; Akbar et al. 1974; Trescott et al. 1976). For a specified head hj,- at the distance r e (Figure la), and with transmissivity T in place of the product Kb, Equation (2) can be rewritten as h . . - h = an i j w 2tt 1 (3) The difference h jj - h v represents the nodal correction to be applied to the computed head in the interior well block. For an unconfined aquifer, the analogous equation is h ? . ij (4) Equations (3) and (4) assume that (a) the aquifer is homogeneous and isotropic; (b) only one fully penetrating well is in the well block; (c) flow is laminar and in steady-state; and (d) well losses are negligible. If h ij is the average hydraulic head computed for the well block, the head in the well may be determined from Equation (3) as long as the rad¬ ius r e can be determined. This equivalent well block radius, r e , is defined as the radius of a hypothetical well for which the average value of hy¬ draulic head for the well block is applicable (Trescott et al. 1976). From Equation (3) it follows that the nodal correction is directly proportional to the equivalent block radius, r e , and to the discharge/transmissivity ratio. The Equivalent Well Block Radius Equations (a) Square grid Herbert and Rushton (1966) and Prickett (1967, 1971) were apparently the first in the hydrological literature to point out the need for correc¬ tions in modeling the true radius of a well. The standard finite-difference equation applied to a square mesh of sides Ax (Figure 2a) and constant transmissivity T, with a discharge well at the node (i,j), has the form h , + h. . + h. +h. i-I, j i+l,j i,j-l J yJ+l (5) The Thiem equation for the flow from an outer circle of radius Ax to an inner circle radius of r e (Figure 1b) is given as „ 2tt( h* - h . .) iJ AX r e (6) 5 * / / / \ _*_ ^ h i,j-1 -«• 9 ~~ ^ ✓ x - ' V - 1 - / r Vi,jf \ / M ve s • \ / >*. m — h i.r**i * h 4 - — H H- 425 -H Fig. 2a. Square grid. Fig. 2b. Rectangular grid. Assuming h* to be the average of h i+J>J -, hj j.j, and hj J+J , and combining Equation (5) and (6), Prickeft (1971*) has’derived the fol¬ lowing expression: n r^XN T1 l—J = g » or r^ = exp (- |)ax s 0.208ax . (8) The same equation can be derived by combining the Thiem equation and Darcy's law (Figures 1b and 1c). In the petroleum literature, among the first to address the problem of equivalent well block radius were van Poollen et al. (1970). Their approach assumed that the computed head of the well block equals the areal average head, leading to the expression: r^ = 0.342AX . (9) Peaceman (1978, 1983) has shown that Equation (9) is incorrect, but that Equation (8) is a good approximation for the equivalent well radius. By solving the same problem numerically, he derived the following equation: 6 r^ = 0.198 Ax . (10) r The constant in Equation (10) is the limit of the ratio lim , where N is Ax 7 the number of nodes. N - ,Ql (b) Rectangular grid For the case when Ax * Ay, (Figure 2b), the literature contains several equations for calculating the equivalent well block radius. Based on the assumption of the areal head average, the equation often used in the petroleum industry is r^ = 0.2(AxAy)^. (11) Trescott et al. ( 1976 ) used the following equation in their two-dimensional ground water flow model: r =0.104( Ax + Ay) . ( 12 ) Peaceman (1983) developed an analytical solution for evaluating r e : S* - r Ax Axi exp I---) Ax i ♦ ($*)’ v AX ’ (13) Performing a numerical analysis of the equivalent radius problem, Peaceman (1983) derived another expression: r = 0.140 (Ax 2 + Ay 2 ) 2 . (14) e The constant in Equation (14) was obtained by deriving the pressure distri¬ bution for an infinite grid and is equal to exp(-y )/4, where y - 0.5772 is Euler's constant. The effects of the various aspect ratios (^) on the equivalent well block radius for the various equations are given in Table 1. The results in this table indicate that the areal head average from Equation (11) underes¬ timates the values of the r e /dx ratio in comparison to the other three equations. For smaller aspect ratios, Equations (12), (13), and (14) give similar values. 7 Table 1. Effects of aspect ratio (Ay/Ax) on equivalent well block radius. *1 AX r /Ax e Eq. (11) Eq. (12) Eq. (13) Eq. (14) 1 0.200 0.208 0.208 0.198 2 0.283 0.312 0.327 0.313 3 0.346 0.416 0.435 0.443 4 0.400 0.520 0.518 0.577 5 0.447 0.624 0.581 0.714 6 0.490 0.728 0.631 0.852 7 0.529 0.832 0.670 0.990 8 0.566 0.936 0.701 1.129 9 0.600 1.040 0.728 1.268 10 0.632 1.143 0.750 1.407 (c) Anisotropic medium Assuming that the principal axes of the transmissivity tensor are parallel to the x and y axes, Peaceman (1983) derived the following expres¬ sion for the nodal correction in an anisotropic medium: h. J h w Q 2tt(T T )* xx yy where T xx is the transmissivity in x direction [L 2 /Tj; T yy is the transmissivity in y direction [L 2 /T]; and (15) For the case T xx = Tyy, Equations (15) and (16) reduce to Equations (3) and (14), respectively. (d) Some other cases Kuniansky and Hillestad (1980) derived equations for the equivalent well block radius for the cases in which the well block is a corner or edge block and in which the well is is not centered in a grid block. Their work shows that the equivalent radius derived by Prickett (1971) for an interior well block, Equation (8), is a good approximation for both edge and corner well blocks. 8 For the case in which the well is not positioned at the center of a block, two different approaches can be used. The first predicts the flow rate with an analytical expression for the well at an arbitrary position; this rate is used to compute the head for the block. To calculate the equivalent well block radius, the flow rate and the computed head are sub¬ stituted into Equation (3). The second approach generates the flow rates with a fine grid so that all the wells are positioned at the grid block centers. These rates are used to calculate the equivalent radii for the subsequent simulations. In both approaches, the calculated rate is assigned to a centered well in the well block. Bennett et al. ( 1982) and McDonald (1985) have extended the equivalent well block radius theory to the multiaquifer or multilayered wells. Ac¬ cording to their work, the presence of the wells changes the nature of the equations that must be solved in a three-dimensional ground water flow simulation. The proposed method allows for calculating the head in the well and individual aquifer discharges to such a well. All previous discussions of well representation in ground water modeling involve finite-difference models. Charbeneau and Street (1979) presented a finite-element numerical method for obtaining an improved distribution of head around a well. Instead of assigning all the discharge of a well to a particular node, they place the well within an element, and the discharge is distributed among the nodes of that element. The numerical results are compared with the analytical solutions for confined and leaky aquifers, and good agreements were found. Example Problem To illustrate the importance of the correct representation of a well in ground water modeling, the two-dimensional finite-difference model (Trescott et al. 1976) was applied to a problem of simulating a single pumping well in an infinite aquifer. This is the classical Theis problem often used to verify a numerical model. The aquifer parameters were chosen from the benchmark problem described by Ross et al. (1982).- The aquifer is homo¬ geneous and isotropic with transmissivity of 0.001 m /s and a storage coef¬ ficient 0.001; the discharge rate is 0.003 m /s and the duration of pumping is 10 days. The constructed numerical model has a square grid design, one pumping well in the center of the model, and no-flow boundaries. A number of runs have been performed with different block sizes. To satisfy the assumption of an infinite aquifer, the no-flow boundaries have been placed for each run far enough from the well so they are not affected by the pumping. The 9 analytical and numerical results are in good agreement for the nodes outside the well block (Figure 3). However, the computed head for the well block is significantly smaller than the head in the well computed by the Theis equa¬ tion. In order to convert the average well block head to the head in the well, the nodal corrections based on Equation (3) are applied. Table 2 shows the effects of the well radius and the block size on the nodal cor¬ rection for the given problem. Table 2. Nodal corrections for the example problem (T = 0.001 m 2 /s, S = 0.001 and Q = 0.003 m 3 /s) Nodal Correction, [m] Ax [m] r = 0.1m w r = 0.25m r w ,= 1.00m 10 1.45 1 .01 0.34 100 2.55 2.11 1.45 200 2.88 2.44 1.78 300 3.07 2.64 1.97 500 3.31 2.88 2.22 1000 3.65 3.21 2.55 2000 3.99 3.54 2.88 It is interesting to note that the approximate equivalent well block radius could also be determined graphically. The analytical solution of the given example problem is plotted in Figure 3. The numerically computed average drawdown for the well block, for the case Ax = Ay = 200 m, is 1.69 m. This drawdown is plotted on the y-axis and a line parallel to the x-axis is extended to intercept the analytical curve. The x value of the intercep¬ tion represents an approximate value of the equivalent well block radius. For the given example it is 40 m, which is close to the value obtained by applying Equation (8). The approximate nodal correction can also be read off the graph. It is the distance on the graph between the analytical curve and the numerically computed average drawdown for the given radius of the well. If, for example, the radius of the well is assumed to be 1 m, the nodal correction from the graph is 1.8 m, which is close to the value given in Table 1. It is important to note that for the given example problem, the values obtained from Equation (3), which represents a steady-state situation, are in good agreement with the values obtained with the Theis nonsteady-state equation. The same problem is used to illustrate the effect of anisotropy on the value of the nodal corrections. The Trescott model uses Equation (3) to compute the nodal correction for a given T xx regardless of the degree of anisotropy. For the given example the nodal correction is 1.78 m. However, with Equations (15) and (16), the nodal correction is 1.29 m and 1.02 m for Tyy/Txx equaling 2 and 3, respectively. 10 M UMOpMDjp Fig. 3. Analytical and numerical solution of example problem. Discussion and Conclusions The head in the well block represents an average for the block and is not the head in the well itseslf. To compute the correct value of the head in the well, two approaches are possible. The first uses an analytical expression for calculating the approximate nodal correction, the magnitude of which depends on the aquifer parameters and on the grid design. A library of analytical solutions for different field situations can easily be incorporated into a code. The other approach to improve modeling of near-well zones develops a localized refined mesh in the finite-difference grid. The radial nature of flow near the wells makes a hybrid approach most suitable. The entire domain is divided into a well region, represented with a cylindrical grid, and a model region represented with a rectangular grid (Pedrosa and Aziz 1986 ). Note that neither of the two methods for calculating the head includes any additional drawdown caused by well losses. Based on study results of the example problem and discussion of the different approaches, it is clear that the near-well zone needs to be mod¬ eled more carefully than is usually done at present. REFERENCES Abou-Kassem, J.H. and K. Aziz. 1985. Analytical Well Models for Reservoir Simulation. Society of Petroleum Engineers Journal , v. 25, no. 4, pp. 573-579. Akbar, A.M., M.D. Arnold, and A.H. Harvey. 1974. Numerical Simulation of Individual Wells in a Field Simulation Model. Society of Petroleum Engineers Journal , v. 14, no. 4, pp. 315-320. Bennett, G.D., A.L. Kontis, and S.P. Larson. 1982. Representation of Mul¬ tiaquifer Well Effects in Three-Dimensional Ground Water Flow Simula¬ tion. Ground Water, v. 20, no. 3, pp. 334-341. Charbeneau, R.J. and R.L. Street. 1979. Modeling Ground Water Flow Fields Containing Point Singularities: A Technique For Singularity Removal. Water Resources Research, v. 15, no. 3, pp. 583-594. Herbert, R. and K.R. Rushton. 1966. Ground water Flow Studies by Resistance Networks. Geotechnique, v. 16, pp. 53-75. Kuniansky, J. and J.G. Hillestad. 1980. Reservoir Simulation Using Bottom- hole Pressure Boundary Conditions. Society of Petroleum Engineers Journal, v. 20, no. 6, pp. 473—486. McDonald, M.G. 1985. Development of a Multi-Aquifer Well Option for a Modular Ground Water Flow Model. Proc. Practical Application of Ground water Models, pp. 786-796. Peaceman, D.W. 1978. Interpretation of Well-Block Pressures in Numerical Reservoir Simulation. Society of Petroleum Engineers Journal, v. 18, no. 3, pp. 183-194. Peaceman, D.W. 1983. Interpretation of Well-Block Pressures in Numerical Reservoir Simulation with Nonsquare Grid Blocks and Anisotropic Perme¬ ability. Society of Petroleum Engineers Journal, V. 23, no. 3, pp. 12 531-543. Pedrosa, O.A., and K. Aziz. 1986. Use of A Hybrid Grid in Reservoir Simula¬ tion. SPE Reservoir Engineering, v. 1, no. 6, pp. 611-621. Prickett, T.A. 1967. Designing pumped well characteristics into electrical analog models. Ground Water , v. 5, no. 4, pp. 38-46. Prickett, T.A. and C.G. Lonnquist. 1971. Selected Digital Computer Tech¬ niques for Ground water Resource Evaluation. Illinois State Water Survey Bulletin 55, 62 pp. Prichett, J.W. and S.K. Garg. 1980. Determination of Effective Well Block Radii for Numerical Reservoir Simulations. Water Resources Research, v. 16, no. 4, pp. 665-674. Ross, B. et al. 1982. Benchmark Problems for Repository Siting Models. U.S. Nuclear Regulatory Comission. NUREG/CR-3097. Trescott, P.C., G.F. Pinder, and S.P. Larson. 1976. Finite-Difference Model for Aquifer Simulation in Two Dimensions with Results of Numerical Experiments. U.S. Geological Survey, Chap. Cl, Bk. 7. van Poollen, H.K., H.C. Bixel, and J.R. Jargon. 1970. Individual Well Pres¬ sures in Reservoir Modeling. Oil and Gas Journal, pp. 78-80. Williamson, A.S. and J.E. Chappelear. 1981. Representing Wells in Numerical Reservoir Simulation: Part 1 - Theory. Society of Petroleum Engineers Journal, v. 21, no. 3, PP- 323-338. IGWMC GROUNDWATER MODELING REPRINT REMEDIAL ACTIONS UNDER VARIABILITY OF HYDRAULIC CONDUCTIVITY by Aly I. El-Kadi presented at The NWWA/IGWMC Conference “Solving Ground-water Problems with Models" February 10-12, 1987 Denver, Colorado GWMI - 87-10 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 REMEDIAL ACTIONS UNDER VARIABILITY OF HYDRAULIC CONDUCTIVITY Aly I. El-Kadi International Ground Water Modeling Center Holcomb Research Institute Butler University Indianapolis, Indiana Abstract Recent research has demonstrated the frequent failure of the classic dispersion equation in describing the mass transport phenomena. The general conclusion of that research has been that the velocity field should be described in greater detail by either a deterministic or a stochastic ap¬ proach. The stochastic approach is applied here to evaluate selected reme¬ dial actions involving recovery wells. A Monte-Carlo technique is adopted in the analysis of the two cases considered, an injection/recovery well, and plume capture by a production well. Variability causes a dispersion-like phenomenon which affects the shape of plume and break-through curves. Variability is highest where the plume advances or contracts. Defining effective dispersivities representing the average stochastic results for use in deterministic analysis fails to recognize the important variability features described by stochastic analysis. In addition, stochastic analysis allows the quantification of uncertainty regarding output results. Management decisions should be based on such uncertainty. Introduction Several physical, chemical, and biological techniques are used to contain spilled or leaked contaminants and to recover and treat ground water. (For details of different techniques see, e.g., U.S. EPA (1985] and Ehrenfeld and Bass [1984]). Containment systems such as recovery wells, interceptor trenches, grout curtains, and slurry walls, interfere with the transport process by altering the flow field. Air stripping, a physical process, removes volatile chemicals from the soil by drawing or venting air through the soil layer, or by passing contaminated water through a packed column or tower with counter-flowing air and water. To increase the effi¬ ciency of the removal process, in situ air stripping is combined with acti¬ vated carbon absorption. 2 Biological methods, performed above ground or in situ, are effective as remediation techniques. Above-ground processes include fixed film treatment such as trickling filters, or suspended-growth systems such as activated sludge (Jensen et al. 1986). In situ biodegradation includes the use of existing soil microorganisms or the addition of microorganisms and nutrients to the contaminated aquifer. The effectiveness of in situ biological treat¬ ment depends on a number of factors such as type and concentration of con¬ taminants, hydrogeology, nutrient availability, dissolved oxygen, pH, tem¬ perature, and salinity (Engineering-Science 1986). Recovery wells are the most commonly used remediation techniques; some of their applications are studied here. In aquifer cleanup, polluted water is extracted and either reinjected after treatment or released to a surface- water body if that is environmentally and economically acceptable. In some situations, injection wells are combined with recovery wells to enhance recovery by altering the hydraulic gradients. The recovery injection system should be designed to intercept the contaminant plume so that no further degradation of the aquifer occurs. Modeling is a very useful tool in the design of such systems (Boutwell et al. 1985). Models to study contamination-related problems, including the design of remedial actions, are based on solution of the classic dispersion-convection equation (e.g., the review by Anderson [1984]). However, a number of re¬ searchers, including Gelhar et al. (1979) and Matheron and de Marsily (1980), have demonstrated that this equation fails to describe contaminant movement near the pollutant source or over short time periods. Application of the equation has resulted in estimates of dispersion coefficient which are both time- and scale-dependent. Although researchers agree that vari¬ ability of hydrologic properties causes such discrepancies, they disagree on ways of dealing with the situation. Generally, two approaches have been adopted. The first includes a detailed description of the velocity, either deterministically or stochastically. In the deterministic approach, hetero¬ geneities, e.g., stratification, are described deterministically (e.g., G*ven et al. 1984, Molz et al. 1983). In the stochastic approach, a random velocity field is employed in the mass transport model, based on a specified spatial and correlation structure of the hydraulic conductivity field. The governing stochastic equation can be solved analytically, as reported in the work of Gelhar et al. (1979), Gelhar and Axness (1983), Bresler and Dagan (1981), Dagan (1982), and Tang et al. (1982). The Monte-Carlo technique, a contrast to closed-form analytical solutions, was also reported by Smith and Schwartz (1980 and 1981a,b). The second approach, based on a modification of the convection-disper¬ sion equation, uses time-dependent dispersivities (e.g., Matheron and de Marsily 1980, Pickens and Grisak 1981). Gelhar et al. (1979) proposed a new one-dimensional equation in a perfectly stratified aquifer. At small times, certain restrictions may invalidate this equation. Various theoretical studies have demonstrated that the classic convection-dispersion equation is valid for large times or large distances if dispersivities are estimated from various statistical characteristics of the hydraulic conductivity distribution (Anderson 1984). The objective of the study is to analyze effects of uncertainty in hydraulic conductivity on the design of remedial actions. Two situations that involve recovery wells are studied: a case of an injection/recovery 3 single well, and plume capture by a production well. The study examines effects of uncertainty in hydraulic conductivity on variability of concen¬ trations and on the cleanup time. The possibility of characterizing the heterogeneous aquifer by an effective dispersion coefficient is also examined. A Monte-Carlo simulation approach is employed in the analysis, considering hydraulic conductivity as a stochastic process. The stochastic model is based on the USGS two-dimensional mass-transport code (MOC), devel¬ oped originally by Konikow and Bredehoeft (1978). Application of MOC to Remedial Actions The Monte-Carlo approach is applied here to the stochastic analysis of the transport problem. MOC is solved for a different set of parameters within each Monte-Carlo run. When numerical techniques are employed in the solution of the problem, care must be taken to minimize numerical errors that may contribute to variability of output results. MOC was tested for certain situations occurring in remedial actions by recovery wells. The three cases investigated were a recharge/recovery single well, a re¬ charge/recovery doublet, and one or two production wells for plume cap¬ ture. (For details of the verification, see El-Kadi (1987)). In the recharge/recovery single-well case, water of known concentration (C Q ) is injected into the well. After some time, the flow is reversed and the contaminated water is pumped out. This approach, known also as the single tracer test, is used in field work to define the dispersive proper¬ ties of aquifers (e.g., G*ven et al. 1985). The situation may also repre¬ sent a cleanup process following extended contamination. Some hypothetical experiments were simulated and the results were compared to the analytical solution of Gelhar and Collins (1971) which solves for the concentration in the well during the withdrawal period. Sensitivity of results to the value of dispersivity, injection time, and well flux were examined. Some fluctuations were noticed in the breakthrough curves, yet the overall behavior of the numerical results was good. Some numerical dispersion occurred due to the radial flow situation; its effect seems most severe for larger well fluxes or longer injection periods. Large relative mass-balance error was noticed, with a maximum value of about -23%. The error value appears to be irrelevant to the accuracy of predic¬ tion. The mass error may be caused by the method of removing solute mass from the aquifer at sink nodes (Konikow and Bredehoeft 1978), rather than by the radial flow situation. The second MOC test involved application to 'a recharge/recovery doublet. The solution of the purely convective transport case was compared with the semi-analytical solution introduced by Javandel et al. (1984). The results of the simulation showed reasonable match of concentrations in the pumping well for a short time period (less than 2.0 years). The two solu¬ tions predicted the same value for the time at which the contaminant reaches the production well. For times larger than 2.0 years the numerical solution for the concentration in the pumping well is not accurate and shows large fluctuations for which the analytical solution represents the upper enve¬ lope. The concentration at the node upstream of the production well showed much less fluctuation than at the node immediately to the right or the left of the production well. The relative mass error balance was reasonably small, approximately -10% to + 2%. The inaccuracy of model results is 4 caused by the arrival of contamination at the sink node, as was also indi¬ cated by Konikow and Bredehoeft (1978) The last test case involved plume capture by one or two production wells. The technique prevents further degradation of the aquifer by using one or more wells. The simulation results were compared with the analytical results provided by Javandel and Tsang (1986). For a steady-state flow situation, the numerical model was run long enough to represent the steady- state condition for mass transport. An interactive procedure, sufficient to capture the plume, was needed to estimate the well flux. MOC was capable of capturing the plume for different values of aH (the difference in hydraulic head value for the upper and lower boundaries), which ranged between 10 and 90 ft. However, the numerical model predicted higher values for well flux over the entire range of aH. The value needed was about 1.5 times the respective analytical value. Sensitivity of results to the mesh size was not studied; as the mesh size decreases, close agreement is possible. The relative concentration in the well showed some fluctuations, yet the be¬ havior of the curves is generally acceptable. The case of plume capture by two wells was also simulated for aH = 20 ft. MOC was also able to capture the plume, yet larger fluctuations in the pumping wells were observed. The well flux was also larger than the theoretical value. The simulation showed that MOC is, in general, accurate in simulating plume capture by recovery wells. The relative mass error for all cases considered was acceptable, -2.7% to -6.4%. Stochastic Analysis In the stochastic analysis of mass transport, the porous medium is assumed to be a realization of a random field. Hydraulic conductivity and other parameters are described as stochastic processes and the flow equation is then solved to define the full distribution of the velocity field or at least its first few moments (i.e., mean and standard deviation or covar¬ iance). The spatial correlation structure of hydraulic conductivity, an important factor in the treatment, must be defined in advance on the basis of field measurements, as are other parameters in the conductivity distri¬ bution. The resulting velocity field, a random variable in this case, constitutes the input to the transport equation. Finally, the transport equation is solved to define the spatial and temporal variability of solute concentration. Other parameters in the equation, such as dispersivity, also may be treated as stochastic processes. In the Monte-Carlo technique, a deterministic problem is solved a large number of times with different sets of generated parameters (called reali¬ zations). Each realization is assumed to be an equally probable represen¬ tation of the actual set. The results are then analyzed statistically to define the distribution of output variables. Hydraulic conductivity reali¬ zations were generated using the technique developed by Mejia and Rodriguez-Iturbe (1974) (see also El-Kadi, 1986). The approach involves the addition of harmonics of random functions that are sampled from the spectral density function. The values of Y, the logarithm of transmissivity, are generated from a knowledge of the mean Uy, the standard deviation ay, and the autocorrelation coefficient y or an increase in n, is similar to an increase in variability. The values of used are °° (i = 0) and 0.0003 (s. = 3333 ft). In fact, due to the finite size of the finite- difference mesh, the case of n less than half of the mesh increment length cannot be simulated, and the smallest correlation length should be i = ax/2 (or Ay/2), i.e., half of the mesh increment length (450 ft). The reason is that values of hydraulic conductivity are taken as constants for each block. The dependency of i can be explained as follows (see also Smith and Freeze 1979, El-Kadi and Brutsaert 1985). If the values of hydraulic con¬ ductivity are highly correlated, a series of high or low values will exist in the block system for a given realization. The resulting concentration will then move further away from the expected value. Over the whole series 7 of Monte-Carlo runs the standard deviation in concentration tends to in¬ crease. However, the effects of increased correlation on the averaged concentration are not large. The reason for these effects can be explained by the influence of the correlation on the sample statistical properties (see Loucks et al. 1981, p. 170). The expected value of the variance of a conductivity realization is smaller than that for the population. The bias in that estimate decreases as the sample size increases. On the other hand, the variance of this estimate is inflated by a factor that depends on the correlation function and whose importance does not decrease with the sample size. For the case shown in Figure 7, the variability in the variance of conductivity realization apparently caused dispersion-like effects; yet the effect was not too large due to the decrease in the expected value of the variance (the sample size in this case is 90; not very large). It can be concluded that increased correlation will cause larger variability on con¬ centration values, yet the dispersion-like effects are not obvious. For a large number of conductivity blocks, it is likely that increasing the cor¬ relation will lead to similar effects of increasing the variability of conductivity. The results regarding the cleanup time are shown in_Table 3. The values in the table are, respectively, the average time, T, the standard deviation, Oy, the 90 percentile, T 90 , the maximum, T max , and the time needed to reach an average concentration of 10%, T. Cleanup-time value was defined as that needed to reach concentration value of 10% of the maximum concentration (henceforth called the cleanup level). In general, due to the mixing effects causjd by variability of conductivity, the expected value of the cleanup time (T) is smaller than the value obtained deterministically (about 2.0 years for all cases). Table 3 shows that different management decisions could be adopted based on different criteria. A decision can be based on the average cleanup time (uncertainty, represented by oj, must be considered); the time needed for the cleanup process to satisry certain probability (e.g., T 90 which represents the time needed in 90% of the cases); the time to reach an average value for the cleanup level over the entire aquifer; and the maximum possible time for cleanup. Decision will be based on a number of factors including available funds for remediation and possible health effects caused by residual concentrations. Table 3 shows also that results are sensitive to the variability of conductivity and its correlation structure. In general, dispersive-like behavior results for highly variable or strongly correlated conductivity fields. The value of the cleanup time depends then on the resulting be¬ havior of the curve as well as the maximum value accepted for concentra¬ tion. For example, for the deterministic solutions with high and low dis- persivity, the relative cleanup time for the two cases should be different if the cleanup level was chosen below or above the point where the two curves intersect. Comparison between Cases l.B and l.E shows that the value of the popu¬ lation mean has practically no effect on the estimates shown in Table 3. As might be expected, considering the microscale dispersion causes additional dispersive effects (compare between Cases l.A and l.B). The highest varia¬ bility in the estimate of the cleanup time has been found for highly vari¬ able or strongly correlated conductivity fields. 8 It is concluded here that variability of conductivity and its corre¬ lation structure have important effects on the results of the single well case; yet the mean of the population does not influence the results in any way. Variability of conductivity leads to variability of results regarding concentration and dispersion-like effects. Case 2: Plume Capture by a Single Well Plume capture is a technique that prevents further degradation of the aquifer through the use of a number of production wells. Recently, Javandel and Tsang (1986) introduced a technique, based on the complex potential theory, to analyze the plume capture by using different aquifer and flow parameters including well flow, aquifer thickness, and Darcy's velocity. The well was located at node (5,5) and a constant concentration source representing the landfill extended over three nodes from (4,2) to 6,2). For the numerical analysis using MOC, the computer time needed for the plume capture case was directly proportional to Darcy's velocity and consequently to the well flux (everything else being fixed). Hence, to reduce the amount of computer time a value of aH = 5 ft was used. The analytical value for the well flux, Q, as estimated from the equation of Javandel and Tsang, was 0.22 cfs. The numerical value for the uniform case (as obtained by using MOC) was about 0.35. A number of tests were performed for the stochastic analysis with different values of well flux. The conductivity values were generated using uy = -1.0, oy = 0.5, and y = «. The results were examined for a few realizations to lest the ability of the well to capture the plume. The model was run for 20 years of simulation time. Figure 8 compares between the 90 percentile of concentration for Q = 0.5 and 1.0 cfs. The plume was captured with Q = 1.0 cfs, indicating that a much higher value of Q is needed to capture the plume (more than 4.0 times the analytical value and about three times the value needed for the uniform case). The reason is the introduction of dispersion-like effects caused by the variability in the conductivity field. To illustrate variability of results after 20 years for the case where Q = 1.0 cfs, Figure 9 shows the coefficient of variation of concentrations superimposed on the results concerning the average relative concentration of 1%. The figure shows that variability is highest at the boundaries of the plume where the front is advancing. The coefficient of variation of concentration over the plume equals about 9.0 or less. Variability is smallest at the source of contami¬ nation where the concentration reached its maximum value. As mentioned before, some numerical inaccuracies may also contribute to variability of results in the low concentration zone. The time change of the average concentration and the coefficient of variation of concentration in the well are shown in Figure 10. Both are practically constant after about 10 years, i.e., when the steady-state situation is reached. The coefficient of variation is highest at small time (about 0.67) and declines to reach an asymptotic value of about 0.35. Some deterministic runs were performed to capture the plume with different dispersivities, in an attempt to match the average plume as ob¬ tained by the stochastic approach. The results concerning the 1% relative concentration are presented in Figure 11 for values of a = 0 and 10 ft. None of the deterministic solutions match closely the average of the sto- chastic results. The plume was not captured for cases with a higher than 10. In other words, it was not possible to match closely the shape of the plume and to capture the plume In the same time. It is concluded that although plume capture can be achieved under variability conditions, a higher value of the well flux is needed. Disper¬ sion-like effects result from the variability in conductivity values. Variability of concentration is highest at plume boundaries where the front is advancing. It is not possible to match the plume shape and to capture the plume by including a high dispersivity value. Sensitivity of results to parameters of the conductivity distribution (i.e., p Y , o y . 4 > y ) was n °t studied. However, additional dispersion-like effects are expected by con¬ sidering a highly variable or strongly correlated conductivity field, as was the case for the single-well case. Conclusions The study demonstrates that variability of conductivity is a very important factor in the analysis of remedial actions by recovery wells. Considering variability results in dispersion-like effects caused by the variations in the velocity fields, represented as a random variable in this case. In addition, uncertainty in concentration results is quantified through the stochastic analysis. Hence, deterministic approaches fail in defining the exact shape of plumes or the break-through curves and also in describing variability of results. The analysis performed considered two situations that commonly exist in remedial actions by recovery wells: a recharge/recovery single well and plume capture by a production well. For the single-well case, variability is maximum at the plume boundaries due to the advancement or contraction of the plume. However, variability in this zone is not important due to the relatively low concentrations. Variability of results does not depend on the expected value of the K population, yet the standard deviation and correlation coefficient are controlling parameters. The cleanup time is influenced by the dispersion-like effects; it also is affected by varia¬ bility and the correlation structure as well as by the prescribed cleanup level (defined as the maximum residual concentration after remediation). The study indicates that different management decisions exist in choosing a remediation strategy based on uncertainty in the cleanup time. The deter¬ ministic solution for the concentration in the well does not predict accur¬ ately the average of the stochastic solution. Although plume capture can be achieved under variability, a higher flux is needed for the production well. For the case simulated, this value was about three times the respective value for uniform aquifers. This value and other results are influenced by the distribution of conductivity and its correlation structure. Again, variability causes the development of disper¬ sion-like effects as well as variability in concentration values. Varia¬ bility of concentrations is highest at the advancing boundary of the plume where the concentration is small. It is not possible to capture the plume and preserve its shape stochastically through the use of a deterministic solution with an altered dispersivity value. 10 The study illustrates that deterministic solutions fail to predict a number of important features provided by the stochastic results and, naturally, to quantify uncertainty in output values. However, as indicated by a number of studies including El-Kadi (1984), parameters of the conduc¬ tivity distribution and the correlation structure should be based on field studies. References Anderson, M.O. 1984. Movement of contaminants in groundwater: Groundwater transport-advection and dispersion. In Groundwater Contamination , National Academy Press, Washington, D.C., pp. 37-45. Boutwell, S.H., S.M. Brown, B.R. Roberts, and Atwood Anderson-Nichols & Co., Inc. 1985. Modeling remedial actions at uncontrolled hazardous waste sites. Office of Solid Waste and Emergency Response, United States Environmental Protection Agency, Washington, D.C. Bresler, E. and G. Dagan. 1981. Convective and pore scale dispersive solute transport in unsaturated heterogeneous fields, water Resources Research , v. 17, no. 6, pp. 1683-1693. Dagan, D. 1982. Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2. The solute transport. water Re¬ sources Research , v. 18, no. 4, pp. 835—848. Ehrenfeld, J. and J. Bass. 1984. Evaluation of Remedial Action Unit Opera¬ tions at Hazardous Waste Disposal Sites. Pollution Technology Review No. 10, Noyes Publications, Park Ridge, New Jersey, 434 pp. El-Kadi, A.I. 1984. Modeling variability in groundwater flow. HRI Paper No. 75a, Holcomb Research Institute, Butler University, Indianapolis, Indiana, 56 pp. El-Kadi, A. I. 1986. A computer program for generating two-dimensional fields of autocorrelated parameters. Ground water, v. 24, no. 5, pp. 663-667. El-Kadi, A.I. 1987. Application of the USGS two-dimensional mass-transport model (MOC) to remedial action by recovery wells. Submitted to Ground Water. El-Kadi, A.I. and W. Brutsaert. 1985. Applicability of effective para¬ meters for unsteady flow in nonuniform aquifers. Water Resources Re¬ search, v. 21, no. 2, pp. 183-198. Engineering-Science. 1986. Cost model for selected technologies for re¬ moval of gasoline components from groundwater. Health and Environ. Sciences Dept., API Pub. No. 4422, American Petroleum Inst., Washington, D.C. Gelhar, L.W. and C.L. Axness. 1983. Three-dimensional stochastic analysis Of macro-dispersion in aquifers. Water Resources Research , v. 19, no. 1, pp. 161-180. Gelhar, L.W. and M.A. Collins. 1971. General analysis of longitudinal dispersion in nonuniform flow. Water Resources Research , v. 7, no. 6, pp. 1511-1521. Gelhar, L.W., A.L. Gutjahr, and R.L. Naff. 1979. Stochastic analysis of macrodispersion in a stratified aquifer. Water Resources Research V. 15, no. 6, pp. 1387-1397. G*ven, 0., R.W. Felta, F.J. Molz, and J.G. Melville. 1985. Analysis and interpretation of single-well tracer tests in stratified aquifers. Water Resources Research , V. 21, no. 5, pp. 676-684. 11 G*ven, 0., F.J. Molz. and J.G. Melville. 1984. An analysis of dispersion In a Stratified aquifer. Water Resources Research , V. 20, no. 10, pp. 1337-1354. Javandel, I., C. Doughty, and C.F. Tsang. 1984. Groundwater Transport: Handbook of Mathematical Models . American Geophysical Union, Water Resources Monograph 10, Washington, D.C., 228 pp. Javandel, I. and C.F. Tsang. 1986. Capture-zone type curves: A tool for aquifer cleanup. Ground water, v. 24, no. 5, pp. 616-625. Jensen, B. s E. Arvin, and A.T. Gundersen. 1986. The degradation of aro¬ matic hydrocarbons with bacteria from oil-contaminated aquifers. Proc. of Petroleum Hydrocarbons and Organic Chemicals in Ground Water-Preven¬ tion, Detection and Restoration. National Water Well Association, Dublin, Ohio. Konikow, L.F. and J.D. Bredehoeft. 1978. Computer model of two-dimensional solute transport and dispersion in ground water. U.S. Geological Sur¬ vey, Techniques of Water-Resources Investigation, bk. 7, ch. C2. Loucks, D.P., J.R. Stedinger, and D.A. Haith. 1981. Water Resources Sys¬ tems Planning and Analysis . Prentice-Hal1, Inc., Englewood Cliffs, New Jersey, 559 pp. Matheron, G. and G. de Marsily. 1980. Is transport in porous media always diffusive? Water Resources Research , V. 16, no. 5, pp. 901—917. Mejia, J.M. and I. Rodriguez-Iturbe. 1974. On the synthesis of random field sampling from the spectrum: An application to the generation of hydrologic spatial processes. Water Resources Research , V. 10, no. 4, pp. 705-711. Molz, F.J., 0. G*ven and J.G. Melville. 1983. An examination of scale- dependent dispersion of coefficients. Ground water, v. 21, no. 6, pp. 715-725. Pickens, J.F. and G.E. Grisak. 1981. Modeling of scale-dependent disper¬ sion in hydrogeologic systems. Water Resources Research , v. 17, no. 6, pp. 1701-1711. Smith, L. and R.A. Freeze. 1979. Stochastic analysis of steady state groundwater flow in a bounded domain, 2. Two-dimensional simulations. Water Resources Research , V. 15, no. 6, pp. 1543—1559. Smith, L.A. and F.W. Schwartz. 1980. Mass transport, 1, A stochastic analysis Of macroscopic dispersion. Water Resources Research , V. 16, no. 2, pp. 303-313. Smith, L. and F.W. Schwartz. 1981a. Mass transport, 2, Analysis of uncer¬ tainty in prediction. Water Resources Research , v. 17, no. 2, 351-369. Smith L. and F.W. Schwartz. 1981b. Mass transport, 3. Role of hydraulic conductivity data in prediction. Water Resources Research , v. 17, no. 5, pp. 1463-1479. Tang, D.S., F.W. Schwartz, and L. Smith. 1982. Stochastic modeling of mass transport in a random velocity field. Water Resources Research , V. 18, no. 2, pp. 231-244. U.S. EPA. 1985. Remedial action at waste disposal sites (Revised). Office of Emergency and Remedial Response, United States Environmental Protec¬ tion Agency, Washington, D.C. Biographic Sketch 12 Aly I. El-Kadi is a research scientist and hydrologist in Holcomb Re¬ search Institute's Water Science Program, Butler University. He received his B.S. and M.S. degrees in Civil Engineering from Ain Shams University, Cairo, Egypt, and his Ph.D. degree in Ground-Water Hydrology from Cornell Univer¬ sity in 1982. His current research includes modeling the effects of para¬ meter variability on chemicals that penetrate the soil in conjunction with water or soil moisture. He has authored or coauthored papers on saturated and unsaturated flow in uniform and fractured porous media, and on stochas¬ tic analysis of flow in heterogeneous porous media. His publications in¬ clude state-of-the-art reports on modeling infiltration and variability studies as they apply to ground-water systems. His present address is Holcomb Research Institute, Butler University, 4600 Sunset Avenue, Indi¬ anapolis, Indiana 46208. FIGURE TITLES 13 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 1. The average of the stochastic analysis, and the 10 and 90 per¬ centile, compared to the deterministic solution (Case l.A). 2. The average of the stochastic analysis, and the 10 and 90 per¬ centile, compared to the deterministic solution (Case l.B). 3. The average of the stochastic solution of Case l.B compared to deterministic runs with a = 0 and 50 ft. 4. The average concentration in the aquifer after 2.0 years (Case l.B). 5. The 1 percent average relative concentration superimposed on a contour map representing the coefficient of variation of concen¬ tration (Case l.B). 6. Sensitivity of results of Case l.B to variation in the standard deviation of Y (the logarithm of transmissivity). 7. Sensitivity of results of Case l.B to variation in the corre¬ lation structure of Y. 8. The 1 percent relative concentration's 90 percentile with Q = 1.0 and 0.5 cfs (plume capture case). 9. The 1 percent average relative concentration superimposed on a contour map representing the coefficient of variation of concen¬ tration (plume capture case). 10. The average and coefficient of variation of concentration in the production well (plume capture case). 11. The 1 percent relative concentration with a = 0 and 10 ft. (deterministic solution) compared to the 1 percent average rela¬ tive concentration for the stochastic solution. IGWMC GROUNDWATER MODELING REPRINT A NEW ANNOTATION DATABASE FOR GROUNDWATER MODELS by Paul van der Heijde and Stan A. Williams presented at The NWWA/IGWMC Conference "Solving Ground-water Problems with Models" February 10-12, 1987 Denver, Colorado GWMI 87-11 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 1 A NEW ANNOTATION DATABASE FOR GROUND WATER MODELS Paul K.M. van der Heijde and Stan A. Williams International Ground Water Modeling Center Holcomb Research Institute, Butler University 4600 Sunset Avenue Indianapolis, Indiana 46208 Abstract The International Ground Water Modeling Center operates as a clearinghouse for Information on ground water models. In 1979, the Center established Its first annotated database of Information on models. The database was Initially implemented on a UNI VAC 9030 mainframe computer using COBOL software. In 1982, the database was transferred to a VAX 11/780 minicomputer and implemented with a database management system called DATATRIEVE. Since installation of the database, the IGWMC staff has continually maintained, updated, and used the annotation system for storage and retrieval of information that is pertinent to specific ground water models. However, recent developments in modeling and in database management systems have generated an awareness of the deficiencies in the current system. Modeling approaches such as optimization methods, stochastic techniques, hydrochemical modeling, and parameter identification modeling are not adequately described in the current annotation system. Also, developments in database management techniques such as hierarchical database organization and the use of expert systems can provide useful tools for Improving the organizational structure and accessibility of the database. In response to new methods in ground water modeling and database management, the International Ground Water Modeling Center has renovated the structure of Its annotation database for ground water models. New features of the database will Include sections on optimization models, stochastic methods, parameter identification models, hydrochemical models, and data processing programs. The new database will also incorporate more complete descriptions of the mechanics of models with sections on model development, solution techniques, boundary conditions, and specific hardware and software requirements. The system is organized In a modular format within a two-tiered hierarchical structure. This structure, which should facilitate accessibility to the database, will be amenable to additions or modifications. It is anticipated that the new database may eventually be Incorporated into a model selection package by coupling it with a complementary expert system. Introduction In 1973, representatives of the Robert S. Kerr Environmental Research Laboratory of the U.S. EPA proposed that SCOPE (the Scientific Committee on Problems of the Environment) initiate an investigation into the state of the art of ground water models. SCOPE requested that Holcomb Research Institute perform this investigation, since the Institute had just completed a critical evaluation into the role of mathematical models in environmental decision-making in the United States. The Institute coordinated the organization of an international steering committee to pursue SCOPE'S original recommended objective. The committee was chaired by John Bredehoeft of the U.S. Geological Survey. Yehuda Bachmat, then Director of Research of the Hydrological Services of Israel, was recruited as project director. One of the principle recommendations of the committee was for the establishment of a central "clearinghouse" which could provide information dissemination, technology transfer, and training in groundwater modeling (van der Heijde 1982). The first project report emanating from the investigations of the steering committee was entitled "Utilization of Numerical Ground Water Models for Water Resource Management," and was produced by Holcomb Research Institute for the U.S. EPA and SCOPE. This report was a .comprehensive review of the state-of-the-art of ground water models. It also presented recomnendations for further developments in groundwater modeling (Bachmat et al. 1980). Four general problem areas were identified: - accessibility of models to potential users - communication between managers and technical staff - inadequacy of data - Inadequacy of modeling effort The report of Bachmat et al. stated that the first of these four problem areas—accessibility of models—should receive the highest priority. Accessibility in this context includes both the quality of available information on models and the level of training of model users. This Initial report emphasized the need for Improvements in the quality and availability of model documentation and descriptive information about models. The recognition of this need provided the early impetus for the establishment of the MARS database of annotated descriptions of ground water models. The Model Annotation and Retrieval System (MARS) was a keystone in the development of the International Ground Water Modeling Center. IGWMC was established in early 1978, and its general mandate was to serve as a cen¬ tralized clearinghouse which would bring together Information on documen¬ tation, application, verification, validation, and availability of ground water models. The Center would also serve as a mechanism for technology transfer by offering short courses, sponsoring conferences on ground water modeling, and generally providing opportunities for interaction among model users, model developers, and those Involved in management of projects related to ground water modeling (van der Heijde 1982). Figure 1 illustrates the functions and organization of IGWMC. Throughout the early IGWMC CLEARINGHOUSE 3 Figure 1. The functions and organization of the International Ground Water Modeling Center. 4 years of IGWMC, the MARS database became an Integral component of the Center. The development of the database provided a means for pursuance of many of the established objectives of IGWMC. The database became a mechanism for improving the accessibility of models, and it indirectly enhanced the quality and credibility of model applications. Development of MARS The development of the MARS database was initiated in 1978 with a major effort at collection of the pertinent information on ground water models. Requests for descriptive information on available models appeared in journals and in the first newsletter distributed by the Center. Also in 1978 IGWMC distributed a questionnaire on model use to users and developers. Results from these Intial efforts were compiled into a proto¬ type version of MARS that was presented for review at a workshop on policies and operational procedures of IGWMC in April 1979. After this initial review, information collection efforts continued, and the database gradually developed until 1982 by which time a total of approximately 400 annotated entries had been collected and verified (van der Heijde 1982). In 1982, the emphasis in the management of MARS was changed from develop¬ ment to review, update, maintenance, and use of the database contents. Although new annotations were continually entered In response to new information, the Center also pursued a strong initiative to evaluate, verify, and update stored information. Many searches were performed for customers, and a set of procedures was established to elucidate and standardize the use of the database (Srinivasan and van der Heijde 1985). Also during this period the working aspects of MARS were studied, with a view toward potential improvements. The annotation form was revised and related database programs and other search mechanisms were investigated. Results of searches for certain types of models were published as documents of possible interest to individuals In the modeling community (van der Heijde 1982). By 1984, the database Included over 600 annotated entries on ground water models. MARS—Organized Under DATATRIEVE In 1982, IGWMC transferred the MARS database from a UNIVAC minicomputer, using in-house COBOL-based software, to a VAX 11/780 minicomputer and reorganized the database under DATATRIEVE, a VAX-based database management system. Within the DATATRIEVE framework, information is coded in a binary format [(0,l)=(no,yes)]. This coded information includes model descriptors on aquifer conditions, boundary conditions, solution techniques, processes modeled, and details about input/output characteristics. Also under DATATRIEVE, several text fields were reserved for information about model development and purpose and for references and remarks pertaining to use of the model (van der Heijde 1982). Several advantages are inherent in the organization of MARS under DATATRIEVE. This system avoids the common "key word" type of search, and is therefore much more flexible than other potential systems. The complete list of descriptors and textual fields is available during the searching process. In this system, no remote dial-up Is necessary. IGWMC staff members perform the searches according to customer requests. The Interaction between the Individual requesting the search and the IGWMC staff member performing the search provides a mechanism for tailoring results to the individual's needs and for assuring the quality of the information sought. MARS—Organized Under RBASE The current renovation of the MARS database will be structured with a format that will be compatible with the RBASE system V database management system. The RBASE system is compatible with a PC environment, and its inherent flexibility should facilitate the new MARS organizational structure. Under the RBASE system, MARS will assume a two-tiered hierarchical structure, and descriptive information will be organized in modules. Access to the new system will be menu-driven. The modular format should ensure greater flexibility than previous systems to make alterations or additions to the system. Figure 2 is a representation of the proposed structure for the new MARS database. As Figure 2 indicates, the new system will not only be structured differently, but will include much more descriptive Information about specific models and types of models. Part I of the structure Includes details about model identification, model development, execution requirements, evaluation, availability, and a general description of the physical system that the model can address. In part II of the structure, descriptive details are organized in modules according to the type of model described. For example, categories for predictive models are fluid flow, solute transport, heat transport, and deformation. Geochemical equilibrium and nonequilibrium models are described under the category for hydrochemistry. The watershed model category includes conjunctive use models, which are usually based on methods for integrating groundwater and surface components. Management models usually combine governing equations for ground water flow or transport with an optimization technique such as linear or quadratic programing (Gorelick 1983). Data processing programs include pre- and postprocessors for models, programs for statistical analysis, and database management systems. Parameter identification models are based on efforts to solve the Inverse problem of defining aquifer parameters through analysis of the dependent variables of the system of interest. In many instances, a model may necessarily fall into two or more of these modules. For example, a management model usually will include some type of predictive model. Recently, some predictive models have been coupled with hydrochemical models. IGWMC staff will monitor such development as they appear. Information within each of these model-type modules is based on the state-of-the-art in model' development for the respective category. Details included within each module describe general characteristics of the model, processes and phenomena addressed in the model, boundary conditions which may be simulated, solution methods, and input/output characteristics. Other types of detail will be addressed according to the category of the model. The Appendix provides specific examples of the details included. A New Database Structure for IGWMC Model Information 6 c CO Figure 2. Proposed structure of the new MARS database. 7 Rationale for New Structure of MARS Several justifications underly the renovation of MARS. The new structure is logical in both the vertical and horizontal dimensions. Vertically, the system follows a hierarchical structure that flows logically from general descriptive information about a model to detailed information about the solution mechanics of a specific model. Horizontally, the modularity of the database facilitates adaptability to dynamic developments in the field of ground water modeling. Any attempt at organization of ground water models must be adaptable to new findings (van der Heijde and Park 1986). The new database will also provide a powerful tool for the improvement of quality control in model applications. Recent research has shown that quality of model applications is often limited by the inaccessibility of information about available models (van der Heijde and Park 1986). Another of the principle reasons for the new organization of the MARS database is compatibility with expert system technology. Developments in artificial intelligence have rendered expert systems accessible to practical applications such as model selection and defining potential solutions to hydrogeologic problems. The conceptual basis of the database is a function of the experience and expertise of the IGWMC staff. The use of this expertise In the categorization and delineation of detailed information on ground water models may be analogous to the logical rules and knowledge base that would be used in an expert system developed for hydrogeologic problems or for ground water model selection. In the future, the MARS system as organized under RBASE may be incorporated into such an expert system. Also, expert systems may eventually be used as mechanisms for determining model reliability and for the Interpretation of simulation results (van der Heijde and Park 1986). Conclusion Since the inception of the International Ground Water Modeling Center in 1978, the development of the MARS database has been an integral component in the pursuit of the Center's objectives. Its most salient attribute is the contribution that the development of MARS has made to the accessibility of detailed Information on ground water models. In the future, the database may also be incorporated into the development of expert systems that may provide powerful mechanisms for problem resolution. The current modifications of MARS will ensure that the database will continue to be an important tool in ground water modeling investigations. Appendix PART I: General Information Model Identification Model name Model category Groundwater flow saturated unsaturated Solute transport Heat transport Subsurface deformation Hydrochemistry Watershed Management—Optimization Data-processing Parameter Identification Abstract (< 5 lines) Model history Date of first release Release dates of updated versions Current version # release date IGWMC check date Built upon an existing model which: ___ (see notes for more info) Part of a program package package name: __ (see notes for more info) Related preprocessing programs Name Purpose Related postprocessing programs Name Purpose Model Development Authors: 1234 Name: Address(l): Telephone: Address(2): Telephone: Original Institution of Model Development Name Address Type of institution Federal/national government State/provincial government Municipal/county administration Planning agency Research institute University Consultant Private industry International organization Other: Code Custodian Name Address Type of institution Federal/national government State/provincial government Municipal/county administration Planning agency Research institute University Consultant Private industry International organization Other: Contact person to obtain code Name (space for 2 contacts) Institution Address Telephone Contact person for model support Name (space for 2 contacts) Institution Address Telephone Program Execution Requlrenents Resident software requirements: __ (e.g. IMSL.MPSX) Hardware Requirements Computer of model residence type Supercomputer Mainframe Minicomputer Microcomputer Make and model Operating system Computer—other implementations (space for several) Type (indicate make and model) Supercomputer: __ (Cray 1, CDC Cyber 205) Operating system: Mainframe: __ (CDC 6600/7600) Operating system: Minicomputer: _(VAX 11/780, Prime) Operating system: Microcomputer: _(IBM PC, AT; Compaq 386) Operating system: _ Storage requirements Core: _ (e.g. 640K) Mass:_(e.g. 10MB) Peripheral hardware Required Optional Magnetic tape unit _ _ Disc unit __ Line printer Plotter _ Math coprocessor _ Graphic board type: _ (e.g. EGA, HerculesJ” Other: Evaluation Documentation available (Y/N) Description status* Theory _ User's instructions Data input _ Application rules _ Program description Variable list _ Flowchart _ Structure _ Example Input/output _ Code listing (Y/N) Documentation reviewed? (Y/N) By whom? _ ★ 1. good 2. sufficient 3. incomplete 4. poor 5. under development 6. not present Internal documentation (comnent statements) Sparse Moderate Comprehensive Model Testing Verification/validation Analytical solutions Hypothetical problems Laboratory experiments Field experiments Code intercomparison Review: (Y/N) By whom: _ (e.g. IGWMC) Level of testing _ (e.g. 1,2,3; IGWMC levels) Code Availability Terms of availability Available (Y/N) Public domain Unrestricted distribution Restricted distribution (e.g. sole source) Proprietary Lease, licensed use Royalty-based use Use as part of consulting service Form of availability _Source code __Compiled code Available as: Tape Diskette Paper listing Telephone transmission General Code Information Language (level/version) Number of statements Number of subroutines General type of code Research code Expert code General use code Educational code Applications Research Field Number of known applications Classroom Third-party users Support Can be used without support Support available Author Third party Level of support Full support Limited or conditional support Support agreement available Additional software capabilities Pre-processing Data storage Data inspection Data formatting Interactive data entry Interactive data editing Digitizing Graphic display of Input data Postprocessing Data inspection Data storage Graphics Screen Plotter Printer Color Remarks: PART I: Physical System Subsurface system Saturated zone Aqulfer(s) hydraulic Single aquifer Single aquifer-aquitard Multiple aquifers/aquitards Hydrodynamic Single layer Multilayered Confined Semi confined Water table Storage-confining layer Porous media Discrete fractures Dual porosity Isotropic Anisotropic Homogeneous Heterogeneous Aquifer deformation Layering Delayed yield from aquifer storage Changing aquifer conditions in space/time Saturated/unsaturated Confined/unconfined Other: Aqultard(s) Homogeneous in depth Heterogeneous in depth Homogeneous in areal extent Heterogeneous in areal extent Surficial Interbedded with aquifers Aquitard compaction Storage In aqultard Other: Unsaturated zone Isotropic Anisotropic Homogeneous Heterogeneous in depth Heterogeneous in areal extent Discrete fractures Macropores Dual porosity (including crusting) Perched water table Tension-saturated zone Other: Surface system Land surface Polders Springs Overland flow Ponding Wetlands Hill slopes Drainage basins Surface water bodies Rivers, canals Lakes, reservoirs, ponds Oceans, seas Interface: subsurface/surface/atmosphere Infiltration Evaporation Evapotranspiration Other: PART II: Fluid Flow General Model Characteristics Temporal Steady state Successive steady states Transient Spatial Subsurface Saturated Parameters: _lumped: _single cell _multice _distributed ID _horizontal vertical 2D _horizontal vertical radial 3D _fully _layered _spherical _axisymmetric Unsaturated 14 Surface Parameters: ID 2D 3D flumped: _single cell __multicell __distributed horizontal _vertical horizontal ^vertical __radial fully _layered _spherical axisynsrietric Parameters: __lumped: __single cell _multicel1 __ distributed ID single channel 2D multichannel 3D ^reservoir Grid design None Orientation Plan or horizontal Cross-section or vertical Preparation Automatic Manual Required Possible Spacing Regular Variable Local refinement Movable Size Predetermined Variable Number of nodes Fixed Variable Maximum? no._ Cell shape Linear Triangular Curved triangular Square Rectangular Quadrilateral Curved quadrilateral Cylindrical Spherical Curvilinear Polygon Cubic Hexahedral Triangular prism Tetrahedral Fluid conditions Physical Homogeneous Heterogeneous Density Units Constant Variable Density-temperature relation Density-concentration relation Viscosity Constant Variable Viscosity-temperature relation Viscosity-concentration relation Compressible Incompressible Multiple fluids Miscible Immiscible Oi1-water Gas(vapor)-water Saltwater-freshwater Multiphase Water Ice Vapor Steam Other: Metric _English Restart capability Updates possible: Parameters Perturbations Boundary conditions Simulation parameters Matrix solution Time sequence Iteration criteria Stability criteria Error criteria Fluid balance over model Sum head/pressure change over model between iterations Maximum head/pressure change at any node Maximum fluid flux change at any boundary node Maximum head/pressure change over a time increment Maximum fluid flux change over a time increment Model Dynamics Flow characteristics Laminar Turbulent Darcy flow Non-Darcy flow Hydrologic processes Diffusion Infiltration Soil evaporation Evapotranspiration Interflow Condensation Condensation Precipitation Capillary uptake Physical phenomena Crusting Freezing/thawing Change-of-phase Hysteresis Buoyancy Deformation Compaction Osmosis Skin effect Consolidation Expansion Boundary Conditions Spatial variability Stage Heads, pressures Springs _ Surface water stage _ Free surface Flux Fluid flux _ Head/pressure- Dependent flux _ No flow Seepage face _ Surface infiltration Groundwater recharge _ Well pumpage/injection Other Moisture content Tidal fluctuations Solution Methods Temporal variability Equations solved (e.g. convection-dispersion): Type of solution Analytical Analytic elements Method of Images Line sinks Dipoles Doublets Vortices Areal sources, sinks Inverse methods Theis type-curve method Cooper-Jacob semi logarithmic method Unconfined aquifers Boulton type-curve method Neuman type-curve method Neuman semi logarithmic method Neuman recovery method Numerical Space approximation Finite difference Block-centered Node-centered Integral finite difference Finite element Variational Galerkin—method of weighted residuals Collocation Numerical integration (e.g. Gauss quadrature) Boundary element Lumped-cell approach Upstream weighting Time approximation Finite difference Strongly implicit Fully implicit Fully explicit Crank-Nicolson Finite element Automatic time increment selection Upstream weighting Matrix-solving technique Iterative Gauss-Seidel (point-successive over-relaxation, PSOR) Line-successive over-relaxation (LSOR) Block-successive over-relaxation (BSOR) Alternating direction (ADI) 18 Iterative alternating direction (IAD I) Predictor-corrector Direct Gauss elimination Cholesky square root Doolittle Thomas Algorithm Point Jacobi Minimum search technique Newton-Raphson Gauss-Newton Steepest descent Iteration criteria Fluid balance over model Total head/pressure change over model between iterations Maximum head change at any node between iterations Maximum flux change at any boundary node Maximum head/pressure change over time increment Maximum flux change over time increment Other: Spatial interpolation Lagrange method Spline functions Kriging Linear, bilinear, trilinear Input/Output Characteristics INPUT Geometry Elevation Ground surface elevation Aquifer/aquitard top Aquifer/aquitard bottom Surface water bed Thickness Aquifer Aquitard Unsaturated zone Root zone Parameters Hydraulic conductivity Transmissivity Intrinsic permeability Porosity Storativity Specific storage Specific yield Hydraulic diffusivity Aquitard leakance Resistance—confining layers Resistance—surface water beds Hydraulic conductivity/moisture content relation Pressure head/moisture content relation Hydraulic conductivity/potential relation Fluid Density Specific weight Viscosity Compressibility Temperature Initial conditions Saturated thickness Head/pressure head/potential distribution Transmissivity Temperature Velocities Position of interface Soil moisture distribution General characteristics Well Characteristics Maximum number of wells _ Fully penetrating Partially penetrating Well bore Diameter Depth Storage Well screen Diameter Elevation/depth of top Elevation/depth of bottom Length Well characteristics (??) Meteorological data Air:—temperature, wind speed, humidity Precipitation Evaporation Evapotranspiration Land-use data Soil cover Impermeable surface area Boundary conditions Stage Heads, pressures Surface water stage Flux Head, pressure-dependent flux Specified flux Well pumpage/injection Surface infiltration Artificial recharge Groundwater recharge 20 Precipitation Other Moisture content Seepage face Free surface Simulation specifications Grid Grid intervals Number of nodes or cells Node locations Number of layers Time Time step sequence Initial time step Number of time steps Matrix solution parameters Relaxation factor Stability criteria Error criteria OUTPltf Echo of input Tabulated output Grid _ Initial heads/pressures/potentials Initial fluxes _ Parameters Hydraulic diffusivity _ Hydraulic conductivity _ Transmissivity _ Storativity _ Specific storage _ Specific yield _ Moisture content _ Resistance-confining layer _ Resistance-beds _ Fluid density _ Fluid viscosity _ Simulation results Head/potential values _ Fluxes Internal _ Boundary _ Velocities _ Pathlines _ Traveltimes _ Isochrones Position of interface _ Stream function _ Water balance _ Precipitation Evapotranspiration Graphic Groundwater recharge Groundwater storage Total Change Surface water balance Other REFERENCES Bachmat, Y., B. Andrews, D. Holtz, and S. Sebastian. 1980. Utilization of Numerical Groundwater Models for Water Resource Management. Report/600/8- 78-012. Environmental Protection Agency, Office of Research and Development, Environmental Research Information Center, Cincinnati, Ohio. Gorelick, Steven M. 1983. A Review of Distributed Parameter Groundwater Management Modeling Methods. Water Resources Research 19:2, pp. 305-319. Srinivasan, P., and P.K.M. van der Heijde. 1985. IGWMC Model Annotation Databases--User's Manual. IGWMC Report no. GWMI 85-26. International Ground Water Modeling Center, Holcomb Research Institute, Butler University Indianapolis, Indiana. van der Heijde, P.K.M.. 1982. Facilitation of General Understanding and Applications of Groundwater Models. IGWMC Report No. GWMI 82-05. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, Indiana. van der Heijde, P.K.M., and Richard Park. 1986. U.S. EPA Groundwater Modeling Policy Study Group: Report of Findings and Discussion of Selected Groundwater Modeling Issues. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, Indiana. van der Heijde, P.K.M. 1986. ITAC and Policy Board meeting notes. Internal memorandum. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, Indiana. ' AVAILABILITY AND DOCUMENTATION OF MODELS IGfefNC GROUNDWATER MODELING REPRINT TECHNOLOGY TRANSFER IN GROUNDWATER MODELING: THE ROLE OF THE INTERNATIONAL GROUND WATER MODELING CENTER by Paul K.M. van der Heijde presented at The NWWA/IGWMC Conference "Solving Ground-water Problems with Models“ February 10-12, 1987 Denver, Colorado GWMI 87-09 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 TECHNOLOGY TRANSFER IN GROUNDWATER MODELING: THE ROLE OF THE'INTERNATIONAL GROUND WATER MODELING CENTER Paul K.M. van der Heijde International Ground Water Modeling Center Holcomb Research Institute, Butler University Indianapolis, Indiana 46208 Abstract The protection of ground water resources has emerged in recent years as a top priority for natural resource management around the world. Antici¬ pating the increased importance of protecting ground water resources, the International Ground Water Modeling Center (IGWMC) was established in 1978 at the Holcomb Research Institute, Indianapolis, Indiana, USA, to advance the use of modeling methodologies by regulatory and management agencies in the development of effective ground water management procedures. To meet Its goals, the IGWMC has developed an extensive technology transfer, re¬ search, and assistance program which Includes dissemination of information about the modeling process and the role of modeling in ground water resource management; model availability; development of selection and testing proce¬ dures; promotion of quality assurance in the application of ground water modeling software; acquisition and distribution of models, supporting soft¬ ware, and documentation; education of model users, managers, and teachers; and publication of the Ground water Modeling Newsletter. A second office opened in Delft, The Netherlands, in 1984, further expands the Center's international activities. Technology Transfer and Training in Ground Water Modeling Technology transfer means dissemination of information on technological advances through communication and education. When applied to ground water modeling, technology transfer includes dissemination of information about the role of modeling in water resource management, model theory, model veri¬ fication and validation, modeling methodology, availability and applic¬ ability of models and related software, model selection, modeling project management, and quality assurance. In a report on the use of models for water resources management, planning, and policy, the Office of Technology Assessment of the U.S. Congress (OTA 1982) considers specific education and training of model 2 developers, users, and managers in various aspects of water resources modeling; such functions are critical components of technology transfer. Other technology transfer functions include the distribution of published, printed, or electronically stored materials such as reports, newsletters, papers, and other communications; information exchanges in meetings, work¬ shops, seminars, conferences, and networking of specialists; and the opera¬ tion of electronic bulletin boards for rapid distribution of announcements, messages, and other dated information. Technology transfer also pertains to the distribution of computer codes, model documentation, and data files, and includes assistance in transferral, implementation, and use of modeling codes. Effective communication and efficient information management form the basis of technology transfer. Communication is often hampered by inadequate comnunication channnels, incompatible language, jargon, different concepts of the same issue, and administrative impediments. Information processing, an important element of technology transfer programs, encompasses a variety of activities including selecting the tech¬ nology and information to be disseminated, ensuring the quality of that information, selecting the appropriate method of transfer, and evaluating the impact of the dissemination approach. Technology transfer Is approached in two ways (van der Heijde and Park 1986): (1) the receiver actively seeks the required Information or tech¬ nology; (2) the receiver has a passive role insofar as supervisors or specialists bring the information or technology to the potential user. Recognizing this difference is crucial to the development of an effective technology transfer program. The most successful technology transfer pro¬ grams stress the first category of information receivers, especially with respect to ground water modeling. Since the late 1970s the International Ground Water Modeling Center has been active in promoting the quality-assured use of modeling in ground water management through its widely recognized and accessed services in ground water modeling technology transfer. Establishment. Goals, and Organizational Structure of the International Ground Water Modeling Center The International Ground Water Modeling Center (IGWMC) was established in 1978 at the Holcomb Research Institute (HRI) of Butler University, Indianapolis, Indiana. In the preceding years HRI was involved in two major modeling research projects, under the auspices of the Scientific Committee on Problems of the Environment (SCOPE) of the International Council of Scientific Unions. The first study focused on problems related to the use of environmental models in decision making (HRI 1976). The second project concentrated on the use of numerical ground water models in water resources management. This project, supported in part by the U.S. Environmental Protection Agency (Bachmat et al. 1978), surveyed the availability and use of simulation models in ground water management. The report identified four problem areas: 3 • accessibility of models for potential users • communication between managers and technical personnel • inadequacies in data • inadequacies in modeling The need was stressed for a centralized modeling information dissemi¬ nation and software distribution facility—a centralized "clearinghouse"—to include all forms of information on models currently available, the problems for which models have been tested and successfully used, as well as public domain software and its documentation. Recommendations made in the report included a detailed outline of the institutional mechanisms and procedures needed to acquire and distribute models; to provide an effective and widely recognized training program for field specialists and ground water project managers; to develop a respected research and development program in support of technology transfer; and to comnunicate efficiently with interested pro¬ fessionals. Based on the experience obtained in its previous modeling projects, guided by the recommendations cited above, and supported by the U.S. Envi¬ ronmental Protection Agency, HRI established the International Ground Water Modeling Center in 1978. The Center's main mission is to promote the cor¬ rect and efficient use of computer-based data analysis and prediction tech¬ niques in support of effective ground water management. The Center accomplishes its mission by: (1) developing and promoting a comprehensive approach to the role and quality-assured use of modeling based decision-support technology in ground water management; (2) assembling, organizing, analyzing, and disseminating information related to the development and qualified use of models and related decision-support technology, in response to changing demands from groundwater management, and benefiting from computer technology development; (3) improving model accessibility through model acquisition from uni¬ versities and from federal and other research agencies, with subse¬ quent review, testing, documentation, and software distribution; and (4) providing federal, state, and local ground water management agencies and the private sector with the,tools and training to analyze existing problems and to screen management options in addressing these problems. Structure of IGWMC To meet its objectives, the Center operates through four divisions and two offices: one in Indianapolis, Indiana, U.S.A., supported by the Holcomb 4 Research Institute of Butler University, and one in Delft, The Netherlands, supported by the Institute of Applied Geoscience of the Dutch research organization TNO. Two of the four IGWMC divisions are directly involved in technology transfer: Clearinghouse, and Training and Education. The Research and Development Division and the Communication Division provide basic support to the Center's mission (Figure 1). The Clearinghouse . The IGWMC emphasizes reducing the time-lag between innovations in ground water data processing and modeling at universities, research laboratories, and major research agencies (USGS, EPA, NRC, USDA, NAS, U.S. Army, DOE) and the availability of these research results to state and private users. To this purpose the Center's clearinghouse makes model and modeling-related information and software available to governmental and private users by using extensive referral-type databases, and by distri¬ buting of a wide collection of management-oriented ground water software. Extensive contacts with researchers and model users, together with the experience of the Center's professional staff, form the basis of these fre¬ quently accessed user-oriented modeling services. The clearinghouse is the framework on which the Center's knowledge base on ground water modeling has been developed and is maintained. Training and Education . To enhance the use of ground water models by qualified personnel, the Center offers a comprehensive annual program of short courses, workshops, and seminars, in which principles, concepts, theories, and applications of ground water models are featured. New approaches in education and training, based on blending educational develop¬ ments with recent advances in computer technology, are being explored. In addition, the Center provides assistance to governmental agencies, educa¬ tional institutions, and private groups in organizing and conducting specially designed training programs. Research and Development . When the Center was established it was realized that the technology transfer in ground water modeling could be successful only if supported by a strong research and development program. Therefore, together with the technology transfer activities, the Center's staff has developed an extensive research and development program. The results of these activities are new water resource management decision- support information, modeling and training methodologies, and related soft¬ ware. The following activities have taken place through the Center's research and development studies: the establishment of guidelines and meth¬ odologies for modeling-related activities such as quality assurance in model development and application, computer program selection, code implementa¬ tion, model evaluation and testing, model documentation, and pre- and post¬ processing; state-of-the-art reviews based on the content of the Center's information bases; and extensive software development for its educational and distribution activities. Communication . Corrcnunication is one basic element of technology trans¬ fer. To ensure adequate communication with all the groups active in ground water management and research, the Center has established a seperate commu¬ nication division. One of the Center's major channels of communication is the quarterly Ground water Modeling Newsletter. 5 international ground water modeling center J Policy Board International Technical Advisory Coamittee 1 ■j International Coordinator Indianapolis Office Delft Office serving serving North, Central, and South Europe, Asia, Africa, and America Australia Holcomb Research Institute TNO-DGv institute of Butler University Appiied Geoscience Indianapolis, Indiana, U.S.A. Delft, The Netherlands Clearinghouse Training and Research and Cosianunicat ions Education Development -knowledge base: -short courses -modeling -information col lection. and workshops methodology services analysis, storage, and dissemination -individual -quality -publications of model information assistance assurance distribution -software evalua- -computer-aided -software -newsletter tion, acquisition, and distribution instruction performance publication -technical assis- -educational -documentation tance and software approaches in center support modeling -model needs and -networking status of modelers Figure 1. Organizational Structure of IGWMC Through its extensive contacts as an intermediary between model devel- 6 opers and models users, the Center is in a unique position to contribute to quantitative approaches to ground water management. Changes in management focus by the states and the private sector can be monitored closely, while the Center's flexible structure allows rapid response to needs for analytic and forecasting tools. International Cooperation through IGWMC In August 1983 HR I and the TNO Institute of Applied Geoscience (DGV/TNO) reached an agreement regarding the establishment of a European IGWMC office in Delft, The Netherlands. The agreement forms the base for expanded access to and benefit from other countries' experiences in ground water modeling. Activities under this agreement include efficient inter¬ office communication and reporting procedures, open exchange of technical information, mutual technical and organizational assistance. Integrating information collected by both offices into a single IGWMC knowledge base, and carrying out joint research and development projects. The Center is governed by a Policy Board representing the participating institutes (HRI and DGV/TNO). The Policy Board supervises the Center's activities and is responsible for setting policies and long-term planning. The agreement also provides for an annual meeting of the International Tech¬ nical Advisory Committee (ITAC) with the Center's Policy Board, and addresses the role of the director of the Center's Indianapolis office as International Coordinator of IGWMC. Future Developments In the past the efficient application of ground water models has often been hampered by limited access to the models and to user information during the selection process; by poorly written and documented software; and by in¬ sufficient knowledge or awareness of the modeling process and a lack of training on the part of the users. Through the establishment of the Inter¬ national Ground Water Modeling Center, many of these problems have been reduced. IGWMC's first six years have emphasized upgrading the quality of ground water modeling through improved access to information and by pro¬ viding extensive training opportunities, both supported by research and development. For the near future, the Center will increase its efforts in assuring high quality ground water modeling through emphasis on internal and external quality assurance approaches and programs in all stages of the modeling process, from model development (coding and documentation), analysis, evaluation, testing, and selection, to application and integration in the resource-management decision process. In considering its role in ground water modeling in the next three to five years, the Center foresees its functions as; continuing to advance the quality of ground water modeling studies through the development of Improved methodologies, procedures, and standards developing and introducing efficient, integrated, computer-aided decision-support methodologies in ground water management promoting the use of quality-assured computer-based technology expanding the IGWMC knowledge base, both vertically (in-depth) and horizontally (to include multimedia transport of pollutants, expo¬ sure and risk analysis, integrated management of surface water and ground water, and nonsimulation computer techniques applicable to ground water management, such as graphics, data processing, kriging, stochastic analysis, optimization, and the use of informa¬ tion theory) continuing to expand clearinghouse services with updated informa¬ tion on modeling methodology and related software, and distribution of selected quality-assured and well-documented computer programs continuing to provide high-quality practical training opportuni¬ ties for ground water professionals and managers, incorporating major new research and technology developments, and regularly adjusting to user needs in terms of topics covered, audience addressed, facilities, and educational methodology continuing to support the clearinghouse and technology transfer functions with research and development activities expanding and improving ways and means to comnunicate with various potential audiences in different parts of the world Acknowledgment The research described in this paper has been funded in part by the U.S. Environmental Protection Agency through Cooperative Agreement ICR- 812603 with The Holcomb Research Institute. It has not been subjected to the Agency peer and policy review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. References Bachmat Y., B. Andrews, D. Holtz, and S. Sebastian. 1978. Utilization of Numerical Groundwater Models for Water Resource Management. EPA-600/8- 78-012, U.S. Environmental Protection Agency, Ada, Oklahoma. HRI. 1976. Environmental Modeling and Decision Making: The United States Experience. New York: Praeger Publishers. OTA. 1982. Use of Models for Water Resources Management, Planning, and Policy. 0TA-0-159, Office of Technology Assessment, U.S Congress, Washington, D.C. van der Heijde, P.K.M., and R.A. Park. 1986. U.S. EPA Ground-water 8 Modeling Policy Study Group; Report of Findings and Discussion of Selected Ground-water Modeling Issues. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, Indiana. Biographical Sketch Director of the Water Science Program, Holcomb Research Institute, and Director, International Ground Water Modeling Center, van der Heijde is trained as a geohydrologist. A native of The Netherlands, he received his M.S. degree in Civil Engineering in 1977 from Delft Technical University, where he specialized in hydraulic engineering and hydrology. His career since 1977 has focused on ground water and quantitative analysis of its interaction with soil and bedrock systems. His current research concentrates on improving the quality of ground water modeling and developing technology-transfer methods and facilities in ground water modeling. Recently, he chaired the EPA Groundwater Modeling Policy Study Group, and is a member of the editorial board of the journal Ground Water. IGWMC PUBLICATIONS IN GROUND WATER MODELING AVAILABLE SOLUTE TRANSPORT MODELS FOR GROUNDWATER AND SOIt WATER QUALITY MANAGEMENT by Paul K.M. van der Heijde and Milovan S. Bel jin GWMI 86-08 August 1986 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 U.S.A. Phone: 317/283-9458 INTRODUCTION Models are useful instrument In understanding the mechanisms of ground- water systems and the processes which influence their quality. Through their predictive capabilities, models provide a means to analyze the consequences of human Intervention in groundwater systems. In managing water resources to meet long-term human and environmental needs, models provide necessary analytic support. Three types of models are frequently used in groundwater quality studies: • Flow models for the analysis of flow patterns and for the determina¬ tion of streamlines, particle pathways, velocities, and traveltimes. • Solute transport models for the prediction of movement, concentra¬ tions, and mass balances of soluable constituents, and for the calculation of radiological doses. • Hydrochemical models, either equilibrium or kinetic, for the calcu¬ lation of chemical constituent concentrations. The flow and solute transport models may be embedded in a management model, describing the system in terms of objective function(s) and constraints and solving the resulting equations through an optimization technique such as linear programming (Gorelick 1983). Two of these model types can be used to evaluate the chemical quality of groundwater: hydrochemical models and solute transport models. In the hydrochemical models, the chemistry is posed independent of any mass transport process. These models, which are general in nature and are used for both ground and surface water, simulate chemical processes which regulate the concentration of dissolved constituents. They can be used to identify the effects of temperature, sped at ion, sorption, and solubility on the concentrations of dissolved constituents. They are described in a separate publication (Rice 1985). Solute Transport Models Solute or mass transport models consider quality in conjunction with quantity. In principle, a mass transport model is based on solving equations for flow and solute transport under given boundary and initial conditions. Under certain conditions such as low concentrations of contaminants and negli¬ gible difference in specific weight between contaminant and the resident water, changes in concentrations do not affect the flow pattern (homogeneous fluid phase). In such cases a mass transport model can be considered as containing two submodels, a flow submodel and a quality submodel. The flow model computes the piezometric heads. The quality submodel then uses the head data to generate velocities for advective displacement of the contaminant, allowing for additional spreading through disperison and for transformations by chemical and microbial reactions. The final result is the computation of concentrations and solute mass balances. In cases of high contaminant concen¬ trations in waste water or saline water, changes in concentrations affect the flow patterns through changes in density and viscosity, which in turn affects the movement and spreading of the contaminant and hence the concentrations 1 (heterogeneous fluid phase). To solve such problems through modeling, simul¬ taneous solution of flow and solute transport equations or Iterative solution between the flow and quality submodels is required (van der Heijde et al. 1985). Mass transport models which handle only convective transport are called irmiiscible transport models, whereas miscible transport models handle mixing resulting from dispersive and diffusive processes. Models which consider both displacements and transformations of contaminants are called nonconservative. Conservative models retain the mass of constituents in liquid form and only simulate convective and dispersive displacements. The transformations in nonconservative models are primarily adsorption, radioactive decay, and biochemical transformations. Thus far, the simplified, linear representation of the the adsorption process has been included princi¬ pally in the nonconservative transport models. In general, current solute transport models assume that the reaction rates are limited and thus depend on the residence time for the contaminant, or that the reactions proceed instantaneously to equilibrium. Various numerical solution techniques are used in solute transport models. They include the finite difference method (FD), the integral finite difference method (IFOM), the finite element method (FE), the collocation method, particle mass tracking methods (e.g., random walk [RW]), and the method of characteristics (MOC). CURRENTLY AVAILABLE SOLUTE TRANSPORT MODELS Although the various processes playing a role in contaminant distribution within groundwater systems are not completely understood, computer codes have been developed for situations which do not require analysis of complex transport mechanisms or chemistry. These codes range from poorly documented research codes to extensively documented and applied program packages. The uses of these programs are generally restricted to conceptual analysis of pollution problems, to feasibility studies in design and remedial action strategies and to data acquisition guidance. IGWMC Model Information Data Bases In the following pages, numerical and semi-analytical mass transport models which are documented to some degree, which have undergone some form of testing or review, and for which the code is available, are listed. This listing is obtained by performing a computerized search in the MARS and PLUTO model information data bases of the International Ground Water Modeling Center. The table is prepared in such a way that it could independently be used as a first step in learning about available mass transport models. Many of the listed models are in the public domain and available at nominal or no cost to the user. Others (marked by in column 4) require special agreements on usage. The columns of the table are explained below. Column 1: Serial number 2 Column 2: Column 3: Column 4: Column 5: Column 6: Column 7: List of authors at the time of model development. Name of organization is given in some cases. Address at which further information on the availability of model is known. When no one's name appears at the top, any one of the authors can be contacted. An asterisk mark follows the name to indicate that the contact address is different from the organization where the model was developed. Name with which the model is referred. Year of latest update of the model is given in parentheses. A mark next to name indicates a special agreement is required for model usage: the models marked "+" are also available from the IGWMC Here, the purposes of the model are such: type of model, aquifer conditions, flow conditions, system-geometry, numerical method, etc. Processes considered in the model for mass transport. It is the last four digits of a number, known as IGWMC-key, by which annotation of each model is stored and retrieved in the IGWMC model information data base. Further Information Complete annotations describing all the model characteristics including program code specifications and the nature of availability (cost, agreement, written permission, etc.) are available at IGWMC at nominal costs. Documentations of the models listed are available at the contact address. The IGWMC appreciates feedback from the users about their experience in trying to acquire the documentation of the models listed, so that the most recent information will be available to future users. 3 REFERENCES Van der Heijde, P.K.M., et al. 1985. Groundwater Management : The Use of Numerical Models . Water Resourc. Monogr. 5, 2nd edition. Washington, D.C.: Am. Geoph. Union. Rice, R. 1985. Listing of Hydrochemical Models which are Documented and Available. Internatioanl Ground Water Modeling Center Publication GWMI 85-15, Holcomb Research Institute, Indianapolis, Indiana. Gorelick, S. 1983. A Review of Distrbuted Parameter Groundwater Management Modeling Methods. Water Resources Research 19(2):305-319. 4 No. Author(s) Contact Address Model Name (1ast update) Model Description Model Processes IGWMC Key 1 . S.W. Ahlstrom H.P. Foote R.J. Serne J.F. Wahburn Battelle Pacific NW Labs P.0. Box 999 Richland, WA 99352 MMT-DPRW (1976) A random-walk model to predict transient, three-dimensional move¬ ment of radio-nuclides and other contaminants in saturated/unsaturated aquifer systems advection dispersion diffusion adsorption decay chemical reactions ion exchange 0780 2. R.G. Baca Rockwel1 Hanford Operations P.0. Box 250 Richland, WA 99352 fectra (1979) A two-dimensioanl, ver¬ tical finite element model to simulate steady or unsteady transport for a given velocity field in saturated or unsaturated porous media advection dispersion diffusion adsorption (chain) decay 0790 3. H.C. Burkholder M.O. Cloninger W.V. Dernier G. Jansen P.J. Liddel1 J.F. Washburn Natl. Energy Software Center* Argonne Natl. Laboratory 9700 S. Cass Avenue Argonne, IL 60439 Tel: 312/972-7250 GETOUT (1979) To predict migration of radionuclides to bio¬ sphere using a steady- state, homogeneous, iso¬ tropic, saturated model of the geosphere advection dispersion diffusion adsorption ion exchange (chain) decay 2080 4. L.A. Davis Water, Waste, and Land, Inc. 1311 S. College Avenue Fort Collins, CO 80524 SEEPV (1980) A finite difference mod¬ el to simulate transient vertical seepage froma tailings Impoundment, including saturated/un- saturated modeling of impoundment with liner, and underlying aquifer advection 2890 5. L.A. Davis Water, Waste, and Lend, Inc. 1311 S. College Avenue Fort Collins, Co 80524 GS2 (1985) A two-dimensional hori¬ zontal or vertical fi¬ nite element model to simulate flow and solute transport in saturated/- unsaturated porous media advection dispersion decay adsorption 2891 6. L.A. Davis G. Segol Water, Waste, and Land, Inc. 1311 S. College Avenue Fort Collins, CO 80524 GS3 (1985) A three-dimensional fi¬ nite element model to simulate flow and solute transport in saturated/- unsaturated porous media advection dispersion 2891 7. D.L. Deangel is G.T. Yeh D.D. Huff G.T. Yeh Oak Ridge National Laboratory Envionmental Sci. Div. Oak Ridge, TN 37830 FRACPORT (1984) An integrated compartmental model for describing the transport of solute in three- dimensional fractured porous medium advection dispersion adsorption decay 3374 8. Delft Hydrau1ics Laboratory J.W. Wesseling Delft Hydraulics Lab. P.0. Box 152 8300 AD Emmeloord The Netherlands Tel: (0)5274-2922 GROWKWA (1982) Transient finite element simulations of two- dimensional, horizontal ground water movement of nonconservative solute transport in a multi¬ layered, anisotropic, hetero-geneous aquifer system advection dispersion diffusion adsorption ion exchange decay chemica1 reactions 2982 5 Mo. Author(s) Contact Address Model Mane (last update) Model Description Model Processes IGWMC Key 9. R. T. 0 i 1 1 on R.M. Cranwel1 R. 8. Lantz S. 8. Pahwa M. Reeves R.M. Cranwel1 Sandia National Labs. Albuquerque, NM 87185 Tel: 509/376-8451 (support only) Code distributed by: Natl. Energy Software Center® Argonne Natl. Laboratory 9700 S. Cass Avenue Argonne, IL 60439 Tel: 312/972-7250 SWIFT A three-dimensional fi¬ nite-difference model for simulation of coup¬ led, transient, density dependent flow and transport of heat, brine, tracers or ra¬ dionuclides in aniso¬ tropic, heterogeneous saturated porous media advection dispersion diffusion adsorption ion exchange decay chemical reactions 3840 10. G.R. Dutt M.J. Shaffer W.J. Moore Bureau of Reclamation U.S. Dept, of Interior 715 S. Tyler, Suite 201 Amari1lo, TX 79101 Salt Trans¬ port in Irrigated Soils (1976) A finite difference mod¬ el for transient one- dimensiona1, simulation of vertical solute transport in the unsa¬ turated zone, coupled with a chemistry model advection ion exchange react ions 2960 1 1 . O.R. Friedrichs C.R. Cole R.C. Arnett O. R. Friedrichs Battelle Pacific NW Labs P. 0. Box 999 Richland, WA 99352 Tel: 509/376-8628/8451 PCP (1977) A semi-analytlcal, ad- veetlve transport model which calculates travel times and paths along an unconfined aquifer for given potential surface 12. S.P. Garabedian L.F. Konikow L.F. Konikow Water Resources Division U.S. Geological Survey 431 National Center Reston, VA 22092 FRONT¬ TRACKING MODEL (1983) A finite dif ference model for simulation of convective transport of a conservative tracer dissolved in groundwater under steady or tran¬ sient flow conditions. The model calculates heads, velocitites and trancer particle posi¬ tions. advection particle tracking 0741 13. M.Th. van Genuchten M. Th. van Genuchten U.S. Salinity Laboratory U.S. Department of Agricu1ture 4500 Glenwood Drive Riverside, CA 92501 SUMATRA-1* (1978) To simulate the simul¬ taneous movement of water and solutes in a one-dimensional satu- rated-unsaturated non- homogeneous soil pro¬ file, including the ef fects of 1inear ad¬ sorption and zero- and first- order decay convection dispersion adsorption ion exchange decay 3430 14. S.K. Gupta C.T. Kinkaid P.R. Meyer C.A. Newbi11 C.R. Cole C.R. Cole Battelle Pacific NW Labs P.0. Box 999 Richland, WA 99352 Tel: 509/376-8451 CFEST (1985) A three-dimensional fi¬ nite element model to simulate coupled tran¬ sient flow, solute- and heat-transport in satu¬ rated porous media advection dispersion diffusion 2070 15. V. Guvanasen T. Chan Applied Geosci. Branch Whiteshel1 Nuclear Research Atmic Enercy of Canada Pinawa Manitoba ROE 110 MOTIF (1986) Finite-element model for one, two and three- demens iona1 saturated/- unsaturated groundwater flow, heat transport, and solute transport in fractured porous media, faci1ifates single- spec ies radionuc1ide transport and solute diffusion from fracture to rock matrix convection dispersion diffusion adsorpfion decay advection 0953 6 Ho. Author(s) Contact Address Model Mane (last update) Model Description Model Processes IGWMC Key 16. S. Hajl-Djafari T. C. Wei Is D'Appolonia Waste Mngmt. Services, Inc. 10 Duff Rd. Pittsburgh, PA 15235 Tel: 412/243-3200 GE0FL0W* (1982) A three-dimensional fi¬ nite element model to simulate coupled tran¬ sient flow, solute- and heat-treasport in satu¬ rated porous media advection dispersion diffusion 3220 17. P. Huyakorn IGWMC* 4600 Sunset Avenue Indianapolis, IN 46208 Tel: 317/283-9458 TRAFRAP+ (1986) A finite element model to study transient, two- dimensional, saturated ground water flow and chemical or radionuclide transport in fractured and unfractured, aniso¬ tropic, heterogeneous, multi-layered porous media advection dispersion diffusion adsorption decay chemica1 react ions 0581 18. P. Huyakorn GeoTrans, Inc. 250 Exchange Place Suite A Herndon, VA 22070 Tel: 703/435-4400 GREASE2* (1982) A finite element model to study transient, multl-dimenslona!, saturated ground water flow, solute and/or energy transport in fractured and unfrac¬ tured, anisotropic, heterogeneous, multi¬ layered porous media advection conduct ion dispersion diffusion buoyancy adsorption 0582 19. P. Huyakorn GeoTrans, Inc. 250 Exchange Place Suite A Herndon, VA 22070 Tel: 703/435-4400 SATURN2* (1982) A finite element model to study transient, two- dimensional variably saturated flow and so¬ lute transport in anisotropic, hetero¬ genous porous media advection conduction dispersion diffusion adsorption decay chemica1 reactions 0583 20. P. Huyakorn GeoTrans, Inc. 250 Exchange Place Suite A Herndon, VA 22070 Tel: 703/435-4400 SEFTRAN* (1985) A two-dimensional finite element model for simu¬ lation of transient flow and transport of heat or solutes in anisotropic, heterogeneous porous media advection dispersion diffusion adsorption decay 0588 21. INTERA Environm. Consult., Inc. K. Kipp* U.S.. Geological Survey Box 25046, mail Stop 411 Denver Federal Center Lakewood, CO 80225 SWIPR (1979) A finite difference model to simulate nonsteady, three- dimensional ground water flow as well as heat and contaminant transport in a heterogeneous aquifer advection conduct ion dispersion diffusion sorption 0692 22. INTERA Environm. ConsuIt., Inc. INTERA Envioronmenta1 Consultants, Inc. 11999 Katy Freeway, Suite. 610 Houston, TX 77079 Tel: 713/496-0993 Hydrologic Contaminant Transport Mode 1 (HCTM)* (1975) A three-dimensional mod¬ el to simulate transient flow and solute trans¬ port in a saturated/- unsaturated, anisotro¬ pic, heterogeneous aqui¬ fer system using finite differences and method of characteristics advection dispersion diffusion sorption decay 0693 23. F.E. Kaszeta C.S. Simmons C.R. Cole Battelle Pacific NW Labs P.0. Box 999 Richland, WA 99352 MMT-1D (1980) To simulate transient, one-dimensional movement of radionuclides and other contaminants in saturated/unsaturated aqulfer systems advection dispersion diffusion sorption (chain) decay chemical reactions 0781 7 No. Author(s) Contact Address Model Name (last update) Model Description Model Processes TEwmC Key 24. K.L. Kipp AERE Harwel1 8336.32 Didcot, Oxfordshire United Kingdom 0X11 ORA Column Transport w i th Sorption (1976) A one-dimensional, steady-state model to simulate vertical mass transport in a soil column and to solve the inverse problem advection diffusion sorption decay 1180 25. T.R. Knowles Texas Dept, of Water Res P.0. Box 13087 Capitol station Austin, TX 78758 Tel: 512/475-3681 GWS1M-11 (1981) A transient, two-dimen¬ sional, horizontal model for prediction of water levels and water quality in an anisotropic, heterogeneous, confined or unconfined aduifer based on finite differ¬ ence method advection dispersion dif fusion 0680 26. L.F. Konikow J.D. 8redehoeft L.F. Konikow* U.S. Geological Survey 431 National Center Reston, VA 22092 Tel: 703/648-5878 USGS-2D-* TRANSPORT/ M0C (1986) To simulate transient, two-dimensional, hori¬ zontal ground water flow and solute transport In confined/semlcon fined or water table aquifers using finite differences and method of character- istics advection disperslon diffusion 0740 27. N.M. Larson M. Reeves Natl. Energy Software Center* Argonne Natl. Laboratory 9700 S. Cass Avenue Argonne, IL 60439 Tel: 312/972-7250 ODMOD (1977) Prediction of coupled, one-dimensional movement of water, and trace con¬ taminants through lay¬ ered, unsaturated soils advection adssorption 2591 28. E. Ledoux Ecole des Mines de Paris Centre d'1nformatique Geologique 35, rue Saint-Honore 77305 - Fontaineble France Tel: (1)422.48.21 NEWSAAM* (1976) A finite difference mod¬ el for transient predic¬ tion of piezo-metric heads and salt transport in a mu 11i-1ayered aquifer advection adsorption 1450 29. I. Miller J. Marlon- Lambert Eileen Poster Golder Associates 2950 Northup Way Bellevue, WA 98004 Tel: 206/827-0777 Golder Groundwater Computer Packaged (1983) A transient finite ele¬ ment model to simulate hydraulic and solute transport characteris¬ tics of two-dimensional, horizontal or axi- symmetric ground water flow in layered aquifer systems advection dispersion diffusion adsorption decay chemical reaction 1010 30. T.N. Narasimhan A.E. Reisenauer K.T. Kay R.W. Nelson C.R. Cole* Battelle pacific NW Labs Water & Land Res. Div. P.0. Box 999 Richland, WA 99352 Tel: 509/376-8451 TRUST* (FLUX/ MULTVL) (1981) A transient integral finite difference model to compute steady and non-steady pressure head distributions in multi¬ dimensional, heterogen¬ eous, variably satu¬ rated, deformable, por¬ ous media with complex geometry advection 0120 31 . R.W. Ne1 son Battelle Pacific NW Labs P.0. Box 999 Richland, WA 99352 Tel: 5Q9/376-8332 PATHS (1978) To evaluate contamina¬ tion problems in un¬ steady, two-dimensional ground water flow sys¬ tems using an analytical solution for the flow equation and the Runge- Kutta method for the pathline equation advection adsorption ion exchange 2120 8 No. Author(s) Contact Address Model Name (last update) Model Description Model Processes IGWMC Key 32. J. Noorishad M. Menran Jahan Noorishad Earth Sciences Division Lawrence Berkeley Lab. Univ. of California Berkeley, CA 94720 ROCMAS-HS (1981) A transient model to solve for two-dimension¬ al dispersive-convective transport of non-conser¬ vative solutes in saturated, fractured porous media for a given velocity field as gener¬ ated by ROCMAS-H convection dispersion diffusion adsorption decay reactions 3081 33. I .L. Nwaogazie I.L. Nwaogazie Dept, of Civil Engnrg. Univ. of Port Harcourt PHB 5323 Port Harcourt, Nigeria S0TRAN (1985) A finite-element solute transport model for two- dimensional unconfined aquifer systems using 1inear or quadratic isoparametric quadri¬ lateral elements and adsorption, biode¬ gradation and radio¬ active decay. dispersion adsorption decay advection 4320 34. D.B. Oakes Water Research Centre Medmenhanm Labs Marlow Buckinghamshire SL7 2HD U.K. NIMBUS (1980) A one-dimensional finite difference model for transient simulations of vertical unsatureted flow and transport of nitrates in soiIs advection dispersion 1221 35. J.F. Pickens G.E. Grisak GTC Geologic Testing Consultants, LTD. 785 Car 1ing Avenue, 4th Floor Ottawa, Ontario Canada K1S 5H7 SHALT* (1980) A finite element model for transient simulation of 2-dimensional, den¬ sity dependent coupled flow and transport of heat and solutes in fractured variably satureated porous media advection conduct ion dispersion diffusion adsorption ion-exchange decay chemica1 reactions 2034 36. G.F. Pinder Princeton Univ. Dept, of Civel Engn. Princeton, NJ 08540 Tel: 609/452-4602 ISOQUAD 2 (1977) A finite element model to solve the transport equation in non-steady, confined, areal, two- dimensional groundwater f 1 ow advection dispersion diffusion 0511 37. T.A. Prickett T.G. Naymik C.G. Lonnquist IL State Water Survey P.0. Box 5050, Sta. A Champaign, IL 61820 Tel: 217/333-6775 Random Walk* (1981) To simulate one-or two- dimensional, steady/non¬ steady flow and solute transport problems in heterogeneous aquifer under water table and/or confined or semi-con¬ fined conditions using a "radom-walk" technique advection dispersion diffusion adsorption decay chemica1 react ion 2690 • 00 K\ A.E. Reisenauer C.R. Cole Battelle Pacific NW Labs Water & Land res. Dlv. P.0. Box 999 Richland, WA 99352 Tel: 509/376-8338/8451 VTT (1979) A transient finite dif¬ ference model to calcu¬ late hydraulic head in con fined/uncon fined, multi-layered aquifer systems and generate streamlines and travel- times advection 2092 • Ov ro B. Ross C. M. Koplik Analytic Sciences Corp. Energy & Environment Div One Jacob Way Reading, MA 01867 Tel: 617/944-6850 WASTE* (1981) To compute one- or two- dimensional horizontal, or one-dimensional ver¬ tical, steady/unsteady transport of radio¬ nuclides in confined or semi-confined, aniso¬ tropic, hetero-geneous mu 11i-aquifer systems advection dispersion diffusion adsorption ion exchange decay 2810 9 No. Author(s) Contact Address Model Name (last update) Model Description Model Processes IGWMC Key 40. A.K. Runchal A.K. Runchal Analytic and Computa¬ tional Research, Inc. 3106 Inglewood Blvd. Los Angeles, CA 90066 PORFLOW II 4 III (1981) Steady or transient, 2-D horizontal, vertical or radial and 3-D simula¬ tion of density depen¬ dent flow heat and mass transport in aniso¬ tropic, hetero-geneous, non-deformab1e saturated prous media with time dependent aquifer and fluid properties convection conduction dispersion diffusion change of phase adsorption decay reactions 3233 41 . B. Sagar B. Sagar Analytic 4 Computational Research, Inc.3106 Inglewood Blvd. Los Angeles, CA 90066 FRACFLOW (1981) Steady and unsteady state analysis of den¬ sity-dependent flow, heat and mass transport in frctured confined aquifers simulating tow- dimensionally the pro¬ cesses in the porous medium and one-dimen¬ sional ly in the frac¬ tures, including time- dependency of properties convection conduction dispersion diffusion consolidation adsorption decay reactions 3232 42. B. Sagar B. Sagar Analytic 4 Computational Research, Inc. 3106 Inglewood Blvd. Los Angeles, CA 90066 FLOTRA (1982) steady or transient, two-dimensiona1, area 1 , cross-sectional or radial simulation of density-dependent flow, heat and mass transport in variable saturated, anisotropic, hetero¬ geneous deformable porous media convection conduction dispersion diffusion consolidation hysteresis adsorption decay reactions 3235 43. R.D. Schmidt U.S. Dept, of the 1nterior Bureau of Mines P.0. Box 1660 Twin Cities, MN 55111 ISL-50 (1979) A three-dimensional semi-ana 1ytica1 model to describe transient flow behavior of leschants and ground water, in¬ volving an arbitrary pattern of injection and recovery wells advection 2560 44. G. Segol G.F. Pinder W.G. Gray G. Segol* Bechtel , Inc. P.0. Box 3965 San Francisco, CA 94119 Tel: 415/768-7159 1NTRUS1 ON (1974) A two-dimensional, ver¬ tical finite element model to simulate tran¬ sient, density dependent flow in a coastal aquifer advection dispersion diffusion 0530 45. H.M. Selim J.M. Davidson H.M. Selim* Louisiana State Univ. Louisiana Agricultural Experimental Station Agronomy Dept. Baton Rouge, LA 70803 Tel: 504/388-2110 ' > NMODEL (1976) Steady or unsteady simu¬ lation of one-dimension¬ al, vertical water and nitrogen transport and nitrogen transformations in saturated and unsatu¬ rated, multi-layered, homogeneous soils advection dispersion diffusion 0290 46. B.J. Travis B.J. Travis Los Almos national Lab. Earth 4 Space Sci. Div. Los Almos, NM 87545 TRACR3D (1984) A three-dimensional finite-difference model of transient two-phase flow and multicomponent transport in deformable, heterogeneous, reactive porous/fractured media dispersion diffusion adsorption decay advection 4270 10 Ho. Author(s) Contact Address Model Name (last update) Model Description Model Processes 1GWMC Key 47. C. 1 . Voss C.l. Voss U.S. Geological Survey 431 National Center Reston, VA 22092 SUTRA* (1984) A finite element simula¬ tion model for two-di¬ mensional, transient or unsteady-state, satu- rated-unsaturated, fluid density dependent ground water flow with trans¬ port of energy or chem¬ ically reactive single species solute transport convection dispersion diffusion adsorption reactions 3830 48. J.W. Warner Colorado State Univ. Civil Engineering Dept. Ft. Collins, CO 80523 Tel: 303/491-5861 RESTOR (1981) A finite element model to calculate the dual changes in concentration of two reacting solutes subject to binary action exchange in flowing ground water by two- dimensional simulation of areal transient or steady ground water flow and transient coupled transport of two solutes in an anisotropic, heterogeneous confined aquifer advection dispersion diffusion ion-exchange 3100 49. G.T. Yeh D.D. Huff Environmental Sci. Div. Oak Ridge National Lab. Oak Ridge, TN 37830 FEMA (1985) A two-dimensional finite element mode 1 to simu¬ late solute transport including radioactive decay, sorption, and biological and chemical degradation. This model solves only solute transport equation and velocity field has to be generated by a flow model dispersion diffusion adsorption decay advection 3376 50. G.T. Yeh D.S. Ward Oak Ridge nat1. Lab. Environmental Sciences Division Oak Ridge, TN 38730 Tel: 615/574-7285 FEMWASTE+ (1981) A two-dimensional cross sectional finite element model for transient simulation of transport of dissolved constitu¬ ents for a given velo¬ city field in a hetero¬ geneous, saturated or unsaturated porous media advection dispersion diffusion adsorption decoy 3371 11 The Fundamentals of Geochemical Equilibrium Models; with a Listing of Hydrochemical Models That Are Documented and Available by Richard E. Rice GWMI 86-04 December 1986 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute Butler University Indianapolis, Indiana 46208 Equilibrium and Multiphase Systems Equilibrium—probably the most fundamental concept of classical thermo¬ dynamics—is defined by Lewis and Randall (1961) as "a state of rest." Rather than implying a cessation of motion at the microscopic level, this definition means either that the macroscopic properties of the system under a given set of external constraints remain unchanged over the course of time, or that the system returns to its original state after the external constraints are momen¬ tarily altered. Mahan (1963) lists the following criteria as necessary for equi1ibriurn: (1) no unbalanced forces acting on or within the system, (2) uniform chemical composition in each phase with no net chemical reactions occurring, and (3) uniform temperature equal to that of the surroundings. The general thermodynamic requirement for this condition is that the change in the appropriate free energy function be zero. The number of external constraints required to determine the state of a system is referred to as the number of degrees of freedom and depends on the number of constituents and phases present. Thus for a system consisting entirely of gases, only one phase can exist at equilibrium, since all gases are infinitely soluble in each other. Completely miscible liquids also form a single phase, but immiscible liquids constitute separate phases. Solids, which generally have only limited solubility in each other, can give rise to a number of different phases at equilibrium. For a single phase such as pure water, for example, two constraints— usually pressure and temperature—are required to determine its state, whereas it is necessary to specify only one constraint for two phases of pure water in equilibrium with each other. For pure water existing in all three phases simultaneously, there are no degrees of freedom; i.e., the so-called triple point of water occurs only at a temperature of 273.16 K and saturation vapor pressure of 6.11 millibars. 1 Mathematically stated, the phase rule (Denbigh 1971) is F = C + 2 - P, (1) where F represents the degrees of freedom, C the number of components, and P the number of phases. For a system in which chemical reactions can occur, C = N - R, (2) i.e., the number of constituents that completely define the system is the dif¬ ference between the total number of chemical species N and the number of inde¬ pendent reactions R relating the various chemical species. An additional restriction arises in the case of electrolytes, since electroneutrality requires that the total number of cations equal the total number of anions, and thus C = N - R - 1. (3) The restrictions imposed by the phase rule must be taken into accout in any multicomponent system, including those in which the number of phases changes through dissolution/precipitation reactions. A system of CaC0 3 , Ca(0H) 2 , and water, for example, can undergo the fol¬ lowing reactions: CaC0 3 ~ Ca 2+ + C0 3 2 ", Ca(0H) 2 <-► CaOH + + Obf, CaOH* Ca 2+ + OH", H 2 0 H* + OH", H + + C0 3 2 " <-+ HC0 3 , H + + HC0 3 H 2 C0 3 . As long as the system contains no gas phase, there are six independent reac¬ tions, though not necessarily this particular set. The second and third reactions above could be replaced by 2 Ca(0H) 2 + H + ■— Ca 2+ + H 2 0, Ca 2+ + H 2 0 <- CaOH + + OH - , without altering the description of the system in terms of N, C, R, and F. Regardless of the set of independent reactions chosen, the total number of different species is ten, and thus C = 3 from eq. (3). Since there are three distinct phases—the aqueous and two solid phases—eq. (1) indicates that the system has only two degrees of freedom; i.e., it is completely char¬ acterized by any two intensive variables, such as temperature, pressure, pH, or the concentrations of dissolved species. In this particular system the number of compounds originally chosen and the number of components are the same, but this is not always true. With the addition of CaO to the above system, the parameters N, R, and P are each increased by one, and the additional independent reaction can be written as CaO + H 2 0 «-► Ca(0H) 2 . Thus the number of components remains three, but the number of degrees of freedom is reduced to one because of the additional solid phase present. In general, the phase rule offers a useful guide in organizing multicomponent systems and characterizing them in the correct thermodynamic terms (Brinkley 1946). Regardless of the number of components or phases, however, true equilib¬ rium can occur only in a closed system, i.e., one that exchanges energy but not matter with its surroundings. For an open system, which can exchange both energy and matter with its surroundings, the time-invariant condition is not equilibrium, but the steady state. The difference between these two condi¬ tions is that the former is characterized by a minimum in free energy, while the latter is not. All natural water systems are of course open systems, so the application of thermodynamic equilibrium models to them should be undertaken with care. Even though the time available for approaching equilibrium in typical ground waters may be on the order of tens to hundreds of years (Morgan 1967), some of the dissolution/precipitation or oxidation/reduction reactions may still occur 3 slowly relative to these time scales. In a particular ground water at any given time, some reactions may be very near equilibrium, while others remain quite far from it. Although thermodynamic calculations cannot provide any information about the speed with which the system is approaching equilibrium, they do yield a description of the state toward which the system is tending. There are, therefore, a number of inherent dangers in uncritically ac¬ cepting the values calculated from any of the numerous equilibrium computer models available. The problems are more fundamental than simply computa¬ tional, and since there have been recent reviews of the different models (Nordstrom et al. 1979, Jenne 1981, Kincaid et al. 1984, Nordstrom and Ball 1984), this report focuses on the varous aspects of the conceptual model as distinguished from the computer code that performs the indicated operations (Mercer et al. 1981). Other recent reviews include one by Plummer et al. (1983), which distinguishes between static and reaction-path models, and a brief one by Potter (1979), which does point out some of the problems with computer models and contains an extensive bibliography. Gibbs Free Energy and Equilibrium Constants As a criterion for the spontaneity of any process, neither the enthalpy nor entropy is entirely satisfactory. A process may be characterized by a large negative value of enthalpy and still not proceed spontaneously, whereas the associated entropy change must include calculations on the surroundings as well as on the system itself. Because of this, the American chemist J. Willard Gibbs developed the free-energy function in the late nineteenth cen¬ tury (Denbigh 1971, Lewis and Randall 1961, Moore 1972). For a process at constant temperature and pressure, the change in free energy G is defined as AG = AH - TAS, (4) where H represents the system's enthalpy, T its absolute temperature, and S its entropy. The advantage of the free-energy function as a measure of sponta¬ neity is that it depends only on the system, not on the surroundings, and incorporates temperature and pressure as its natural independent variables. 4 The change in Gibbs free energy can also be written in the form AG = q ' q rev’ (5) where q and q rev are the actual and the reversible heats, respectively, asso¬ ciated with a given process at constant temperature and pressure (Mahan 1963). For a process occurring under equilibrium conditions, q and q rev are equal, and AG = 0. If the process proceeds irreversibly, however, q = q.. < Q rev , and AG < 0. Therefore AG can never be greater than zero, and for a system tending irreversibly toward equilibrium, the free energy decreases until finally reaching its minimum value at equilibrium. Thus the general chemical reaction aA + pB *—> yC + 6D, (6) where the Latin majiscules represent chemical species and the Greek miniscules the appropriate stoichiometric coefficients, is defined as having reached equilibrium when the total Gibbs free energy of the products (the final state) minus that of the reactants (the initial state) is zero, i.e., when AG = yG^ + 6Gp - cfG^ ~ pGg = 0. (7) Each of the individual molar free energies is related to the activity a^ of the particular species i by the expression G. = G? + RT£n a i , (8) where G° represents the free energy of the species in some standard state and R is the gas constant. The expression for AG may be rewritten from eqs. (7) and (8) as AG = AG° + RT£n a Y a 6 a C a D a"a P a A a B (9) At equilibrium the ratio of activities is equal to the equilibrium constant K, 5 so AG° = -RT£n K, (10) which is the well-known relationship between standard free energy and the equilibrium constant. Each reaction of the set chosen for a particular model must be charac¬ terized by an equilibrium constant. In any geological environment there is an extremely large number of possible reactions, and this is reflected by the data bases of many of the models, some of which consist of several hundred reactions. These include not only reactions occurring solely in the aqueous phase, but also heterogeneous reactions between dissolved species and solid phases, such as precipitation/dissolution and ion exchange, as well as oxidation/reduction and degradation reactions that may be catalyzed by microorganisms in the soil. At least three fundamental problems are associated with such tabu¬ lations of thermodynamic data. A particular species may simply be omitted from the data base, so even though it is present in the system being modeled, it will obviously not appear in the final speciation results nor will its effect on the speciation of other elements. The program WATEQ3 (Ball et al. 1981), for example, is an extension of WATEQ2 (Ball et al. 1979) through the addition of several uranium species, but the expanded data base does not include vanadium, which frequently occurs naturally with uranium, and thus the influence of minerals containing both elements cannot be taken into account. Even when the data base does contain particular minerals, thermochemical data for them may not be known with very great precision. This problem is frequently compounded by other uncertainties such as nonstoichiometry, solution-dependent composition with respect to replaceable cations, metastable forms, and variation in free energy and solubility with the degree of crys¬ tallinity (Stumm and Morgan 1981). The tabulated thermodynamic data is also usually not checked for internal consistency. Because the data for a particular reaction may come from more than one source (and may thus be determined by different methods), there is no 6 guarantee that all calculations were made with consistent values of the neces¬ sary auxiliary quantities or that the data satisfies the appropriate thermo¬ dynamic relationships. Pressure Dependence of Equilibrium Constants Of all the available computer models, only SOLMNEQ (Kharaka and Barnes 1973) contains a pressure correction for the equilibrium constants, and at moderate temperatures and pressures this is usually quite small. From eq. (10) and the thermodynamic relationship 3G. ( ar> T = V < n > where V. is the partial molar volume of species i, the result is ,9£nKx ^ 9P ; T AV RT’ ( 12 ) where AV represents the difference in partial molar volumes between products and reactants. The term AV is ordinarily less than 30 cm 3 unless the reaction is char¬ acterized by a net change in the number of covalent bonds; an increase in the number of bonds decreases AV and vice versa (Moore 1972). Thus for reactions that exhibit a large change in AV and take place in deep ground-water systems (where the actual pressure can indeed be very large), the pressure dependence of K may no longer be insignificant. Temperature Dependence of Equilibrium Constants The effect of temperature on the equilibrium constant is much greater than that of pressure, and only a few of the computer models do not include a subroutine for temperature correction. The temperature dependence of the Gibbs free energy at constant pressure P is related to the enthalpy change for the reaction (AH°) through the Gibbs-Helmholtz equation (Lewis and Randall 1961): 7 (13) r 3(AG°/T) 1 3T The substitution of eq. (10) leads to the van't Hoff equation, , 3£n K . _ AH° ^ 3T ; P RT 2 ’ (14) or in integral form, t 2 AH° f, ^ dT . (15) In those cases for which the heat capacity (Cp) of each reactant and product is known as a function of temperature, AH° can be determined as a function of temperature, since dAH° dT ACp = a + bT + cT 2 + * * (16) Equation (15) may thus be integrated directly, and the resulting temperature- dependent expression for the equilibrium constant is frequently given in the form log K t = A + BT + C/T + D log T . (17) As Table I shows for the WATEQ series, such empirical expressions are avail¬ able for only a small fraction of the reactions included in computer models. TABLE I. NUMBER OF EMPIRICAL EXPRESSIONS FOR CALCULATING log K IN WATEQ SERIES Model # Empirical Expressions Total # Reactions Reference WATEQ 9 157 Truesdell and Jones (1974) WATEQF 15 191 Plummer et al. (1976) WATEQ2 17 526 Ball et al. (1979) WATEQ3 22 588 Ball et al. (1981) 8 For the remaining reactions, the van't Hoff equation is used. This is obtained from eq. (15) with the assumption that AH° is constant over the desired temperature range. The reference temperature is usually 25°C, so the integrated form is log K y = log K 2g8 AH 298 1 . 2.303R A 298 ; (18) Whether this is a good approximation or not depends on how constant AH° is over the particular temperature range; usually the smaller the range, the better the assumption. But a good approximation or not, the van't Hoff equation is often the only means for computing equilibrium constants at tem¬ peratures other than 25°C. As a comparison between log K values calculated from eq. (17) and those from eq. (18), Table II contains these values over the temperature range 0-100°C for all the reactions in WATEQ (Truesdell and Jones 1974) for which there are empirical expressions. The values from eq. (17) are presumably more accurate than those of eq. (18), since the former expressions are de¬ rived from measurements over a range of temperatures, though the WATEQ program does not indicate the range of applicability for any of these expressions. Table II provides a number of interesting comparisons. Rx #25. The equilibrium constant for the hydrolysis of boric acid in¬ creases with increasing temperature according to the van't Hoff equation, but actually decreases with increasing temperature according to the empirical expression. Rxs #35 & 68. The log K's calculated from the empirical expressions both exhibit maxima, but the van't Hoff equation can obviously predict only monotonic changes with temperature. Rx #72. The constants B and D are both set equal to zero in the em¬ pirical expression, which thus has exactly the same form as the van't Hoff equation. The agreement between log K values cal- 9 TABLE II. VALUES OF LOG K FROM WATEQ, CALCULATED BY EQS. (12) and (13) FOR COMPARISON o O CO CO 00 <3- m <71 CM PM cn PM O rH in rH CO CD t-H co CO 00 CD O rH rH co co O o o oo rH in rH co 00 CD CM CM CD pm • *$- *7 o rH CO CO CO <71 PM CM pm • • • • • • o • • • • • • • • • co co CO 00 rM O oo rH pm CD CD o <71 rH rH CM CM CD CD | 9 rH rH rH i I i • 1 rH 1 I i i • t x: r>» CO 00 00 m o |M. r-~ CD rH rH CD O CM *7 CO CD o co CD 00 00 o CM co CM r-'. rH n PM PM CO O cn CO 00 H cn cn CM cn vf cn CO rH rH cn r—i rH CO m ' 3 ' => 3 - • • ro d 00 00 00 d ad PM pm* CD CD d oi rH rH CM CM CD CD l i rH rH rH 1 i t 1 rH i 9 l i i i • CD CO 00 o CO pm rH <71 PM cn u TO CM m m I TO O <_> x CO CO CM PM 9 ** o CO I TO-* o co co X m oo co TO 0 l For the case of more than one solute in solution, all the solutes must simultaneously conform to the limit in eq. (21); such ideal behavior may also be stipulated in the limit as the mole fraction of water goes to unity. If the expression for activity in eq. (20) is substituted into eq. (8), the result may be written G. = G° + RT£n m. + RT£n y., (22) where the terms G° + RT£n m. represent the free energy of component i in an ideal solution, i.e., one that follows Henry's Law over the entire range of concentrations. Thus the term involving the activity coefficient is a measure of the real solution's deviation from this ideality and represents the extent of interaction between ions of the same kind. Since it is not possible to separate the effects of cations and anions in an electrically neutral solution, the properties of individual ions cannot be determined experimentally. It is necessary then to relate the laboratory value of the mean activity coefficient of an electrolyte, which represents an average over both cations and anions, to the calculated values of single-ion activity coefficients. This requires an assumption, often the mean-salt or 12 ACTIVITY (a) Figure 1. Activity vs. molality for HC1 solutions. The dotted line represents Henry's Law behavior. MOLALITY 1 + bl (29) which not only has the correct limiting form and obeys the condition in eq. (27) (because b is a universal empirical parameter), but also fits experi¬ mental data better than the conventional Debye-Huckel term given by eq. (23). 16 CALCIUM CARBONATE SOLUBILITY IMOLAL) Figure 2. CaC0 3 solubility as a function of NaCl concentration, experimental values and those calculated by four different equilibrium models (Kerrisk 1981). 17 Although the specific-interaction model is more complicated mathematical¬ ly, it has the distinct advantage of not explicitly including ion pairs for ions that are only weakly associated, such as Ca 2+ and Cl . Instead, the second virial coefficient accounts for these weak associations through its dependence on the ionic strength (Weare et al. 1982). Weare and his coworkers (Harvie and Weare 1980, Eugster et al. 1980, Harvie et al. 1982, Harvie et al. 1984) have begun applying this model to simple electrolyte systems—the most complicated thus far is one containing only 11 different ionic species—but the preliminary results appear to be a significant improvement over calcula¬ tions based on ion pairing. Figure 3 compares experimental values of CaS0 4 solubility with those calculated with the ion-pair model (Plummer et al. 1976, Kharaka and Barnes 1973) and with the ion-interaction model (Harvie and Weare 1980). There is still considerable work to be done before the spe¬ cific-interaction model can be applied to ground waters in general, but it clearly has the advantage of being able to treat more concentrated solutions than ion-pair theory. Pitzer's equations have already been or are currently being incorporated into at least three geochemical models: EQ3NR (Wolery 1983), SOLMNEQ (Kharaka and Barnes 1973), and PHREEQE (Parkhurst et al. 1980). Oxidation-Reduction Reactions Of all the reactions included in any of the computer models, only a small fraction consists of oxidation-reduction reactions. The model REDEQl-UMD (Harriss et al. 1984), for example, lists only twenty-two redox couples, and its authors caution that the kinetics of many oxidation-reduction reactions may be slow. The emf or Nernst potential (E) for any reaction involving electron transfer can be determined from the expression (Moore 1972) E = E° - (30) where n is the number of electrons transferred, F is the faraday, and the chemical notation refers to the general reaction in eq. (1). The term E° is 18 Figure 3. CaSO^ • ZH^O solubility as a function of Na 2 S0 4 concentration at two different NaCl concentrations (Weare et al. 1982). 19 the standard emf of the redox reaction and can be calculated from the standard electrode potentials of the half reactions that sum to the overall reaction (Latimer 1952). The fact that oxidation-reduction reactions can be characterized electro- chemically in this manner has led to the idea that a ground-water system's "redox state" can be described in terms of a single parameter, either an overall Nernst potential, usually designated Eh (Freeze and Cherry 1979), or the negative logarithm of the electron activity designated pe (Truesdell 1968) in analogy with pH. The idea that a single parameter like pe or Eh can char¬ acterize an entire system is based on the assumption that all the oxidation- reduction reactions occurring in the system are at equilibrium. That this is not true has been stated again and again (Morris and Stumm 1967, Jenne 1981, Wolery 1983), but apparently with little effect, since suggestions for some particular redox couple as an overall indicator of the system continue (Liss et al. 1973, Cherry et al. 1979). Lindberg and Runnel Is (1984) have quantitatively demonstrated the futil¬ ity in trying to characterize an entire ground-water system by a single redox parameter. The field-measured Eh value for each of approximately 600 water analyses was compared with the Nernst potential calculated from the data on ten different redox couples by means of the computer model WATEQFC (RunnelIs and Lindberg 1981). As these same authors (Lindberg and Runnels 1984) state: "The profound lack of agreement between the data points and the dashed line [which represents equilibrium points] shows that internal equilibrium is not achieved. Further, the computed Nernstian Eh values do not agree with each other. ... If any measured Eh is used as input for equilibrium calcu¬ lations, the burden rests with the investigator to demonstrate the reversi¬ bility of the system." Because many of the important oxidation-reduction reactions are very slow and some are even irreversible, it is virtually impossible that any natural- water system can reach equilibrium with respect to all of its redox couples. Improvements in this area of computer modeling will require the inclusion of experimental data for each of the major redox couples in the water system under study. 20 Conclusion Combining concepts from thermodynamics and electrochemistry, equilibrium models can be a valuable tool in predicting the behavior of complex geochem¬ ical systems. They contain pitfalls, however, and an understanding of—or at least familiarity with—the underlying conceptual model as well as the computa¬ tional methods should aid in properly using a particular equilibrium model and in making sound scientific judgements about its input and output. On the other hand, an unsuspecting user can be lulled into a false sense of security. These models calculate concentrations to what appears to be extremely great accuracy, yet there are two major cautions: 1) equilibrium is a state that open systems (i.e., real geochemical systems) can never achieve, and 2) the results of any model cannot be any better than the assumptions and raw data that go into it. In spite of these limitations, equilibrium models can provide a useful frame of reference for the knowlegeable user. 21 REFERENCES Ball, James W. , Everett A. Jenne, and Mark W. Cantrell. 1981. "WATEQ3—A Geochemical Model with Uranium Added.' 1 Open-File Report 81-1183. U.S. Geological Survey, Menlo Park, California. Ball, James W. , Everett A. Jenne, and Darrell Kirk Nordstrom. 1979. "WATEQ2—a computerized chemical model for trace and major element speciation and mineral equilibria of natural waters." In Chemical Model¬ ing in Aqueous Systems . Ed. Everett A. Jenne. ACS Symposium Series 93. Washington, D.C.: American Chemical Society, pp. 815-835. Barta, Leslie, and Daniel J. Bradley. 1985. "Extension of the specific interaction model to include gas solubilities in high temperature brines." Geochim. Cosmochim. Acta 49:195—203. Bjerrum, N. 1926. "Ionic association. I. Influence of ionic association on the activity of ions at moderate degrees of association." Kgl. Danske Videnskab. Selskab. Math.-fys. Medd. 7(9):1-48. Bockris, John O'M., and Amulya K.N. Reddy. 1970. Modem Electrochemistry. 2 vols. New York: Plenum Press. Brinkley, Stuart R. , Jr. 1946. "Note on the conditions of equilibrium for systems of many constituents." J. Chem. Phys. 14:563-564, 686. Cherry, John A., Ali U. Shaikh, D.E. Tallman, and R.V. Nicholson. 1979. "Arsenic species as an indicator of redox conditions in groundwater." J. Hydrol. 43:373—392. Davies, C.W. 1967. Electrochemistry. London: Philosophical Library. Denbigh, Kenneth. 1971. The Principles of Chemical Equilibrium. 3rd ed. Cambridge: Cambridge University Press. 22 Eugster, Hans P. , Charles E. Harvie, and John H. Weare. 1980. "Mineral equilibria in a six-component seawater system, Na-K-Mg-Ca-S0 4 -Cl-H 2 0, at 25°C." Geochim. Cosmochim. Acta 44:1335—1347. Freeze, R. Allan, and John A. Cherry. 1979. Groundwater. Englewood Cliffs: Prentice-Hal1. Fuoss, Raymond M. 1958. "Ionic association. III. The equilibrium between ion pairs and free ions." J. Am. Chem. Soc. 80:5059-5061. Fuoss, Raymond M. , and Charles A. Krauss. 1933. "Properties of electrolytic solutions. II. The evaluations of A 0 and of K for incompletely disso¬ ciated electrolytes." J. Am. Chem. Soc. 55:476-488. Garrels, Robert M. , and Charles L. Christ. 1965. Solutions, Minerals, and Equilibria. San Francisco: Freeman, Cooper. Harriss, Donald K., Sara E. Ingle, David K. Taylor, and Vincent R. Magnuson. 1984. "REDEQL-UMD, A Users Manual for the Aqueous Chemical Equilibrium Modeling Program REDEQL-UMD." University of Minnesota, Duluth. Harvie, Charles E., Hans P. Eugster, and John H. Weare. 1982. "Mineral equilibria in the six-component seawater system, Na-K-Mg-Ca-S0 4 -Cl-H 2 0 at 25°C. II. Compositions of the saturated solutions." Geochim. Cosmochim. Acta 46:1603-1618. Harvie, Charles E. , Nancy Miller, and John H. Weare. 1984. "The prediction of mineral solubilities in natural waters: the Na-K-Mg-Ca-H-Cl-S0 4 -0H- HC0 3 -C0 3 -C0 2 -H 2 0 system to high ionic strengths at 25°C." Geochim. Cosmochim. Acta 48:723-751. Harvie, Charles E. , and John H. Weare. 1980. "The prediction of mineral solubilities in natural waters: the Na-K~Mg-Ca-Cl-S0 4 -H 2 0 system from zero to high concentrations at 25°C." Geochim. Cosmochim. Acta 44:981-997. 23 Ingle, Sara E. , Marcus D. Schuldt, and Donald W. Schults. 1978. "A User's Guide for REDEQL.EPA: A Computer Program for Chemical Equilibria in Aqueous Systems. 11 Report EPA-600/3-78-024. U.S. Environmental Protec¬ tion Agency, Corvallis, Oregon. Jenne, Everett A. 1981. "Geochemical Modeling: A Review." Report PNL-3574. Pacific Northwest Laboratory, Richland, Washington. Kerrisk, Jerry F. 1981. "Chemical Equilibrium Calculations for Aqueous Geothermal Brines." Report LA-8851-MS. Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Kharaka, Yousif K. , and Ivan Barnes. 1973. "SOLMNEQ: Solution-Mineral Equilibrium Computations." Report PB-215-899. U.S. Geological Survey, Menlo Park, California. Kincaid, C.T., J.R. Morrey, and J.E. Rogers. 1984. "Geohydrochemical Models for Solute Migration. Volume 1. Process Description and Computer Code Selection." Report EA-3417. Electric Power Research Institute, Palo Alto, California. Latimer, Wendell M. 1952. The Oxidation States of the Elements and Their Potentials in Aqueous Solutions. 2nd ed. New York: Prentice-Hall. Lewis, Gilbert Newton, and Merle Randall. 1961. Thermodynamics. 2nd ed. Revised by Kenneth S. Pitzer and Leo Brewer. New York: McGraw-Hill. Lietzke, M.H., R.W. Stoughton, and T.F. Young. 1961. "The bisulfate acid constant from 25 to 225° as computed from solubility data." J. Phys. Chem. 65:2247-2249. Lindberg, Ralph D. , and Donald D. Runnells. 1984. "Ground water redox reac¬ tions: an analysis of equilibrium state applied to Eh measurements and geochemical modeling." Science 225:925-927. Liss, P.S., J.R. Herring, and E.D. Goldberg. 1973. "The iodide/iodate system in seawater as a possible measure of redox potential." Nature Phys. Sci 242:108-109. 24 Ma, Y.H., and C.W. Shipman. 1972. “On the Computation of Complex Equilib¬ ria." AIChE J. 18:299-304. Maclnnes, Duncan A. 1919. "The activities of the ions of strong electro¬ lytes." J. Am. Chem. Soc. 41:1086—1092. Mahan, Bruce H. 1963. Elementary Chemical Thermodynamics. Menlo Park: Benjamin. Mercer, J.W., C.R. Faust, W.J. Miller, and F.J. Pearson, Jr. 1981. "Review of Simulation Techniques for Aquifer Thermal Energy Storage (ATES)." Report PNL-3769. Pacific Northwest Laboratory, Richland, Washington. Mesmer, R.E., C.F. Baes, Jr., and F.H. Sweeton. 1972. "Acidity measurements at elevated temperatures. VI. Boric acid equilibria." Inorg. Chem. 11:537-543. Millero, F.J. 1982. "Use of models to determine ionic interactions in natural waters." Thalassia Jugoslavica 18:253—291. Moore, Walter J. 1972. Physical Chemistry. 4th ed. Englewood Cliffs: Prentice-Hal1. Morgan, James J. 1967. "Applications and limitations of chemical thermo¬ dynamics in natural water systems." In Equilibrium Concepts in Natural Water Systems. Ed. Werner Stumm. Advances in Chemistry Series 67. Washington, D.C.: American Chemical Society, pp. 1-29. Morris, J. Carrell, and Werner Stumm. "Redox equilibria and measurements of potentials in the aquatic environment." In Equilibrium Concepts in Na¬ tural Water Systems. Ed. Werner Stumm. Advances in Chemistry Series 67. Washington, D.C.: American Chemical Society, pp. 270-285. Nordstrom, Darrell Kirk, and James W. Ball. 1984. "Chemical models, computer programs and metal complexation in natural waters." In Complexation of Trace Metals in Natural Waters. Ed. C.J.M. Kramer and J.C. Duinker. The Hague: Martinus Nijhoff/Dr. W. Junk Publishers, pp. 149-164. 25 Nordstrom, D.K., et al. 1979. "A comparison of computerized chemical models for equilibrium calculations in aqueous systems." In Chemical Modeling in Aqueous Solutions. Ed. Everett A. Jenne. ACS Symposium Series 93. Washington, D.C.: American Chemical Society, pp. 857-692. Parkhurst, David L., Donald C. Thorstenson, and L. Niel Plummer. 1980. "PHREEQE—A Computer Program for Geochemical Calculations." Water- Resources Investigations 80-96. U.S. Geological Survey, Reston, Virginia. Pitzer, Kenneth S. 1973. "Thermodynamics of electrolytes. I. Theoretical basis and general equations." J. Phys. Chem. 77:268-277. Pitzer, Kenneth S., and Janice J. Kim. 1974. "Thermodynamics of electro¬ lytes. IV. Activity and osmotic coefficients for mixed electrolytes." J. Am. Chem. Soc. 96:5701-5707. Plummer, L. Niel, Blair F. Jones, and Alfred H. Truesdell. 1976. "WATEQF—A Fortran IV Version of WATEQ, a Computer Program For Calculating Chemical Equilibrium of Natural Waters." Water-Resources Investigations 76-13. U.S. Geological Survey, Reston, Virginia. Plummer, L. Niel, David L. Parkhurst, and Donald C. Thorstenson. 1983. "Development of reaction models for ground-water systems." Geochim. Cosmochim. Acta 47:665—686. Potter, Robert W., II. 1979. "Computer modeling in low temperature geochem¬ istry." Rev. Geophgs. Space Phys. 17:850-860. Robinson, R.A., and R.H. Stokes. 1959. Electrolyte Solutions. 2nd ed. London: Butterworths. Runnells, Donald D., and Ralph D. lindberg. 1981. "Hydrogeochemical explora¬ tion for uranium ore deposits: use of the computer model WATEQFC." J. Geochem. Explor. 15:37-50. 26 Sposito, Garrison, and Shas V. Mattigod. 1980. "GEOCHEM: A Computer Program for the Calculation of Chemical Equilibria in Soil Solutions and Other Natural Water Systems." University of California, Riverside. Stumm, Werner, and James J. Morgan. 1981. Aquatic Chemistry. 2nd ed. New York: John Wiley. Truesdell, A.H. 1968. "The advantage of using pe rather than Eh in redox equilibrium calculations." J. Geol. Educ . 16:17-20. Truesdell, Alfred H. , and Blair F. Jones. 1974. "WATEQ, a computer program for calculating chemical equilibria of natural waters." J. Res. U.S. Geol . Surv. 2:233-248. Weare, John H. 1981. "Geothermal-Brine Modeling—Prediction of Mineral Solubilities in Natural Waters: the Na-K-Mg-Ca-H-C 1 -S 0 4 - 0 H-HC 0 3 -C 03 -C 02 “ H 2 0 System to High Ionic Strengths at 25°C." Report D0E/SF/11563-T1. U.S. Department of Energy. Weare, John H. , Charles E. Harvie, and Nancy Mdller-Weare. 1982. "Toward an accurate and efficient chemical model for hydrothermal brines." Soc. Pet. Eng . J. 22:699-708. Whitfield, M. 1975. "Improved specific interaction model for sea water at 25°C and 1 atmosphere total pressure." Mar. Chem. 3:197-213. Wolery, Thomas J. 1983. "EQ3NR, A Computer Program for Geochemical Aqueous Speciation-Solubility Calculations: User's Guide and Documentation." Report UCRL-53414. Lawrence Livermore National Laboratory, Livermore, Cal i form* a. 27 Model Name (Last Update) [Language] Author(s) Contact Address Model Description Model Processes balance D.L. Parkhurst L.N. Plummer Using the chemical compositions of mass balance on (1982) L.N. Plummer U.S. Geological Survey water samples from two points along a elements I FORTRAN) D.C. Thorstenson Water Resources Division flow path and a set of mineral phases mixing end- 12201 Sunrise Valley Drive hypothesized to be the reactive con- member waters Reston, VA 22092 stituents in the system, the program redox reactions calculates the mass transfer necessary isotope balance to account for the observed changes in no thermodynamic composition between the two water constraints on samples. the reactions EQ3NR/6 T.J. Wolery T.J. Wolery EQ3NR is a geochemical aqueous specia- redox reactions (1983) Lawrence Livermore National tion/solubi1ity program that can be need not be at new version of Laboratory used alone or in conjunction with EQ6, equiIibrium EQ3NR expected P.0. Box 808, L-204 which performs reaction-path calcula- in 1987 Livermore, CA 94550 tions. Accomodates up to 40 elements, IFTN end CFT 300 aqueous species, 15 gases, and 275 FORTRAN for the minerals. CDC 7600 and Cray-1, as well as portability to IBM, UNIVAC, and VAX) EQUILIB J.R. Morrey Vasel W. Roberts Models chemical equilibria in geo- redox reactions (1978) D.W. Shannon Electric Power Research thermal brines at various elevated I FORTRAN IV for 1nstitute temperatures. Contains 26 elements. CDC and VAX) 3412 Hi II view Avenue 200 aqueous species, 7 gases, and 186 Palo Alto, CA 94304 minerals. GEOCHEM G. Sposito G. Sposito A program for predicting the equilib- mass balance for (1980) S.V. Matt 1 god Department of Soi1 and rium distribution of chemical species each species not currently Environmental Sciences in soil solution and other natural redox reactions available, new University of California water systems. Includes 45 elements. cation adsorp- version expected Riverside, CA 92521 1853 aqueous species, 42 organic tion and in 1987 ligands, 3 gases, and 250 minerals and exchange 1 FORTRAN IV for solids. IBM 370 and VAX) MINEQL2 J.C. Westa11 F.M.M. Morel A program for the calculation of chem- mass balance (1980) J.L. Zachary Dept, of Civil Engineering ical equilibria in aqueous systems. redox reactions (FORTRAN) F.M.M. Morel Massechusetss Institute of surface adsorp- Technology tion Cambridge, MA 02139 MINTED A.R. Felmy David Disney A program for calculating geochemical mass balance for (1904) D.C. Glrvin ADP Section equilibria, containing the WATEQ3 data- each component ] (FORTRAN IV for E.A. Jenne Environmental Research Lab. base. Includes 31 elements, 373 redox reactions UNIVAC 1100 and U.S. Environmental aqueous species, 3 gases, and 328 ion exchange VAX) Protection Agency solids. six surface com- College Station Road plexation Athens, GA 30613 mode 1s PHREEQE D.L. Parkhurst L.N. Plummer An equi1ibriurn mode 1 that can calculate mass balance (1980) D.C. Thorstenson U.S. Geological Survey mass transfer as a function of stepwise redox reactions (FORTRAN IV) L.N. Plummer Water Resources Division temperature change or dissolution. for 3 elements 12201 Sunrise Valley Drive Includes 19 elements, 120 aqueous ion exchange Reston, VA 22092 species, 3 gases, and 21 minerals. PROTOCOL G. Pickre11 D.D. Jackson A coupled kinetic/equilibrium program kinetic sub- (1984) D.D. Jackson Lawrence Livermore National for calculating dissolution reactions mode 1s (FORTRAN for Laboratory of inorganic solids in aqueous solu- -empirica1 CDC-7600 end P.0. Box 808, L-329 tion, with specific application to cor- -dissolution of Cray-1 ) Livermore, CA 94550 rosion of vitrified nuclear waste by silica groundwater. Incorporates equilibrium -surface routines from MINEQL. coverage Model Name (Last Update) (Language] Author(s) Contact Address Model Description Model Processes REDEQL.EPA (1978) (FORTRAN) S.E. Ingle M.D. Schuldt D.W. Schults D.W. Schults Hatfield Marine Sci. Cntr. U.S. Environmental Protection Agency Newport, OR 97365 A program to compute aqueous equilibria for up to 20 metals and 30 ligands in a system. Includes 46 elements, 94 aqueous species, 2 gases, and 13 minerals/solIds. mass balance redox reactions- comp location REDEQL-UMD (1984) (FORTRAN) D.K. Harriss S.E. Ingle D.K. Taylor V.R. Magnuson V.R. Magnuson Department of Chemistry University of Minnesota Duluth, MN 55812 A program to compute equilibrium dis¬ tributions of species concentrations in aqueous systems. Standard version includes 53 elements, 109 aqueous species, 2 gases, and 27 mixed solids redox reactions surface complex- ation adsorp¬ tion model SOLMNEQ (1973) not currently available, new version expected during 1987 I PL/1) Y.K. Kharaka 1. Barnes Y.K. Kharaka U.S. Geological Survey, MS/427 345 Middiefieid Road Menlo Park, CA 94025 A program for computing the equilibrium distribution of species in aqueous solution. Includes 26 elements, 162 aqueous species, and 158 minerals. mass balance of each element redox reaction SOLMNQ (1983) (FORTRAN) B.W. Goodwin M. Munday B.W. Goodwin Atomic Energy of Canada Ltd. Whiteshell Nuclear Research Estab1ishment Pinawa, Manitoba ROE 1L0 Canada Art interactive chemical spec i at ion program that calculates equilibrium distributions for inorganic aqueous species often found in groundwater, a FORTRAN version of SOLMNEQ. Includes 28 elements, 239 aqueous species, and 181 sol ids. mass balance of each element aqueous species of uranium and plutonium added redox reactions WATEQ2 (1980) (PL/I and FOR¬ TRAN. A PC FORTRAN version, desig¬ nated WATEQ4F, is also avaitable) J.W. Ball E.A. Jenne D.K. Nordstrom J.W. Bal1 U.S. Geological Survey, MS/21 345 Midd1efield Road Menlo Park, CA 94025 A chemical equilibrium model for calculating aqueous spec!at ion of major and minor elements among naturally occurring ligands. mass balance redox reactions WATEQ3 (1981) IPL/l i J.W. Bal1 E.A. Jenne M.W. Cantrel1 J.W. Bal1 U.S. Geological Survey MS/21 345 Middiefield Road Menlo Park, CA 94025 The WATEQ2 model with the addition of uranium species. mass balance redox reactions WATEQF (1984) (FORTRAN 77) 4 L.N. Plummer B.F. Jones A.H. Truesdel1 1 L.N. Plummer U.S. Geological Survey Water Resources Division 12201 Sunrise Valley Drive Reston, VA 22092 A program to model the thermodynamic speciat ion of inorganic ions and com¬ plex species in solution for a given water analysis. A FORTRAN version of the original WATEQ (1973) in PL/1. mass balance redox reactions international ground water modeling center Price List of Publications and Services Available from IGWMC July 1987 Holcomb Research Institute Butler University Indianapolis. Indiana 46208 USA TNO-DGV Institute of Applied Geoscience P 0 Box 285. 2800 AQ Oelft Tha Netherlands Contents Page 1. General Ordering Information.1 2. IGWMC Publications.2 3. IGWMC Groundwater Modeling Software 3.1 FORTRAN Programs.6 3.2 BASIC Programs.13 3.3 Hewlett-Packard HP-41 C Programs.18 4. IGWMC's Groundwater Model Information Retrieval System (MARS and PLUTO databases).21 i GMSBOOl 1. General Ordering Information ORDERS ARE FILLED AS ITEMS ARE AVAILABLE. CUSTOMER WILL BE INVOICED AFTER THE ORDER IS COMPLETED . REMITTANCE IN U.S. DOLLARS SHOULD BE FORWARDED AFTER RECEIPT OF INVOICE. Postage and handling fees are not included in prices. Postage and handling fees will be based on actual costs. Shipment abroad will be by surface rate unless otherwise requested. All listed prices are in U.S. dollars. Prices are subject to change without notice. Allow 2 to 3 weeks for preparation of software shipments. IGWMC updates this price list monthly. Please contact us for the latest listing. 1 2. I6WMC Publications 1983 Unit Price GWMI 83-02/2 Walton, W.C. Handbook of Analytical Ground Water Model Codes for Radio Shack TRS-80 Pocket Computer and Texas Instruments TI-59 Hand-held Programmable Calculator. Notes: Short Course April 11-15, 1983. GWMI 83-09 El-Kadi, A.I. Modeling Infiltration for Water Systems. HRI Paper No. 21. GWMI 83-10 El-Kadi, A.I. and P.K.M. van der Heijde. A review of infiltration models: identification and evaluation. Paper 83-2506, Winter Meeting Am. Soc. of Agric. Eng., December 13-16, 1983, Chicago, Illinois. Reprint. GWMI 83-11 van der Heijde, P.K.M. and P. Srinivasan. Aspects of the Use of Graphic Techniques in Ground Water Modeling. Proc. UCOWR Annual Meeting, July 24-27, 1983, Columbus, Ohio. Reprint. 1984 GWMI 84-06 GWMI 84-10 GWMI 84-12 GWMI 84-13 GWMI 84-14 GWMI 84-15 GWMI 84-17 Walton, W.C. Handbook of Analytical Ground Water Models. Notes: Short Course April 9-13, 1984. El-Kadi, A.I. Modeling Variability in Groundwater Flow. HRI Paper No. 31, June 1984. El-Kadi, A.I. Automated Estimation of the Parameters of Soil Hydraulic Properties. Huyakorn, P.S. et al. Testing and Validation of Models for Simulating Solute Transport in Ground- Water: Development, Evaluation, and Comparison of Benchmark Techniques. The International Ground Water Modeling Center. van der Heijde, P.K.M. Availability and Applica¬ bility of Numerical Models for Ground Water Resources Management. Presented at the NWWA/IGWMC conference, August 15-17, 1984, Columbus, Ohio. Reprint. Srinivasan, P. PIG-A Graphic Interactive Pre¬ processor for Ground Water Models. Presented at the NWWA/IGWMC conference, August 15-17, 1984, Columbus, Ohio. Reprint. El-Kadi, A.I. Stochastic Versus Deterministic Modeling of Ground Water Flow. Presented at the NWWA/IGWMC conference, August 15-17, 1984, Columbus, Ohio. Reprint. 10.00 15.00 free free 60.00 8.50 3.50 25.00 free free free 2 IGMMC Publications (continued) Unit Price 1985 GWMI 85-01 GWMI 85-06 GWMI 85-07 GWMI 85-08 GWMI 85-12 GWMI 85-16 GWMI 85-17 GWMI 85-28 GWMI 85-29 GWMI 85-30 GWMI 85-31 Beljin, M.S. Selected Bibliography on Solute Transport Processes in Groundwater, January 1985. 7.50 van der Heijde, P.K.M. Utilization of Numerical Models in Groundwater. Presented at the ASCE Computer Applications in Water Resources Con¬ ference in Buffalo, New York, June 10-12, 1985. free van der Heijde, P.K.M., P.S. Huyakorn and J.W. Mercer. Testing and Validation of Ground Water Models. Presented at the NWWA/IGWMC Conference, "Practical Applications of Ground Water Models," August 19-20, 1985, Columbus, Ohio. free van der Heijde, P.K.M. Groundwater Contamination Following a Nuclear Exchange. Report of the SCOPE/ENUWAR Workshop in Delft, The Netherlands, October 3-5, 1984. free van der Heijde, P.K.M. and M.S. Beljin. Listing of Heat Transport Models which are Documented and Available. 2.50 van der Heijde, P.K.M. and M.S. Beljin. Listing of Models to Simulate Location and Movement of Fresh Water-Salt Water Interfaces in Groundwater. August 1984. 2.50 van der Heijde, P.K.M. Review of DYNFLOW and DYNTRACK Groundwater Simulation Computer Codes. Report findings for the U.S. ERA, Washington, DC. free van der Heijde, P.K.M. Modeling Contaminant Transport in Groundwater. Presented at the 1985 Washington Conference on Groundwater Protection and Cleanup, November 12-13, 1985, Arlington, Virginia. free van der Heijde, P.K.M., Spatial and Temporal Scales in Groundwater Modeling. Presented at the SCOPE/ INTECOL/ICSU workshop, "Spatial and Temporal Varia¬ bility of Biospheric and Geosheric Processes," October 27-November 1, 1985, St. Petersburg, Florida. free Beljin, M.S. Listing of Microcomputer Graphic Software for Groundwater Industry. November 1985. free Beljin, M.S. Analytical Modeling of Solute Transport. Presented at the NWWA/IGWMC Conference, "Practical Applications of Ground Water Models," August 19-20, 1985, Columbus, Ohio. free 3 IGWMC Publications (continued) Unit Price GWMI 85-32 1986 GWMI 86-04 GWMI 86-05 GWMI 86-06 GWMI 86-07 GWMI 86-08 GWMI 86-13 1987 GWMI 87-01 GWMI 87-02 GWMI 87-03 GWMI 87-04 Huyakorn, P.S., P.F. Andersen, and H.O. White, Jr., and P.V.M. van der Heijde. Testing and Application of a Finite-Element Groundwater Flow and Transport Model. Presented at the International Syposium on Management of Hazardous Chemical Waste Sites, Winston-Salem, October 9-10, 1985. Rice, R.E. The Fundamentals of Geochemical Equilibrium Models; with a Listing of Hydrochemical Models that are Documented and Available. December 1986. van der Heijde, P.K.M. Listing of Review Publi¬ cations and Textbooks in Groundwater Modeling. Beljin, M.S. Microcomputers in the Analysis of Pump Test Data. Presented at the Southeastern Ground Water Symposium, October 30-31, 1986, Orlando, Florida. van der Heijde, P.K.M. and M.S. Beljin. Selected References on Programs for Hand-held Calculators. van der Heijde, P.K.M. and M.S. Beljin. Available Solute Transport Models for Groundwater and Soil Water Quality Management. August 1986. El-Kadi, A.I. and L. Smith. Stochastic and Geo- statistical Analysis for Groundwatr Modeling: Part II. Notes: Short Course 15-19, 1986. van der Heijde, P.K.M and A.I. El-Kadi. Short Course Notes: Basics of Groundwater Modeling. March 18-20, 1987 Mercer, J.W., P.F. Andersen, and L. Konikow. Applied Groundwater Modeling. Notes: Short Course March 23-27, 1987. van der Heijde, P.K.M. and P. Srinivasan. Summary Listing of Groundwater Models for Mainframe and Minicomputers (MARS data base). van der Heijde, P.K.M. and P. Srinivasan. Selected Summary Listing of Available and Documented Ground- water Models for Mainframe and Minicomputers (MARS data base). free 3.50 free free 2.50 2.50 35.00 75.00 75.00 25.00 20.00 4 IGfeftC Publications (continued) Unit Price GWMI 87-05 van der Heijde, P.K.M. and P. Srinivasan. Listing of Available Groundwater Models for Microcomputers (PLUTO data base). GWMI 87-06 Bel jin, M.S. Representation of Individual Wells in Two-dimensional Groundwater Modeling. Presented at the NWWA/IGWMC Conference, "Solving Groundwater Pro¬ blems with Models," February 10-12, 1987, Denver, Colorado. 20.00 free 5 3. IGWMC Groundwater Modeling Software 3.1 FORTRAN Computer Programs for Mainframe and Microcomputers The IGWMC distributes a rapidly increasing number of FORTRAN programs related to groundwater modeling. All FORTRAN codes are implemented on a DEC VAX 11/780. Unless specified differently, the programs are also available for IBM PC/XT/AT microcomputers and compatibles. Copies of the software are provided on a magnetic tape in user-specified formats. PC versions are provided on 5.25" or 3.5" diskettes (includes source code, executable version and sample data). Unless indicated otherwise, the programs are public domain. IGWMC software comes complete with documentation, including pertinent reports, user's instructions, program listing, and example problems. Support Policy The Center provides limited support for the software it distributes. Assistance in implementation is available in the form of written or telephone response to user's questions. For some models, IGWMC refers the inquiring party to model developer(s) for optional code acquisition and/or support. Software support provided by the Center pertains only to programs obtained directly from the Center. The user of IGWMC software accepts and uses the program material as it is at the user's own risk, relying solely upon his/her own inspection of the program material and without reliance upon any representation or description concerning the program material. Neither HRI nor its individual staff members make any expressed or implied warranty of any kind with regard to the program material. Therefore, neither HRI nor its individual staff members shall be liable for any damages in connection with the furnishing, use, or performance of the program material. The Center maintains a list of its software purchasers. Users receive notification of updates regarding distributed programs. For users who obtained the program from the Center, updated versions of the software are available at cost. System Requirements: IBM PC or compatibles with 640K, Math Coprocessor 8087/80287 and printer. A list of currently available FORTRAN programs is given on the following pages. 6 INTERNATIONAL GROUND WATER FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC MODELING CENTER • Holcomb Research Institute Butler University Indianapolis, Indiana 46208 USA Tel :317/283-9458 • TNO-DGV institute of Applied Geoscience P.O. Box 285, 2600 AG Delft The Netherlands TeI:15/569330 AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE S ORDER I C. Su, R.H. Brooks FP (1.0 11/85) 6170 To determine the parameters of the retention function (the soil-water characteristic func¬ tion) from experimental data 70 F0S01 I . Javande 1 , C. Doughty, C.F. Tsang AGU-10 Single pack¬ age with 6310, 6312, 3940, 6313, and 6383 120 F0S05 LTIRD (1.0 02/85) 6310 A semi-analyticai solution to radial dispersion in porous media, calculating the dimen¬ sionless concentration of a particular solute, injected into an aquifer, as a function of dimensionless time for dif¬ ferent values of dimensionless radius. OOAST (1.0 02/85) 6312 An analytical solution for one- dimensional solute transport including convection, disper¬ sion, decay, and adsorption in porous media. RESSQ (1.0 02/85) 3940 A semi-ana Iytica 1 model for calculation of streamlines lo¬ cation of contaminant fronts and calculation of concentra¬ tion of sinks through simula¬ tion of two-dimensional, advec- tive contaminant transport with adsorption in a homogeneous, isotropic confined aquifer with steady-state regional flow. RT (1.0 02/85) 6313 This program converts a time series of concentration data from one or more observation wells into a spatial concentra¬ tion distribution in the aqui¬ fer at various times. The mod¬ el can be used for cases when regional flow can be neglected and a single production well creates a radial flow field in the aquifer . Preprocessor and Post¬ processor incIuded (see ART) M. S. Beijin ART (1.0 10/86) 6383 A pre- and post-processor for RT. This is an interactive program for inputting new data and storing them in a file, editing existing data. ART also can be used to graphically display resuits of RT. IBM PC graph i c board (CGA) required 7 FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE $ ORDER # J.V. Tracy MOCNRC (1.0 03/85) 0740M A modified version of The original Konikow/Bredehoeft MOC model to include radioactive decay and adsorption (Linear, Langmuir and Freundich isotherm). Single package with MOCNRC and MOCNRCM 150 F0S06 D.C. Kent, l. leMaster, J. Wagner MOCNRCM (1.0 12/86) This is a modified MOCNRC model that can simulate water-table aquifer conditions. In addi¬ tion the model has option to simulate only the head distri¬ bution, anq to create an output for use with SAS graphics pro¬ grams. Preprocessor i nc1uded Part of F0S06 L.F. Konikow, J.D. Bredehoeft MOC (2.3 04/87) 0740 A model to simulate transient, two-dimensional, horizontal groundwater flow and solute transport in confined aquifers using the method of character¬ istics and including flow simu¬ lating subroutines. The model has option to simulate radio¬ active decay and linear adsorp¬ tion. The user can specify up to 16 particles per cell. Two versions of the model are in¬ cluded in the package: one using interative ADI, and one using SlP numerical solution technique. Preprocessor PREM0C (3.1 04/87) i nc1uded 200 F0S07 M.G. McDonald, A.W. Harbaugh mooflow (1.1 05/86) 3980 A modular finite-difference groundwater model to simulate two-dimensional and quasi- or fu11y-three-dimensiona1, tran¬ sient flow in anisotropic, het¬ erogeneous, layered aquifer systems. 120 F0S08 J.C. Parker, M.Th.van Genuchten CXTFIT (1.0 05/85) 3432 An inverse model to determine values for the one-dimensional analytical solute transport parameters using a nonlinear least-squares method. 70 F0S09 D. L. Parkhurst, L. N. Piummer , D.C. Thorstenson BALANCE (1.0 05/86) 3400 A chemical equilibrium model for calculation of the mass transfer along a flow path and including redo* reactions. 50 F0S10 J.N. Plummer, B.F. Jones, A.H. Truesdell WATIN/WATEQF ( 1.0 08/86) 3620 A program for chemical equili¬ brium calculations and specia- tion including redox reactions. Preprocessor i ncIuded 120 F0S1 1 T.A. Prickett, C.G. Lonnquist PLASM (1.1 06/86) 0322 A finite difference model to simulate two-dimensional tran¬ sient, saturated flow in an anisotropic, heterogeneous sin¬ gle- or multi- layered aquifer system with water table and/or confined or leaky confined con- ditions . Preprocessor avaiiable for mainframe version 95 F OS 1 2 8 FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE $ T.A. Prickett, T.G. Naymik, C.G. Lonnquist RANDOM WALK (1.0 H/85) 2690 A model to simulate one- or two-dimensional steady or un¬ steady solute transport pro¬ blems in heterogeneous aquifers under water table or confined conditions based on the random walk method and including flow simulating subroutines. Preprocessor avaiiabie for mainf rame version 95 P.C. Trescott, S.P. Larson, L.J. Torak USGS-3D-F LOW (1.0 1982) 0770 A finite difference model to simulate transient, quasi- and fully three-dimensional, satu¬ rated flow in anisotropic, het¬ erogeneous groundwater systems. Main f rame version only 95 P.S. Trescott, G.F. Pinder, S.P. Larson USGS-2D-FL0W (1.0 1976) 0771 A finite difference model to simulate transient, two-dimen¬ sional horizontal or vertical flow in an anisotropic and het¬ erogeneous confined, leaky-con¬ fined or water-table aquifer. Main frame vers ion only 95 M.Th. van Genuchten SOHYP (1.0 04/86) 6226 An analytical model for calcu¬ lation of the unsaturated hy¬ draulic conductivity function using the soil moisture reten¬ tion data via Muaiem or the Burdine theories. 70 M.Th. van Genuchten UNSAT1 (1.0 07/85) 3431 A fininte element model to sim¬ ulate one-dimensional satu- rated-unsaturated flow in het¬ erogeneous SOiIs. 70 M.Th. van Genuchten, W. J . Alves ONE-D (1.0 07/85) 6220/ 24 A package of 5 analytical solu¬ tions to tne one-dimensional convective-dispersive transport equation with linear adsorp¬ tion, zero-order production, and f i rst order decay. 70 M. Vauc l i n (modified by A.1. El-Kadi) INFIL (1.0 06/83) 3570 To solve a one-dimensional in¬ filtration into a deep homogen¬ eous soil using finite differ¬ ences; output includes water content profile and amount and rate of infiltration at differ¬ ent simulation times. Soil properties need to be expressed in mathematical form. 70 G.T. Yeh, D.S. Ward FEMWASTE-1 (1.0 1981) 3371 A two-dimensional finite ele¬ ment model for transierrt simu¬ lation of areal or cross-sec¬ tional transport of dissolved constituents for a given velo¬ city field m an anisotropic, heterogeneous porous medium. The velocity field is generated by the FEMWATER-1 code. Main frame vers ion only 95 ORDER # FOS13 FOS15 F0S16 FOS17 FOS18 F0S19 F0S20 F0S21 9 FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE $ ORDER # G.T. Yeh, D.S. Ward FEMWATER-1 (1.0 1981) 3370 A two-dimensional finite ele¬ ment model to simulate tran¬ sient, cross-sectional flow in saturated-unsaturoted aniso¬ tropic, heterogeneous porous media. Ma inf rame version on 1 y 95 F0S22 A.1. E1-Kad i stati (1.0 06/85) 6333 A program for basic statistical analysis of data. Various mo¬ ments of the statistical dis¬ tribution are estimated: the mean, median, standard devia¬ tion, coefficient of variation, variance, standard error, maxi¬ mum, minimum, and range. 70 F0S24 A.i. El-Kadi ST2D (1.0 1985) 4160 This program is for the sto¬ chastic analysis of gravity drainage via the Monte-Carlo technique. It consists of three sections: a generator for hydraulic conductivity realiza¬ tions, a finite-element simu¬ lation program, and a statis¬ tical analysis routine. Mainframe version on y 120 FOS25 J.B. Kool, J.C. Parker, M.Th.van Genuchten ONESTEP (1.1 10/85) 3433 This program will estimate parameters in the van Genuchten soil hydraulic property model from measurements of cumulative outflow with time during one- step experiments. The program combines a nonlinear optimiza¬ tion routine with a Galerkin finite element model. 70 FOS26 C.l. Voss SUTRA (1.0 1985) 3830 A two-dimensional model to sim¬ ulate density dependent fluid movement under saturated or unsaturated conditions and transport of either energy or dissovied substances in a sub¬ surface environment employing a hybrid finite-element and integrated-finite-difference method. Mainf rame version on iy 120 F0S27 S.A. Williams , A.1. E1-kadi COVAR (1.0 01/86) 6334 A program for generating two- dimensional fields of autocor- reiated parameters which are log-normally distributed (e.g., hydraulic conductivity). The program uses a technique based on matrix decomposition. the generated parameter field rep¬ resents a major requirement in the stochastic analysis vie Monte-Carlo techniques. 50 F0S29 M.Th. van Genuchten SUMATRA-1 (1.0 02/86) 3430 A one-dimensional finite- element model to simulate the simultaneous movement of water and solutes in satureted-unsa- turated and non-homogeneous soil. The effects of linear adsorption and zero- arid first- order decay are included. 70 F0S30 10 FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE S ORDER # P.R. Schroeder , et a 1. HELP (1.0 01/87) Hydrologic Evaluation of Landfill Performance program for estimation of surface runoff, subsurface drainage, and leachate that may be expected from operation of a variety of possible landfill designs. The program models the effects of precipitation , surface storage, runoff, infiltration, percolation, evapotranspirat ion , soil moisture storage,and lateral drainage. 120 FOS32 P.S. Huyakorn, et a 1. TRAFRAP (1.0 03/86) 0589 A two-d i mens i ona 1 finite ele¬ ment code for simulating fluid flow and transport of radio¬ nuclides in fractured and un¬ fractured porous media. The code can be used to model both groundwater flow and solute transport, or either process separately. The TRAFRAP model accounts for 1) fluid interac¬ tions between the fractures and porous matrix blocks; 2) advec- t i ve-dispersive transport in the fractures and diffusion in the porous matrix blocks and fracture skin; and 3) chain re¬ actions of radionuclide compon¬ ents. A major advantage of TRAFRAP is the capability to model the fractured sustem us¬ ing either the dua1-porosity or the discrete-fracture modeling approach or a combination. Main f rame vers ion only 250 FOS33 M.A. Butt, C.D. McElwee VARQ (1.0 04/86) 6082 A program to calculate aquifer parameters by automatically fitting pump test data and Theis type curve. The program allows variable discharge rate during the test. 50 FOS34 M.Th. van Genuchten CF1T IM (1.0 11/85) 6227 A program for estimation of non-equilibrium solute trans¬ port parameters from miscible displacement experiments. De¬ pending upon the exact form of the transport model, the pro¬ gram allows up to five differ¬ ent parameters to be estimated. 70 FOS35 W.E. Sanford, L.F„ Konikow MOCDENSE (1.0 01/87) 0742 A numerical model to simulate solute transport and dispersion of either one or two consti¬ tuents in groundwater where there is two-dimensional, den¬ sity-dependent flow. The model is a modified version of the Konikow and Bredehoeft Model MOC (1978), which uses finite- difference methods and the method of characteristics to solve the flow and transport equations. 120 FOS36 11 FORTRAN MAINFRAME AND MICROCOMPUTER SOFTWARE AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE $ ORDER # G.T. Yeh AT123D (1.0 6/87) 6120 An analytical solution for transient 1-, 2-, or 3-dimen- sionel transport in a homo¬ geneous, anisotropic aquifer. The program allows for retarda¬ tion and decay. Simulation of a variety of source configura¬ tions and boundary conditions is possible. 95 FOS38 12 IGWMC Groundwater Modeling Software - (continued) 3.2 BASIC Programs for Microcomputers BASIC programs for IBM PC/XT/AT microcomputers and compatibles are avail¬ able from IGWMC. The programs come with a documentation and include code listing and example problems. A copy of the BASIC program is provided on 5V diskette (includes source code, executable version, and document file). System Requirements: IBM PC or compatibles with 128K and printer. Some programs require IBM compatible graphic board (or HERCULES graphic board) and HP7475A plotter. For more information contact IGWMC. For ordering information see page 1. 13 INTERNATIONAL GROUND WATER BASIC MICROCOMPUTER PROGRAMS AVAILABLE FROM IGWMC MODELING CENTER • Holcomb Research Institute Butler University Indianapolis, Indiana 46208 USA Te I :317/283-9458 • TNO-DGV Institute of Applied Geoscience P.0. Box 285, 2600 AG Delft The Netherlands TeI:15/569330 NAME AUTHORS (Version IGWMC PURPOSE REMARKS PRICE ORDER Date) KEY S # P.K.M. van der Heijde PLUME 6020 An analytical model to calcu- 35 BAS01 (1.0 10/83) late three-dimensional concen¬ tration distribution in a homo¬ geneous aquifer with a contin¬ uous solute injection in a one¬ dimensional flow field. P.K.M. van der Heijde, PLASM 6010 A finite difference model for A teaching 35 BAS02 P. Srinivasan (4.0 01/86) simulating two-dimensional tool. For transient saturated flow in rea1-world confined aquifers. mode Iing see #F0S12 P.K.M. van der Heijde, RANDOM WALK 601 1 To simulate one- or two- dimen- A teaching 35 BAS03 P. Srinivasan (3.3 06/86) sional, steady or unsteady flow tool. For and transport problems in homo- rea1-wor1d geneous aquifers under confined modeling see conditions. #F0S13 P.K.M. van der Heijde THWELLS 6022 An analytical model to calcu- IBM PC 50 BAS04 (2.0 02/87) late drawdown or buildup in graphic non-steady groundwater flow in board an isotropic homogeneous non- leaky confined aquifer with multiple pumping and injection wells. Boundary effects can be required included through use of image wells. Results are displayed in tabular form, time-drawdown curves, and contour plots. The program has options to read from and write to an external file. Metric or English units can be used. P.K.M. van der Heijde THEISFIT 6080 To calculate aquifer parameters 35 BAS05 (1.0 09/83) by automatically fitting type curve and pump test data from pumping an isotropic homogen¬ eous nonleaky confined aquifer. P.K.M. van der Heijde GWFLOW2 6023 A menu-driven series of 7 rou- IBM PC 50 BAS06 (2.0 04/87) tines each containing an analy- graphic tical solution to a groundwater board flow problem. Results are displayed in tabular form or time-drawdown curves. Metric or English units can be used. required A.1. El-Kadi INF 1 L 6335 To calculate infiltration rate IBM PC 35 BAS07 (1.0 12/83) and amount and water content VAX 11/780 profile at different times us¬ ing the Philip series solution of a one-dimensional form of the Richards equation. MICROVAX 14 BASIC MICROCOMPUTER PROGRAMS AVAILABLE FROM IGWMC (continued) NAME (Version Oate) PLUME2D ( 1.2 01 / 86 ) 35 Micro¬ computer programs (1.1 03/85) PESTRUN (1.1 01/85) RADFLOW (1.0 09/84) TETRA (1.1 09/85) BASIC GWF (1.1 01/87) SOIL (1.0 04/85) IGWMC KEY 6024 6350 6280 6064 6430 6030 6330 PURPOSE An analytical model to calcu¬ late the tracer concentration distribution in a homogeneous, nonleaky contained aquifer with uniform regional flow. The program uses the we I I-function for solute advection and dis¬ persion in a system with con¬ tinuously injecting, full, pen¬ etrating wells. It includes options for retardation and radioactive decay. A series of analytical and sim¬ ple numerical programs to anal¬ yze flow and transport of so¬ lutes and heat in confined, leaky confined, and water table aquifers with simple geometry. A simple pesticide runoff model to approximate runoff values to identify watersheds which need attention to evaluate effects of different conservation prac¬ tices. A finite difference model for transient radial flow towards a well in a homogeneous, isotro¬ pic aquifer. The model allows for switching from confined to unconfined conditions when wa¬ ter levels are drawn beneath top of aquifer. Program in¬ cludes restart capabilities for varying pumping schedules. A simple program to calculate velocity components in three dimensions from hydraulic head measurements. Groups of four observation points are con¬ nected to form tetrahedrons, and a linear interpolation is used to calculate head gra¬ dients for each tetrahedron. Application of Darcy's Law then yields velocity components. Analysis of plane, steady or unsteady groundwater flow in an isotropic, heterogeneous, con¬ fined or unconfined 'aquifer by the finite element method. To estimate soil hydraulic pro¬ perties using a non-linear least-square analysis; major input to the code includes pairs of measured water content and suction. REMARKS PRICE S 35 l BM PC TRS-80/1I I APPLE lie 70 35 35 35 Shareware: 10 price does not include con tribution to author IBM PC 35 VAX 11/780 MICR0VAX ORDER # BAS08 BAS09 BAS 10 BASH BAS 12 BAS 13 BAS 1 4 15 BASIC MICROCOMPUTER PROGRAMS AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE REMARKS PRICE $ ORDER # M.S. Belj in SOLUTE (1.0 01/85) 6380 A program package of 8 analy¬ tical models for solute tran¬ sport in groundwater, a metric- to-English unit conversion pro¬ gram, and a subroutine to cal¬ culate error functions. The user-friendly, menu-driven pro¬ grams come with optional screen end printer graphics. IBM PC or HERCULES graphic board optiona1 70 BAS 15 P.K.M. van der Heijde based on FORTRAN program by C.D. McE1 wee TSSLEAK (1.2 09/85) 6081 This program fits the Hantush and Jacobs equation to experi¬ mental pump test data to obtain the "best" values for storage c oeffiecient, transmissivity, leakage coefficient, and aqui- terd permeability by using a least-squares procedure. 35 BAS 16 M.S. Be 1jin PUMPTEST (1.0 06/86) 6382 An interactive, menu-driven program package to calculate transmissivity and storage coefficient from time-drawdown, distance-drawdown, or recovery pump test data. Jacob's method and regression analysis are applied to the user's specified portion of the data curve. The program has options for inter¬ active input of data, reading data from an external file, an editor, screen graphics, plot¬ ting on an HP7475A plotter, and displaying results on the screen or the printer. English or metric system of units can be used. IBM PC or HERCULES graphic board ana HP7475A plotter optional 70 BASIS K.R. Bradbury, E.R. Rothschi1d TGUESS (1.0 01/86) 6450 A program for estimating trans¬ missivity from specific capac¬ ity data. The program will correct the data for partial penetration and well losses. 35 BAS 19 G.N. Paudya I , A. Das Gupta OPTP/PTEST (1.0 10/86) 6570 A fully interactive package consisting of two programs for determining the optimal well discharge. PTEST computes the coefficients and exponent of the nonlinear drawdown equation using data from a step-drawdown test. 0PTP computes the opti¬ mal discharge using a single¬ variable constrained nonlinear programming algorithm. 35 BAS20 P.K.M. van der Heijde THCVFIT (1.0 01/87) 6025 An interactive program to de¬ termine transmissivity and storage coefficient from pump¬ ing tests. it replaces tradi¬ tional curve fitting by on screen graphic rendition of the Theis well function and the field drawdown dare. The pro¬ gram has option to read from end write to an external file. IBM PC graphic board (CGA) required 35 BAS21 16 BASIC MICROCOMPUTER PROGRAMS AVAILABLE FROM IGWMC (continued) AUTHORS NAME (Version Date) IGWMC KEY PURPOSE remarks PRICE $ ORDER # D.8. Thompson TlMELAG (1.0 05/87) 6580 A program to estimate hydraulic conductivity from time-lag tests for most wel 1 configura¬ tions. The method involves instantaneously raising and lowering the water level in a we 11 (Hvorslev 1951). Pub 1ished in Ground Water 25T7T7 10 BAS22 17 IGWMC 6roundwater Modeling Software - (continued) 3.3 Hewlett-Packard HP-41C Documented programs for Hewlett-Packard HP-41C are available from IGWMC. The programs come with complete documentation and include code listing. Prerecorded magnetic cards for each program are also available. IGWMC differentiates between four product types. Unit Price IGWMC Standard Program Documentation including Program Listing (paper copy) $ 5.00 IGWMC Standard Program Documentation and pre¬ recorded magnetic cards $ 10.00 Special Reports (Documentation and prerecorded magnetic cards), e.g., AQTST $25.00 HP-41C Program Package (documentation and prerecorded magnetic cards of 11 selected programs) $55.00 For ordering information see page 1. 18 INTERNATIONAL GROUND WATER HEWLETT-PACKARD HP-41C PROGRAMS AVAILABLE FROM IGV#4C MODELING CENTER • Holcomb Research Institute Butler University Indianapolis, Indiana 46208 USA Tel: 317/283-9458 • TNO-DGV Institute of Applied Geoscience P.O. Box 285, 2600 AG Delft The Netherlands Tel: 15/569330 IGWMC-Key TITLE Proqram Name HPS-01 Hantush "Well-Function" in the Pocket Calculator WURL 1 $ HPS-02 Theis Condition Well Field NWELLS 2 T HPS-03 One-Dimensional Non-Steady Ground Water Flow EDELMAN't HPS-04 Steady Radial Ground Water Flow in a Finite Leaky Aquifer ISLE 1 T HPS-05 Streamlines and Traveltimes for Regional Ground Water Flow Affected by Sources and Sinks F LOP-2 1 T HPS-06 Advection and Dispersion from a Stream with Regional Flow STRDISP 2 T HPS-07 Advection and Dispersion from a Solute Injection Well RADDISP 2 * HPS-08 Analysis of Various Flow in a Single Aquifer Including Leakance Problems and Recharge aqmodl^t HPS-09 Evaluating Theis Parameters from a Pumping Test TFIT3J HPS-10 Inverse Solutions of the Theis Equation I - Single Parameter Calculation THINV-1 3 T HPS-11 Evaluation of Well Characteristics from Step- Drawdown Test Data FASTEP 5 * HP S-12 Economically Optimal Well Discharge Rate QOPTIM 5 * 'English version by IGWMC Original program by IGWMC Modified by IGWMC ^Original version ^Documented in: Helweg, O.J. et al. (1983), Improving Well Pump Efficiency. American water works Assoc., 6666 West Quincy Ave.„ Denver, CO 80235. Phone: 303/ 794-7711. ONLY PRE-RECORDED MAGNETIC CARDS AVAILABLE FROM IGWMC. ♦Nonstandard pricing ^Included in 11-program package 19 Hewlett-Packard HP-41C Programs (continued) IGWMC-Key TITLE __ Program Name HPS-13 Benefit-Cost Analysis for Replacement or PRA * * * S Rehabilitation of Pump HPS-14 Metric-English Units Conversions MECONV 3 HPS-15 Inverse Solutions of the Theis Equation THINV-2 3 T II - Aquifer hydraulic constants HPS-21 Aquifer Test Analysis with a Hand-held Calculator AQTST 4 * HPS-22 Two-Dimensional Flow to a Horizontal Drain in a DRAIN* Confined Aquifer HPS-23 Ground Water Unit Step and Rectangular Pulse PULSE 3 Response HPS-24 HPS-25 HPS-26 HPS-27 HPS-28 HPS-29 Parameter Analysis and Drawdown Calculation ANSTPY 3 for Anisotropic Confined Aquifers An Idealized Ground Water Flow and Chemical S-PATHS 4 Transport Model Simulation of Well Pumping and Recovery in a THEIS 3 * 6 Confined or Unconfined Aquifer Calculating Drawdown for a Single Well in a BOUN 4 * 6 Confined/Unconfined Aquifer Bounded by Two Parallel Impermeable Boundaries Two-dimensional Pollution Plume in a Homogeneous PLUME2D 3 Aquifer with a Uniform Horizontal Flow Field Including Dispersion and Retardation List of Groundwater Reserves LGWRES 4 HPS-30 HPS-31 HPA-32 Solute Transport of a Contaminant from WMPLUME 4 Multiple Point Sources A Program to Calculate Aquifer Transmissivity SPCAP 4 from Specific-Capacity Data A Program to Calculate Mounding due to Asymmetric Recharge INVHAN 1 * ‘English version by IGWMC Original program by IGWMC Modified by IGWMC 4 0riginal version s Documented in: Helweg, O.J. et al. (1983), Improving well pump efficiency. Am. Water Works Assoc., only pre-recorded magnetic cards available from IGWMC 6 2HPS 26 and 27 are documented in a single standard priced note ♦Nonstandard pricing ^Included in 11-program package 20 4. IGWMC's Groundwater Model Information Retrieval System The IGWMC's data bases, MARS and PLUTO, are designed to facilitate rapid accessibility to information on groundwater models for mainframe and microcomputers, respectively. Each model is described by a set of annotations of its operating characteristics, capabilities and availability. An extensive checklist, the model annotation form, is developed to describe each model as completely and consistently as possible. This list is used by IGWMC staff to enter the model information in one of the databases. For retrieval of specific model information from the databases a model annotation retrieval form is filled out by requestor or at the Center. A computer search is then executed by IGWMC staff, identifying those models which are suited for requestor's problem. Information on these models is printed either in suimary form or as a listing of the complete annotations. Before sending it to the requestor the search results are evaluated by the Center's technical staff. For the rapidly expanding category of microcomputer software, IGWMC has recently developed the PLUTO database. PLUTO and MARS are based on the same concepts, but the presentation of model information differs in that PLUTO has more emphasis on software compatibility and hardware specifications. For both databases the following services are available. Search and Retrieval: Price MARS 1. Selected Sunmary listing (GWMI 87-04) 20.00 2. Summary listing of all stored models (GWMI 87-03) 25.00 3. Complete annotation without search one annotation 5.00 each additional annotation 1.00 4. Executing search 15.00 each selected complete annotation 1.00 each selected suronary, per 20 annotations 1.00 PLUTO 5. Summary listing of available models (GWMI 87-05) 20.00 6. Executing search each selected annotation 15.00 .50 Special selections from the databases are possible. Contact IGWMC. For ordering information see page 1. 21 vy November 1986 U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP Report of Findings and Discussion of Selected Ground-water Modeling Issues by Paul K.M. van der Heijde and Richard A. Park International Ground Water Modeling Center Holcomb Research Institute Butler University Indianapolis, Indiana 46208 Project Officers: Joseph F. Keely Clint W. Hall Scott R. Yates Office of Research and Development, R.S. Kerr Environmental Research Laboratory, Ada, Oklahoma 74820 This study was conducted under Cooperative Agreement CR-812603 with the U.S. Environmental Protection Agency, R.S. Kerr Environmental Research Laboratory, Ada, Oklahoma 74820 INTERNATIONAL GROUND WATER MODELING CENTER Holcomb Research Institute, Butler University, Indianapolis, Indiana 46208 CONTENTS 1. Executive Summary.. Introduction.. Issues.1 Code Selection and Acceptance.2 Review and Procurement of Modeling Studies...2 Research Needs.....3 Information Exchange.4 Staff.5 Recruitment and Retention. 5 Training.5 Workload...5 2. The U.S. ERA Ground-water Modeling Policy Study Group.7 Introduction.7 Definition of Terms.7 Responsibilities and Objectives.8 Authority.8 Reporting.8 3. Role of Ground-water Models in U.S. EPA.10 Mathematical Ground-water Models. 10 Ground-water Models in U.S. EPA.11 Site-specific Modeling.12 Generic Modeling.13 Development of Regulations and Policies.13 Permitting. 14 Remedial Action.15 Ground-water Modeling in Program Offices...16 Office of Drinking Water. .16 Office of Health and Environmental Assessment.16 Office of Pesticide Programs.16 Office of Policy, Planning and Evaluation.18 Office of Toxic Substances.18 Office of Solid Waste.18 Office of Waste Programs Enforcement.18 4. Modeling Concerns.20 Program Office Concerns. 20 Office of Drinking Water...20 Office of Pesticide Programs.20 Office of Policy, Planning and Evaluation.21 Office of Toxic Substances.21 Office of Waste Programs Enforcement.22 Office of Research and Development.23 Office of Health and Environmental Assessment.23 Envirorvnental Research Laboratory, Athens, Georgia.23 Concerns of Regional Offices.23 Model Use in Regional Activities.24 Regional Staff.25 Quality Assurance.26 Procurement of Modeling Studies.27 Technology Transfer and Training.27 Facilities and Resources.28 Legal Concerns.29 Summary of Regional Concerns...29 5. Discussion of Issues.30 Adequacy of Modeling Theory and Data.30 Ground-water Code Review and Testing.33 Model Evaluation...34 Model Review.34 Model Examination.34 Evaluation of Documentation.35 Evaluation Ease of Use.35 Computer Code Inspection.35 Model Verification.36 Model Validation.36 Validation Scenarios.38 Sensitivity Analysis.38 Proprietary Codes versus Public Domain Codes and Acceptance Criteria.39 Banning the Use of Proprietary Codes.39 Continuing the Use of Proprietary Codes.40 Options for U.S. EPA.41 Quality and Usefulness of Model Studies.43 Code Selection.43 Quality Assurance in Ground-water Modeling Studies.46 Definition and Role of Quality Assurance in Ground-water Modeling.46 Current EPA Quality Assurance Policies.47 EPA Quality Assurance Options for Ground-water Modeling.49 Model Development...50 Model Application.51 Technology Transfer and Training to Sustain and Improve Expertise of Agency Personnel.52 Technology Transfer and Training In EPA.53 Information Exchange on Ground-water Modeling.54 Training.56 Recruitment and Retention.57 References. 59 Appendix: Composition of Study Group.62 i v SECTION 1 EXECUTIVE SUMMARY INTRODUCTION In late 1985, the Office of Environmental Processes and Effects Research of the U.S. EPA invited the International Ground Water Modeling Center at Holcomb Research Institute to coordinate and lead a Study Group, charged with examining issues related to U.S. EPA use of ground-water models and associated constraints. This task has been pursued through meetings, thorough fact¬ finding documented in a series of interim reports, and a final report sunmarizing the findings of the Study Group and presenting the most prominent issues. The Study Group included representatives of various EPA Program Offices and external ground-water modeling experts, who met with technical and managerial staff of the Program Offices and selected Regions. Interim reports and final findings were circulated to a group of corresponding members from Program and Regional Offices for their comment Ground-water models are mathematical tools to aid in organizing information pertinent to complex ground-water systems, and in evaluating alternative options for efficient mangement of ground-water resources. It is within such a decision support framework, pertinent to EPA's mission, that the Study Group meetings were held and modeling related issues explored. ISSUES The Study Group has examined the Agency's uses of ground-water flow and contaminant transport models, and its associated needs and capabilities. Specifically, the use of models in regulatory decision making (e.g., banning), permitting, and enforcement actions was discussed in the context of potential liabilities and the subsequent need for Agency policies. Mathematical models are often efficient means for EPA to develop its ground-water protection programs. Currently available models may be used to test hypotheses about site-specific and generic problems and to assist analysis of alternative courses of action in solving ground-water protection problems. Models are also used in research to develop a fuller understanding of the physical, chemical, and biological processes that affect ground-water quality. The latter use is aimed at developing models which accurately represent complex situations such as those involving imniscible fluids, dense plumes, or fractured rock aquifers. The role of ground-water-flow and contaminant-transport models in the development of policies and regulations, and in permitting and in planning monitoring and remedial action, is continuing to grow within the EPA. 1 However, the Study Group found that this growth does not seem to occur in a structured and coordinated manner. At present, no criteria or policies provide Agency-wide guidance in the use of models for regulatory planning and decision-making purposes. EPA's ground-water modeling needs currently seem to outpace its actual use of models in virtually all program areas. Areas where official policy statements on aspects of ground-water model use may be advisable include the use of proprietary models (as opposed to public domain software) for enforcement support work, minimal documentation and quality control procedures for the development and application of models, and possibly the promotion of "standard" models. Such policy statements should be consistent with the policies adopted for surface water and air modeling. Suggestions and recommendations are organized under the following five major topics: code selection and acceptance, review and procurement of modeling studies, research needs, information exchange, and staff. CODE SELECTION AND ACCEPTANCE In recent years both development and use of ground-water models in studies performed by or for the U.S. EPA have become the focus of professional criticism, public discussion, and even adversary legal procedures. Therefore, determining code reliability, establishing code acceptance criteria, and providing guidance in model selection have become increasingly important. In establishing an Agency mechanism for such guidance, proper attention should be given to definition of study objectives, determination of modeling scenarios, system conceptualization, and formulation of selection criteria. The reliability of codes should be established by adopting a widely accepted review and testing procedure. Agency acceptance of a model should be based on technical and scientific soundness, user friendliness, and legal and administrative considerations. A list should be compiled of reviewed and validated computer codes acceptable to the Agency, and the Agency should advocate the use of such codes. Proprietary codes important to the Agency's mission should, where possible, be brought into the public domain. The Agency should assess the use of "expert systems" for assistance in selection and use of acceptable models. Such a system should be oriented to solving problems rather than identifying systems and processes. Options include a meeting of specialists to detail courses of action, and a pilot study to explore the potential of this new technology. REVIEW AND PROCUREMENT OF MODELING STUDIES One of the major issues emerging from the Study Group concerned the quality of ground-water modeling studies carried out by or for the U.S. EPA, and the usefulness of the study results in the Agency's decision-making process. 2 Agency decisions should be based on the use of technically and scientifically sound data collection, information processing, and interpretation methods. Within an overall Agency quality assurance (QA) program, a procedural and organizational framework for QA in ground-water modeling should be established. All projects that involve modeling should have an adequate QA plan defined. Such a plan should specify goals for the quality of resulting data and processed information acceptable to the user; should contain detailed descriptions of the measures to be taken to achieve prescribed quality objectives; and should assign responsibility for achieving the goals. The plan should also contain procedures for documenting the activities within a project in order to establish an administrative record supporting Agency decision making. EPA should have effective review and auditing procedures in place to monitor QA performance of the modeling project teams. More than in the past, attention should be given to applying such quality assessment procedures during a project, and not just at the end of it. The Study Group considers it important that a more direct mechanism for reviewing and directing the work of outside contractors be established. Current procurement procedures limit the effectiveness of EPA project managers, especially in the Regions. RESEARCH NEEDS Appropriate models do not yet exist for all types of ground-water problems because many of the assumptions and simplifications common to existing models do not allow faithful simulations in unusual situations, and because the natural processes that affect fluid and contaminant movement are not yet fully understood. This is especially true for chemical and biological processes. Improvements are needed, concurrently, in several major areas. Data acquisition methods and interpretive models are needed that can examine to an unprecedented degree the physical, chemical, and biological processes controlling the transport and fate of ground-water contaminants. Unfortunately, few of the constants and coefficients needed to incorporate chemical and biological processes into contaminant transport evaluations are available presently. Development is need for simulation of flow and transport in fractured and dual-porosity media and in multimedia. Further, representation of stochastic processes in predictive modeling, and incorporation of economic factors in modeling to improve estimation of clean-up costs, should be studied. Models are needed for management of ground-water contamination plumes, as well as risk assessment and risk management. Special attention should be given to research that includes volatilization, multiphase flow, density-dependent flow, and immiscible flow in ground-water models. Fundamental research supporting ground-water modeling is considered necessary in such areas as 3 retardation. transient behavior of process parameters (e.g., hydraulic conductivity) desorption for nonhydrophobic chemicals multicomponent transport and chemical interaction transport of silt with sorbed chemicals in aquifers improved numerical accuracy, stability, and efficiency INFORMATION EXCHANGE In recent years modeling for ground-water protection has become a rapidly growing area of technology. As a result, information on technological and scientific advances has become increasingly available for ground-water management. Disseminating this information through communication and education is the goal of technology transfer. In its broadest sense, technology transfer includes the distribution of modeling codes and documentation, and providing training and assistence in model use. The Study Group found that many improvements in information exchange, training, and software distribution can be made within the Agency. The U.S. EPA should establish a systematic technology transfer program with ground-water modeling as an integral component. Such a program should be based on an active approach in providing information and should be flexible enough to disseminate research results quickly. Furthermore, a program should be developed to train, on a continuous basis, agency personnel in ground-water modeling as an integral part of their involvement with ground-water quality issues. Such a training program should incorporate recent scientific and technological advances and provide opportunity to share practical experience. Both information exchange and training should reach each staff member involved in ground-water projects. For ad hoc consultation on specific problems, project managers should have access to experts such as (1) in-house Regional experts, (2) experts within ORD, perhaps located at EPA labs, (3) contractors, or (4) experts from other agencies. In addition, a networking mechanism needs to be developed to promote increased communication and sharing of experiences among staff of Regional Offices and Program Offices. Results from relevant research projects have not been disseminated effectively to Regions and some Program Offices. Reports on ground-water modeling should be distributed to a targeted mailing list that is updated frequently. Many publications are in the open literature, and provision should be made for distributing these as reprints. Each division or branch Involved in ground-water modeling should have its own working library of pertinent publications. 4 STAFF Recruitment and Retention The difficulty in attracting and retaining a skilled staff with a background in ground-water model application, and the high turnover rates among staff, are serious problems in all EPA Offices. The Study Group suggests three steps to alleviate this situation: • The Office of Human Resources should establish the position of "hy¬ drogeologist." • A career path for hydrogeologists should be established through the GS-15 level. • The Agency should encourage staff to take short courses and graduate- level courses in ground-water geology and modeling conditionally at the Agency's expense. Traininq Training of Agency personnel in the effective use of ground-water models continues to be a problem. Many staff members with degrees in engineering and environmental science, need additional basic training in ground-water geology. Physical geology and ground-water geology should be offered through the EPA Training Institute and other institutions, and should be required prior to taking a ground-water modeling course. The training of EPA staff in ground-water modeling should be aimed mainly at providing skills needed to evaluate the effectiveness of model codes and modeling work, since only a few of the staff will be (or should be) in a position to become modeling experts. Training should be based on the realities of needs, staff backgrounds, administrative structures and constraints, and potential changes in staff. Management should be sensitive to the financial and time requirements necessary for adequate training. (One does not become a competent modeler by completing a one-week short course or training program.) Workload The workload problems identified at the Study Group meetings in Regions I and III clearly limit the scientifically sound use of the analytical, planning, and design methods provided by efficient modeling. Furthermore, to incorporate efficient modeling into projects, project managers should be sensitized to the potential and proper role of models in their work. Staff with training and experience in ground-water modeling should be available in each Region. Because of the time and effort required to characterize a ground-water system before a suitable remedy can be selected, management needs to recognize 5 that it often takes a significant amount of time to properly perform studies in which modeling is an integral part. The Study Group finds it extremely important, especially under conditions of severe time and budget constraints, that models be used at an early stage of the project in order to optimize data collection, data analysis, and design of management alternatives. 6 SECTION 2 THE U.S. EPA GROUND-WATER MODELING POLICY STUDY GROUP INTRODUCTION During 1984 and 1985, a series of issues related to the selection and use of ground-water models within the U.S. Environmental Protection Agency (EPA) was brought forward by EPA staff in several Program Offices and Regional Offices. These issues included assessment of the validity of computer codes, criteria for selection of appropriate models for specific applications, and review procedures for determining the applicability and validity of models used by third parties. Another important issue brought forward was the need for quality assurance in modeling projects carried out by or for the Agency. The Office of Environmental Processes and Effects Research of EPA/ORD (OEPER) and its Robert S. Kerr Environmental Research Laboratory (RSKERL) in Ada, Oklahoma, held wide-ranging discussions on these issues to determine the best way to resolve them. In early 1985, a consensus led to the formation of a Study Group charged with examining issues related to EPA use of ground-water models and associated needs and constraints. Specifically, the Study Group was to conduct a thorough fact-finding, documented in a series of working papers and summarized in a report intended to inform EPA managers of the issues and their significance for various offices and programs. The working papers might later be modified and adopted as official guidance documents. Accordingly, the Office of Environmental Processes and Effects Research asked the International Ground Water Modeling Center at Holcomb Research Institute, Indianapolis, to coordinate the activities of the Study Group and invite other Program Offices and model users in Regional Offices to partici¬ pate in the Study Group activities. Formal requests for designated participa¬ tion went out to the Assistant Administrators for Solid Waste and Emergency Response, for Water, for Pesticides and Toxic Substances, and for Policy, Planning and Evaluation. Several offices responded and named representatives; the persons participating in the Study Group activities are listed in Appendix 1. DEFINITION OF TERMS The general objectives of ground-water management can be characterized as the optimal and efficient utilization of ground-water resources and the protection of those resources for sustained and future utilization. Because most of the modeling-related ground-water management issues important to EPA pertain to specific programs as administered by the various Offices, the discussions were to be focused on those elements which are or should be included in an Agency-wide approach to modeling. For the purposes of this study, ground-water models are defined as restricted to the mathematical framework describing a ground-water system and 7 its inherent processes and stresses, and the subsequent implementation of that mathematical framework in a computer code. Physical models and screening and ranking models (e.g., DRASTIC) are not included. Traditionally, ground-water was understood to encompass the saturated zone of the subsurface. Because of the close relationship between transport and fate of contaminants in the unsaturated and saturated zones of the subsurface, and their equal importance in addressing ground-water protection, the ground-water modeling issues discussed in this report relate to both unsaturated and saturated zones. RESPONSIBILITIES AND OBJECTIVES The Study Group was to review problems and issues relating to ground-water modeling practices in EPA Program Offices and Regions, to evaluate model needs and uses in existing programs, and to present approaches and provide guidance to improve model use and solve the problems identified. The Group was responsible for conducting thorough fact-finding (including site visits to Agency Offices in Region I, III, and X, and a meeting with staff of Program Offices at EPA Headquarters), and for summarizing its findings in a series of interim reports and a final report containing selected working or position papers. AUTHORITY Established by OEPER/RSKERL, the Study Group has operated under the auspices of the International Ground Water Modeling Center at Holcomb Research Institute, which is responsible for delivering a report on the Group's activities and findings. An IGWMC staff member (van der Heijde) served as Chairperson. The Study Group's authority to produce working papers for the EPA was specifically limited to fact-finding, reporting, and suggesting new policies or changes in current Agency policies, and in providing guidance to Agency staff. REPORTING The Study Group met five times, initially at the Holcomb Research Institute, Indianapolis, Indiana, and subsequently in EPA offices in Boston, Philadelphia, Seattle, and Washington, D.C. A report of the findings of each meeting was prepared and distributed to Study Group members. The present text is the final report and includes the major elements of the individual meeting reports. The report is divided into four parts: (1) the role of ground-water models in U.S. EPA; (2) concerns of various offices and individuals within the EPA with respect to a variety of modeling issues; (3) discussion of major modeling issues, including adequacy of modeling theory and data reliability, and acceptance of ground-water simulation codes, quality and usefulness of model studies, and technology transfer and training of EPA staff, and (4) various ways to resolve current porblems and improve the qualified use of models for decision-making procedures within the Agency. The third section contains the four issue papers prepared initially by the Study Group. The report starts with an executive sunfnary. 8 Before submitting the final report to the EPA, participating and corresponding members were asked to comment on a draft version. Corments received have led to a rearrangement of topics and an expansion of policy- related discussions. All Study Group documents and written communications are filed at the Holcomb Research Institute. 9 SECTION 3 ROLE OF GROUND-WATER MODELS IN U.S. EPA MATHEMATICAL GROUND-WATER MODELS The analysis of ground-water flow and contaminant transport cannot yet be thought of as exact science. Although the physical processes involved obey known mathematical and physical principles, precise aquifer and contaminant characterization is hard to obtain and often makes even plume definition a difficult task. However, where these characteristics have been reasonably established, ground-water models may provide a viable, if not the only, method to predict contaminant transport, locate areas of potential environmental risk, and assess possible remediation/corrective actions. Mathematical models are used to help organize the essential details of complex ground-water management problems so that reliable solutions are obtained. Applications include a wide range of technical, economic, and sociopolitical aspects of ground-water supply and quality (Holcomb Research Institute 1976; Bachmat et al. 1978; Mercer and Faust 1981; U.S. Office of Technology Assessment 1982; Javandel et al. 1984; van der Heijde et al. 1985). Existing models can be categorized by their technical uses, as follows (Bachmat et al. 1978; van der Heijde et al. 1985): (1) parameter identification models, (2) predictive models, (3) resource management models, and (4) data manipulation codes. Parameter identification models are most often used to estimate the aquifer coefficients for fluid flow and contaminant transport characteristics, such as annual recharge (Puri 1984), coefficients of permeability and storage (Shelton 1982; Khan 1986a, 1986b), and dispersivity (Guven et al. 1984; Strecker and Chu 1986). Predictive models are the most numerous because they are the primary tools for testing hypotheses (Andersen et al. 1984; Mercer and Faust 1981; Krabbenhoft and Anderson 1986). Resource management models are combinations of predictive models, constraining functions (e.g., total pumpage allowed), and optimization routines for objective functions (e.g., optimization of well-field operations for minimum cost or minimum drawdown/pumping lift). Very few of these are so well developed and fully supported that they may be considered practicable, and there does not appear to be an extensive effort to improve the situation (van der Heijde 1984a, 1984b; van der Heijde et al. 1985). Data manipulation codes also have received little attention until recently. They are now becoming increasingly popular because they simplify input preparation (as "preprocessors") for increasingly complex models, and because they facilitate the production of graphic displays (as 10 "postprocessors") of the model outputs (van der Heijde and Srinivasan 1983; Srinivasan 1984; Moses and Herman 1986). Other software packages are available for routine and advanced statistics, specialized graphics, and database management needs (Brown 1986). GROUND-WATER MODELS IN U.S. EPA A policy of resource protection based on monitoring is by its very nature always reactive, not preventive; however, model-based policies and regulations can be both preventive and reactive. Because adequate on-site monitoring is not always feasible due to costs, available manpower, or site accessibility, models can provide a viable and effective alternative. An optimal approach to the management of ground-water resources includes the integrated use of modeling and monitoring strategies. Mathematical models can be helpful to EPA in managing ground-water protection programs. Currently available models may be used to test hypo¬ theses about site-specific and generic problems, and to develop a fuller understanding of the physical, chemical, and biological processes that affect ground-water quality. The former use is self-evident, but the latter use is also quite important because many improvements are necessary before models can accurately represent complex situations such as those involving immiscible fluids, dense plumes, or fractured rock aquifers. Few aspects of the Agency's ground-water protection programs can function efficiently without the use of mathematical models. Any activity requiring some estimate of ground-water flow or contaminant transport, including data gathering and interpretation, can benefit from the judicious use of ground- water models. Some of the principal areas where mathematical models can now be used to assist in the management of EPA's ground-water protection programs are: • development of regulations and policies • planning and design of corrective actions and waste storage facil¬ ities • problem conceptualization and analysis • development of guidance documents • design and evaluation of monitoring and data collection strategies • enforcement Specifically, ground-water modeling plays or could play a role in: • determining or evaluating the need for regulation of specific waste disposal, agricultural, and industrial practices 11 • analyzing policy impacts such as evaluating the consequences of setting regulatory standards and banning rules, and of delisting actions • assessing exposure, hazard, damage, and health risks • evaluating reliability, technical feasibility and effectiveness, cost, operation and maintenance, and other aspects of waste-disposal facility designs and of alternative remedial actions • providing guidance in siting of new facilities and in permit issuance and petitioning • detecting pollutant sources • developing aquifer or well-head protection zones • assessing liabilities such as post-closure liability for disposal sites These activities can be broadly categorized as either site-specific or generic modeling efforts, and these categories can be further subdivided into point-source or nonpoint-source problems. The success of these modeling efforts depends on the accuracy and efficiency with which the natural processes controlling the behavior of ground water, and the chemical and biological species it transports, are simulated. The accuracy and efficiency of the simulations, in turn, depend heavily on the applicability of the assumptions and simplifications adopted in the model(s), and on subjective judgments made by the modeler and management. SITE-SPECIFIC MODELING Whether for permit issuance, investigation of potential problems, or remediation of proven contamination, site-specific models are necessary for the Agency to fulfill its mandate under a number of major environmental statutes. The National Environmental Policy Act of 1970 stipulates a need to show the impact of major construction activities in Environmental Impact Statements; potential impacts are often projected successfully by the use of mathematical models. Some of the most difficult site-specific problems facing the Agency involve hazardous waste sites falling under the purviews of RCRA and CERCLA/Superfund. Associated with most of these sites is a complex array of chemical wastes and the potential for ground-water contamination. The hydrogeologic settings of such sites usually appear quite Intricate when examined at scales appropriate for technical assessments and remediation efforts (e.g., hundreds to thousands of feet). In all phases of these analyses, ground-water models are useful Instruments. 12 GENERIC MODELING In a number of instances where the Agency has limited data or other constraints, site-specific modeling is not feasible. As a result, many decisions are made with the assistance of generic models. Such models are more often analytical than numerical, in contrast to site-specific models. This is a logical consequence of the simplified mathematics of analytical models, the significantly greater data requirements of numerical models, and the higher costs of numerical simulations. The Agency has many statutory responsibilities that benefit from generic modeling, including the estimation of potential environmental exposures and their integration with dose-response models to yield health-based risk assessments. These assessments are necessary, for example, in issuing compound-specific rulings on products subject to preregistration requirements under the Toxic Substances Control Act (TSCA) and the Federal Insecticide, Fungicide, and Rodenticide Act (FIFRA). More generalized policy formulation activities also benefit from generic modeling; examples include policy decisions about land disposal "banning," setting Alternate Concentration Limits, preparing Technical Enforcement Guidance Documents (i.e., for moni¬ toring network designs), and "delisting" under RCRA. DEVELOPMENT OF REGULATIONS AND POLICIES Evaluation of the impacts (economic, health-risk, and otherwise) of regulations or policy scenarios requires process-oriented, generic models. Some specific uses of such models in the evaluation of proposed and existing policies and regulations within the U.S. EPA include: • testing the efficacy of standards such as meeting 10-6 health risk levels (versus background or detection limits) • evaluation of conservation tillage through the use of linked surface water, unsaturated-zone, and saturated-zone ground-water models • developing guidance for well-setback with pesticide applications, using uncertainty analysis • evaluating the seriousness of various "failures" of injection wells through the use of sensitivity analyses Generic models are often used to provide a technical rationale for policy development, as illustrated by the following examples: • An analytical contaminant transport model coupled with Monte Carlo analysis has been used to provide the technical justification for restricting the land disposal of hazardous wastes, under the Hazardous and Solid Waste Amendments of 1984 (HSWA 1984) of RCRA. The hazardous disposal ban decisions are based on the results of model simulations for a wide range of site-specific hydrogeologic characteristics. 13 • A model has been used for the analysis of potential failure scenarios of waste injection in deep wells in four different regional hydrogeologic settings. The results of the study will be used to set policy and develop regulations under HSWA 1984 (Section 3004 f and 9 ). • Long-term fate of hazardous waste injected into deep saline ground- water environments has been studied by means of a hydrogeochemical simulation model. The results will be used to aid in setting policy and siting criteria for the petition process of the hazardous waste injection well ban under HSWA 1984. • A computer model is planned for use in evaluating the need and effectiveness of ground-water monitoring programs for hazardous waste injection wells. The results will be used to help develop regula¬ tions under HSWA 1984 (Section 3004 f and g) for the ground-water monitoring of hazardous waste injection wells. Sometimes, models are used as an integral part of EPA policies and regulations. Such models are often published in the Federal Register as part of the rule-making process. Examples are the delisting model used to delist wastes, and the banning model in the land disposal restriction rule. Models are being used increasingly to implement policies and regulations pertinent to hazardous waste facilities, such as to prove or disprove a CERCLA endangerment, and to determine clean-up levels. PERMITTING In discussing the role of modeling in the permitting process, the Study Group differentiated between permit applications having a site-specific character, and EPA product permit review procedures where the models are used generically for screening purposes. These different types of usage require different types of models and expertise. Reliable data on actual transport and fate of chemicals are often 'lacking, especially in the case of nonpoint-source releases, as for pesticides. Under such constraints, generic models are used to evaluate the potential for pollution and contaminant migration. The lack of data often prevents the use of complex numerical models, thus forcing the permit writer to make rigorous assumptions regarding the system under study. However, permit writers may not have the expertise to evaluate such model usage adequately. In the permitting process for hazardous waste facilities, ground-water models can be used on a site-specific basis by owners/operators of hazardous waste facilities, to show compliance with the permit requirements, and by regulatory agencies to validate the information provided for permitting purposes. The permitting agency could be the EPA or a corresponding state agency. These models can be used to evaluate site characteristics, to 14 determine the optimal location of monitoring wells, to estimate the transport and fate of contaminants, and to assess corrective action plans. REMEDIAL ACTION Ground-water models are used increasingly in the CERCLA response process for remediation of hazardous substances releases. The current state of ground-water modeling practices for remedial response analyses is highly variable from site to site. A typical model application for Superfund- financed or enforcement-related remedial response actions includes site investigation to assist in problem definition and system conceptualization (thereby guiding data collection and data analysis), to identify the contam¬ ination source, and to predict future contamination and health risks. Models are also used for development and evaluation of remedial alternatives during the remedial investigation/feasibility study (RI/FS) stages, and for analysis of design specifications for the chosen remedial action alternative. The use of ground-water models is fairly standard for the design of pump and treat types of remedial alternatives; however, they are not widely used for other types of remedial alternatives. Further, models are sometimes used to assess required clean-up levels, the extent of required source removal, and the projected performance characteristics of remedial action designs, as well as to formulate postoperation and closure requirements. Models contribute to justifying the basis for Agency action (i.e., exposure analysis as part of public health risk assessment procedures or as part of an enforcement endangerment assessment). Some examples of source identification with models are contained in the literature for Superfund sites as are several examples where ground-water models are being used to assist in the interpretation of monitoring data after the implementation of the remedial alternative. Discretion as to when to use a computer code and which code to use in a remediation project is often left to the EPA contractors and/or the responsible parties who perform the RI/FS. Some impediments to model applications in remediation analysis result from the segmented nature of the overall remedial response process, with different activities being conducted in discrete steps, at different times, and often with different contractors. Thus, during a specific step ground- water modeling may not be implemented due to time or cost constraints, or models may be selected for only a few of the potential uses rather than for multiple uses. Few, if any, comprehensive ground-water model applications exist from the start to the finish of a site-remedial response. 15 GROUND-WATER MODELING IN PROGRAM OFFICES Office of Drinking Water The Underground Injection Control (UIC) Program, which originated in the Safe Drinking Water Act (SDWA) (1974) and is now subject to provisions of the Resource Conservation and Recovery Act (1984 Amendments), requires an evaluation of the potential for excessive pressure build-up and contaminant movement out of the injection zone. Mathematical models are the primary mechanism for the required evaluation, due in part to the difficulty of installing monitoring wells several thousand feet deep. Because of the character of the injected waste and because most underground waste injection takes place in deep sedimentary basins, the models and assumptions required for the UIC program (for example, saline aquifers at 5,000-foot depth) differ from those common to most other Offices. Particularly, UIC uses complex models such as the three-dimensional finite- difference density-dependent flow and transport simulators SWIPR and SWIFT, which were developed for saline waste injection problems. Because of the chemical characteristics of the waste, interaction of the Injected waste with the resident ground water is often of major concern. Geochemical equilibrium models (such as MINTEQ, WATEQF, and EQ3/EQ6) are currently being used by the Agency and its consultants to represent the chemical processes occurring in the subsurface when injected waste interacts with the resident ground water. The regulations also call for determinations of which aquifers serve, or could serve, as underground sources of drinking water (USDW), based on a lower quality limit of 10,000 ppm total dissolved solids. Here, modeling has been found to be a useful adjunct to gathering and interpreting field data, as in the U.S. Geological Survey's efforts to assist EPA in determining USDW (e.g., the Regional Aquifer System Analysis [RASA] program). Another USDW program, for the designation of Sole Source Aquifers(SSA), has frequently used models for establishing and managing water quality goals. Designation of the Spokane Valley-Rathdrum Prairie SSA, for instance, included an evaluation of nonpoint nitrate sources with a ground-water model developed for EPA by the USGS. Office of Health and Environmental Assessment The Office of Health and Environmental Assessment (OHEA) is developing guidance for exposure and health risk assessments. This guidance will be used to support the various provisions of the RCRA Amendments and CERCLA. Current focus is on selection criteria for ground-water fate and transport models to be used in exposure assessments. Office of Pesticide Programs The primary ground-water-related concern in the Office of Pesticide Programs (OPP) is the assessment of pesticide leaching and contamination of underlying aquifers resulting from normal use of registered pesticides, and 16 evaluation of new pesticides for registration. In conjunction with data received from registrants as required by the Federal Insecticide, Fungicide, and Rodenticide Act (F1FRA), OPP uses models to assess the leaching potential of pesticides. Past and present efforts have focused on predicting whether various pesticides are likely to leach to ground water following normal use, rather than on their spreading potential within aquifers. This focus is due to the large areal, nonpoint-source loading aspect of the OPP problem, as opposed to localized point sources often of concern to other Program Offices. Also, the current policy of OPP is to protect all potable sources of ground water, not only those which are currently used for drinking water. Therefore, occurrence of pesticide residues in potable ground water is the issue of concern, rather than dilution and degradation prior to arrival of residues at well heads. Initially, OPP used the PESTAN model; however, it has been replaced with the more accurate and time-varying Pesticide Root Zone Model (PRZM) developed by the EPA Environmental Research Laboratory (AERL), Athens, Georgia. Models such as PRZM will not be used as the sole basis for regulation, but they can provide important information for the regulatory process. For example, predictions of significant concentrations at a point deep in the unsaturated zone can imply that a pesticide has the potential to contaminate ground water and this can be an important piece of evidence for the regulatory process. Other issues that can be addressed using models include; the effect of rate and timing of applications on leaching, comparisons between use sites, and relative ranking of pesticides. For example, using PRZM, it was shown that April applications of aldicarb in Florida reduced leaching in comparison to June applications. In contrast, earlier application in Long Island would be subject to heavy spring rains, and application in early summer would reduce leaching. Based on this analysis, current Florida regulations state that aldicarb must be applied prior to April 1. Models that link an unsaturated zone portion with a saturated portion are the next step for modeling. Such linked models should give a more accurate estimate of ground-water pesticide concentrations than unsaturated zone models, which predict only concentrations above the water table. Prediction of migration from a use site is of less concern because of OPP's policy to protect all potable ground water. Care must be taken when using linked models to avoid the unrealistic belief that predicted concentrations represent reality. Two primary reasons for this are difficulties in measuring and/or estimating system parameters, and lack of databases to validate the models. Currently, OPP is helping to fund a project by the EPA R.S. Kerr Environmental Research Laboratory (RSKERL), Ada, Oklahoma, and Oklahoma State University, which will link an unsaturated zone model, most likely PRZM, with a saturated zone model. One unresolved problem in pesticide modeling is the influence of "macropore flow" on pesticide leaching. This type of flow can be described as the initial rapid downward leaching of water and solute through preferential flow paths (such as cracks or empty root channels) in the soil at the onset of a storm. PRZM and similar models assume the classic water front flow, with solute appropriately retarded due to adsorption. With this assumption, 17 pesticide leaching can be underestimated. The dynamics and quantification of macropore flow need to be studied and implemented in models. Office of Policy, Planning and Evaluation For the Office of Policy, Planning and Evaluation (OPPE), ground-water models are used as part of the policy analysis, together with surface water and air models. Under the new ground-water protection policy, regional staff members expect to use models in the selection and management of Class 2A aquifers. Office of Toxic Substances Currently, the role of ground water as pathway for exposure to hazardous contaminants is being studied by the office of Toxic Substances under the Toxic Substance Control Act—TSCA, and various scenarios leading to exposure via ground water are being modeled. Existing ground-water models are considered adequate, especially for evaluating generic situations. Office of Solid Waste The Office of Solid Waste (OSW) must review ground-water models submitted to the Agency by hazardous waste disposal facilities seeking Part B permits. As an example, OSW recently reviewed the modeling effort by SCA-Chemical Waste Management for the New York Model City facility, which resulted in the design of an acceptable ground-water monitoring system. Yet because regulations require that requestors submit only certain information, evaluations of permit requests are sometimes hampered by lack of data, and this often makes Agency model use impractical for this purpose. Office of Waste Programs Enforcement RCRA Enforcement Division— The Office of Waste Programs Enforcement (OWPE) has developed a single analytic framework for comparing risks from different ground-water contamina¬ tion sources occurring in a wide variety of climatic and hydrogeologic settings. This framework is based on the Office of Solid Waste's (OSW) liner location model that has been developed over the last few years. OWPE modified this model slightly and supplied six additional source terms in addition to the hazardous waste sources developed by OSW. OWPE currently models the following source types: sanitary landfills, municipal, industrial and mining surface impoundments, underground storage tanks, septic tanks, agricultural feedlots, road de-icing, hazardous waste landfills, and hazardous waste surface impoundments. Each source type is divided into three to five subcategories, based on such factors as size and constituents. Releases from each source type are profiled over time; for instance, the water balance method is used for municipal landfills. Seventy-two environmental settings are used in the model, each composed of a different combination of values for (1) depth to ground water, (2) net-recharge rate, (3) aquifer configuration, and (A) ground-water velocity. Several variables such as fraction of organic carbon in soil are held constant across all environments. 18 The subsurface transport portions of the liner location model are composed of an algorithm which estimates the amount of time for contaminants to reach ground water (the McWhorter-Nelson wetting front model), and a satu¬ rated zone model which estimates the time for contaminants to reach a well. Next steps are adding pesticides as a source, and improving analysis of hydrogeologic variables. Because of the importance of resource loss to the ground-water protection strategy, OWPE is also re-evaluating all sources in terms of impacts on resource loss (i.e., volume of aquifer contaminated by each source type). CERCLA Enforcement Division— As a result of the following factors, the frequency of ground-water model applications at Superfund sites will increase rapidly under the reauthorized Superfund law: • expansion of the Superfund program (i.e., with a 2.5- to 5-fold increase in funds) • increased emphasis on permanent measures, such as in situ treatment and ground-water restoration, which will require better understanding of the interaction of remedial technology with ground-water systems • the need to address contaminated aquifer "sites" with multiple sources, such as the San Gabriel Basin aquifer, and other complex sites • more sites will be undergoing actual design and implementation of the remedial response alternative • the need to reduce uncertainty in remedial response analyses • the need to quantify the performance (effectiveness) of remedial response alternatives rather than rely only on field data and best¬ engineering judgment OWPE is investigating the use of ground-water modeling for fund-financed CERCLA actions, but with a focus on the use of simple, desk-top fate and transport calculations to predict the effects that leaching from residual soils at Superfund sites could have on ground-water receptors. For example, the VHS model of Domenico and Palciauskus (developed by the Office of Solid Waste for delisting applications) was used at a site in Maine to predict initial soil cleanup targets for trichloroethylene (TEC). An important development is the increase of model use by Potentially Re¬ sponsible Parties (PRPs), usually large companies with considerable amounts of money and liability at stake. Often they employ models to contest Agency decisions or to propose certain remedial actions. 19 SECTION 4 MODELING CONCERNS PROGRAM OFFICE CONCERNS Two major kinds of modeling issues are likely to be of concern to EPA: those most frequently encountered by the national Program Offices, and those of particular interest to Regional Offices. These two Study Group foci provide structure to the following overview of the issues. Office of Drinking Mater Most of the Office of Drinking Water's (ODW) ground-water modeling is related to the Underground Injection Control (UIC) Program. The UIC Program's use of ground-water models is unique in the Agency because of the type of geology involved, the physical and chemical characteristics of the injected waste, and the pressure buildup during injection. Models and assumptions required to simulate this type of environment differ from those of interest to other EPA Program Offices. Therefore, many of the solute transport models currently used at EPA are not suited for the UIC Program. Instead, special models such as those developed for saline waste injection problems (e.g., SWIPR), are used. It is important, therefore, to determine the adequacy and adaptability of existing transport models in meeting these specific needs of the UIC program, and to establish a program for model enhancement and development for UIC use. The Office of Drinking Water is also concerned about the applicability of geochemical equilibrium model codes (such as MINTEQ, WATEQF, and EQ3/EQ6) to address the chemical processes that occur in the subsurface when the waste interacts with the resident ground water. These types of codes are currently being used by the Agency and Its consultants, but their validity for application to most of the problems encountered in the UIC program has not been established. Hence, there is an urgent need to evaluate these models for application to UIC program conditions. Office of Pesticide Programs The Office of Pesticide Programs (OPP) uses models to assess the leaching potential of pesticides. Thusfar, OPP has focused on the modeling of unsaturated zone processes. Currently, it is cofunding the development of linkage of its pesticide transport model PRZM with a saturated zone transport model. The main concern of OPP is to avoid the unrealistic belief that predicted concentrations represent reality. Two prime reasons for the uncertainty occurring in predictions are the difficulties in measuring and/or estimating system parameters, and the lack of databases to validate the models. This uncertainty is further aggravated by the unresolved problem of modeling pesticide transport in the presence of macropores (e.g., empty root channels, cracks in soil). 20 Office of Policy, Planning and Evaluation The main concern of the Office of Policy, Planning and Evaluation (OPPE) relates to the establishment of a set of ground-water models as "standards" for various policy purposes. OPP considers this an important Issue because use of a ground-water model as part of a policy analysis requires considerable time and effort in demonstrating to reviewers that the particular model is appropriate and gives accurate results in the considered case. Typically, OPPE does not have this problem with surface-water dispersion models and air models, for which there are well-established Agency standards. Given the increased attention ground-water contamination issues will demand in the near future, they find it important that performance standards and review criteria be established for ground-water models. Regardless of whether certain models are deemed acceptable, or whether performance standards for different models are set based on their planned use, criteria will need to be developed for evaluating the models. In this regard OPPE finds it useful to have a ground-water modeling catalog at hand similar to, but less extensive than, the Agency's wasteload allocation handbook. A set of administrative and scientific criteria that OPPE finds particularly important includes: • trade-offs between costs of running a model and accuracy • profile of model user and definition of required user-friendliness • accessibility in terms of effort, cost, and restrictions • acceptable temporal and spatial scale and level of aggregation allowed or required • classification of types of contaminants (organics, metals, etc.) the model can handle • description of the model input variables that can be varied (and by how much) and the factors that are considered constant \ • data requirements of the model in the context of the cost for data col lection Office of Toxic Substances Although most regulations are not yet based on exposure to hazardous contaminants via ground water, such considerations are increasingly used in the evaluation of new policies. If ground-water exposure is expected to be an important pathway, various scenarios of exposure via ground water are modeled. To do so efficiently, the existing database needs to be expanded both with generic data and with regional or site-specific data. The use of models for evaluation of new chemicals is currently also hampered by lack of product data, often because of their proprietary nature. 21 Office of Waste Programs Enforcement Recent experiences by the Office of Waste Programs Enforcement (OWPE), particularly within the CERCLA (Comprehensive Environmental Response, Compensation and Liability Act or Superfund) Enforcement Division, have led to recognition of a critical gap in Agency procedures regarding the selection and application of ground-water computer models used for simulating flow and contaminant transport. The primary modeling concern is the lack of any EPA policy on the use of proprietary models by or for EPA. This has been an especially thorny issue in several CERCLA enforcement actions and is likely to surface in the RCRA Program as well. To remedy this situation, two efforts are currently underway in OWPE. The first entails revising the modeling sections of the RCRA Technical Enforcement Guidance Document to promote the use of nonproprietary models. The second is the development of a policy memo concerning this issue that the Director of OWPE will distribute to EPA Regional Offices. It is expected that this memo will strongly discourage the use of proprietary models. OWPE has specifically requested the Ground-water Modeling Study Group to address this issue and to make recommendations confirming or modifying the policy decisions being made by OWPE. Furthermore, it is OWPE's belief that it should not be the sole referee or arbiter of ground-water computer codes as they are encountered in the enforcement process; OWPE has neither the resources nor the broader Agency responsibility to establish unilateral criteria by which a code may be judged acceptable to the Agency. However, OWPE feels strongly that such criteria must be developed. Other modeling issues that need to be considered are closely tied to the proprietary model issue discussed above. These include definition of what constitutes "acceptance" of a model by the technical community, establishment of an adequate level of "peer review," and establishing quality assurance protocols for model development, selection, and application. Because most of the modeling projects to be reviewed by the Agency are site-specific analyses where calibration data may or may not exist, a well- defined set of guidelines for model calibration and predictive phases is necessary. There evaluating lacking. simulation evaluation is general agreement that many of the parameters required for contaminant transport (especially spatially varying parameters) are In the absence of complete information, the most meaningful results are those expressed in probabilistic terms. Guidelines for of field data and simulation results are necessary. 22 Office of Research and Development Office of Health and Environmental Assessment— The Office of Health and Environmental Assessment (OHEA) is developing guidance for health and exposure assessments, including development of mathematical selection criteria for ground-water fate and transport mod¬ eling. This guidance will be used to support the various provisions of the RCRA Amendments and CERCLA. The exposure assessment guidelines, proposed in the Federal Register on November 23, 1984 and soon to be published in final format, outline the next phase of guidance to be developed by OHEA. As part of this effort, the Exposure Assessment Group met with the model users in several Program Offices in early 1985 to discuss the approach being used in developing selection criteria for ground-water transport modeling and the needs of Program Offices. Subsequently, the Office of Waste Programs EnfdPcewent initiated several meetings to bring issues relating to the Agency procedures and criteria for model selection and use of computer codes to the attention of concerned parties. A workgroup has been formed to review Agency procedures and criteria for ground-water model selection, particularly directed toward exposure assessment. Environmental Research Laboratory, Athens, Georgia— Guidelines to determine the adequacy of proprietary versus public domain software, quality control measures, and liability are integral parts of any modeling assessment; however, correct application procedures and data evaluation for regulatory decision making may be equally important. For example, most modeling studies utilize lumped parameters that are a reflection of the mass balance approach for advective-dispersive models. Although these types of approaches may be appropriate for problems in ground-water systems where the contaminant is distributed over the entire ground-water basin, they may not be adequate where dispersion is important. Guidelines relating to the appropriate methodology used should be made available to managers. CONCERNS OF REGIONAL OFFICES The Regions have varied interests in model development, selection, and use, and in training in modeling. In discussions with the staff, the following major issues surfaced: • limited knowledge of model availability • the need for assistance in selecting and usvng available models for a specific site • guidance in model reliability and interpretation of simulations • need for additional models for multiphase flow and contaminant behavior in the vadose zone • improved interaction and coimunication with technical staff in other Regional Offices, Headquarters, and EPA labs 23 • training in basic processes (geology, hydrology, fate and transport, etc.) for the project officers as well as modeling training for the technical experts in the Region. (In some Regions It was stated that EPA "prefers to rely on Agency expertise rather than external consul¬ tants, because of the 'burden of proof' needs.") • hiring and retaining technical staff who have received special training in modeling • need for ground-water modeling policies consistent with those for surface water modeling Model Use in Regional Activities Because the Study Group visited only three of EPA's ten Regions, the fol¬ lowing discussion is somewhat limited. However, the Study Group considers these findings indicative of the general Regional situation in ground-water modeling. EPA Region I is involved with a number of major Superfund sites as well as other hazardous waste disposal sites. The staff is actively involved in site investigations (RI/FS) and regulatory/enforcement actions involving PRPs (Potentially Responsible Parties). In the Study Group meeting in Boston it was stated that PRPs, usually large companies with considerable amounts of money and liability at stake, employ models to contest the cases or propose certain remedial actions. In Region III models have not been used to the extent that they have become controversial; there are as yet no cases in which they are contested. Staff members pointed out that under the new ground-water protection policy, they expect to use models in the selection and management of class 2A aquifers. For permit requests under RCRA, numerical models are not yet used. This is partly due to the extensive karstic limestone aquifers in the Region, which make modeling impractical, although various analytical models are used in the permit review process. Because regulations are rather strict with respect to the type of information to be submitted, evaluations of permit requests are sometimes hampered by lack of data, and this too makes the use of models impractical. Further guidelines for data collection and use of models seem necessary. It appears that in Region X numerical ground-water models have been used for only one Superfund site and for no RCRA sites. At the beginning of what is perhaps a three- to five-year project life, it is difficult to anticipate staffing, data, and modeling needs. Project funding tends to be incremental, and therefore data collection and analysis are often short-sighted. The approach sometimes becomes ad hoc, when a stepwise approach would be preferable. (On the other hand, additional funding may not be forthcoming, discouraging use of a stepwise approach.) If litigation is anticipated, the project needs to be carefully executed and 24 documented. Project managers need to know what the tools are, what the cost will be, and how results can be used. Often data are available only as hard copy; significant costs are involved in transferring such data into machine processable form. An example is an ongoing Superfund demonstration in Region X where $400,000 has been spent to assemble and digitize existing data for a county. Regional Staff The rapid expansion of responsibilities under Superfund has forced the Regions in recent years to expand significantly their project staffs. It is apparent to the Study Group that in many Regions, project managers (primarily those working under the jurisdiction of CERCLA) are so involved in project work itself that little time is left to do anything other than meet administrative requirements and deadlines. In general, managers are instructed to perform RI/FS work in less than one year, with completion of the Rl-phase within four to six months. This does not allow time for adequate data collection, and much less for extensive modeling, which is therefore often considered an unaffordable luxury. Because of the deadlines and workload, project managers, who are often untrained in ground-water modeling, have no time to keep abreast of devel¬ opments in modeling and, therefore, are often not qualified to introduce modeling into the project or to evaluate modeling done by contractors or PRPs. Model application and interpretation of results is very subjective and may be the core of an expert's testimony in court. The expert's interpretation of results represents the culmination of months of technical work. EPA staff must be capable of providing the expert with both policy and technical oversight, e.g., in the quality assurance of the project. If this oversight is lacking, the expert's work may be misdirected or poor in quality. That this is a significant problem is illustrated by the modeling deficiencies frequently displayed by EPA contractors. Regional staff in the Hazardous Waste divisions are almost all gener¬ alists with degrees in environmental science or environmental engineering. In the three regions none of the project managers had formal training as a hydrogeologist (nor is there a "hydrogeologist" position in the Agency). A broader multidisciplinary team is viewed as mandatory. There is a tendency to underestimate staffing needs; and even with breadth, staff tends to be spread too thin. Internal capabilities can be provided by the Environmental Services Division, present in some of the Regions, but are not always used optimally. If a project gets too complex, EPA staff is often pulled off the project and the job is given to a contractor. An additional problem facing the regions is that most of the good people eventually go to consulting firms, once they have experience, resulting in a high turnover. In a number of Regions, the rapid expansion of Regional project staff and the high turnover rate have led to a situation where many of the project managers have less than two years' experience on the job. 25 There is also a significant difference between Regions in their management approach to optimal use of the limited human resources available. Various measures have been implemented to acconmodate the new tasks required by recent legislation. For example. Region III staff has established a pool of in-house experts (three hydrogeologists and three toxicologists) who are available to help the project officers. In addition, each site has a formal review "team" that includes a hydrogeologist, a toxicologist, and an administrator. Regular in-house technical meetings provide an environment for good communication among staff concerned with ground water. A strong interest was expressed, expecially by the Regions, in having advice and guidance available for mdel selection and use, given site-specific conditions. This can be achieved through the establishment of a blanket agreement with a nonprofit agency to provide a quick means of bringing in capable outside experts for advice on specific cases. Quality Assurance The aforementioned conditions force Regional staff to rely heavily on their contractors for modeling in Superfund projects. Contractors are selected through competitive bidding for large contracts. Modeling, which is often only a small part of these contracts, is sometimes done by the contractors themselves, but frequently by subcontractors who are not chosen by the Region. Thus, Regional staff has little control over who performs the work and must use the "national" contractors because of existing procurement procedures. In addition, there has been little quality assurance (QA) in the past over modeling work performed by the contractors, partly because of the tight schedules under which the work must be carried out, and partly because of the shortage of experienced staff. Once a project starts, often no formal review takes place until the project reaches its final stages. The project officer at EPA is the only person who might review the study while it is in progress, and in most cases the final QA is conducted within the Region itself by personnel in divisions that are involved administratively. Sometimes Regional staff forms a technical review team for such purposes. In some Regions part of the modeling work is reviewed by a USGS modeler available to the Agency through an interagency arrangement, but no formal review meetings are held. Linder such conditions, many of the modeling studies in the Regions may be of limited usefulness due to Incorrect siting or data collection; incorrect use of available data; inadequate modeling of the physical system, such as flow in fractured bedrock; or invalid boundary conditions. Major constraints in addressing these problems are procurement policies mandated by government units outside EPA, specifically the U.S. Office of Management and Budget, and the lack of in-house expertise. High turnover in project managers, together with the inability to review the degree of success or failure in earlier projects, leaves little institutional memory for learning from previous studies. 26 Procurement of Modeling Studies Regional problems include a rapid expansion of the EPA responsabi1ities as well as high turnover of personnel. One solution is to have contracts to meet specific needs, as opposed to large-scale contracts for general support. For example, a contractor might be engaged to provide modeling support for all Regions. (However, at the present time the Agency does not have enough technical expertise to review its contract work.) Procurement procedures also could be changed so that Regional Offices have more control over the selection of their contractors. Technical exper¬ tise should be specified and given greater weight in the selection of a con¬ tractor. At the very least, it should be recognized that ground-water con¬ cerns tend to drive hazardous waste remediation (e.g., the RI/FS process) in terms of time and effort, both to characterize the problem and to clean up the site. This should be more emphatically stressed in the selection of "national" contractors so that qualified ground-water contractors are chosen. There is also a need for guidelines in selecting modeling contractors. The modeler or someone familiar with the application (e.g., the modeler's supervisor) should be available to serve as an expert witness. In addition, the quality of the presentation of the results of a modeling exercise to management, and to the judge in the courtroom, is important for the ultimate success of the modeling effort. Because model use in enforcement and litigation is likely to continue to grow with time, continuing attention should be given to issues related to such applications. Study Group participants also considered it important to establish a more direct mechanism for reviewing and directing the work of outside contractors. This would involve the establishment of thorough QA/QC procedures for modeling studies, and would include a detailed review process to be conducted throughout the modeling process, with stop/go decisions at each critical point. Postmortem analyses of selected cases with all staff should be encour¬ aged, so the lessons learned can be communicated and applied to current and future site investigations. Managers should require computer-processible data; a protocol for database management systems and better data-processing techniques should be adopted. This will significantly improve the efficiency of modeling-based data analyses needed for the resolution of many ground-water issues. Technology Transfer and Training The Study Group has found only a limited understanding among Regional staff of what software is available. As Regional staff anticipates an 27 increased use of models, there is a need for improved technology transfer and training. Most of the technical staff indicated strong interest in (but expressed their concerns and frustrations over the lack of opportunities for) structured training or self-study in modeling. However, many of those in need of additional training should first be trained in general hydrogeology, hydrochemistry, and data analysis, before focusing on modeling. Because model use is expected to increase in the future, the development of in-house exper¬ tise, by whatever means, appears to be a major priority. Most of the tech¬ nical and managerial personnel recognize that they need not become modeling "experts" and only want sufficient training to be knowledgeable users or competent judges of the appropriateness of models used by PRPs and contracted consultants. A strong interest was expresses, expecially by the Regions, in having advice and guidance available for model selection and use, given site- specific conditions. This can be achieved through the establishment of a blanket agreement with a nonprofit agency to provide a quick means of bgringing in capable outside experts for adivce on specific areas. A network of staff concerned with ground water also is needed so that experience can be shared. Technology transfer is ineffective if it simply consists of reports sent to Regional libraries; a better environment is needed in which state-of-the-art technology is distributed and used. Regional staff indicated the need for a better institutional relationship between Regional Offices and EPA Laboratories. If communication is facilitated, the synergisms would work to the advantage of the Regions and would more than justify the expenditure of travel money. Facilities and Resources Some Regions have a rather extensive collection of modeling software available internally, either in their Program Offices or from their Environ¬ mental Support Division. Other Regions have some Incidental software, but are not aware of sources of additional models. Some Regions consider proprietary models an acceptable alternative if source code is available and if the model is well-tested and properly documented; others use only public domain software. Computer facilities available in the Regions for ground-water assessments include microcomputers, mostly IBM PCs or compatibles, as well as terminal access to the Region's administrative minicomputer facilities. Some Regions have MicroVAX super microcomputers. A few Regions have indirect links with local, external computer facilities such as a university campus or a USGS District Office. EPA facilities in Triangle Park are not used, however, either because the need has not been identified, or because the use of these facilities is relatively cumbersome in terms of access time, response time for printed output, and the like. Procurement regulations sometimes inhibit adequate expansion of existing computer systems in terms of hardware and 28 software to serve efficient data acquisition and processing as well as modeling needs, thus limiting efficient software selection and model use. Local expertise, which is readily available in some Regions (e.g., from Boston-area universities in Region I), is seldom fully exploited. Short courses or special classes are sometimes arranged, but they are often con¬ sidered relatively unsuccessful because they are too theoretical or too condensed. Legal Concerns During the discussions at Boston, a number of legal concerns were brought up. In general, legal procedures become important when technical regulations are not available or do not cover the issue under consideration, and also when existing regulations are not followed and need to be enforced. In accord with earlier findings of the Study Group, it was mentioned that a model's use by the Agency anywhere in the country results in de facto acceptance of that model in litigation, and this presents a major legal problem. There is also a lack of information exchange within the Agency about which models are available; where and under what conditions they have been used; what the results from the models provide in terms useful to management; and what administrative, technical, and legal problems are encountered. Related to the legal problems is the Agency's need to retain its expert witnesses for litigation. The high turnover rate within the Agency causes continuity problems in staffing (which are sometimes avoided by working with consultants). However, consultants may get certain jobs by promising the assistance of senior modelers, while the actual work is done by junior modelers. This can result in an unsatisfactory modeling effort, and in court problems when the senior modeler, who has not been directly involved in the detailed work, is presented as the expert witness. Summary of Regional Concerns In summary, many issues raised with respect to ground-water modeling are identical to those raised by the national Program Offices. This is especially true in the areas of training needs, QA/QC in modeling, and legal require¬ ments. Additional attention has been drawn to administrative and technical constraints in the Regions. The management of technical contracts is a problem because project managers (on-site coordinators) do not have the time to remain technically qualified, partly due to lack of support from regional management for independent study. This problem is particularly critical because ground-water modeling is sometimes performed by unqualified con¬ tractors and because model use will continue to increase in the future. 29 SECTION 5 DISCUSSION OF ISSUES ADEQUACY OF MODELING THEORY AND DATA Appropriate models do not yet exist for all ground-water problems because many of the assumptions and simplifications corrmon to existing models do not allow faithful simulations, and because the natural processes that affect fluid and contaminant movement have yet to be fully understood. This is especially true for chemical and biological processes. Although large advances have been made concerning the behavior of individual contaminants, studies of the interactions among contaminants are still in their infancy. These studies include, for example, the ability of certain solvents to increase dramatically the mobility of ordinarily slow-moving pesticides, of polynuclear aromatic hydrocarbons, and of others. Other areas where substantial progress is needed lie in understanding the immiscible flow and transport considerations so crucial to solving the problems of leaking underground storage tanks, and the manner in which contaminant transport in fractured rocks differs from transport through porous sediment. Finally, certain well-understood phenomena pose unresolved difficulties for mathematical formulations, such as the dynamic operation of partially penetrating wells in unconfined aquifers. Improvements are needed, concurrently, in several major areas. Models need to be improved mathematically so that errors arising from computational techniques (e.g., numerical approximations) are minimized. While continuing research in this area has been a well-recognized need, other topics just as important have received less than adequate attention. The theories on which models are based need to be developed further so that proper representations of the true influences of various natural pro¬ cesses can be incorporated into model applications. Theories that have been used for solving regional water supply problems are generally applicable to localized water-quality problems such as hazardous waste sites. However, certain specialized needs are peculiar to chemically complex problems. The highly variable distributions of contaminants at hazardous waste sites, for instance, create a number of practical difficulties. These result in part from incomplete understanding of chemical Interactions and the role that microbiota may play in the transport and fate of subsurface contaminants. Such difficulties also result from limitations of available data, of field methods, and of quantitative tools. As a result, data acquisition methods and interpretive models are needed that can examine to an unprecedented degree the physical, chemical, and biological processes controlling the transport and fate of ground-water contaminants. Unfortunately, few of the constants and coefficients needed to incorporate chemical and biological processes into contaminant transport 30 evaluations are available presently. This does not mean, however, that some indication of their contributions cannot be estimated; much of the existing information can be used in a semiquantitative manner (i.e., sensitivity analyses and "worst-case" scenarios). Efforts to collect field data and to estimate natural-process parameters must be expanded and improved so that model applications will be more physically based and thereby capable of more accurate predictions. For example, so few data are available concerning the exact location, volume, composition, and timing of chemical releases at existing sites that it is very difficult for modelers to determine the appropriate configuration of what is referred to as the "source term" in modeling. One prevailing misconception in this regard is the idea that additional chemical sampling of monitoring wells can provide definitive clues to the missing historical data, but this is true only superficially. Although indication of the source term can be obtained from the patterns of chemical movement, there is no guarantee that causal relationships can be discovered or that the patterns will remain constant. A common misconception is that all field methods and tools necessary for obtaining data to run the models are available, if not in optimal form, at least in a useful form. In fact, however, direct measurements are unreliable or cannot be obtained for a number of parameters such as ground-water flow velocity and direction, rates of sorption and desorption (retardation) of organic chemicals, and the potential for biotransformation. This misconception parallels the mistaken idea that all necessary theories have been worked out and that further refinements are needed only so that better precision and accuracy can be obtained. The integration of geologic, hydrologic, chemical, and biological processes into usable subsurface flow and transport models is possible only if the data and concepts invoked are sound. The data must be representative as well as accurate and precise. The degree to which the data are representative is relative to the scale of the problem and the concepts guiding data collection and interpretative efforts. Careful definition of these concepts is crucial; special attention should be given to the spatial and temporal variations involved. The use of newly developed theories to help solve field problems is often a frustrating exercise. Most theoretical advances call for some data not yet practically obtainable (e.g., chemical interaction coefficients and relative permeabilities of immiscible solvents and water). In addition, the incorporation of theoretical relationships into mathematical models typically is made possible by invoking certain assumptions and simplifications that alter the intended relationship. Therefore, theoretical advances in modeling ground-water problems must be accompanied by improvements in data collection and mathematical representation efforts. The Study Group has identified a variety of new models and modeling approaches as important to EPA's mission: • simulation of flow and transport in multimedia (e.g., coupled models for surface water/ground-water interaction) 31 • representation of stochastic processes in predictive modeling, and improving the applicability of geostatistical models • improved modeling of hydrochemical speciation • simulation of flow and transport in fractured and dual-porosity media, including diffusion in dead-end pores • simulation of flow and transport in soils containing macropores • determination of effects of concentration-dependent density on ground-water flow and pollutant transport • determination of effects of alteration of geologic media on hydro- logical and chemical characteristics (e.g., dehydration of clay when attacked by solvents, change in sorptive capacity of material when heated) • representation of the three-dimensional effects of partially pene¬ trating wells on water table aquifers • development of models for management of ground-water contamination plumes • development of expert systems (artificial intelligence) for such tasks as selecting appropriate submodels or subroutines for specific problems • application of parameter identification models to be used with site studies • further development of pre- and postsimulation data processors • continued development of risk assessment and management models • modeling of volatilization, multiphase flow, and immiscible flow • incorporation of economic factors to improve estimation of clean-up costs • development of generic and site-specific parameter databases Fundamental research supporting ground-water modeling is considered necessary in such areas as: • transient behavior of process parameters (e.g., retardation, hydraulic conductivity) • desorption for nonhydrophobic chemicals • multicomponent transport and chemical interaction 32 • enhanced transport mechanisms (e.g., piggy-backing on more mobile chemicals) • transport of silt with sorbed chemicals in aquifers • improved numerical accuracy, stability, and efficiency Although an extensive research and development program exists at ORD, more attention should be brought to bear on the mechanisms by which those needs can be satisfactorily addressed. GROUND-WATER CODE REVIEW AND TESTING The last few years have seen a significant increase in the use of ground- water models in situations leading to litigation, congressional hearings, or extensive public discussion. In such cases, both the theoretical foundation and coding of a model, as well as its application, may be contested. This situation is typical for many of the hazardous waste landfills, impoundments, and spill sites investigated by EPA under Superfund legislation. Ground-water models are used to determine the extent of present ground-water contamination, to identify the source of that contamination, to predict future contamination and health risk, and to assess remedial action alternatives. These determinations often form the basis for the principal findings of a site's Remedial Investigation and Feasibility Study (RI/FS) under CERCLA. Discretion as to when to use a computer code and which specific computer code to use is often left to the EPA contractors and/or the responsible parties who perform the RI/FS. Criticism of the modeling aspect of the RI/FS can take three forms: (1) use of an erroneous conceptual model of the physical system to which the computer code is applied; (2) inappropriate application (or use) of the computer code; and (3) errors in the code's formulations, assumptions, and coding which renuer it unreliable. The first two points may be apparent from a critical review of the RI/FS and may be sharply debated by technical experts. The last point is not apparent from an evaluation of the RI/FS and can only be determined after a detailed assessment and extensive use of the computer code. The resulting confusion, especially among nonmodelers (judges, attorneys, regulatory agency staff, and legislators), has led to doubts about the utility of the modeling methodology in general. Such controversies can be avoided by applying adequate quality assurance (QA) to all stages of the modeling project. Selecting an appropriate and well-tested model is another significant measure that can be taken. Model testing is an important aspect of QA in model development. Evidence of a code's technical soundness can be established prior to any legal proceeding. Where a computer code has not been peer-reviewed and independently tested, the criticism of the code itself becomes a valid way to attack both the computer code and the RI/FS which may have relied on it. However, it should be realized that code testing requires a large amount of time and effort. Often, time and resources are limited for post-simulation review efforts. 33 Adequate model testing and validation should be an integral part of all research and development projects resulting in modeling software to be used in support of the Agency's regulatory mission. Model Evaluation Before a ground-water model is used as a planning and decison-making tool by ERA or a cooperating agency, its credentials should be established, independently of its developers. This can be done by systematically testing and evaluating the characteristics of the model. Code testing is generally considered to encompass verification and validation of the model (Adrion et al. 1982). To evaluate ground-water models in a systematic and consistent manner, some institutions have developed model review, verification, and validation procedures (van der Heijde et al. 1985; Moran and Mezga 1982). Sometimes, independent review and testing is sought. Generally, the review process is qualitative in nature, while code testing results can be evaluated by quantitative performance standards. Model Review— A complete review procedure comprises examination of model concepts, governing equations, and algorithms chosen, as well as evaluation of documentation and general ease-of-use, and examination of the computer coding (Huyakorn et al. 1984; van der Heijde et al. 1985). If the model has been verified or validated by the author, the review procedure should include evaluation of this verification and validation process. To facilitate thorough review of the model, detailed documentation of the model and its developmental history is required. In addition, to ensure independent evaluation of the performed verification and validation, the computer code should be available for implementation on the reviewer's computer facilities, together with a file containing the original test data used in the code's verification and validation. Review should be performed by experienced modelers knowledgable in theoretical aspects of ground-water modeling. Because review is rather subjective in nature, selection of the reviewers is a sensitive and critical process. Model Examination— Model examination involves determining whether anything fundamental was omitted in the initial conceptualization of the model. Such a procedure will determine whether the concepts of a model adequately represent the nature of the system under study, and will identify the processes and actions pertinent to the model's intended use. The examination is also intended to determine whether the equations representing the various processes are valid within the range of the model's applicability, whether these equations conform mathematically to the intended range of the model's use, and whether the selected solution approach is the most appropriate. Finally, model examination should determine the appropriateness of the selected initial and boundary conditions and establish the applicability range of the model. 34 For complex models, detailed examination of the implemented algorithms is required to determine whether appropriate numerical schemes, in the form of a computer code (ASTM 1984), have been adopted to represent the model. This step should disclose any inherent numerical problems such as non-uniqueness of the numerical solution, inadequate definition of numerical parameters, incorrect or nonoptimal values used for these parameters, numerical dispersion, numerical instability such as oscillations or divergent solution, and problems regarding conservation of mass. In addition, the specific rules for proper application of the model should be analyzed from the perspective of its intended use. These rules include data assignment according to node-centered or block-centered grid structure for finite-difference methods; size and shape of elements in integrated finite-difference and finite-element methods; grid size variations; treatment of singularities such as wells; approach to vertical averaging in two-dimensional horizontal models or layered three-dimensional models; and treatment of boundary conditions. Consideration is also given to the ease with which the mathematical equations, the solution procedures, and the final results can be physically interpreted. Evaluation of Documentation— The documentation is evaluated through visual inspection, comparison with existing documentation standards and guidelines, and use as a guide in preparing for and performing verification and validation runs. Good documentation includes a complete treatment of the equations on which the model is based, underlying assumptions, boundary conditions that can be incorporated in the model, method used to solve the equations, and limiting conditions resulting from the chosen method. The documentation must also Include a user's manual containing instructions for operating the code and preparing data files, example problems complete with input and output, programmer's instructions, computer operator's instructions, and a report of the initial code verification. Evaluating Ease of Use— The data files provided by the model developer are used to evaluate the operation of the code and the user's guide through a test-run process. In this stage special attention is given to the rules and restrictions ("tricks", e.g., to overcome restrictions in applicability) necessary to operate the code, and to the code's ease-of-use aspects (van der Heijde 1984). Computer Code Inspection— Part of the model review process is the inspection of the computer code. In this inspection attention is given to the manner in which modern programming principles have been applied to code structure, optimal use of the programming language, and internal documentation. 35 Model Verification The objective of the verification process is twofold: (1) to check the accuracy of the computational algorithms used to solve the governing equations, and (2) to assure that the computer code is fully operational. To check the code for correct coding of theoretical principles and for major programming errors ("bugs"), the code is run using problems for which an analytical solution exists. This stage is also used to evaluate the sensitivity of the code to grid design, to various dominant processes, and to a wide selection of parameter values (Huyakorn et al. 1984; Sykes et al. 1983; Ward et al. 1984; Gupta et al. 1984). Although testing numerical computer codes by comparing results for simplified situations with those of analytical models does not guarantee a fully debugged code, a well-selected set of problems ensures that the code's main program and most of its subroutines, including all of the frequently called ones, are being used in the testing. In the three-level test procedure developed by the International Ground Water Modeling Center (IGWMC), this type of testing is referred to as level I (van der Heijde et al. 1985). Hypothetical problems are used to test special features that cannot be handled by simple close-form solutions, as in testing irregular boundary conditions and certain heterogeneous and anisotropic aquifer properties; this is the IGWMC level II testing. For both level I and level II testing, sensitivity analysis is applied to further evaluate code characteristics. Model Validation Model validation is defined as the comparison of model results with numerical data independently derived from laboratory experiments or observations of the environment (ASTM 1984). Complete model validation requires testing over the full range of conditions for which the model is designed. Model development is an evolutionary process, responding to new research results, developments in technology, and changes in user requirements. Model review and validation needs to follow this dynamic process and should be applied each time the model is modified. The objective of model validation is to determine how well the model's theoretical foundation describes the actual system behavior in terms of the "degree of correlation" between model calculations and actual measured data for the cause-and-effect responses of the system. Obviously, a comparison to field data is required. Such a comparison may take either of two forms. One form, calibration, is the weaker form of validation insofar as it tests the ability of the code (and the model) to fit the field data, with adjustments of the physical parameters (Ward et al. 1984). Some researchers prefer to classify calibration as a form of verification rather than a form of validation. The other form of validation is that of prediction. This is a test for the ability of the model to fit the field data with no adjustments of 36 the physical parameters. In principle, this is the correct approach to validation. However, unavailability and inaccuracy of field data often prevent such a rigid approach. Typically, a part of the field data is designated as calibration data, and a calibrated site-model is obtained through reasonable adjustment of parameter values. Another part of the field data is designated as validation data; the calibrated site model is used in a predictive mode to simulate similar data for comparison. The quality of such a test is therefore determined by the extent to which the site model is "stressed beyond" the calibration data on which it is based (Ward et al. 1984). In the IGWMC testing procedure, this apnroaeh is referred to as level III testing. For many types of ground-water models, a complete set of test problems and adequate data sets for the described testing procedure are not yet available. Therefore, testing of these models is generally limited to extended verification, using existing analytical solutions. Whether a model is valid for a particular apolicaticn can be assessed through the use of performance criteria, sometimes called validation or acceptance criteria. If various uses in planning and decision making are foreseen, different performance criteria might he defined. The user should then carefully check the validity of the model for the intended use. Three levels of validity can be distinguished (ASTM 1984): (1) Statistical Validity—using statistical measures to check agreement between two different distributions, the calculated one and the measured one; validity is established by using an appropriate performance or validity criterion (2) Deviative Validity—If not enough data are available for statistical validation, a deviation coefficient D can be established, e.g., D = l(x-y)/x]•100 % where x = predicted value and y = measured value. The deviation coefficient might be expressed as a summation of relative deviations. If ED is a deviative validity criterion supplied by subjective judgment, a model can considered to be valid if D < ED; (3) Qualitative Validity—using a qualitative scale for validity levels representing subjective judgment: e.g., excellent, good, fair, poor, unacceptable. Qualitative validity is often established through visual inspection of graphic representations of calculated and measured data (Cleveland and McGill 1985). The aforementioned tests apply to single variables and determine local- or-single variable validity; if more than one variable is present in the model, then the model should also be checked for global validity and for validity consistency (ASTM 1984). For a model with several variables to be globally valid, all the calculated outputs should pass validity tests. Validity consistency refers to the variation of validity among calculations having different input or comparison data sets. A model might be judged valid under one data set but not under another, even within the range of conditions for which the model has been designed or is supposedly applicable. Validity 37 consistency can be evaluated periodically when models have seen repeated use. Often, the data used for field validation are not collected directly from the field but are processed in an earlier study. Therefore, they are subject to inaccuracies, loss of information, interpretive bias, loss of precision, and transmission and processing errors, resulting in a general degradation of the data. As noted earlier, for many types of ground-water models no field data sets are available to execute a complete validation. One approach sometimes taken is that of code intercomparison, where a newly developed model is compared with existing models designed to solve the same type of problems as the new model. If the simulation results from the new code do not deviate significantly from those obtained with the existing code a relative or comparative validity is established. It is obvious that as soon as adequate data sets become available, all the involved models should be validated with those data. Further development of databases for field validation of solute transport models is necessary. This is also the case for many other types of ground- water models. These research databases should represent a wide variety of hydrogeological situations and should reflect the various types of flow, transport, and deformation mechanisms present in the field. The databases should also contain extensive information on hydrogeological, soil, geochemical, and climatological characteristics. With the development of such databases and the adoption of standard model-testing and validation procedures, the reliability of models used in field applications can be improved considerably. Validation Scenarios— Often, various approaches to field validation of a model are viable. Therefore, the validation process should start with defining validation scenarios. Planning and conducting field validation should include the following steps (Hern et al. 1986): • define data needs for validation and select an available data set or arrange for a site to study • assess the data quality in terms of accuracy (measurement errors), precision, and completeness • define model performance or acceptance criteria • develop strategy for sensitivity analysis • compare model performance with established acceptance criteria Sensitivity Analysis— An important characteristic of a model is its sensitivity to variations or uncertainty in input parameters. Sensitivity analysis defines quantitatively or semiquantitatively the dependence of a selected model performance 38 assessment measure (or an intermediate variable) on a specific parameter or set of parameters (Intera 1983). Model sensitivity can be expressed as the relative rate of change of selected output caused by a unit change in the input. If the change in the input causes a large change in the output, the model is sensitive to that input. Sensitivity analysis is used to identify those parameters most influential in determining the accuracy and precision of model predictions. This information is of importance to the user, as he must establish required accuracy and precision in the model application as a function of data quantity and quality (Hern et al. 1986). In this context the use of a sensitivity index as described by Hoffman and Gardner (1983) is of interest. It should be noted that if models are coupled, such as in multimedia transport of contaminants, the propagation of errors and the increase in uncertainty through the subsequent simulations must be analyzed as part of the sensitivity analysis. PROPRIETARY CODES VERSUS PUBLIC DOMAIN CODES AND ACCEPTANCE CRITERIA Is the use of proprietary codes in solving ground-water problems for or by U.S. EPA acceptable, or should they be banned in favor of publicly available codes? Deciding this policy issue has become imperative to the Agency in recent years, as an increasing number of modeling-based analyses performed by consultants in regulatory compliance cases are contested in the courtroom, and Agency decision-making processes in general are subject to increasing public scrutiny. A proprietary code consists of computer software that is sold, leased, or used on a royalty basis, which generally conditions its use and limits its distribution. Some proprietary codes are publicly accessible, but restricted in transfer and use. According to this definition, proprietary codes used for solving ground-water problems could include: (1) ground-water simulation codes, (2) databases, (3) statistical packages, and (4) graphical packages. Public domain codes consist of software and documentation that can be used, copied, transferred, distributed, modified, or sold without any legal restrictions such as violation of copyrights. There are various reasons why the use of proprietary codes is regarded problematic by EPA: lack of peer review and validation; problems with intercomparison and reproducabi1ity of results; administrative complications; and lack of access to software and theoretical basis. On the other hand, owners of proprietary code rights often propagate the use of these codes for comnercial, scientific, and other reasons. In this section the concerns of the Agency are reviewed and advantages and disadvantages of the use of proprietary codes for Agency purposes are discussed. Finally, it lists the elements which should form the basis of an Agency policy regarding the use of models, both proprietary and publicly accessible. Banning the Use of Proprietary Codes 39 The Office of Waste Program Enforcement (OWPE) prefers the use of public codes if litigation is anticipated, assuming such code is available, even if the code is less efficient than an alternative proprietary code. Banning of proprietary codes is expected to eliminate some of the problems encountered in court cases. One of these problems is related to the notion that the code itself and its theoretical foundation might become contested. Unrestricted access to the computer code and documentation is considered crucial in such cases. However, if adequate model selection guidelines existed, including requirements for code review, validation, and documentation and were applied consistently, such problems might be less significant. The inaccessibility of some proprietary codes and documentation can cause other problems. EPA regulations (40 CFR 124.11 and 124.12) provide a mechanism for formal public hearings during the RCRA permit process. All aspects of EPA's decision making, including the use of ground-water models, are subject to public scrutiny. EPA use of models not accessible by the public may complicate the proceedings and result in EPA having to duplicate the modeling effort with a publicly accessible model. EPA's continued use of nonpublicly accessible models increases the likelihood of Federal Open Information Act litigation. EPA policy restricting the use of nonpublicly accessible models may reduce this likelihood. Another consideration brought to bear in this issue is that Regional staff does not have enough time and expertise to evaluate models or to go through a proper model selection process without support from model experts. This support can be provided indirectly by establishing a list of reviewed and validated models acceptable to the Agency, and through various forms of guidance such as reports, and by expert systems. Not all proprietary codes are publicly inaccessible. The private sector may control the use and dissemination of its computer models through copyright protection, patent protection, trade secret protection, or through a contractual or license agreement. Most of the issues discussed above result from attempts by some companies to maintain tight controls over their models through trade secret protection. The rationale is that a model contains some formulation that makes it superior to those generally available, and that this formulation gives the company an advantage over its competitors. Exercising control through copyright protection and contractual agreements might be more difficult to enforce than trade secret protection. However, they allow for greater and quicker acceptance of the model by the technical community. Continuing the Use of Proprietary Codes Several reasons have been brought forward for the continued use of proprietary ground-water simulation codes: • Use of proprietary codes encourages code development for solving new problems. If proprietary codes are banned, this Incentive will be removed, greatly inhibiting fpture code development in the private sector. Because it is difficult for private companies to obtain funding for code development, the main justification for investing 40 corporate funds in code development is the anticipation that some development costs will be recovered through code sales or value-added use. Capital gain is a major incentive for code development. • Use of proprietary codes encourages private companies to enhance codes originally developed in the public domain. Many research- oriented codes developed in universities have been generalized, made user-friendly, and have been documented more fully by enterprising private developers. If the profit incentive is removed, further development and enhancement of public domain codes for applied purposes will be discouraged. • Proprietary codes provide solutions to some problems that publicly available codes cannot solve. In some cases—for example, complex three-dimensional transport problems—publicly available codes are not adequate. Banning proprietary codes would eliminate the use of some of the more sophisticated codes. An alternative is to bring such a code into the public domain through outright purchasing, and Installing a service and support agreement with the code developers. • Proprietary codes are often subject to rigorous internal QA. This is not always the case with public domain codes. If a code has problems (bugs), it will develop a bad reputation and will not sell, or will compromise the reputation of the company; a company Is not likely to profit by selling inferior codes. In practice, however, many distributors of proprietary ground-water modeling software are relatively small companies, some of which do not apply QA to model development and documentation. An Agency model review and validation policy applied to public domain codes should cull out the inferior ones. • User support might be available for proprietary codes. Once a proprietary code has been sold, it is in the interest of the seller to help the user apply the code in the best possible manner. Again, this does not always accur, as the smaller ground-water modeling software distributors often have limited resources for support and maintenance of their software. Often, consistent user support is not available for public domain codes. • The use of proprietary databases, statistical packages, and graphical packages is rather widely accepted; the current regulatory questions focus specifically on the use of ground-water simulation codes. Policies applied to ground-water modeling software should be consistent with those established for other software. Options for U.S. EPA From the Study Group discussions, the U.S. EPA it became apparent that needs to establish a consistent policy concerning selection and use of well tested and validated ground-water models in all projects carried out by or for the Agency. Such a policy should address the Issues of model acceptance and use of proprietary codes and should be consistent with policies regarding surface water and air models. It should focus on the basic needs and concerns of the Program Offices at headquarters as well as the Regional Offices, and 41 should be realistic with respect to the Agency's capabilities to implement it. With respect to proprietary codes opinions of the Study Group varied from out-right banning to ensuring a proper place for them in Agency policy. In general. Study Group members agreed on the importance of the following elements for an Agency policy regarding model acceptance: • Establish a formal Agency mechanism to review and validate models for use for and by EPA and define model acceptance criteria. This approach should be restricted to publicly available, noncopyrighted codes and should prevent shifting the burden of testing and peer reviewing proprietary codes from the contractor to the Agency. • The Agency should compile and distribute a list of reviewed and validated computer codes acceptable to the Agency, and require contractors (and urge PRPs) to use them. (This approach is comparable to the one employed by the Agency with respect to simulation software for surface water and air systems.) • EPA should identify proprietary codes that it regards important to the Agency's mission; such codes should be brought into the public domain, after passing the Agency review and validation process. • A special list should be compiled of those proprietary codes that have passed a comparable review and validation process; however, their use for Agency purposes should be restricted to noncontroversial issues. • Wherever possible the Agency should advocate the use of publicly accessible ground-water modeling software. The Study Group recommends that a general framework of nondiscriminatory criteria should be established by the Agency to apply to both public domain and proprietary codes. These criteria should include: • publication and peer review of the conceptual and mathematical framework • full documentation and visibility of the assumptions • testing of the code according to prescribed Agency methods; this should include verification (checking the accuracy of the computational algorithms used to solve the governing equations), and validation (checking the ability of the theoretical foundation of the code to describe the actual system behavior) • trade secrets (unique algorithms that are not described) should not be permitted if they might affect the outcome of the simulations; proprietary codes are already protected by the copyright law In establishing a model use policy the Agency might take a hard line requiring that acceptable public domain computer codes, where available, be used in situations where a contractor or PRP's nonpeer-reviewed proprietary code is unavailable for full EPA review or where such a review would be burdensome to the Agency (e.g., documentation is poor, Agency resources 42 scarce, etc.)* Establishing such a prophylactic policy that would require all computer codes to have passed peer review, be validated, and publicly available before EPA would rely on their results, allows all parties (PRPs, EPA, State, Intervenors, etc.) access to the same code to perform similar analyses. Another way to deal with the problem is to develop test procedures and validation criteria acceptable to the Agency, but leave the actual review and validation process to be initiated by the contractor or PRP. In that case precautions need to be taken to assure that the contractor and PRP cannot influence the outcome of the review and validation. This approach allows unique proprietary codes to be used when called for. These particular codes would have to be addressed on a case-by-case basis. In the interim, before any Agency policy takes shape. Regional and Headquarter project officers need to be sensitized to the potential problems of approving the use of proprietary, undocumented, nonpeer-reviewed, nonvalidated computer codes. In case an Agency policy includes acceptance of proprietary codes provisions should be made regarding distribution of program copies of licensed software and documentation to the Agency for purposes of regulatory compliance, as well as provisions for reasonable use by third parties (at reasonable cost) of code documentation and an executable version of the program code, or, at a minimum, access to the use of the code. Unreasonable cost to a group, such as a public interest group, could violate the provisions of the Freedom of Information Act. For proprietary codes, the Agency might also require from a contractor proof of copyright, ownership, or license to perform, display and use the code. In case the Agency intends to use the software internally, a license should be obtained to perform, display, use, and reproduce the code and related documentation in all parts of the Agency. Bids to the Agency can be written so that code is acceptable foregoing criteria are met. Other code users, such as Potentially Parties (PRPs), are not restricted to these options, but they take being challenged if they do not adhere to them. only if the Responsible the risk of QUALITY AND USEFULNESS OF MODEL STUDIES Code Selection Using models to analyze alternative solutions to ground-water problems requires a number of steps, each of which should be taken conscientiously and reviewed carefully. After the decision to use a model has been made, the selection process is initiated. Selection of a model is the process of matching a detailed description of the modeling needs with well-defined. 43 quality-assured characteristics of existing models. In selecting an appropriate model, both the model requirements and the characteristics of existing models must be carefully analyzed. Major elements in evaluating modeling needs are: (1) formulation of the management objective to be addressed and the level of analysis sought; (2) description of the system under study; and (3) analysis of the constraints in human and material resources available for the study. Model selection is partly quantitative and partly qualitative. Many subjective decisions must be made, often because there are insufficient data in the selection stage of the project to establish the importance of certain characteristics of the system to be modeled. Definition of modeling needs is based on the management problem at hand, questions asked by planners and decision makers, and on the understanding of the physical system, including the pertinent processes, boundary conditions, and system stresses. The major criteria in selecting a model are: (1) that the model is suited for the intended use; (2) that the model is thoroughly tested and validated for the intended use; and (3) that the model code and documentation are complete and user-friendly. If different problems must be solved, more than one model might be needed or a model might be used in more than one capacity. In such cases, the model requirements for each of the problems posed have to be clearly defined at the outset of the selection process. To a certain extent this is also true for modeling the same system in different stages of the project. Growing understanding of the system and data availibility might lead to a need for a succession of models of increasing complexity. In such cases, flexibility of the model or model package might become an important selection criterion. It should be realized that a perfect match rarely exists between desired characteristics and those of available models. Many of the selection criteria are subjective or weakly justified. If a match is hard to obtain, reassessment of these criteria and their relative weight in the selection process is necessary. Hence, model selection is very much an Iterative process. Special attention should be given to model selection procedures because there is a strong interest, especially in the Regions, in having advice and guidance available for model selection and use, given site-specific conditions. Selecting an appropriate model is crucial to the success of a modeling project. Special measures are necessary to assure a proper selection process. A major problem in model use is model credibility. In the selection process special attention should be given to assure the use of qualified models that have undergone adequate review and testing. The various elements of model selection should be addressed in an Agency guidance document. In a related activity ORD/OHEA is currently involved in 44 the development of model selection guidance focussed on exposure assessment. Further Information on ground-water model selection is presented in Rao et al. (1981), Kincaid et al. (1984), Boutwell et al. (1985), and Simmons and Cole (1985). In standardizing model selection, three major approaches are employed in characterizing the validation of numerical models. In one, the model is tested according to established procedures; when accepted, the model is prescribed in regulations for use in cases covered by those regulations. This approach dues not leave much flexibility for incorporating the advances of recent research and technological development. The second approach includes the establishment of a list of ground-water simulation codes as "standard" codes for various generic and site specific management puposes. To be listed, a code should pass a widely accepted review and test procedure. However, establishing "standard models" will not prevent discussion of the appropriateness of a selected model for analysis of a new policy nor of its use in a particular decision-making process. In discussions at OPPE the following related issues emerged: 1. Is it better to establish standard models for the Agency, or should cri¬ teria or guidelines be developed by which EPA analysts can evaluate use of models. In considering this issue, questions have been raised such as: • Are there legal liabilities for setting up certain models as accept¬ able? (For instance, if the Agency certifies a model for use, can the Agency no longer criticize an Industry's use of that model in a Superfund case?) • Does certification squelch the development of new, better models? • What balance should there be between using the newer, faster models and using mature models already subjected to peer review? 2. Different types of models are appropriate for different uses. Their role in Agency ground-water concerns seems to be divided between the models used for national, priority-setting purposes (EPA ground-water strategy), or regulatory purposes, and those models used for site-specific purposes (Superfund sites, leaking underground storage tanks). Separate sets of criteria, or different types of models, must be established for each of these classes. A third approach is to prescribe a review-and-test methodology in the regulations and require the model development team to show that the model code satisfies the requirements. This approach leaves room to update the codes as long as each version is adequately reviewed and tested. An example of this last approach is the quality assurance program for models and computer codes of the U.S. Nuclear Regulatory Commission (Silling 1983). The Agency should assess the use of "Expert Systems" for assistance in selection and use of available models for problem oriented needs. Such an expert system might contain a database of "acceptable" models, and might 45 include options for system conceptualization, code selection and project evaluation. A select group of Agency experts and outside experts (in modeling and artificial intelligence) could be brought together with a selection of potential users for such an assessment and to recommend a course of action. In addition, a demonstration project could be initiated to explore the potential of expert systems in providing the required guidance. QUALITY ASSURANCE IN GROUND-WATER MODELING STUDIES Definition and Role of Quality Assurance in Ground-water Modeling One of the major issues emerging from the Study Group concerned the quality of modeling studies carried out by or for the U.S EPA, and the usefulness of the study results In the Agency's decision-making process. As the Agency's decisions are contested, often in litigation, the analytical framework on which these decisions are based also can become contested. Hence, Agency decisions should be based on the use of technically and scientifically sound data collection, information processing, and interpretation methods. In litigating a case where the expert witness relies on a ground-water model, the model (and its results) must be "of a type reasonably relied upon by experts in the particular field in forming opinions or inferences upon the subject" (Rule 703; Federal Rules of Civil Procedure). Although no criteria have been established within the technical community, it follows logically that the models must show some minimal level of credibility and reliability before they may be relied upon by experts. Such credibility can be established by documentation of the program, publication of the model's underlying theory, review, and validation, and its widespread use within the technical community. EPA runs the risk of undermining its case if the models it uses cannot be shown to be credible. In addition, where EPA continues to use models not generally accepted or used by the technical community, it will continue to be faced with litigation on the formulations of such models. If on the other hand EPA uses models that are generally accepted by the technical community, model credibility is more likely to be recognized by opposing parties. Furthermore, concerns raised in the Regions and by QWPE regarding the often unsatisfactory use of models in field studies, are identicative of the need for guidance and QA policies in model application. Comprehensive quality assurance, as applied to data collection, data processing, and modeling, and sound model selection policies provide the mechanisms to ensure that decisions are based on the best available data synthesis and problem analysis methods. Quality assurance (QA) in ground-water modeling is the procedural and operational framework put in place by the organization managing the modeling study, to assure technically and scientifically adequate execution of all project tasks included in the study. QA in ground-water modeling should be applied to both model development and model application projects. The two major elements of QA are quality control (QC) and quality assessment. Quality control refers to the procedures that ensure the quality of the final product. These procedures include the use of appropriate methodology. 46 adequate validation, and proper usage (Taylor 1985). To monitor the quality control procedures and to evaluate the quality of the study products, quality assessment is applied. Quality assessment consists of two elements: auditing and technical review. Audits are procedures intended to assess the degree of compliance with QA requirements, commensurate with the level of QA prescribed for the project. Compliance is measured in terms of traceability of records, accountability (approvals from responsible staff), and fulfillment of commitments in the QA plan. Technical review implies independent evaluation of the technical and scientific bases of a project and of the usefulness of its results. Various phases of quality assessment exist for both model development and application. First, review and testing is performed by the author, and sometimes by other employees not involved in the project, or by invited experts from outside the organization. Also to be considered is the quality assessment by the organization for which the project has been carried out. Again, three levels can be distinguished: project or product review or testing by the project officer or project monitor, by technical experts within the funding or controlling organization, and by an external peer review group. Although the U.S. EPA has a centrally managed QA program in place, pertinent regulations and guidance are very much oriented to the quality of measurements and monitoring techniques. Data synthesis methods such as modeling are not well covered by the current policy. At present, ad hoc QA requirements are being developed for individual modeling projects by various project officers in cooperation with quality assurance officers, and might differ from Office to Office. These requirements are called for in part by the Administrative Procedures Act, which compels EPA to maintain an administrative record to support its regulatory decision making. If computer models are used by EPA in decision making, data input records should be maintained as part of the administrative record. The Study Group strongly suggests extending current EPA QA approaches to data synthesis and problem analysis methods, especially modeling. Such a policy, following the approach taken in the QA policy for environmental measurements, Is outlined In this section. The approach presented applies to the quality assurance and quality assessment of both modeling research and development projects as well as to studies in which existing- models are used to assist management in decision making. Some related issues such as model testing and the use of proprietary codes are discussed in separate sections of this report. Current EPA Quality Assurance Policies In May 1979 EPA initiated an Agency-wide quality assurance program to ensure that all environmental measurements conducted by the regional offices, program offices, EPA laboratories, contractors, grantees, or other sources would result in scientifically valid and defensible data of documented precision, accuracy, representativeness, comparability (standardization), and completeness (Stanley and Verner 1985). The Administrator delegated to the Office of Research and Development (ORD) the authority and responsibility for developing, coordinating, and Implementing the mandatory Agency-wide QA program. 47 ORD established the Quality Assurance Management Staff (QAMS) to serve as the central management authority for this program. The QAMS activities involve: (1) the development of policies and procedures; (2) coordination for and direction of the implementation of the Agency QA program; and (3) review, evaluation, and audit of program activities involving environmental monitoring and other types of data generation. In an effort to ensure consistency of the QA program with the Agency's mission and objectives, the requirement was established that a single QA Officer be designated for each Agency organization involved in the program, and that adequate data documentation would be prepared for each measurement activity to ensure that the results were of known quality and defensible. In April 1984, EPA Order 5360.1, "Policy and Program Requirements to Implement the Mandatory Quality Assurance Program," was issued and, for the first time, provided a regulation basis for the Agency QA program. The order specifies certain activities crucial to the implementation of a QA program. These activities include the following: • Development and regular updating of a QA project plan for all projects and tasks involving environmentally related measurements, in accordance with approved guidelines • Assuring implementation of QA for all contracts and financial assis¬ tance involving environmentally related measurements, as specified in applicable EPA regulations, including subcontracts and subagreements • Conducting audits (technical system, performance evaluations, data quality bench studies, etc.) on a scheduled basis of organizational units and projects involving environmentally related measurements • Developing and adopting technical guidelines for estimating data quality in terms of precision, accuracy, representativeness, com¬ pleteness, and comparability, as appropriate, and incorporating data quality requirements in all projects and tasks involving environ¬ mentally related measurements • Establishing achievable data quality limits for methods cited in regulations, based on results of method evaluations arising from the methods standardization process (e.g.» ASTM Standard D2777-77) • Implementing corrective actions, based on audit results, and incorpor¬ ating this process into the management accountability system • Providing appropriate training based on perceived needs, for all levels of QA management, to assure that QA responsibilities and requirements are understood at every stage of project implementation Each Agency organization engaged in environmentally related measurement is required to submit a QA Management Plan (QAMP) for approval by the Agency's QA Management Staff. The QAMP sets forth the management philosophy of the organization with respect to quality assurance/quality control. It identifies 48 the key elements of the QA program, how they are to be implemented, and who is responsible for the implementations. The QA project plan is a specific delineation of an offeror's approach for accomplishing the QA specification in a Statement of Work. When a QA project plan is not required as a part of the technical proposal, the Con¬ tracting Officer may require the QA project plan as a deliverable under the contract. The plan should address the following: • A statement of policy concerning the organization's commitment to implement a QA program to assure generation of measurement data of adequate quality to meet the requirements of the Statement of Work • An organizational chart showing the position of the QA function or person within the organization. It is highly desirable that the QA function or person be independent of the functional groups that generate measurement data • A delineation of the authority and responsibilities of the QA function or person and the related data quality responsibilities of other functional groups of the organization • The type and degree of experience in developing and applying QA procedures to the proposed sampling and measurement methods needed for performance of the Statement of Work • The background and experience of the personnel proposed to accomplish the QA specifications in the Statement of Work • The offeror's general approach for accomplishing the QA specifications in the Statement of Work EPA Quality Assurance Options for Ground-water Modeling The primary goal of an EPA ground-water modeling QA program should be to ensure that all modeling-based problem analysis supported by the Agency is of a known and scientifically acceptable quality, and is verifiable and defensible. Decisions by management rest on the quality of environmental data and data analysis; therefore, program managers should be responsible for: (1) specifying the quality of the data required from environmentally related measurements; (2) indicating the level of problem-solving-oriented data analysis; (3) specifying the quality required from the tools used in the analysis (e.g., models); and (4) providing sufficient resources to assure an adequate level of QA. All routine or planned projects or tasks carried out for the U.S. EPA or from which the results will form the basis of Agency action and which involve environmentally related measurements, information processing, and modeling, should be undertaken with an adequate QA plan. Such a plan should specify goals for the quality of resulting data and processed information acceptable to the user, contain detailed description of the measures to be taken to achieve prescribed quality objectives, and assign responsibility for achieving 49 the goals (QC). It should also contain procedures for documenting the activities within a project in order to provide evidence that standards of quality have been met. Different levels of QA can be distinguished, dependent on use of the modeling results; e.g., if results are expected to be used in litigation, a high level of QA (including quality assessment!) Is required. EPA should have effective quality assessment procedures in place to monitor the QA performance of the modeling project teams. QA should be applied to all stages of the modeling project, not just at the end as is currently often the case; QA should be an integral part of all projects. It should not drive or manage the direction of a project nor is it intended to be an after-the-fact filing of technical data. The Agency's quality assessment process should be conducted throughout the modeling process, with stop/go decisions at each critical point. A paper trail for QA in model development and application is required by the Administrative Procedures Act. However, at present there are no guidelines for modeling projects, nor Is there a central document room for material collected on a case. As an example, a report on a modeling project should give: • assumptions • parameter values and sources • boundary and initial conditions • nature of grid and grid design justification • changes and verification of changes made in code • actual input used • output of model runs and interpretation • validation (or at least calibration) of model In addition, depending on the level of QA required, the following files may be retained (in hard-copy and, at higher levels, in digital form): • version of the source code used • verification input and output • validation input and output • application input and output If any modifications are made to the model coding for a specific problem, the code should be tested again; all QA for model development should again be 50 applied including accurate recordkeeping and reporting. All new input and output files must be saved for inspection and possible reuse. Model Development— Ideally, QA should be applied to all ongoing and yet-to-be-developed codes, and should include such aspects as verification of the mathematical framework, field validation, benchmarking, and code comparison. At a minimum the theory should be peer-reviewed; a fully documented version should be available for testing. Different types of QA are required for numerical and analytical models; in particular, such a procedure should call for the verification of the assumptions underlying the use of an analytical model and for validation of numerical models. A detailed dicussion of testing and validation of ground-water models is presented in the section on ground-water model testing and validation. QA for code development and maintenance should include complete recordkeeping of the model development, of modifications made in the code, and of the code validation process. Model Application— A lack of consistency in model acceptance and use prevails across the Offices. This is of particular concern because acceptance of a model in one Office confers legal acceptability for all the Agency. ORD's role should be to establish criteria for evaluation and use of models and to provide technical expertise as needed. QA in model application should address all facets of the model application process: • correct and clear formulation of problems to be solved • project description and objectives • modeling approach to the project • is modeling the best available approach and if so, is the selected model appropriate and cost-effective? • conceptualization of system and processes, including hydrogeologic framework, boundary conditions, stresses, and controls • explicit description of assumptions and simplifications • data acquisition and interpretation • model selection or justification for choosing to develop a new model • model preparation (parameter selection, data entry or reformatting, and gridding) • the validity of the parameter values used in the model application 51 • protocols for parameter estimation and model calibration, to provide guidance, especially for sensitive parameters • level of information in computer output (variables and parameters displayed, formats, layout) • identification of calibration goals and evaluation of how well they have been met • sensitivity analysis • postsimulation analysis (including verification of reasonability of results, interpretation of results, uncertainty analysis, and the use of manual or automatic data processing techniques, as for contouring) • establishment of appropriate performance targets (e.g., a 6-foot head error should be compared with a 20-foot head gradient or drawdown, not with the 250-foot aquifer thickness!); these targets should recognize the limits of the data • presentation and documentation of results • evaluation of how closely the modeling results answer the questions raised by management In addition, a project QA plan should contain: • title page with provision for approval signatures • table of contents • project organization(s) and responsibilities • quality assurance objectives for modeling, in terms of validity, uncertainty, accuracy, completeness, and comparability • internal quality assessment checks and frequency • quality assurance performance audits and their frequency • quality assurance reports to management • specific procedures used in routinely assessing validity, uncertainty, completeness, and comparability of modeling studies • corrective action TECHNOLOGY TRANSFER AND TRAINING TO SUSTAIN AND IMPROVE EXPERTISE OF AGENCY PERSONNEL In 1982 the Office of Technology Assessment of the U.S. Congress (OTA) published a report on the use of models for water resources management, planning, and policy. As discussed in a previous section of this report, the 52 Study Group found that many of OTA's conclusions and recommendations on technology transfer and training apply to ground-water modeling at the EPA. These include such OTA findings as the following: • Levels of communication between decision makers and modelers are low, and little coordination of model development, dissemination, or use occurs within individual federal agencies. • Developing and using models is a complex undertaking, requiring personnel with highly developed technical capabilities, as well as adequate budgetary support for computer facilities, collecting and processing data, and such support services as user assistance. Technology transfer in general means dissemination of information on technological advances through communication and education. When applied to ground-water modeling, technology transfer Includes dissemination of information about the role of modeling in water resource management, model theory, the process and management of modeling, availability and applicability of model software, information on quality assurance, model selection, and model testing and verification. It also pertains to the distribution of computer codes and documentation, and includes assistance in transferral, implementation, and use of codes. The OTA report considers specific education and training of model de¬ velopers, users, and managers in aspects of water resources modeling, as a critical component of technology transfer. Other technology transfer mechanisms include the distribution of published, printed, or electronically stored materials, such as reports, newsletters, papers, computer codes, data files, and other communications, and discussions and information exchanges in meetings, workshops, and conferences. Effective communication forms the basis of technology transfer. Com¬ munication is often hampered because of insufficient communication channels, incompatible language or jargon use, existence of different concepts, and administrative impediments. Despite recent examples of successful modeling use in developing ground- water protection policies in the United States and abroad, managers still do not rely widely on modeling for decision-making. One of the major obstacles is the inability of modelers and program managers to communicate effectively. An ill-posed problem yields answers to the wrong questions. Sometimes, this is the result of managers and modelers speaking different jargon. On another level of coimnunication, managers should appreciate how dif¬ ficult it is to explain the results of complicated models to nontechnical audiences such as in public meetings and courts of law. One useful means of overcoming these limitations in communication involves the effective use of audio-visual aids. 53 i Technology Transfer and Training in EPA Ground water and ground-water modeling expertise is disjunct within EPA. Program Offices display different approaches to the role of modeling In their activities and show different levels of sophistication in model use, either in a generic sense during the development of policies and regulations, or in the preparation of site-specific guidance. Consequently, many inconsistencies in ground-water modeling have resulted. The Study Group concluded that among the modeling issues addressed during its meetings, retaining and improving the expertise of EPA personnel deserve a high priority. Because it is expected that model use will increase in the future, the development of in-house expertise, by whatever means, appears to be a major priority. A major Impediment to meeting the modeling needs of the Agency is the inadequacy of the current levels of model-related training and information exchange. If models are to be used effectively in water resources analysis, training in basic concepts of modeling and in proper interpretation of model results must be offered to decision makers at all levels of the Agency. Further, there is a need for specific training in the use of individual models, and a need for continuingly informing of and educating users and managers in research developments, new regulations and policies, and field experience. Most EPA technical and managerial personnel do not need to become modeling "experts," but should have sufficient training to be knowledgeable users or at least competent judges of the appropriateness of models used by PRPs and contracted consultants. The return on investments made in applying mathematical models to ground- water problems depends a great deal on the training and experience of the technical support staff involved in their use. Managers should be aware that specialized training and experience are necessary to develop and apply mathematical models, and that relatively few technical support staff can be expected to have such skills. This is due in part to the need for familiarity with a number of scientific disciplines, so that the model may be structured to faithfully simulate real-world problems. Managers should have some working knowledge of the sciences involved so that they might put appropriate questions to specialists. In practice this means that ground-water modelers, technical staff, and their supervisors should become involved in continuing education efforts, and managers should expect and encourage this. They should be sensitive to the financial and time requirements necessary for adequate training. (One does not become a competent modeler in a course of a week; that takes years and is a combination of training and experience.) Unfortunately, because of the staff's professional background and expertise and because of the nature of the activities in which they are involved, modeling often has a lower priority than some other ground-water- related training needs, such as general hydrogeology, data collection and analysis, and the administrative and legislative framework in which the work is carried out. However, knowledge of basic hydrogeology is a necessary preface to effective modeling. 54 Information Exchange on Ground-water Modeling There is an urgent need for expanding existing and developing new mechanisms to disseminate and exchange technical information. Two different approaches exist to information exchange: (1) the receiver actively seeks the required information or technoloyg; (2) the receiver has a passive role insofar as supervisors or internal or external specialists bring the information or technology to the potential user. EPA has various mechanisms in place to facilitate both approaches. Reports on research and development carried out with funding from the Agency are published and distributed either through EPA's Center for Environmental Research Information (CERI), in Cincinnati, or through the National Technical Information Services (NTIS), in Springfield, Virginia. Furthermore, an extensive technology transfer program for ground-water modeling has been developed by the International Ground Water Modeling Center (IGWMC) with major funding from EPA, and includes information exchange, software distribution, training activities, and model user assistance. Finally, the EPA-supported National Ground Water Information Center, operated by the National Water Well Association (NWWA), provides access to a large information base on ground- water literature. Despite these efforts, information from research projects often is not disseminated effectively to potential users in the national Program Offices or the Regional Offices, and the staff is not aware of the existence of certain EPA guidance documents and software. Technology transfer is ineffective if it simply consistes of reports sent to Regional or Program Office libraries. To resolve this problem, steps should be taken to promote increased conmunlcation and sharing of experiences among staff through: • compiling a list of all potential model users and "modeling experts" within the Agency; such a list can be maintained by ORD and should be accessible via computer/phone networks to all Agency staff; this list should be provided to organizations such as the IGWMC and NWWA to enable them to reach out efficiently to all EPA staff interested in ground-water modeling • establishing an electronic bulletin board on ground-water modeling that is open to all staff interested in modeling • organizing regular exchange sessions for personnel involved with similar modeling-related activities throughout the Agency; these sessions would permit staff members to keep track of developments in each other's projects and to institutionalize their experience with applications of particular models to particular sites Meetings of Regional and ORD people should be held to address questions such as: • What support is available to Regions? 55 • What is the experience with particular problems in the various Re¬ gions? (Turnover of personnel may be partly due to the feeling that they are alone, "out on a limb.") For assistance with specific problems, project managers and other staff should have access to senior hydrogeologists with extensive modeling experience, such as (1) in-house Regional experts, (2) experts within ORD, perhaps located at EPA labs; (3) contractors, or (4) experts from other agencies. In addition, adequate use should be made of federally supported organizations such as the International Ground Water Modeling Center, and the National Center for Groundwater Research. When particular models are applied at particular sites, the Agency needs to institutionalize the experience. A central clearinghouse should be created for keeping records on models used in the Agency. It should have readily available information on: (1) where and under what conditions they have been used; (2) what results were provided in terms of usefulness to management; and (3) what administrative, technical, and legal problems were encountered. This information should include contact persons, site descriptions, model modifications, and details on QA procedures. Each program office, regional office, division, or branch involved in ground-water modeling should have its own working library of pertinent publications maintained by a coordinator who will ensure that the library is up-to-date and that each staff member involved in ground-water projects receives important information in a timely way. Each Regional and Program Office should have a ground-water library and subscribe to at least the major ground-water journals. Wide distribution of EPA manuals, guidance documents, and research reports within the Agency is crucial. The "EPA model user" mailing list mentioned above could be used for the distribution. Many publi¬ cations are in the open literature, and provision should be made for distri¬ buting their reprints. Training To address effectively the issue of improving Agency expertise in ground- water modeling, an evaluation should be made of the existing modeling expertise of Agency personnel and the requirements in terms of educational background and level of experience for the technical, administrative, and management staff involved in modeling projects. Based on such an analysis, a training program for staff of Regional Offices and Headquarters should be developed. An Agency training program should be based on the realities of needs, staff backgrounds, administrative structures and constraints, and changes in staff. Such training should be mainly to provide skills needed to evaluate adequately the effectiveness of model codes and modeling work, as only a few of the staff will be (or should be) in a position to become experts in modeling. There are many ways to obtain education on such advanced ground-water topics as mathematical modeling. Existing and potential approaches Identified include: 56 • ongoing training workshops at selected Regional Offices and at Headquarters; these should be expanded to include all Offices • training courses at the EPA research laboratories • presentation of courses by EPA experts (in the EPA Institute) • attendance at such external training programs as professional short courses organized by universities, the International Ground Water Modeling Center, the National Water Well Association, and the U.S. Geological Survey There should be satisfactory performance on competency tests in the short courses and an achievement of A or B in academic courses, or the staff member should pay back the Agency for money spent. Other possible approaches include: • stationing Regional or Headquarters staff at one of the Agency research laboratories or the USGS for a few months in order to become familiar with current research in general, and modeling in particular • 3- to 6-month positions at EPA for university faculty on sabbatical leave • courses offered at EPA by outside consultants • lecture tours through the Regions • structured self-study • seminars on particular case studies • part- or full-time academic study conditional to long-term public service committment. • the use of television in workshops presented simultaneously in all Regions and Offices. Generally, the more contact time with instructors, the greater the benefit in terms of increased problem-solving capabilities. Unfortunately, the greater the contact time with instructors in formal education settings, the greater the disruption in terms of increased absence from the job. EPA's current ground-water training efforts tend to be of short contact time. This is aggravated by a lack of in-house programs to reinforce training and education received in formal settings. A promising alternative to formal education is self-study coupled with obtaining experience, under the guidance of a senior modeling specialist. Review of work in progress by someone capable of acting as a foil for ideas and approaches would substantially improve model applications. Given the shortage of senior modeling specialists. It would seem advantageous to maximize the time they spend on model applications, and to minimize the time they spend on other duties. Workshops should include case studies of model application to actual problems encountered by Regional and Program Offices. Manuals should be prepared to 57 explain when and how models should be used and when a geohydrological modeler should be consulted. Recruitment and Retention Difficulty in attracting and retaining staff with suitable backgrounds for applying ground-water models, along with high turnover rates among such staff, are serious problems in all EPA Offices. Losing trained personnel to consulting firms seems to be the result of the significantly higher salaries offered and the recognition given by private firms to the special qualifications of hydrogeologists. Three steps should be taken to improve this situation: • The Office of Human Resources should establish the position of "hydro¬ geologist." OTA's Superfund Strategy report lists establishment of the position as its highest priority, and the Study Group concurs. • A career path for hydrogeologists should be established through the GS- 15 level. The GS-15 slot would be established for a "National Expert" as defined by Civil Service so that EPA can retain senior-level experience comparable to that found in consulting firms. • The Agency should encourage staff to take short courses and graduate- level courses in ground-water geology and modeling at the Agency's expense, and the graduate courses should be allowed for degree credit. As in some other federal agencies, EPA could require a certain number of years of service with the Agency for a given amount of additional training at the Agency's expense. This would control the rate at which trained staff are lost, and would provide a continual supply of trained younger staff. Another major way to maintain an adequate level of in-house expertise in ground-water modeling is by assuring that properly trained, experienced technical staff can continue to perform in a technical role without losing opportunities for career advancement (i.e., avoiding promotions of good technical people to administrative positions as the only available career path). 58 REFERENCES Adrion, W.R., M.A. Branstad, and J.C. Cherniasky, 1982. Validation, verification, and testing of computer software. ACM Computing Surveys, Vol.14(2), p.159—192. Andersen, P.F., Faust, C.R. and J.W. Mercer, 1984. Analysis of conceptual designs for remedial measures at Lipari Landfill. Ground water 22 ( 2 ): 176-190. ASTM, 1984, Standard practices for evaluating environmental fate models of chemicals. Annual book of ASTM standards, E 978-84, Am. Soc. for Testing and Materials, Philadelphia, PA. Bachmat, Y., B. Andrews, D. Holtz, and S. Sebastian, 1978. Utilization of Numerical Groundwater Models for Water Resource Management. U.S. EPA Report no. EPA-600/8-78-012. U.S. Environmental Protection Agency, R.S. Kerr Environmental Research Laboratory, Ada, OK. Boutwell, S.H., S.M. Brown, B.R. Roberts, and D.F. Atwood, 1985. Modeling remedial actions at uncontrolled hazardous waste sites. EPA/540/2-85/001, U.S. Environmental Protection Agency, 0SWER/0RD, Washington, DC. Brown, J., 1986. 1986 Environmental software review. Pollution Engineering 18(1):18-28. Cleveland, W.S., and R. McGill, 1985. Graphical perception and graphical methods for analyzing scientific data, science 229:828-833. Gupta, S.K., C.R. Cole, F.W. Bond, and A.M. Monti, 1984. Finite-element three-dimensional ground-water (FE3DGW) flow model: Formulation, computer source listings, and user's manual. ONWI-548, Battelle Memorial Inst., Columbus, OH. Guven, 0., F.J. Molz, and J.G. Melville, 1984. An analysis of dispersion in a Stratified aquifer. Water Resources Research 20(10):1337-1354. Hern, S.C., S.M. Melancon, and J.E. Pollard, 1986. Generic steps in the field validation of vadose zone fate and transport models. In: Hern, S.C., and S.M. Melancon (eds.), Vadose Zone Modeling of Organic Pollutants. Lewis Publishers, Inc., Chelsea, MI. pp. 61-80. Hoffman, F.D., and R.H. Gardner, 1983. In: J.E. Till and H.R. Meyer (eds.), Radiological Assessment. NUREG/CR-3332, ORNL-5968, U.S. Nuclear Regulatory Commission, Washington, DC. Holcomb Research Institute, 1976. Environmental Modeling and Decision Making. Praeger Publishers, New York, N.Y. Huyakorn, P.S., A.G. Kretschek, R.W. Broome, J.W. Mercer, and B. H. Lester. 1984. "Testing and Validation of Models for Simulating Solute Transport in Groundwater: Development, Evaluation, and Comparison of Benchmark Techniques." GWMI 84-13, International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN 46208, 420 pp. Intera Environmental Consultants, Inc., 1983. A proposed approach to uncertainty analysis. ONWI-488, Battelle Memorial Inst., Columbus, OH, 68 pp. Javendel, I., C. Doughty, and C.F. Tsang, 1984. Groundwater Transport: Handbook of Mathematical Models. AGU Water Resources Monograph no. 10. American Geophysical Union, Washington, DC Khan, I.A., 1986a. Inverse problem in ground water: model development. Ground water 24(1):32-38. Khan, I.A., 1986b. Inverse problem in ground water: model application. Ground Water 24(1)39-48. 59 Kincaid, C.T., J.R. Morrey, and J.E. Rogers, 1984. Geohydrological models for solute migration; Vol.l: Process description and computer code selection. EA 3417.1, Electric Power Research Inst., Palo Alto, CA. Krabbenhoft, D.P., and M.P. Anderson, 1986. Use of a numerical groundwater flow model for hypothesis testing. Ground water 24(1):49—55. Mercer, J.W., and C.R. Faust, 1981. Groundwater Modeling. National Water Well Association, Worthington, OH. Moran, M.S., and L.J. Mezga, 1982. Evaluation factors for verification and validation of low-level waste disposal site models. D0E/0R/21400-T119, Oak Ridge National Laboratory, Oak Ridge, TN, 10 pp. Moses, C.O., and J.S. Herman, 1986. Computer notes—WATIN—a computer program for generating input files for WATEQF. Ground water 24(1):83—89. Puri, $., 1984. Aquifer studies using flow simulations. Ground water 22(5):538-543. Rao, P.S.C., R.E. Jessup, and A.C. Hornsby, 1981. Simulation of nitrogen in agro-ecosystems: criteria for model selection and use. In: Nitrogen cycling in ecosystems of Latin America and the Caribbean. Proceed. Internat. Workshop, Cali, Colombia, March 16-21, 1981, pp. 1-16. Silling, S.A. 1983. Final technical position on documentation of computer codes for high-level waste management. NUREG/CR-0856, U.S. Nuclear Regulatory Commission, Washington, DC, 11 pp. Siimons, C.S., and C.R. Cole. 1985. Guidelines for selecting codes for ground- water transport modeling of low-level waste burial sites. Volume 1—Guideline approach. PNL-4980 Vol. 1. Battelle Pacific NW Laboratory, Richland, WA. Shelton, M.L., 1982. Groundwater management in basalts. Ground water 20(1):86-93. Srinivasan, P., 1984. PIG—A Graphic Interactive Preprocessor for Ground- Water Models. IGWMC Report no. GWMI 84-15. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN. Stanley, T.W., and S.S. Verner, 1985. The U.S. Environmental Protection Agency's Quality Assurance Program. In J.K. Taylor and T.W. Stanley (eds.) "Quality Assurance for Environmental Measurements." ASTM Special Technical Publication 867, Am. Soc. for Testing and Materials, Philadelphia, PA. Strecker, E.W. and W. Chu., 1986. Parameter identification of a groundwater contaminant transport model. Ground water 24(1):56—62. Sykes, J.F., S.B. Pahwa, D.S. Ward, and R.B. Lantz. 1983. The validation of SWENT, a geosphere transport model. In scientific Computing, R. Stepleman et al. (eds.). IMACS/North-Holland Publishing Company, New York, NY. Taylor, J.K., 1985. What is Quality Assurance? In J.K. Taylor and T.W. Stanley (eds.) "Quality Assurance for Environmental Measurements." ASTM Special Technical Publication 867, Am. Soc. for Testing and Materials, Philadelphia, PA, pp. 5-11. U.S. Office of Technology Assessment, 1982. Use of Models for Water Resources Management, Planning, and Policy. U.S. OTA, for Congress of the United States. U.S. Government Printing Office, Washington, DC. van der Heijde, P.K.M., 1984a. Availability and Applicability of Numerical Models for Ground Water Resources Management. IGWMC Report no. GWMI 84- 14. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN. 60 van der Heijde, P.K.M., 1984b. Utilization of Models as Analytic Tools for Groundwater Management. IGWMC Report no. GWM1 84-19. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN. van der Heijde, P.K.M., 1984. Availability and applicability of numerical models for groundwater resources management. In: "Practical Applications of Ground Water Models," proceedings NWWA/IGWMC conf., Columbus, OH, August 15-17, 1984. van der Heijde, P.K.M., and P. Srinivasan, 1983. Aspects of the Use of Graphic Techniques in Ground Water Modeling. IGWMC Report no. GWMI 83- 11. International Ground Water Modeling Center, Holcomb Research Institute, Butler University, Indianapolis, IN. van der Heijde, P.K.M., Y. Bachmat, J. Bredehoeft, B. Andrews, D. Holtz, and S. Sebastian, 1985. Groundwater Management: The Use of Numerical Models , 2nd edition. AGU Water Resources Monograph no. 5. American Geophysical Union, Washington, DC. van der Heijde, P.K.M., P.S. Huyakorn, and J.W. Mercer, 1985. Testing and validation of groundwater models. In: "Practical Applications of Groundwater Modeling," proceedings NWWA/IGWMC conf., Columbus, OH, August 19-20, 1985. Ward, D.S., M. Reeves, and L.E. Duda. 1984. Verification and field comparison of the Sandia Waste-Isolation Flow and Transport Model (SWIFT). NUREG/CR-3316, U.S. Nuclear Regulatory Conmission, Washington, DC. 61 APPENDIX COMPOSITION OF STUDY GROUP The Study Group was composed of three EPA modeling experts, three EPA model users, three non-Agency modeling experts, and the Chairperson. Members were as follows: Chairperson: Paul K.M. van der Heijde International Ground Water Modeling Center Holcomb Research Institute Butler University Indianapolis, Indiana EPA experts: Douglas Ammon Hazardous Waste Engineering Research Lab Cincinnati, Ohio Robert Carsel Environmental Research Laboratory Athens, Georgia Joseph F. Keely (until July 1, 1986) Clinton W. Hall (after July 1, 1986) R.S. Kerr Environmental Research Laboratory Ada, Oklahoma EPA users: Peter Ornstein Office of Waste Programs Enforcement Washington, D.C. David Morganwalp Office of Drinking Water Washington, D.C. Zubair Saleem Office of Solid Waste Waster Management and Economics Division Washington, D.C. Non-Agency experts: James W. Mercer GeoTrans, Inc. Herndon, Virginia 62 Richard A. Park Holcomb Research Institute Butler University Indianapolis, Indiana Suresh C. Rao Department of Soil Science University of Florida Gainsville, Florida In order to ensure the broadest possible participation and representation by Program Offices, but without increasing the size of the Study Group to unmanageable proportions, a number of Agency personnel (listed below) were designated as Corresponding Members. These individuals were copied on all correspondence and were asked to comment on r the interim and final documents produced by the Study Group. Some of the Corresponding Members attended one or more of the Study Group meetings. Study Group Corresponding EPA Members: James Bachmaier Office of Solid Waste Waste Management and Economics Division Washington, D.C. Stuart Cohen Office of Pesticide Programs Exposure Assessment Branch Arlington, Virginia Norbert Dee Office of Ground Water Protection Washington, D.C. Michael Gruber Office of Policy Analysis Washington, D.C. Stephen C. Hern Exposure Assessment Research Division Environmental Monitoring Systems Laboratory Las Vegas, Nevada Seong T. Hwang Office of Health and Environmental Assessment Washington, D.C. David Kyllonen Underground Injection Control Section Region IX San Francisco, California 63 Matthew Lorber Office of Pesticide Programs Exposure Assessment Branch Arlington, Virginia Tom Merski Office of Groundwater Region III, Philadelphia, Pennsylvania Lee Mu 1 key Environmental Research Laboratory Athens, Georgia Wi11iam A. Mullen Office of Groundwater Region X, Seattle, Washington Annett Hold Office of Pesticides and Toxic Substances Washington, D.C. Hope Pillsbury Office of Policy Analysis Washington, D.C. Herbert Quinn Office of Research and Development Water and Land Division Washington, D.C. Paul B. Schumann Office of Solid Waste and Emergency Response Hazardous Site Control Division Washington, D.C. Carol Wood Office of Ground Water Region I, Boston, Massachusetts EPA project officers for the Study Group activities were: Keely (ORD/RSKERL, Ada, 0K)(until July 1, 1986), Clinton W. Hall Ada, OK)(after July 1, 1986); Coordinator for ORD, Washington, D Cordle (ORD/Water and Land Div.) Joseph F. (ORD/RSKERL, C. was Steve 64 United States Robert S. Kerr Environmental Environmental Protection Research Laboratory Agency Ada OK 74820 Research and Development EPA/600/8-87/003 Jan. 1987 &EPA The Use of Models in Managing Ground-Water Protection Programs ■ E P A /600/8-87/003 January 1987 THE USE OF MODELS IN MANAGING GROUND-WATER PROTECTION PROGRAMS Joseph F. Keely, Ph.D, P.Hg. Robert S. Kerr Environmental Research Laboratory U.S. Environmental Protection Agency Ada, Oklahoma 74820 Office of Research and Development U.S. Environmental Protection Agency Ada, Oklahoma 74820 / DISCLAIMER The information in this document has been funded wholly or in part by the United States Environmental Protection Agency. It has been subjected to the Agency’s peer and administrative review, and it has been approved for publication as an EPA document. 11 FOREWORD The U.S. Environmental Protection Agency was established to coordinate administration of the major Federal programs designed to protect the quality of our environment. An important part of the Agency's effort involves the search for information about environmental problems, management techniques and new technologies through which optimum use of the Nation's land and water resources can be assured and the threat pollution poses to the welfare of the American people can be minimized. EPA's Office of Research and Development conducts this search through a nationwide network of research facilities. As one of the facilities, the Robert S. Kerr Environmental Research Laboratory is the Agency's center of expertise for investigation of the soil and subsurface environment. Personnel at the laboratory are responsible for management of research programs to: (a) determine the fate, transport and transformation rates of pollutants in the soil, the unsaturated zone and the saturated zones of the suburface environment; (b) define the processes to be used in characterizing the soil and subsurface environment as a receptor of pollutants; (c) develop techniques for predicting the effect of pollutants on ground water, soil and indigenous organisms; and (d) define and demonstrate the applicability and limitations of using natural processes, indigenous to the soil and subsurface environment, for the protection of this resource. This report contributes to that knowledge which is essential in order for EPA to establish and enforce pollution control standards which are reasonable, cost effective and provide adequate environmental protection for the American public. Clinton W. Hall Director Robert S. Kerr Environmental Research Laboratory 111 ABSTRACT Because ground-water quality protection is emerging as a major National environmental problem of this decade, there is increasing pressure on regulators and the regulated to identify, assess or even anticipate situations involving ground-water contamination. Site-specific and generic mathematical models are increasingly being used by EPA to fulfill its mandates under a number of major environmental statutes which call for permit issuance, investigation of potential problems, remediation activities, exposure assessment and a myriad of other policy decisions. Mathematical models can be helpful tools to managers of ground-water protection programs. They may be used for testing hypotheses about conceptualizations and to gather a fuller understanding of important physical, chemical and biological processes which affect ground-water resources. The possible outcomes of complex problems can be addressed in great detail, if adequate data are available. The success of these efforts depends on the accuracy and efficiency with which the natural processes controlling the behavior of ground water, and the chemical and biological species it transports, are simulated. Success also depends heavily on the expertise of the modeler and the communication with management so that the appropriateness, underlying assumptions, and limitations of specific models are appreciated. IV CONTENTS Foreword iii Abstract iv Figures vii Tables ix Acknowledgments x 1. The Utility of Models 1 Introduction 1 Management Applications 3 Modeling Contaminant Transport 6 Categories of Models 7 Chapter Summary 8 2. Assumptions, Limitations, and Quality Control 9 Introduction 9 Physical Processes 9 Advection and Dispersion 10 Complicating Factors 13 Considerations for Predictive Modeling 14 Chemical Processes 15 Chemical/Electronic Alterations 15 Nuclear Alterations 16 Chemical Associations 16 Surface Interactions 17 Biological Processes 19 Surface Water Modeling Analogy 19 Ground-Water Biotransformations 20 A Ground-Water Model 20 Analytical and Numerical Models 22 Quality Control 23 Chapter Summary ' 25 3. Applications in Practical Settings 29 Stereotypical Applications 29 Real-World Applications 29 Field Example No. 1 30 Field Example No. 2 32 Practical Concerns 44 Chapter Summary 52 v 4. Liabilities, Costs, and Recommendations for Managers 55 Introduction 55 Potential Liabilities 55 Economic Considerations 56 Managerial Considerations 64 Chapter Summary 66 References 69 vi FIGURES Number Page 1-1 Small’sand tank’physical aquifer model 2 1-2 Laboratory column housed in constant-temperature environmental chamber 2 1-3 Electric analog aquifer model constructed by Illinois State Water Survey 3 1- 4 Typical ground-water contamination scenario and a possible contaminant transport model grid design for its simulation 4 2- 1 The influence of natural processes on levels of contaminants downgradient from continuous and slug-release sources 11 2-2 Examples of plots prepared with the Jacob's approximation of the Theis analytical solution to well hydraulics in an artesian aquifer 24 2- 3 Mathematical validation of a numerical method of estimating drawdown, by comparison with an analytical solution 26 3- 1 Location map for Lakewood Water District wells contaminated with volatile organic chemicals 31 3-2 Schematic illustrating the mechanism by which a downgradient source may contaminate a production well 33 3-3 Location map for Chem-Dyne Superfund Site 35 3-4 Chem-Dyne geologic cross-section along NNW-SSE axis , 36 3-5 Chem-Dyne geologic cross-section along WSW-ENE axis 37 3-6 Shallow well ground-water contour map for Chem-Dyne 38 Vll Number Page 3-7 Typical arrangement of clustered, vertically- separated wells installed adjacent to Chem-Dyne and the Great Miami River 39 3-8 Estimates of transmissivity obtained from shallow and deep wells during Chem-Dyne pump test 41 3-9 Distribution of total volatile organic chemical contamination in shallow wells at Chem-Dyne during October 1983 sampling 42 3-10 Distribution of tetrachloroethene in shallow wells at Chem-Dyne during October 1983 sampling 45 3-11 Distribution of trichloroethene in shallow wells at Chem-Dyne during October 1983 sampling 46 3-12 Distribution of trans-dichloroethene in shallow wells at Chem-Dyne during October 1983 sampling 47 3-13 Distribution of vinyl chloride in shallow wells at Chem-Dyne during October 1983 sampling 48 3-14 Distribution of benzene in shallow wells at Chem- Dyne during October 1983 sampling 49 3-15 Distribution of chloroform in shallow wells at Chem- Dyne during October 1983 sampling 50 3- 16 General relationship between site characterization costs and clean-up costs as a function of the characterization approach 54 4- 1 Average price per category for ground-water models from the International Ground Water Modeling Center 57 4-2 Price ranges for IBM-PC ground-water models available from various sources 59 4-3 Costs of sustaining ground-water modeling capabilities at two different computing levels, for a five-year period 61 Vlll TABLES Number Page 2- 1 Natural processes that affect subsurface contaminant transport 10 3- 1 Chem-Dyne pump test observation network 43 3-2 Conventional approach to site characterization efforts 51 3-3 State-of-the-art approach to site characterization efforts 52 3- 4 State-of-the-science approach to site characterization efforts 53 4- 1 Desired backgrounds and salary ranges advertised for positions requiring ground-water modeling 60 4-2 Screening-level questions to help ground-water managers focus mathematical modeling efforts 65 4-3 Conceptualization questions to help ground-water managers focus mathematical modeling efforts 66 4-4 Sociopolitical questions to help ground-water managers focus mathematical modeling efforts 67 ACKNOWLEDGMENTS The author is indebted to the many fine scientists, engineers, and support staff at the Robert S. Kerr Environmental Research Laboratory for their assistance. In particular, Dr. Marvin D. Piwoni and Dr. John T. Wilson made substantial contributions to Chapter 2 in the chemical and biological sections, respectively. Ms. Carol House and Ms. Renae Daniels typed many drafts of the document. Ms. Kathy Clinton prepared most of the illustrations. The author is grateful to Drs. William F. McTernan and Douglas C. Kent of Oklahoma State University for their technical reviews of the manuscript. Thanks also go to Ir. Paul van der Heijde of the International Ground Water Modeling Center for his readings of early drafts of many sections, and for the use of certain photographs. Mr. Marion R. (Dick) Scalfs guidance and encouragement as EPA Project Officer on this project are deeply appreciated. Comments and suggestions from the readers are welcome; the author assumes all responsibility for any errors, omissions, or misstatements. CHAPTER 1 THE UTILITY OF MODELS INTRODUCTION Every time man attempts to simulate the effects of natural phenomena, he is engaging in the scientific art of modeling. Models are nothing more than simplified representations of reality, and their creation and use involves a considerable degree of subjective judgment, as well as an attempt to incorporate known scientific facts. There are many forms of models, each having specific advantages and disadvantages compared with the remainder. Physical models, such as sand-tanks used to simulate aquifers (Figure 1-1) and laboratory columns used to study the relative motion of various contaminants flowing through aquifer materials (Figure 1-2), provide an element of reality which is enlightening and satisfying from an intuitive viewpoint. Their main disadvantage relates to the extreme efforts and time required to generate a meaningful amount of data. Other difficulties relate to the care required to obtain samples of subsurface material for construction of these models, without significantly disturbing the natural condition of the samples. Analog models are also physically based, but their operating principle is one of similarity, not true-life representation. A typical example is the electric analog model (Figure 1-3), in which capacitors and resistors are able to closely replicate the effects of the rate of release of water from storage in aquifers. The clear disadvantage is that ’a camel is not a horse', even if both can carry a load. As is the case with other physically based models, data generation is slow and there is little flexibility for experimental design changes. Mathematical models are non-physical, relying instead on the quantification of relationships between specific parameters and variables to simulate the effects of natural processes (Figure 1-4). As such, mathematical models are abstract and provide little in the way of a directly observable link to reality. Despite this lack of intuitive grace, mathematical models can generate powerful insights into the functional dependencies between causes and effects in the real world. Large amounts of data can be generated quickly, and experimental modifications can be made with minimal effort, so 1 wmsmgm&sgmsmmi? mam m - . "t Figure 1-1. Small "sand tank" physical aquifer model. Three Pumping wells (A.B.C) penetrate the homogeneous sand to study the effects of well hydraulics on plume movements. Vials in foreground contain various concentrations of water-active dyes. Figure 1-2. Laboratory column housed in constant-temperature environmental chamber. Contaminated solutions are injected into column through inlet tubing in top, by action of hydraulic press in foreground. Samples of the advancing front are withdrawn through ports visible on right-hand side and bottom of column. 2 ANALOG MODEL OF GROUND WATER UNDERLYING EAST ST. LOUIS . &; • ■ ' v ' . V Figure 1-3. Electric analog aquifer model constructed by Illinois State Water Survey. The regular array of resistors and the two electric "pumps" shown are hard-wired into a board covered with the appropriate geologic maps. that many possible situations can be studied in great detail for a given problem. MANAGEMENT APPLICATIONS Mathematical models can and have been used to help organize the essential details of complex ground-water management problems so that reliable solutions are obtained (Holcomb Research Institute, 1976; Bachmat and others, 1978; U.S. Office of Technology Assessment, 1982; van der Heijde and others, 1985). Some principal areas where mathematical models are now being used to assist in the management of ground-water protection are: (1) appraising the physical extent, and chemical and biological quality, of ground-water reservoirs (e.g., for planning purposes), (2) assessing the potential impact of domestic, agricultural, and industrial practices (e.g., for permit issuance), (3) evaluating the probable outcome of remedial actions at waste sites, and aquifer restoration techniques generally, and (4) providing health-effects exposure estimates. The success of these efforts depends on the accuracy and efficiency with which the natural processes controlling the behavior of ground water, and the chemical and biological species it transports, are simulated (Boonstra and de Ridder, 1976; Mercer 3 Figure 1-4. Typical ground-water contamination scenario and a possible contaminant transport model grid design for its simulation. Values for natural process parameters would be specified at each node of the grid in performing simulations. The grid density is greatest at the source and at potential impact locations. 4 and Faust, 1981; Wang and Anderson, 1982). The accuracy and efficiency of the simulations, in turn, are heavily dependent on subjective judgements made by the modeler and management. In the current philosophy of ground-water protection programs, the value of a ground-water resource is bounded by the most beneficial present and future uses to which it can be put (U.S. EPA, 1984). In most instances, physical appraisals of ground-water resources are conducted within a framework of technical and economic classification schemes. Classification of entire ground- water basins by potential yield is a typical first step (Domenico, 1972). After initial identification and evaluation of a ground-water resource, strategies for its rational development need to be devised. Development considerations include the need to protect vulnerable recharge areas and the possibility of conjunctive use with available surface waters (Kazmann, 1972). Ground-water rights must be fairly administered to assure adequate supplies for domestic, agricultural, and industrial purposes. Because basinwide or regional resource evaluations normally do not provide sufficent resolution for water allocation purposes, more detailed characterizations of the properties and behavior of an aquifer, or of a subdivision of an aquifer, are usually needed. Hence, subsequent classifications may involve local estimation of net annual recharge, rates of outflow, and the pumpage which can be substained without undesirable effects. The consequences of developments which might affect ground- water quality may be estimated initially by employing generalized classification schemes; for example, classifications based on regional hydrogeologic settings have been presented (Heath, 1982; Aller and others, 1985). Very detailed databases, however, must be created and molded into useful formats before decisions can be made on how best to protect and rehabilitate ground-water resources from site- specific incidents of natural and manmade contamination. The latter are ordinary ground-water management functions which benefit from the use of mathematical models. There are other uses, however, which ought to be considered by management. The director of the International Ground Water Modeling Center discussed the role of modeling in the development of ground-water protection policies recently, noting its success in many policy formulation efforts in the Netherlands, the United States, and Israel. Nevertheless, he concluded that modeling was not widely relied upon for decision-making by managers; the primary obstacle has been an inability of modelers and program managers to communicate effectively (van der Heijde, 1985). The top executives of a leading high-tech ground-water contamination consulting firm made the same point clearly, going on to highlight the need for qualified personnel appreciative of the appropriateness, underlying assumptions, and limitations of specific models (Faust and others, 5 1981). Because these views are widely held by technical professionals, it will be emphasized herein that mathematical models are useful only within the context of the assumptions and simplifications on which they are based. If managers are mindful of these factors, however, mathematical models can be a tremendous asset in the decision- making process. MODELING CONTAMINANT TRANSPORT Associated with most hazardous waste sites is a complex array of chemical wastes and the potential for ground-water contamination. The hydrogeologic settings of these sites are usually quite complicated when examined at the scale appropriate for technical assessments and remediation efforts (e.g., 100's of feet). As a result, data acquisition and interpretation methods are needed that can examine to an unprecendented degree the physical, chemical, and biological processes that control the transport and fate of ground-water contaminants. The methods and tools that have been in use for large-scale characterizations (e.g., regional water quality studies) are applicable in concept to the specialized needs of hazardous waste site investigations; however, the transition to local-scale studies is not without scientific and economic consequences. In part, this stems from the highly variable nature of contaminant distributions at hazardous waste sites; but it also results from the limitations of the methods, tools, and theories used. Proper acknowledgement of the inherent limitations means that one must project the consequences of their use within the framework of the study at hand. Assessments of the potential for contaminant transport require interdisciplinary analyses and interpretations. Integration of geologic, hydrologic, chemical, and biological approaches into an effective contaminant transport evaluation can only be possible if the data and concepts invoked are sound. The data must be accurate, precise, and appropriate at the intended problem scale. Just because a given parameter (e.g., hydraulic conductivity) has been measured correctly at certain points with great reproducibility, is no guarantee that those estimates represent the volumes of aquifer material assigned to them by a modeler. The degree to which the data are representative, therefore, is not only relative to the physical scale of the problem, it is relative to the conceptual model to be used for interpretation efforts. It is crucial, then, to carefully define and qualify the conceptual model. In so doing, special attention should be given to the possible spatial and temporal variations of the data that will be collected. To circumvent the impossibly large numbers of measurements and samples which would be needed to eliminate all uncertainties regarding the true relationships of parameters (e.g., hydraulic conductivity) and variables (e.g., contaminant concentrations and rates of movement), more comprehensive theories are constantly 6 under development. The use of newly developed theories to help solve field problems, however, is often a frustrating exercise. Most theoretical advances call for some data which are not yet practically obtainable (e.g., chemical interaction coefficients, relative permeabilities of immiscible solvents and water, etc.). The 'state-of- the-art' in contaminant transport assessments is necessarily a compromise between the sophistication of 'state-of-the-science' theories, the current limitations regarding the acquisition of specific data, and economics. In addition, the best attempts to obtain credible data are limited by natural and anthropogenic variabilities; and these lead to the need for considerable judgement on the part of the professional. Despite these technical limitations, how well the problem is conceptualized remains the most serious concern in modeling efforts. For example, researchers recently produced dramatic evidence to show that, in spite of detailed field measurements, extrapolations of two-dimensional model results to a truly three- dimensional problem lead to wildly inaccurate projections of the actual behavior of the system under study (Molz and others, 1983). Therefore, it is incumbent on model users to recognize the difference between an approximation and a misapplication. Models should never be used strictly on the basis of familiarity or convenience; an appropriate model should always be sought. CATEGORIES OF MODELS The foregoing is not meant to imply that appropriate models exist for all ground-water problems, because a number of natural processes have yet to be fully understood. This is especially true for ground-water contaminant transport evaluations, where the chemical and biological processes are still poorly defined. For, although great advances have been made concerning the behavior of individual contaminants, studies of the interactions between contaminants are still in their infancy. Even the current understanding of physical processes lags behind what is needed, such as in the mechanics of multiphase flow and flow through fractured rock aquifers. Moreover, certain well-understood phenomena pose unresolved difficulties for mathematical formulations, such as the effects of partially penetrating wells in unconfined (water-table) aquifers. The technical-use categories of models are varied, but they can be grouped as follows (Bachmat and others, 1978; van der Heijde and others, 1985): (1) parameter identification models, (2) prediction models, (3) resource management models, and (4) data manipulation codes. 7 Parameter identification models are most often used to estimate the aquifer coefficients determining fluid flow and contaminant transport characteristics, like annual recharge (Puri, 1984), coefficients of permeability and storage (Shelton, 1982; Khan, 1986a and b), and dispersivity (Guven and others, 1984; Strecker and Chu, 1986). Prediction models are the most numerous kind of model, and abound because they are the primary tools for testing hypotheses about the problem one wishes to solve (Andersen and others, 1984; Mercer and Faust, 1981; Krabbenhoft and Anderson, 1986). Resource management models are combinations of predictive models, constraining functions (e.g., total pumpage allowed) and optimization routines for objective functions (e.g., optimization of wellfield operations for minimum cost or minimum drawdown/pumping lift). Very few of these are so well developed and fully supported that they may be considered practically useful, and there does not appear to be a significant drive to improve the situation (van der Heijde, 1984a and 1984b; van der Heijde and others, 1985). Data manipulation codes also have received little attention until recently. They are now becoming increasingly popular, because they simplify data entry (’preprocessors') to other kinds of models and facilitate the production of graphic displays (’postprocessors’) of the data outputs of other models (van der Heijde and Srinivasan, 1983; Srinivasan, 1984; Moses and Herman, 1986). Other software packages are available for routine and advanced statistics, specialized graphics, and database management needs (Brown, 1986). CHAPTER SUMMARY Mathematical models can be helpful tools to managers of ground-water protection programs. They may be used for testing hypotheses about conceptualizations and to gather a fuller understanding of important physical, chemical, and biological processes which affect ground-water resources. The possible outcomes of complex problems can be addressed in great detail, if adequate data are available. Mathematical modeling is neither simple nor impossible, but its successful application relies heavily on the expertise of the modeler and the degree of communication with management. The merits of any problem solving technique need to be judged by many criteria, the most important of which may not relate to mathematical sophistication. Qualitative judgements by prior experience, “back of the envelope’ calculations, analytical models, and other non-numerical modeling methods should be considered for a reason which deserves emphasis; the data available or obtainable may not justify extensive numerical model analyses (Javendel and others, 1984). After all, it is neither pretty nor efficient to 'use a silver sledgehammer to drive a thumbtack'. 8 CHAPTER2 ASSUMPTIONS, LIMITATIONS, AND QUALITY CONTROL INTRODUCTION There are many natural processes that affect chemical transport from point to point in the subsurface. These natural processes can be arbitrarily divided into three categories: physical, chemical, and biological (Table 2-1). Conceptually, contaminant transport in the subsurface is an undivided phenomenon composed of these processes and their interactions (Figure 2-1). At this level the transport process may be gestalt: the sum of its parts, measured separately, may not equal the whole because of interactions between the parts. In the theoretical context, a collection of scientific laws and empirically derived relationships comprise the overall transport process. The universally shared premise that underlies theoretical expressions is that there are no interactions, measureable or otherwise. Significant errors may result from the discrepancy between conceptual and theoretical appproaches. Also the simplifications of theoretical expressions used to solve practical problems can cause substantial errors in the most careful analyses. Assumptions and simplifications, however, must often be made in order to obtain mathematically tractable solutions. Because of this, the magnitude of errors that arise from each assumption and simplification must be carefully evaluated. The phrase, "magnitude of errors", is emphasized because highly accurate evaluations usually are not possible. Even rough approximations are rarely trivial exercises because they frequently demand estimates of some things which are as yet ill-defined. PHYSICAL PROCESSES Until recently, ground-water scientists studied physical processes to a greater degree than chemical or biological processes. This bias resulted in large measure from the fact that, in the past, ground-water practitioners dealt mostly with questions of adequate water supplies. As quality considerations began to dominate ground-water issues, the need for studies of the chemical and biological factors, as well as more detailed representations of the physical factors, became apparent. 9 Table 2-1. Natural processes that affect subsurface contaminant transport. PHYSICAL PROCESSES Advection (porous media velocity) Hydrodynamic Dispersion Molecular Diffusion Density Stratification Immiscible Phase Flow Fractured Media Flow CHEMICAL PROCESSES Oxidation-Reduction Reactions Radionuclide Decay Ion-Exchange Complexation Co-Solvation Immiscible Phase Partitioning Sorption BIOLOGICAL PROCESSES Microbial Population Dynamics Substrate Utilization Biotransformation Adaptation Co-metabolism There are two complimentary ways to view the physical processes involved in subsurface contaminant transport: the piezometric (pressure) viewpoint and the hydrodynamic viewpoint. Ground-water problems of yesterday could be addressed by the former, such as solving for the change in pressure head caused by pumping wells. Contamination problems of today also require detailed analyses of wellfield operations, for example, pump-and- treat plume removals. However, solutions to such problems depend principally on hydrodynamic evaluations, such as computing ground-water velocity (advection) distributions and dispersion estimates for migrating plumes. Advection and Dispersion Ground-water velocity distributions can be approximated if the variations in hydraulic conductivity, porosity, and the strength and location of recharge and discharge sources can be estimated. While there are several field and laboratory methods for estimating hydraulic conductivity, these are not directly comparable because different volumes of aquifer material are 10 Relative Concentration _► Relative Concentration DISTANCE FROM CONTINUOUS CONTAMINANT SOURCE Figure 2-1. The Influence of natural processes on levels of contaminants downgradient from continuous and slug-release sources. 11 affected by different tests. Laboratory permeameter tests, for example, obtain measurements from small core samples and thus give point value estimates. These tests are generally reliable for consolidated rock samples, such as sandstone, but can be highly unreliable for unconsolidated samples, such as sands, gravels, and clays. Pumping tests give estimates of hydraulic conductivity that are averages over the entire volume of aquifer subject to the pressure changes induced by pumping. These give repeatable results, but they are often difficult to interpret. Tracer tests are also used to estimate hydraulic conductivity in the field, but are difficult to conduct properly. Regardless of the estimation technique used, the best that can be expected is order-of-magnitude estimates for hydraulic conductivity at the field scale appropriate for site-specific work. Conversely, porosity estimates that are accurate to better than a factor of two can be obtained. Estimation of the strength of nonpoint sources of recharge to an aquifer, such as infiltrating rainfall and leakage from other aquifers, is another order-of-magnitude effort. Similarly, nonpoint sources of discharge, such as aquifer losses to gaining streams, are difficult to quantify. Estimation of the strength of point sources of recharge or discharge (injection or pumping wells) can be highly accurate. Consequently, it is not possible to generalize the quality of velocity distributions. They may be accurate to within a factor of two for very simple aquifers, but are more often accurate to an order-of-magnitude only. This situation has changed little over the past 20 years because better field and laboratory methods for characterizing velocity distributions have not been developed. This, however, is not the primary difficulty associated with defining the advective part of contaminant transport in the subsurface. The primary difficulty is that field tests for characterizing the physical parameters that control velocity distributions are not incorporated into contamination investigations on a routine basis. The causes seem to be: a perception that mathematical models can *back-out' an approximation of the velocity distribution (presumably eliminating the need for field tests); unfamiliarity with field tests by many practitioners; and a perception that field tests are too expensive. A more field oriented approach is preferable because the non¬ uniqueness of modeling results has been amply demonstrated, and this leads to uncertain decisions regarding the design of remedies. Dispersion estimates are predicated on velocity distribution estimates and their accuracy is therefore directly dependent on the accuracy of the estimated hydraulic conductivity distribution. Tracer tests have been the primary method used to determine dispersion coefficients until recently. Presently there are suggestions that any field method capable of generating a detailed understanding of the spatial variability of hydraulic conductivity, which in turn could give an accurate representation of the velocity 12 distribution, may be used to derive estimates of dispersion coefficients. The manner in which data from field tests should be used to derive estimates of dispersion coefficients, however, is a controversial issue. There are both deterministic and stochastic schools of thought, and neither has been conclusively demonstrated in complex hydrogeological settings. Complicating Factors Certain subtleties of the spatial variability of hydraulic conductivity must be understood because of its key role in the determination of velocity distributions and dispersion coefficients. Hydraulic conductivity is also known as the coefficent of permeability because it is comprised of fluid factors as well as the intrinsic permeability of the stratum in question. This means that a stratum of uniform intrinsic permeability (which depends strictly on the arrangement of its pores) may have a wide range of hydraulic conductivity because of differences in the density and viscosity of fluids that are present. The result is a dramatic downward shift in local flow directions near plumes that have as little as a \% increase in density relative to uncontaminated water. Such density contrasts frequently occur at landfills and waste impoundments. It is often necessary to correct misimpressions of the direction of a plume because density considerations were not addressed. Many solvents and oils are highly insoluble in water, and may be released to the subsurface in amounts sufficient to form a separate fluid phase. Because that fluid phase will probably have viscosity and density different from freshwater, it will flow at a rate and, possibly, in a direction different from that of the freshwater with which it is in contact. If an immiscible phase has density approximately the same or less than that of ground water, this phase will not move down past the capillary fringe of the ground water. Instead, it will flow along the top of the capillary fringe in the direction of the maximum water-level elevation drop. If the density of an immiscible phase is substantially greater than the ground water, the immiscible phase will push its way into the ground water as a relatively coherent blob. The primary direction of its flow will then be down the dip of the first impermeable stratum encountered. There is a great need for better means of characterizing such behavior for site-specific applications. Currently, estimation methods are patterned after multiphase oil reservoir simulators. One of the key extensions needed is the ability to predict the transfer of trace levels of contaminants from the immiscible fluid to ground water, such as xylenes from gasoline. Anisotropy is a subtlety of hydraulic conductivity which relates to structural trends of the rock or sediments of which an aquifer is composed. Permeability and hydraulic conductivity are directionally dependent in anisotropic strata. When molten material from deep underground crystallizes to form granitic or 13 basaltic rocks, for instance, it forms cleavage planes which may later become the preferred directions of permeability. Marine sediments accumulate to form sandstone, limestone, and shale sequences that have much less vertical than horizontal permeability. The seasonal differences in sediments that accumulate on lakebeds, and the stratification of grain sizes deposited by streams as they mature, give rise to similar vertical-to- horizontal anisotropy. Streams also cause anisotropy within the horizontal plane, by forming and reworking their sediments along a principal axis of movement. These structural variations in permeability would be of minimal concern except that ground water does not flow at right angles to water-level elevation contours under anisotropic conditions. Instead, flow proceeds along oblique angles, with the degree of deviation from a right-angle pathway proportional to the amount of anisotropy. This fact is all too often ignored and the causes again seem to be a reluctance to conduct the proper field tests, combined with an over-reliance on mathematical modeling. If the pathways created by cleavage planes and fractures begin to dominate fluid flow through a subsurface stratum, the directions and rates of flow are no longer predictable by the equations used for porous rock and sediments. There have been a number of attempts to represent fractured flow as an equivalent porous medium, but these tend to give poor predictions when major fractures are present and when there are too few fractures to guarantee a minimum degree of interconnectedness. Other representations that have been studied are various dual porosity models, in which the bulk matrix of the rock has one porosity and the fracture system has another. Further development of the dual porosity approach is limited by the difficulty in determining a transfer function to relate the two different porosity schemes. Research in this area needs to be accelerated because there is a great likelihood of fractured flow in just those situations commonly believed to be the most suitable for disposal of hazardous wastes, such as building landfills on 'impermeable' bedrock. Considerations for Predictive Modeling Equations for the combined advection-dispersion process are used to estimate the time during which a nonreactive contaminant will travel a specific distance, the pathway it will travel, and its concentration at any point. The accuracy of most predictions is only fair for typical applications, because of the complexity of the problems and the scarcity of site-specific hydrogeologic data. The lack of such data can be improved on with much less effort than is commonly presumed, especially when the cost of another round of chemical sampling is compared with the costs of additional borings, core retrievals, geophysical logging, or permeability testing. 14 Equations that assume a nonreactive contaminant have limited usefulness, because most contaminants react with other chemical constituents in subsurface waters and with subsurface solids in a manner that affects the rate at which they travel. Nevertheless, nonreactive advection-dispersion equations are often used to generate ’worst-case' scenarios, on the presumption that the maximum transport velocity is obtained (equal to that of pure water). This may not be as useful as it first seems. Remedial action designs require detailed breakdowns of which contaminants will arrive at extraction wells and when, how long contaminants will continue their slow release from subsurface solids, and whether the contaminants will be transformed into other chemical species by chemical or biological forces. To address these points, special terms must be added to the advection-dispersion equations. CHEMICAL PROCESSES As difficult as the foregoing complications may be, predicting how chemical contaminants move through the subsurface is a relatively trivial matter when the contaminants behave as ideal, nonreactive substances. Unfortunately, such behavior is limited to a small group of chemicals. The actual situation is that most contaminants will, in a variety of ways, interact with their environment through biological or chemical processes. This section focuses on the dominant chemical processes that may ultimately affect the transport behavior of a contaminant. As with the physical processes previously discussed, some of the knowledge of chemical processes has been translated into practical use in predictive models. However, the science has, in many instances, advanced well beyond what is commonly practiced. Furthermore, there is considerable evidence that suggests that numerous undefined processes affect chemical mobility. Most of the deviation from ideal nonreactive behavior of contaminants relates to their ability to change physical form by energetic interactions with other matter. The physical-chemical interactions may be grouped into: alterations in the chemical or electronic configuration of an element or molecule, alterations in nuclear composition, the establishment of new associations with other chemical species, and interactions with solid surfaces. ChemicaL/Electronic Alterations The first of these possible changes is typified by oxidation- reduction or redox reactions. This class of reactions is especially important for inorganic compounds and metallic elements because the reactions often result in changes in solubility, complexing capacity, or sorptive behavior, which directly impact on the mobility of the chemical. Redox reactions are reasonably well understood, but there are practical obstacles to applying the known science because 15 it is difficult to determine the redox state of the aquifer zone of interest and to identify and quantify the redox-active reactants. Hydrolysis, elimination, and substitution reactions that affect certain contaminants also fit into this classification. The chemistry of many organic contaminants has been well defined in surface water environments. The influence of unique aspects of the subsurface, not the least of which is long residence time, on such transformations of important organic pollutants is currently under investigation. There is also a need to investigate the feasibility of promoting in-situ abiotic transformations that may enhance the potential for biological mineralization of pollutants. Nuclear Alterations Another chemical process interaction, which results in internal rearrangement of the nuclear structure of an element, is well understood. Radiodecay occurs by a variety of routes, but the rate at which it occurs is always directly proportional to the number of radioactive atoms present. This fact seems to make mathematical representation in contaminant transport models quite straightforward because it allows characterization of the process with a unique, well defined decay constant for each radionuclide. A mistake that is often made when the decay constant is used in models involves the physical form of the reactant. If the decay constant is applied to the fluid concentrations and no other chemical interactions are allowed, then incorporation of the constant into the subroutine which computes fluid concentrations will not cause errors. If the situation being modeled involves chemical interactions such as precipitation, ion-exchange, or sorption, which temporarily remove the radionuclide from solution, then it is important to use a second subroutine to account for the non-solution-phase decay of the radionuclide. Chemical Associations The establishment of new associations with other chemical species is not as well understood. This category includes ion- exchange, complexation, and co-solvation. The lack of understanding derives from the nonspecific nature of these interactions which are, in many instances, not characterized by the definite proportion of reactants to products (stoichiometry) typical of redox reactions. While the general principles and driving mechanisms by which these interactions occur are known, the complex subsurface matrix in which they occur provides many possible outcomes and renders predictions uncertain. Ion-exchange and complexation reactions heavily influence the mobility of metals and other ionic species in the subsurface in a reasonably predictable fashion. Their influence on organic contaminant transport, however, is not well understood. Based on 16 studies of pesticides and other complex organic molecules, natural organic matter (such as humic and fulvic materials) can complex and thereby enhance the apparent solubility and mobility of synthetic organic chemicals. Research is needed to define the magnitude of such interactions, not only with naturally occurring organic molecules but also with man-made organics present in contaminated environments. Research is also needed to determine if these complexes are stable and liable to transport through the subsurface. Examination of the degree to which synthetic organic chemicals complex toxic metals is also necessary. There is no theoretical objection to such interactions, and there is ample evidence that metals are moving through the subsurface at many waste sites. Co-solvation occurs when another solvent is in the aqueous phase at concentrations that enhance the solubility of a given contaminant. This occurs in agricultural uses, for example, where highly insoluble pesticides and herbicides are mixed with organic solvents to increase their solubility in water prior to field application. There is every reason to expect similar behavior at hazardous waste sites, where a variety of solvents are typically available. At present, prediction of the extent of the solubility increases that might occur at disposal sites in the complex mixture of water and organic solvents is essentially impossible. Researchers have started examining co-solvation as an influence on pollutant transport, by working on relatively simple mixed solvent systems. This research will be extremely useful, even if the results are limited to a qualitative appreciation for the magnitude of the effects. At the extreme, organic solvents in the subsurface may result in a phase separate from the aqueous phase. In addition to movement of this separate phase through the subsurface, contaminant mobility that involves partitioning of organic contaminants between the organic and aqueous phases must also be considered. The contaminants will move with the organic phase and will, depending on aqueous phase concentrations, be released into the aqueous phase to a degree roughly proportional to their octanol-water partition coefficients. An entire range of effects is possible, from increasing to slowing the mobility of the chemical in the subsurface relative to its migration rate in the absence of the organic phase. The equilibrium partitioning process increases the total volume of ground water affected by contaminants, by releasing apportion of the organic phase constitutents into adjacent waters. It may also interfere with transformation processes by affecting pollutant availability for reaction, or by acting as a biocidal agent to the native microflora. 17 Surface Interactions Of those interactions that involve organic chemicals in the environment, none has been as exhaustively studied as sorption. Sorption studies relate, in terms of a sorption isotherm, the amount of contaminant in solution to the amount associated with the solids. Most often the sorption term in transport models is estimated for simplicity from the assumption that the response is linear. This approximation can produce serious mass balance errors. Typically, the contaminant mass in the solution phase is underestimated and contaminant retardation is thereby overestimated. In practical applications, this means that high contaminant levels can be detected at a monitoring well long before they were predicted. To resolve the discrepancy between predicted and actual transport, most practitioners arbitrarily adjust some other poorly- characterized model parameter, for example, dispersion. This leads to the creation of a model that does not present various natural process influences in proper perspective. The predictions from such models are likely to be qualitatively, as well as quantitatively, incorrect. More widespread consideration should be given to accurate representation of non-linear sorption, particularly in transport modeling at contaminated sites. The time dependency of the sorption process is a related phenomenon that has also been largely ignored in practical applications of sorption theory. Most models assume that sorption is instantaneous and completely reversible. A growing body of evidence argues to the contrary, not only for large organic molecules in high-carbon soils and sediments, but also for solvent molecules in low-carbon aquifer materials. Additionally, there must be some subtle interplay between sorption kinetics and ground-water flow rates which gains significance in pump-and-treat remediation efforts, where flow rates are routinely substantially increased. Constant pumpage at moderate-to-high flow rates may not allow contaminants that are sorbed to solids sufficient times of release to increase solution concentrations to maximum (equilibrium) levels prior to their removal from the aquifer. Hence, treatment costs may rise substantially due to the prolonged pumping required to remove all of the contaminants and due to the lowered efficiency of treatment of the less contaminated pumped waters. Evidence from Superfund sites and ongoing research activities suggests that contaminant association with a solid surface does not preclude mobility. In many instances, especially in glacial tills that contain a wide distribution of particle sizes, fine aquifer materials have accumulated in the bottom of monitoring wells. Iron-based colloids have been identified in ground water downgradient from a site contaminated with domestic wastewater. If contaminants can associate with these fine particles, their mobility through the subsurface could be markedly enhanced. To determine the 18 significance of particle transport to pollutant movement, studies must be performed at such contaminated sites. Although knowledge about chemical processes that function in the subsurface has been significantly expanded in recent years, this information is only slowly finding its way into practical interpretations of pollutant transport at contaminated sites. Evidence from field sites suggests that much remains to be learned about these processes. BIOLOGICAL PROCESSES Many contaminants that enter the subsurface environment are biologically reactive. Under appropriate circumstances they can be completely degraded to harmless products. Under other circumstances, however, they can be transformed to new substances that are more mobile or more toxic than the original contaminant. Quantitative predictions of the fate of biologically reactive substances are at present very primitive, particularly compared to other processes that affect pollutant transport and fate. This situation resulted from the ground-water community's choice of an inappropriate conceptualization of the active processes: subsurface biotransformations were presumed to be similar to biotransformations known to occur in surface water bodies. Only very recently has detailed field work revealed the inadequacy of the traditional view. Surface Water Model Analogy As little as five years ago ground-water scientists considered aquifers and soils below the zone of plant roots to be essentially devoid of organisms capable of transforming contaminants. As a result, there was no reason to include terms for biotransformations in transport models. Recent studies have shown that water-table aquifers harbor appreciable numbers of metabolically active microorganisms, and that these microorganisms frequently can degrade organic contaminants. It became necessary to consider biotransformation in transport models. Unfortunately, many ground-water scientists adopted the conceptual model most frequently used to describe biotransformations in surface waters. The presence of the contaminant was assumed to have no effect on microorganism populations that degrade it. It was also assumed that contaminant concentration does not influence transformation kinetics, and that the capacity to transform the contaminant is uniformly distributed throughout the body of water under study. These assumptions are often appropriate for surface waters: contaminant concentration is usually too low and the residence time too short to allow adaptation of the microbial community to the contaminant, and the organisms that are naturally pre-adapted to the contaminant are mixed throughout the water body by 19 turbulence. Consequently, utilization kinetics can conveniently be described by simple first-order decay constants. In surface waters these constants are usually obtained by monitoring contaminant disappearance in water samples. Ground-Water Biotransformations These circumstances rarely apply to biotransformation in ground water. Contaminant residence time is usually long, at least weeks or months, and frequently years or decades. Further, contaminant concentrations that are high enough to be of environmental concern are often high enough to elicit adaptation of the microbial community. As a result, the biotransformation rate of a contaminant in the subsurface environment is not a constant, but increases after exposure to the contaminant in an unpredictable way. Careful field work has shown that the transformation rates in aquifers of typical organic contaminants, such as alkylbenzenes, can vary as much as two orders of magnitude over a meter vertically and a few meters horizontally. This surprising variability in transformation rates is not related in any simple way to system geology or hydrology. It is difficult to determine transformation rates in subsurface material. Most microbes in subsurface material are firmly attached to solid surfaces; usually less than \% of the total population is truly planktonic. As a result, the microbes in a ground-water sample grossly underrepresent the total microbial population in the aquifer. Thus, contaminant disappearance kinetics in a ground-water sample do not represent the behavior of the material in the aquifer. It is therefore necessary to do microcosm studies with samples representative of the entire aquifer system - a formidable technical challenge. A Ground-Water Model These concerns have prompted re-examination of assumptions about biotransformation implicit or explicit in traditional modeling approaches, with the realization that no one qualitative description of biotransformation can be universally applicable. Field experience has shown that the relationships that describe the biological fate of contaminants actually change within aquifers in response to geochemical constraints on microbial physiology. Rather than describing biotransformation with a continuous function applicable at all points in the aquifer, it may be more realistic to examine key geochemical parameters and to use that information to identify the relationship for biotransformation that applies at any particular point. These key parameters could include the contaminant concentration, oxygen or other electron-acceptor concentration, redox state, pH, toxicity of the contaminant or co-occurring 20 materials, and temperature. One such model has been evaluated in the field. The model described an alkylbenzene and polynuclear aromatic hydrocarbon plume in a shallow water-table aquifer. Microcosm studies showed that organisms in the aquifer had adapted to these contaminants, and would degrade them very rapidly when oxygen was available. As a result of this adaptation, the hydrocarbon biodegradation rate was not controlled by any inherent property of the organisms. Rather, the physical transport processes of diffusion and dispersion seemed to dominate by controlling oxygen availability to the plume. Because the biotransformation rate was controlled by physical processes, the actual model was very simple. Oxygen and hydrocarbon transport were simulated as conservative solutes using the U.S. Geological Survey method-of-characteristics code. A subroutine examined oxygen and hydrocarbon concentrations at each node (point located at interesecting grid lines of the model) and generated new concentrations based on oxidative metabolism stoichiometry. When the model was used to simulate the growth of the plume over time, it illustrated an important property of many such plumes. The plume grew until the rate of admixture of oxygen balanced the rate of release of hydrocarbons from the source. Afterward, the extent of the plume was at steady-state (stopped growing). The body of field experience that can be drawn on to properly assign laws for biotransformation is growing rapidly. Transport- limited kinetics may commonly apply to releases of petroleum hydrocarbons and other easily degradable materials such as ethanol or acetone in oxygenated ground water. On the other hand, materials that can support a fermentation may follow first-order kinetics. Unfortunately, many important biotransformations in ground water are still mysteries. Rapid field methods to determine if adaptation has occurred at a site are needed. Tools to predict whether adaptation can be expected, and to estimate the time required for adaptation if it does occur, are also needed. For systems that are limited by transport processes, field methods to estimate the aquifer processes that control mixing, such as transverse dispersion and exchange processes across the water table, are required. For systems that are limited by the intrinsic biotransformation rate, new laboratory test methods (possibly, improved microcosms) that will provide reliable estimates of the kinetic parameters are required. In addition to being sufficiently accurate and precise, these new methods should provide estimates that are truly representative of the hydrologic unit being simulated. Because contaminants typically have long residence times in aquifers, slow transformation rates can have environmental significance. The test methods should 21 therefore be sufficiently sensitive to measure transformation rates that are significant in the hydrologic context being simulated. Finally, there is a need for models that go beyond simple prediction of contaminant concentrations at points in the aquifer, and forecast the concentrations produced by production wells. ANALYTICAL AND NUMERICAL MODELS One of the more subtly involved decisions that must be made is whether to use an analytical or a numerical model to solve a particular problem. Analytical models provide exact solutions, but many simplifying assumptions must be made for the solutions to be tractable. This places a burden on the user to test and justify the underlying assumptions and simplifications (Javendel et al, 1984). For example, the Theis equation is an analytical expression which is used to predict the piezometric head changes for pumping or injection wells in confined aquifers (Freeze and Cherry, 1979; Todd, 1980): s = [Q/(4*n*T)] * [-0.5772 - ln(u) + u - (u 2 /(2*2!)) 4- ( U 3/(3*3!)) - (u 4 /(4*4!))..] where: 's' is the change in piezometric head, 'Q' is the flowrate of the well, T' is the transmissivity of the aquifer, and V = (r 2 * s) / (4 * T * t); V is the radial distance from the well, ‘S’ is the storage coefficient of the aquifer, and 't' is the length of time the well has been operating. Here the principal assumptions are (Lohman, 1972): (1) the aquifer is homogenous and isotropic, (2) the aquifer is of infinite areal extent, relative to the effects of the well (no nearby boundaries), (3) the well is screened over the entire saturated thickness of the aquifer, (4) the saturated thickness of the aquifer does not vary as a result of the operation of the well, (5) the well has an infintesimal diameter so that waters in storage in the casing represent an insignificant volume, and (6) water is removed from or injected into the aquifer with an instantaneous change in the piezometric head. Evaluation of the foregoing equation, which incorporates an infinite Taylor series representing the well function integral, can be accomplished graphically using type curves (Walton, 1962; Lohman, 1972). Alternatively, a simplification can be made so that the Theis equation is directly solvable (Cooper and Jacob, 1946). This is done by dropping all terms in the Taylor series with powers greater than one, and is strictly valid for cases where 'u' has a value less than 0.01 (e.g., Figure 2-2). Physically, this corresponds to a limitation on the 22 predictive power of the modified Theis equation; head changes predicted at locations far from the well are inaccurate, except for long durations of pumpage (i.e., approaching equilibrium or steady- state conditions). Numerical models are much less burdened by these assumptions and are therefore inherently capable of addressing more complicated problems, but they require significantly more data and their solutions are inexact (numerical approximations). For example, the assumptions of homogeneity and isotropicity are unnecessary due to the ability to assign point (nodal) values of transmissivity and storage. Likewise, the capacity to incorporate complex boundary conditions obviates the need for the 'infinite area extent' assumption. There are, however, difficult choices facing the user of numerical models; i.e. time steps, spatial grid designs, and ways to avoid truncation errors and numerical oscillations must be chosen (Remson and others, 1971; Javendel and others, 1984). These choices, if improperly made, may result in errors unlikely to be made with analytical approaches (e.g.; mass imbalances, incorrect velocity distributions, and grid-orientation effects). QUALITY CONTROL These latter points signify a greater need for quality control measures when contemplating the use of numerical models. Three levels of quality control have been suggested previously (Huyakorn and others, 1984): (1) validation of the model's mathematics by comparison of its output with known analytical solutions to specific problems, (2) verification of the general framework of the model by successful simulation of observed field data, and (3) benchmarking of the model's efficiency in solving problems by comparison with other models. These levels of quality control address the soundness and utility of the model alone, and do not treat questions of its application to a specific problem. Hence, at least two additional levels of quality control appear justified: (4) critical review of the problem conceptualization to ensure that the modeling effort considers all physical, chemical, and biological processes which may affect the problem, and (5) evaluation of the specifics of the application; e.g., appropriateness of the boundary conditions, grid design, time steps, etc. Validation of the mathematical framework of a numerical model is deceptively simple. The usual approach for ground-water flow models involves a comparison of drawdowns predicted by the Theis 23 Flowrate = 100,000cu. ft./d Transmissivity = 10,000 sq. ft./d Storage coefficient = 0.0001 Drawdown, ft 0 200 400 600 800 1,000 Radius of Observation, ft Figure2-2. Example of plots prepared with the Jacob's approximation of the Theis analytical solution to well hydraulics in an artesian aquifer. 24 analytic solution to those obtained by using the model, such as depicted in Figure 2-3. The 'deceptive' part is the foreknowledge that the Theis solution can treat only a very simplified situation as compared with the scope of situations addressable by the numerical model. In other words, analytical solutions cannot test most of the capabilities of the numerical model in a meaningful way; this is particularly true with regard to simulation of complex aquifer boundaries and irregular chemical distributions. Field verification of a numerical model consists of first calibrating the model using one set of historical records (e.g., pumping rates and water levels from a certain year), and then attempting to predict the next set of historical records. In the calibration phase, the aquifer coefficients and other model parameters are adjusted to achieve the best match between model outputs and known data; in the predictive phase, no adjustments are made (excepting actual changes in pumping rates, etc.). Presuming that the aquifer coefficients and other parameters are known with sufficient accuracy, a mismatch means that either the model is not correctly formulated or that it does not treat all of the important phenomena affecting the situation being simulated (e.g., does not allow for leakage between two aquifers when this is actually occurring). Field verification exercises usually lead to additional data gathering efforts, because existing data for the calibration procedure are often insufficient to provide unique estimates of key parameters. This means that a 'black box' solution may be obtained, which may be good only for the records used in the calibration. For this reason, the blind prediction phase is an essential check on the uniqueness of the parameter values used. In this regard, field verification of models using datasets from controlled research experiments may be much more achievable practically. Benchmarking routines to compare the efficiency of different models in solving the same problem have only recently become available (Ross and others, 1982; Huyakorn and others, 1984). Much more needs to be done in this area, because some unfair perceptions continue to persist regarding the ostensibly greater utility of certain modeling techniques. For example, it has been said many times that finite element models (FEM's) have an inherent advantage over finite difference models (FDM's) in terms of the ability to incorporate irregular boundaries (Mercer and Faust, 1981); the number of points (nodes) which must be used by FEM's is considerably less due to the flexible nodal spacings that are allowed. Benchmarking routines, however, show that the much longer computer time required to evaluate FEM nodes causes there to be little, if any, cost advantage for simulations of comparable accuracy. 25 Drawdown, ft Flowrate = 100,000cu. ft./d Duration of Pumpage, days Figure 2-3. Mathematical validation of a numerical method of estimating drawdown, by comparison with an analytical solution. 26 CHAPTER SUMMARY Mathematical models of subsurface contaminant transport are simplified representations of reality that incorporate a number of theoretical assumptions about the natural processes governing the transport and fate of contaminants. In order for solutions to be made tractable, further simplifications are made in applications of theory to practical problems. Hence, mathematical models produce representions that are not faithful to the true complexities of real world situations. This limitation can be partially compensated for, however, by detailed mathematical testing to determine the magnitude of errors generated by the assumptions and simplifications involved. The application of mathematical models is also subject to considerable error in practical situations due to the lack of appropriate field determinations of natural process parameters. This source of error is not adequately addressed by sensitivity analyses or stochastic techniques for estimating uncertainty, contrary to popular beliefs. Rather, the high degree of hydrogeologic, chemical, and microbiological complexity typically present in field situations demands characterization of the magnitude of influences from various natural processes by actual field determinations. Both the mathematics describing models and the parameter input to models must be subjected to rigorous quality control procedures. Otherwise, results from field applications of models are likely to be qualitatively, as well as quantitatively, incorrect. Quality control methodologies must recognize that accuracy and precision determinations are insufficient measures of quality. The correctness of the problem conceptualization, and the representativeness of parameter values are essential quality considerations. 27 ' CHAPTERS APPLICATIONS IN PRACTICAL SETTINGS STEREOTYPICAL APPLICATIONS As stated in preceeding sections, models are simplifications of reality that may or may not faithfully simulate the actual situation. Typically, attempts are made to mimic the effects of hydrogeologic, chemical, and biological processes in practical applications of models. These almost always involve idealizations of known or suspected features of the problem on hand. For example, the stratification of alluvial, fluvial, and glacial deposits may be assumed to occur in uniformly thick layers, despite the great variability of stratum thicknesses found in actual settings. Large blocks of each stratum are assumed to be homogeneous. Sources of chemical input are commonly assumed to have released contaminants at constant rates over the seasons and years of operational changes that the sources were active. The areal distribution of rainfall and the actual schedules of pumpage from production wells are also artificially homogenized in most modeling exercises. All these idealizations are made necessary by a lack of the appropriate historical records and field-derived parameter estimates, and all reduce the reliability of predictions made with models. The degree of usefulness of a model is therefore directly dependent on the subjective judgements that must be made in data collection and preparation efforts prior to attempting mathematical simulations. This is true not only in a quantitative sense, but also in a qualitative sense because it is the data gathering phase of a project that begets the conceptualization on which the model will be based. REAL-WORLD APPLICATIONS To illustrate this point, the highlights of two very different contamination problems will be described. The first involves a relatively limited contamination incident arising from a very small source and having few contaminants. The second involves a major contamination incident arising from the operation of a chemical reprocessing facility that handled dozens of different contaminants in large amounts. The common theme that is shared by the two cases, as should also apply to virtually all cases, is one of seeking to define the relative influences of natural processes affecting contaminant transport in order to optimize the assessment 29 and remediation of the problem. It is the validity of the conceptual model of what is happening at these sites that is most important, not the application of a particular mathematical model. Field Example No. 1 The Lakewood Water District in Lakewood, Washington operates a number of wells for drinking water supply purposes. Some of the wells operated by the District, such as the two primary wells at its Ponders' Corner site (Figure 3-1), have been contaminated with low levels of volatile organic chemicals (VOC's). During the course of the investigations at the Ponder's Corner site, a number of cost-saving sampling alternatives were chosen. These related principally to the field use of a portable gas chromatograph (Organic Vapor Analyzer) for the screening of water samples and soil extracts taken while drilling monitoring wells, and to the use of selective analyses (VOCs only) of ground-water samples when initial results showed only a narrow group of contaminants to be present. The lowered analytical costs, in part, allowed for increased expenditures for geotechnical characterization of the site (Wolf and Boateng, 1983). The geotechnical efforts, particularly the pump tests that were conducted, led to a realization that the source of the contaminants was to be found regionally downgradient (Keely and Wolf, 1983). The pumping strength of the water-supply wells, when operating, was sufficient to pull contaminants over 400 feet back upgradient of the wells they affect. This behavior was somewhat unexpected. A unique feature of the field investigation was the taking of the ground-water samples from the pumping wells concurrent with drawdown measurements obtained during pump tests (Keely, 1982). The pump tests yielded estimates of local transmissivity and storage coefficients. They also confirmed the presence of a major aquifer boundary nearby; a buried glacial till drumlin just west of the site parallels the general direction (northwest) of regional flow. The pump tests clearly showed some anisotropy of the sediments as well; drawdown contours produced an elliptical cone of drawdown, the major axis of which was aligned with the regional flow to the north. This information resulted in modifications to the original plans, which called for drilling and constructing several monitoring wells west of the site. Instead, more monitoring wells were drilled along the north-south axis. Chemical analysis of the samples taken concurrent with drawdown measurements formed a time-series of contaminant concentrations that provided a clue to where the contaminant source was located (Keely, 1982). The time-series showed that the well nearest the downgradient edge of the wellfield was exposed to increasing contaminant levels as pumping continued, whereas the upgradient pumping well remained largely unaffected (Keely and Wolf, 1983). 30 South Tacoma Way/Pacific Highway Figure 3-1. Location map for Lakewood Water District wells contaminated with volatile organic chemicals. 31 The hydrogeologic parameter estimates obtained from the pump tests strengthened the conceptualization of contaminants being drawn back against the regional flow because the capture zones of the pumping wells were sufficiently distorted by the local anisotropy to more than encompass the contaminant source. Without considering the anisotropic bias along the regional flow path, the estimated boundaries of the capture zone for either of the two wells marginally reached the distance to the contaminant source. The mechanism by which the two wells became contaminated seemed to be understood from a hydraulic point of view (Figure 3-2), but the chemical information did not seem to provide a consistent picture. The source of contamination, a septic tank at a dry-cleaning facility, was found to have received large amounts of tetrachloroethylene and trichloroethylene, but no known amounts of cis- or trans-dichloroethylene; whereas the contaminated wells had relatively high concentrations of dichloroethylene. Initially it was thought that other sources might also be present and would explain the high concentrations of dichloroethylene. However, it soon became clear that recent research results regarding the potential for biotransformation of tetrachloroethylene and trichloroethylene (Wilson and McNabb, 1981) would more satisfactorily explain the observations. Simulations of this kind of problem could be adequately performed only by contaminant transport models capable of incorporating the effects of the pumping wells on the regional flow field. More sophisticated approximations would also require the ability to account for the anisotropic and heterogeneous character of the site, the retardation of the VOC's by sorption, and their possible biotransformations. Given the highly localized nature of the contaminant source and limited extent of the plume, however, there was insufficient justification for pursuing such efforts. The resolution of the problem was possible by relatively simple source removal techniques (excavation of the septic tank and elimination of discharges). Field Example No. 2 Similar experience with special use of geotechnical methods and state-of-the-art research findings occurred at the 20-acre Chem- Dyne solvent reprocessing site in Hamilton, Ohio (Figure 3-3). During operation of the site (1974-1980), poor waste handling practices such as onsite spillage of a wide variety of industrial chemicals and solvents, direct dischage of liquid wastes to a stormwater drain beneath the site, and mixing of incompatible wastes were engaged in routinely. These caused extensive soil and ground-water contamination, massive fish kills in the Great Miami River, and major onsite fires and explosions, respectively. The stockpiling of liquid and solid wastes resulted in thousands of badly 32 0 Well H-2 Well H-1 Water Table m — Figure 3-2. Schematic illustrating the mechanism by which a downgradient source may contaminate a production well, and by which a second well may isolate the source through hydraulic interference. 33 lcorroded eaking drums that posed a long-term threat to the environment (CH 2 M-Hill, 1984). The seriousness of the ground-water contamination problem became evident during the initial site survey (1980-1981), which included the construction and sampling of over twenty shallow monitoring wells (Ecology and Environment, 1982). The initial survey indicated that the contaminant problem was much more limited than was later shown to be the case (Roy F. Weston Inc., 1983, CH 2 M-Hill, 1984). A good portion of the improvement in delineating the plume was brought about by an improved understanding of the natural processes controlling transport of contaminants at the site. The initial site survey indicated that ground-water flow was generally to the west of the site, toward the Great Miami River, but that a shallow trough paralleled the river itself as a result of weak and temporary stream influences. The study concluded that contaminants would be discharged from the aquifer into the river (Ecology and Environment, 1982). That study also concluded that the source was limited to highly contaminated surface soils, and that removal of the uppermost three feet of the soil would essentially eliminate the source. That conclusion was, however, based on faulty soil sampling procedures. The soil samples that were taken were not preserved in air-tight containers, so that most of the VOC's leaked out prior to analysis. That the uppermost soil sample showed high VOC levels is probably explained by the co-occurrence of viscous oils and other organic chemicals that may have served to entrap the VOC's. The more viscous and highly retarded chemicals did not migrate far enough into the vertical profile to exert a similar influence on samples collected at depths greater than a few feet. Subsequent studies of the site corrected these misinterpretations by producing data from proper soil samplings and by incorporating a much more detailed characterization of the fluvial sediments and the natural flow system. In those studies vertical profile characterizations were obtained from each new borehole drilled by continuous split-spoon samples of subsurface solids; and clusters of vertically-separated monitoring wells were constructed. The split-spoon samples helped to confirm the general locations of interfingered clay lenses and clearly showed the high degree of heterogeneity of the sediments (Figures 3-4 and 3-5). While an extensive network of shallow wells confirmed earlier indications of general ground-water flow toward the river (Figure 3- 6), the clusters of vertically separated wells revealed that dramatic downward gradients existed adjacent to the Great Miami River (Figure 3-7). This finding indicated that the migrating plume would not be discharged to the river, but would instead flow under the river. 34 I Figure 3-3. Location map for Chem-Dyne Superfund Site. 35 SSE T3 5 (/) 6 Cn ■o c <0 CO 0) > rt> o> £ m u <5 O' I L. cn >H 0> y ra U o to CTl in co in r- in to in m in ^r in cn in CM in in a in (1SH *»as») NOI1VA333 Figure 3-4. Chem-Dyne geologic cross-section along NNW-SSE axis. 36 Clayey silt, silty clay Sandy gravel, gr. sand ENE a CD a a a CD CD CD CD a CD C3 CD CD r- ID m CO C\J »— CD lO in in in in in in in in in in (1SW NOI1VA313 Figure 3-5. Chem-Dyne geologic cross-section along WSW-ENE axis. 37 Figure 3-6. Shallow well ground-water contour map for Chem- Dyne. Flow is generally to the river (west) and down the valley (southwest). 38 SHALLOW WELL GROUND SURFACE INTERMEDIATE WELL DEEP WELL Figure3-7. Typical arrangement of clustered, vertically- separated wells installed adjacent to Chem-Dyne and the Great Miami River. 39 The presence of major industrial wells on the other side of the river supported this conclusion. The plume would be drawn to greater depths in the aquifer by the locally severe downward gradient, but whether the industrial wells would actually capture the plume could not be determined. That determination would require careful evaluation of the hydrogeologic features beneath the river; something that has not been attempted because of the onset of remedial actions designed to stop the plume from reaching the river. The field characterization efforts did, however, include the performance of a major pump test so that the hydrogeologic characteristics of the contaminated portion of the aquifer could be estimated. The pump test was difficult to arrange, because the pumping well had to be drilled onsite for reasons of potential liability and lack of property access elsewhere. The drillers were considerably slowed in their work by the need to don air-tanks when particularly contaminated subsoils were encountered because the emission of volatile fumes from the borehole presented unacceptable health risks. Since the waters which would be pumped were expected to be contaminated, it was necessary to construct ten large temporary holding tanks (100,000 gallons each) onsite to impound the waters for testing and possible treatment prior to being discharged to the local sewer system (CH 2 M-Hill, 1984). The costs and difficulty of preparing for and conducting the test were worth the effort, however. The water levels in thirty-six monitoring wells were observed during the test and yielded a very detailed picture of transmissivity variations (Figure 3-8), which has been used to help explain the unusual configuration of the plume (Figure 3-9) and which were used to guide the design of a pump-and- treat system. Storage coefficients were also estimated; and though the short duration of the test (14 hours) did not allow for definitive estimates to be obtained, it was clear that qualitative confirmation of the generally non-artesian (water-table) nature of the aquifer beneath the site was confirmed. An automated data acquisition system (computer-controlled pressure transducer system) was used to monitor the water levels and provide real-time drawdown plots of 19 of the 36 wells (Table 3-1), greatly enhancing the information obtained with only minimal manpower requirements. The benefits from conducting the pump test cannot be overemphaiszed; qualitative confirmation of lithologic information and semi- quantitative estimation of crucial parameters were obtained. Finally, the distribution patterns of contaminant species that emerged from the investigations at Chem-Dyne were made understandable by considering research results and theories regarding chemical and microbiological influences. Once again there seemed to be evidence of transformation of tetrachloroethene (Figure 3-10) to less halogenated daughter products such as trichloroethene (Figure 3-11), dichloroethene (Figure 3-12), and vinyl chloride / monochloroethene (Figure 3-13). The relative rates 40 Figure 3-8. Estimates of transmissivity obtained from shallow and deep wells during Chem-Dyne test. 41 Figure 3-9. Distribution of total volatile organic chemical contamination in shallow wells at Chem-Dyne during October, 1983 sampling. 42 Table 3-1. Chem-Dyne Pump Test Observation Network. Obs. Radial Init. Wtr. Method of Well No. Dist. Level Measurement (feet) (ft, MSL) (type, field unit) MW1 957 563.68 manual, electric probe MW2 965 563.74 manual, electric probe MW3 848 563.96 automatic, float-type MW4 537 563.27 manual, electric probe MW5 313 563.31 manual, electric probe MW6 420 564.40 manual, electric probe MW7 480 563.30 manual, electric probe MW8 740 563.01 manual, electric probe MW9 487 563.08 manual, electric probe MW10 186 563.29 auto., pressure transducer MW11 502 562.90 auto., pressure transducer MW12 232 562.39 auto., pressure transducer MW13 232 563.19 auto., pressure transducer MW 14 701 - dry - no data collected MW15 611 563.10 auto., pressure transducer MW16 1275 562.47 manual, electric probe MW17 1518 560.03 manual, electric probe MW18 692 562.67 manual, electric probe MW19 1204 559.80 automatic, float-type MW20 1225 562.10 auotmatic, float-type MW21 1259 561.29 manual, electric probe MW22 1261 559.95 auto., pressure transducer MW23 298 563.07 auto., pressure transducer MW24 398 563.07 auto., pressure transducer MW25 53 563.04 auto., pressure transducer MW26 62 562.96 auto., pressure transducer MW27 272 563.13 auto., pressure transducer MW28 248 562.99 auto., pressure transducer MW29 167 563.23 auto., pressure transducer MW30 993 561.25 manual, electric probe MW31 465 562.78 automatic, float-type MW32 1236 559.56 auto., pressure transducer MW33 690 562.06 auto., pressure transducer MW34 454 562.29 auto., pressure transducer MW35 651 562.93 auto., pressure transducer MW36 696 562.69 manual, electric probe Pumping Well 0 (ref. pt.) 562.97 auto., pressure transducer 43 of movement of these contaminants, as well as other common solvents like benzene (Figure 3-14) and chloroform (Figure 3-15), generally conformed to predictions based on sorption principles. The remediation efforts also made use of these contaminant transport principles in estimating the capacity of the treatment system needed and the length of time necessary to remove residuals from the aquifer solids (CH^M-Hill, 1984). During the latter stages of negotiations with the Potentially Responsible Parties (PRP's), government contractors prepared mathematical models of the flow sytem and contaminant transport at Chem-Dyne (GeoTrans, 1984). These were used to estimate the possible direction and rate of migration of the plume in the absence of remediation, the mass of contaminants removed during the various remedial options, and the effects of sorption and dispersion on those estimates. Because of the wide range of sorption properties associated with the variety of VOC's found in significant concentrations, it was necessary to select values of retardation constants that represented the likely upper- and lower-limits of sorptive effects. It was also necessary to estimate or assume the values of other parameters known to affect transport processes, such as dispersion coefficients. While the developers of the models would be the first to acknowledge the large uncertainties associated with those modeling efforts due to lack of information about the actual history of chemical inputs and other important data, there was agreement between the government and PRP technical experts that the modeling efforts had been very helpful in assessing the magnitude of the problem and in determining minimal requirements for remediation. Consequently, modeling efforts will continue at Chem- Dyne. Data generated during the remediation phase will be used to refine models in an ongoing process so that the effectiveness of the remedial action can be evaluated properly. PRACTICAL CONCERNS In many ways, there may be too much confidence among those not directly involved in ground-water quality research regarding current abilities to predict transport and fate of contaminants in the subsurface. The discussions in the preceeding sections should place in proper perspective the admittedly remarkable advances that have been made in recent years by illustrating the practical and conceptual uncertainties that remain unresolved. Continuing research efforts will eventually resolve these uncertainties, but those efforts will be considerably slower if existing results are not routinely incorporated into practical situations. Research results must be tested in real-world settings because there is no alternative mechanism for validating them. Just as importantly, there are economic arguments for incorporating research findings and state- 44 Figure 3-10. Distribution of tetrachloroethene in shallow wells at Chem-Dyne during October, 1983 sampling. 45 Figure 3-11. Distribution of trichloroethene in shallow Chem-Dyne during October, 1983 sampling. wells at 46 / Figure 3-12. Distribution of trans-dichloroethene in shallow wells at Chem-Dyne during October, 1983 sampling. 47 Figure 3-13. Distribution of vinyl chloride in shallow wells at Chem-Dyne during October, 1983 sampling. 48 Figure 3-14. Distribution of benzene in shallow wells at Chem- Dyne during October, 1983 sampling. 49 CP 10 Figure 3-15. Distribution of chloroform in shallow wells at Chem- Dyne during October, 1983 sampling. 50 of-the-art techniques into routine contaminant investigations and remediations. Additional effort devoted to site-specific characterizations of natural process parameters, rather than relying almost exclusively on chemical analyses of ground-water samples, can significantly improve the quality and cost effectiveness of remedial actions at such sites. To underscore this point, condensed summaries are provided of the principal activities, benefits, and shortcomings of three possible site characterization approaches: conventional (Table 3-2), state-of-the-art (Table 3-3), and state-of-the-science (Table 3-4). To further illustrate this, a qualitative assessment of desired trade¬ offs between characterization and clean-up costs is presented in Figure 3-16. Table 3-2. Site Characterization Conventional Approach. ACTIONS TYPICALLY TAKEN Install a few dozen shallow monitoring wells Sample and analyze numerous times for 129 + pollutants Define geology primarily by driller's log and cuttings Evaluate hydrology with water level maps only BENEFITS Rapid screening of problem Moderate costs involved Field and lab techniques standardized Data analysis relatively straightforward Tentative identification of remedial options possible SHORTCOMINGS True extent of problem often misunderstood Selected remedial alternative may not be appropriate Optimization of remedial actions not possible Clean-up costs unpredictable and excessive Verification of compliance uncertain and difficult As illustrated there, some investments in specialized equipment and personnel will be necessary to make transitions to more sophisticated approaches, but those investments should be more than paid back in reduced clean-up costs. The maximum return on increased investments is expected for the state-of-the-art approach, and will diminish as the state-of-the-science approach is reached because highly specialized equipment and personnel are not widely available. It is vitally important that this philosophy be considered, 51 Table 3-3. Site Characterization State-of-the-Art Approach. RECOMMENDED ACTIONS Install depth-specific well clusters Sample and analyze for 129 + pollutants Analyze selected contaminants in subsequent samplings Define geology by extensive coring/split-spoon samples Perform limited tests on solids (grain size, clay content) Conduct geophysical surveys (resistivity soundings, etc.) BENEFITS Conceptual understanding of problem more complete Better prospect for optimization of remedial actions Predictability of remediation effectiveness increased Clean-up costs lowered, estimates improved Verification of compliance soundly based, more certain SHORTCOMINGS Characterization costs somewhat higher Detailed understanding of problem still difficult Full optimization of remedial actions not likely Field tests may create secondary problems Demand for specialists increased because the probable benefits in lowered total costs, health risks, and time for effective remediations can be substantial. CHAPTER SUMMARY It is well understood that models may be used to examine the significance of various natural processes controlling the behavior of ground water and the chemical and biological species it transports. Literally thousands of laboratory and field experiments have been conducted and interpreted with the assistance of mathematical models. Virtually all of these, however, are performed in idealized settings and are tightly controlled. There still exists the need to truly evaluate the predictive accuracy and utility of mathematical models for real-world contamination problems. Managers are increasingly asking technical support staff to use models to interpret field data within the context of developing and optimizing alternative solutions to their problems. That this cannot be done in a meaningful way, without serious efforts to characterize natural process parameters at actual sites, must also be fully appreciated. 52 Table 3-4. Site Characterization State-of-the-Science Approach. % IDEALIZED APPROACH Assume 'State-of-the-Art Approach* as starting point Conduct tracer-tests and borehole geophysical surveys Determine % organic carbon, exchange capacity, etc. of solids Measure redox potential, pH, DO, etc. of fluids Evaluate sorption-desorption behavior using select cores Identify bacteria and assess potential for biotransformation BENEFITS Thorough conceptual understanding of problem obtained Full optimization of remedial actions possible Predictability of remediation effectiveness maximized Clean-up costs lowered significantly, estimates reliable Verification of compliance assured SHORTCOMINGS Characterization costs significantly higher Few previous field applications of advanced theories Field and laboratory techniques not yet standardized Availability of specialized equipment low Demand for specialists dramatically increased 1 53 Figure 3-16. General relationship between site characterization costs and clean-up costs as a function of the characterization approach. 54 CHAPTER 4 LIABILITIES, COSTS, AND RECOMMENDATIONS FOR MANAGERS INTRODUCTION There are many texts available that describe the derivation of the theories underlying mathematical models, the technical applications of models, and related technical topics (e.g., data collection and parameter estimation techniques). Few texts treat the nontechnical issues that managers face when evaluating the possible uses of models, such as potential liabilities, costs, and communications between the modeler and management. These are, however, important considerations because many modeling efforts fail as a consequence of insufficient attention to them. This section is therefore directed to those issues. POTENTIAL LIABILITIES Some of the liabilities attending the use of mathematical models relate to the degree to which predictive models are relied on to set conditions for permitting or banning specific practices or products. If a model is incapable of treating specific applications properly, there may be substantially incorrect decisions made. Depending on the application, unacceptable environmental effects may begin to accumulate long before the nature of the problem is recognized. Conversely, unjustified restrictions may be imposed on the regulated community. Inappropriate or inadequate models may also cause the 're-opening clause' of a negotiated settlement agreement to be invoked when, for instance, compliance requirements that were guided by model predictions of expected plume behavior are not met. Certain liabilities relate to the use of proprietary codes in legal settings, where the inner workings of a model may be subject to disclosure in the interests of justice. The desire for confidentiality by the model developer would likely be subordinate to the public right to full information regarding actions predicated on modelingresults. The mechanisms for protection of proprietary rights do not currently extend beyond extracted promises of confidentiality by reviewers or other interested parties. Hence, a developer of proprietary codes still assumes some risk of exposure of innovative techniques, even if the code is not pirated outright. 55 Yet other liabilities may arise as the result of misapplication of models or the application of models later found to be faulty. Frequently, the choices of boundary and initial conditions for a given application are hotly contested; misapplications of this kind are undoubtedly responsible for many of the reservations expressed by would-be model users. It has also happened many times in the past that a widely used and highly regarded model code was found to contain errors that affected its ability to faithfully simulate situations for which it was designed. The best way to minimize these liabilities is to adopt strict quality control procedures for each application. ECONOMIC CONSIDERATIONS The nominal costs of the support staff, computing facilities, and specialized graphics' production equipment associated with numerical modeling efforts can be high. In addition, quality control activities can result in substantial costs; the determining factor in controlling these costs is the degree to which a manager must be certain of the characteristics of the model and the accuracy of its output. As a general rule, costs are greatest for personnel, moderate for hardware, and minimal for software. The exception to this ordering relates to the combination of software and hardware purchased. An optimally outfitted business computer (e.g., VAX 11/785 or IBM 3031) costs upwards of $100,000; but it can rapidly pay for itself in terms of dramatically increased speed and computational power. A well complimented personal computer (e.g., IBM-PC/AT or DEC Rainbow) may cost $10,000; but the significantly slower speed and limited computational power may infer hidden costs in terms of the inability to perform specific tasks. For example, highly desirable statistical packages like SAS and SPSS are unavailable or available only with reduced capabilities for personal computers; many of the most sophisticated mathematical models are available in their fully capable form only on business computers. Figure 4-1 gives a brief comparison of typical costs for software for different levels of computing power. Obviously, the software for less capable computers is cheaper, but the programs are not equivalent; so managers need to thoroughly think through what level is appropriate. If the decisions to be made are to be based on very little data, it may not make sense to insist on the most elegant software and hardware. If the intended use involves substantial amounts of data and sophisticated analyses are desired, it would be unwise to opt for the least expensive combination. Based on experience and observation, there does seem to be an increasing drive away from both ends of the spectrum and toward the middle; that is, the use of powerful personal computers is 56 Ave. Price U.S. Dollars Ground-Water Modeling Software Categories 1 = Mainframe/business computer models 2 = Personal computer versions of mainframe models 3 = Original IBM-PC and compatibles' models 4 = Handheld microcomputer models (e.g., Sharp PCI 500) 5 = Programmable calculator models (e g., HP41-CV) Note: Prices include software and all available documentation, reports, etc. Source of data: International Ground Water Modeling Center Figure 4-1. Average price per category for ground-water models from the International Ground-water Modeling Center. 57 I increasing rapidly, whereas the use of small programmable calculators and large business computers alike is declining. In part, this stems from the significant improvements in the computing power and quality of printed outputs obtainable from personal computers. In part, it is due to the improved telecommunications capabilities of personal computers; they are now able to emulate the interactive terminals of large business computers so that vast computational power can be accessed and the results retrieved with no more than a phone call. Most importantly for ground-water managers, many of the mathematical models and data packages have been 'down sized' from mainframe computers to personal computers; many more are being written directly for this market. Since it is expected that most managers will want to explore this situation a bit more, Figure 4-2 has been prepared to provide some idea of the costs of available software and hardware for personal computers. Table 4-1 lists salary ranges and desired backgrounds for the technical support staff needed to operate such systems, based on advertisements posted in the past two years. Figure 4-3 is an attempt to place all of the nominal costs in perspective. The technical considerations discussed in previous sections indicate that the desired accuracy of the modeling effort directly affects the total costs of mathematical simulations. Thus managers will want to determine the incremental benefits gained by increased expenditures for more involved mathematical modeling efforts. There are many economic theories which can be helpful in determining the incremental benefits gained per increased level of investment. The most straightforward of these are the cost-benefit approaches commonly used to evaluate the economic desirability of water resource projects. There are two generalized approaches in common practice: the 'benefit/cost ratio' method and the 'net benefit' method. The benefit/cost ratio method involves tallying the economic value of all benefits and dividing that sum by the total cost involved in generating those benefits (i.e.; B/C = ?). A ratio greater than one is required for the project to be considered viable, though there may be sociopolitical reasons for proceeding with projects that do not meet this criterion. Consider the example where a project is about to get underway and has gained considerable social or political momentum when the initial cost estimates begin to prove too low. Not proceeding or substantially altering the work may be economically wise; however, such a decision may be viewed as a breach of faith by the public. Regardless of how this kind of situation evolves, it is not uncommon for certain costs to be forgiven or subsidized and this muddies the picture for incremental benefits or trade-off analyses. 58 Ave. Price U.S. Dollars Min. ♦ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Vendors of Ground-Water Models 1. International Ground Water Modeling Ctr. 2. Computapipe Co. 3. Data Services, Inc. 4. GeoTrans, Inc. 5. Hydrosoft, Inc. 6. In Situ, Inc. 7. Irrisco Co. 8. Koch and Assoc. 9. KRS Enterprises, Inc. 10. Michael P. Spinks Co. 11. RockWare, Inc. 12. Solutech Corp. 13. T. A. Prickett & Assoc. 14. James S. Ulrick Co. 15. Watershed Research, Inc. Figure 4-2. Price ranges for IBM-PC ground-water models available from various sources. 59 Table 4-1. Desired backgrounds and salary ranges advertised for positions requiring ground-water modeling. Position Title: Hydrogeologist (Argonne Nat’l. Lab.) Salary Offered: $31,619 - $48,876 Desired Bkgrnd: familiarity and experience in field testing and monitoring of ground-water flow and the use of numerical models Position Title: Hydrologic Modeler (Univ. of Wisonsin) Salary Offered: $23,000 - $25,000 Desired Bkgrnd: primary strength in application of numerical models to ground-water flow and chemical transport; strong chemical background Position Title: Soil Scientist (U S. EPA) Salary Offered: $31,619 - $41,105 Desired Bkgrnd: knowledge of (i) soil physics, (ii) processes governing transport and fate of chem. and bio. species, (iii) math, statistics and geostat.; and the ability to develop computer codes Position Title: Ground-Wtr .Hydrol. (Inyo Co. Wtr Dept., CA) Salary Offered: starting up to $32,000 Desired Bkgrnd: at least three years experience including field work, surface/ground-water resource eval., environ, assess., flow modeling, FORTRAN Position Title: Hydrogeologist (S.W. Texas State Univ.) Salary Offered: $24,444 - $30,096 Desired Bkgrnd: academic training in hydrogeology, min. 2 years experience, knowledge of limestone aquifers and computer operations Position Title: Geochemist (U S. Nuclear Regulatory Comm.) Salary Offered: $23,170 - $41,105 Desired Bkgrnd: knowledge of solute and radionuclide transport, including speciation, attenuation (sorption) numerical modeling Position Title: Hydrogeol./Civil.Engr. (typical consulting firm) Salary Offered: 'conmmensurate with experience’ Desired Bkgrnd: strong background in applied ground-water flow and contaminant transport modeling, knowledge of federal/state regulations 60 Ground-Water Models ($5K) Benefits- 25% Salary, Service, Expendables, and Optional Software (S10K) IBM-PGAT & Accessories w/Financing ($15K) Salary: 1.0 FTE -$40K/yr Service, Expendables, and Optional Software ($25K) Salary: 1.0 FTE - $40K/yr Benefits - 25% Salary Ground-Water Models ($15K) VAX 11/785 & Accesories w/Financing (S125K) Figure4-3. Costs of sustaining ground-water modeling capabilities at two different computing levels for a five-year period. 61 The 'net benefit’ method involves determining the arithmetic difference of the total benefits and total costs (i.e.; B-C = ?). Here the obvious criterion is that the proposed work result in a situation where total benefits exceed total costs. This approach is most often adopted by profit-making enterprises, because they seek to maximize the difference as a source of income. The ratio method, by contrast, has long been used by government agencies and other non¬ profit organizations because they seek to show the simple viability of their efforts irrespective of the costs involved. In a very real sense, then, these two general economic assessment methods stem from different philosophies. They share many common difficulties and limitations, however. For example, there is a need to predict the present worth of future costs and to amortize benefits over the life of a project. The mechanics of such calculations are well known, but they necessarily involve substantial uncertainties. For example, the present worth of a series of equal payments for equipment or software can be computed by (White and others, 1984): P = A*[((l + i)n-l)/(i*(l + i) n )] where: P is the present worth, A is the series payment each interest period, i is the interest rate per period, and n is the number of interest periods. Note, however, that the interest rate must be estimated; this has fluctuated widely in the past two decades as a result of inflationary and recessionary periods in our economy. The significance of this is that a small difference in the interest rate results in tremendous differences in the present worth estimate because of the exponential nature of the equation. It is also possible to compute the future worth of a present investment, to calculate the percentage of worth annually acquired through single payments or serial investments, and so on. One should be aware that these methods of calculating costs belong to the general family of 'single-objective', or 'mutually-exclusive alternative' analyses which presuppose that the cost of two actions is obtained by simple addition of their singly-computed costs. In other words, the efforts being evaluated are presumed to have no interactions. For some aspects of ground-water modeling efforts this assumption may not be valid; e.g., one may not be able to specify software and hardware costs independently. In addition, these methods rely on the 'expected value concept', wherein the expected value of an alternative is viewed as the single product of its effects and the probability of their occurence. This means that high-risk, low-probability alternatives and low-risk, high-probability alternatives have the same expected value. 62 To overcome these difficulties it is necessary to use methods which can incorporate functional dependencies between various alternatives and which do not rely on the expected value concept, . such as multi-objective decision theories (Asbeck and Haimes, 1984; Haimes and Hall, 1974; Haimes, 1981). A conceivable use would be the estimation of lowered health risks associated with various , remedial action alternatives at a hazardous waste site. In such a case the output of a contaminant transport model would be used to provide certain inputs (i.e., water levels, contaminant concentrations, etc.) to a health effects model, and it would convert these into the inputs for the multiobjective decision model (e.g., probability of additional cancers per level of contaminant). The primary difficulty with these approaches to cost-benefit analyses is in clearly formulating the overall probabilities of the alternatives, so that the objectives which are to be satisfied may be ranked in order of importance. A related difficulty is the need to specify the functional form of the inputs (e.g., the 'population distribution function' of pumpage rates or contaminant levels). Historical records about the inputs may be insufficient to allow their functional forms to be determined. Another problem compounding the cost-benefit analysis of mathematical modeling efforts relates to the need to place an economic value on intangibles. For example, the increased productivity a manager might expect as a result of rapid machine calculations replacing hand calculations may not be as definable in terms of the improved quality of judgements made as it is in terms of time released for other duties. Similarly, the estimation of improved ground-water qaulity protection benefits may necessitate some valuation of the human life and suffering saved (rather nebulous quantities). Hence, there is often room for considerable 'adjustment' of the values of costs and benefits. This flexibility can be used inappropriately to improve otherwise unsatisfactory economic evaluations. Lehr (1986) offers a scathing indictment of the Tennessee Valley Authority for what he described as an 'extreme injustice', perpetrated by TVA in the form of hydroelectric projects which have ’incredibly large costs and negative cost benefit ratios'. Finally, some costs and benefits may be incorrectly evaluated because the data on which they are based are probabilistic and this goes unrecognized. For instance, we often know the key parameters affecting ground-water computations (i.e., hydraulic conductivity) only to within an order-of-magnitude due to data collection limitations. In these situations great caution must be exercised. On the one hand, excessive expenditures may be made to ensure that the model 'accurately' simulates observed (though inadequate) data. On the other, the artistic beauty of computer generated results * sometimes generates its own sense of what is 'right', regardless of apparent clashes with common sense. The reason the basic data are uncertain is very important. Costs are not uncertain just because of 63 lack of information about future interest rates; many times expectations are not realized because of societal and technological changes. Miller (1980) noted that EPA overestimated the cost of compliance with its proposed standard for vinyl chloride exposure by 200 times the actual costs. MANAGERIAL CONSIDERATIONS The return on investments made to use mathematical models rests principally with the training and experience of the technical support stafT applying the model to a problem and on the degree of communication between those persons and management. In discussing the potential uses of computer modeling for ground- water protection efforts, Faust and others (1981) summarized by noting that 'the final worth of modeling applications depends on the people who apply the models'. Managers should be aware that a fair degree of specialized training and experience are necessary to develop and apply mathematical models, and relatively few technical support staff can be expected to have such skills presently (van der Heijde and others, 1985). This is due in part to the need for familiarity with a number of scientific disciplines, so that the model may be structured to faithfully simulate real-world problems. What levels of training and experience are necessary to apply mathematical models properly? Do we need 'Rennaissance' specialists or can interdisciplinary teams be effective? The answers to these questions are not clear-cut. From experience it is easy to see that the more informed an individual is, the more effective he or she can be. It is doubtful, however, that any individual can master each discipline with the same depth of understanding that specialists in those fields have. What is clear is that some working knowledge of many sciences is necessary so that appropriate questions may be put to specialists, and so that some sense of integration of the various disciplines can evolve. In practice this means that ground-water modelers have a great need to become involved in continuing education efforts. Managers should expect and encourage this because the benefits to be gained are tremendous, and the costs of not doing so may be equally large. An ability to communicate effectively with management is essential also. Just as is the case with statistical analyses, an ill-posed problem yields answers to the wrong questions ('I know you heard what I said, but did you understand what I meant?'). Some of the questions managers should ask technical support staff, and vice versa, to ensure that the solution being developed is appropriate to the actual problem are listed in Tables 4-2 through 4-4. Table 4-2 consists of 'screening level' questions, Table 4-3 addresses the need for correct conceptualizations, and Table 4-4 is comprised of sociopolitical concerns. 64 Table4-2. Screening-level questions to help ground-water managers focus mathematical modeling efforts. General Problem Definition (1) What are the key issues; quantity, quality, or both? (2) What are the controlling geologic, hydrologic, chemical and biological features? (3) Are there reliable data (proper field scale, quality controlled, etc.) for preliminary assessments? (4) Do we have the model(s) needed for appropriate simulations? Initial Responses Needed (1) What is the time-frame for action (imminent or long-term) (2) What actions, if taken now, can significantly delay the projected impacts? (3) To what degree can mathematical simulations yield meaningful results for the action alternatives, given available data? (4) What other techniques or information (generic models, past experiences, etc.) would be useful for initial estimates? Strategies for Further Study (1) Are the critical data gaps identified; if not, how well can simulations determine the specific data needs? (2) What are the trade-offs between additional data and increased certainty of the simulations? (3) How much additional manpower and resources are necessary for further modeling efforts? (4) How long will it take to produce useful simulations, including quality control and error-estimation efforts? On another level of communication, managers should appreciate how difficult it will be to explain the results of complicated models to non-technical audiences such as in public meetings and courts of law. Many scientists find it a trying exercise to discuss the details of their labors without the convenience of the jargon of their discipline. Some of the more useful means of overcoming this limitation involve the production of highly simplified audio-visual aids, but this necessarily involves a great deal of work. The efforts which will be required to sell purportedly self-explanatory graphs from computer simulations may rival the efforts spent on producing the simulations initially. 65 Table 4-3. Conceptualization questions to help ground-water managers focus mathematical modeling efforts. Assumptions and Limitations (1) What are the assumptions made, and do they cast doubt on the model's projections for this problem? (2) What are the model's limitations regarding the natural processes controlling the problem; can the full spectrum of probable conditions be addressed? (3) How far in space and time can the results of the model simulations be extrapolated? (4) Where are the weak spots in the application, and can these be further minimized or eliminated? Input Parameters and Boundary Conditions (1) How reliable are the estimates of the input parameters; are they quantified within accepted statistical bounds? (2) What are the boundary conditions, and why are they appropriate to this problem? (3) Have the initial conditions with which the model is calibrated been checked for accuracy and internal consistency? (4) Are the spatial grid design(s) and time-steps of the model optimizedfor this problem? Quality Control and Error Estimation (1) Have these models been mathematically validated against other solutions to this kind of problem? (2) Has anyone field verified these models before, by direct applications or simulation of controlled experiments? Have these models been mathematically validated against other solutions to this kind of problem? (3) How do these mdoels compare with others in terms of computational efficiency, and ease of use or modification? (4) What special measures are being taken to estimate the overall errors of the simulations? CHAPTER SUMMARY Mathematical models can be helpful tools to managers of ground-water resource programs. Models may be used for testing hypotheses about conceptualizations and to generate better understanding of important physical, chemical, and biological processes which affect ground-water resources. The possible outcomes of complex problems can be addressed in great detail, if 66 Table4-4. Sociopolitical questions to help ground-water managers focus mathematical modeling efforts. Demographic Considerations (1) Is there a larger population endagered by the problem than we are able to provide sufficient responses to? (2) Is it possible to present the model's results in both non¬ technical and technical formats, to reach all audiences? (3) What role can modeling play in public information efforts? (4) How prepared are we to respond to criticism of the model(s)? Political Constraints (1) Are there non-technical barriers to using this model, such as 'tainted by association' with a controversy elsewhere? (2) Do we have the cooperation of all involved parties in obtaining the necessary data and implementing the solution? (3) Are similar technical efforts for this problem being undertaken by friend or foe? (4) Can the results of the model simulations be turned against us; are the results ambiguous or equivocal? Legal Concerns (1) Will the present schedule allow all regulatory requirements to be met in a timely manner? (2) If we are dependent on others for key inputs to the model(s), how do we recoup losses stemming from their non-performance? (3) What liabilities are incurred for projections which later turn out to be misinterpretations originating in the model? (4) Do any of the issues relying on the application of the model(s) require the advice of attorneys? adequate data are available. Mathematical modeling is neither simple nor impossible. Its successful application, however, relies heavily on the expertise of the modeler and the degree of communication with management. The use of mathematical models in the decision-making process means that the user will inevitably incur certain liabilities. Anticipation of problem areas and some sensitivity to the possible misuses of models will greatly minimize potential liabilities, as will rigorous quality control programs. 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