s 14.GS: CIR 224 c. 1 STATE OF ILLINOIS WILLIAM G. STRATTON, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director STUDIES OF WATERFLOOD PERFORMANCE II. TRAPPING OIL IN A PORE DOUBLET Walter Rose Paul A. Witherspoon DIVISION OF THE ILLINOIS STATE GEOLOGICAL SURVEY JOHN C FRYE, Chief URBANA CIRCULAR 224 1956 Digitized by the Internet Archive in 2012 with funding from University of Illinois Urbana-Champaign http://archive.org/details/studiesofwaterfl224rose STUDIES OF WATERFLOOD PERFORMANCE II. TRAPPING OIL IN A PORE DOUBLET by Walter Rose and Paul A. Witherspoon ABSTRACT This paper discusses the pore doublet, a parallel arrangement of a small- and large -diameter capillary tube, as a model of reser- voir rock. The displacement of oil by water is analyzed for the pore -doublet system, and from the results we have developed re- vised notions about waterflood character and consequences. The conclusions reported are not all in accord with previous assertions of other authors, but we believe them to be consistent with expec- tations. Subjects specifically discussed include: 1) displacement ef- ficiency as affected by viscosity ratio, pore texture, and capillary versus pressure -gradient driving forces; 2) explanations for the occurrence of subordinate phase production and the efficiency of imbibition waterflooding; and 3) concepts about fingering phenomena. The main conclusion, which is at variance with what has been pre- viously asserted, is that oil tends to be trapped in the smaller (rather than the larger) tube of the pore doublet. INTRODUCTION The Illinois State Geological Survey has undertaken, as part of its research program in petroleum engineering, a series of studies on the flow of fluids through porous media. It is hoped that the results of such studies will provide a better understanding of the waterflooding method of improving oil recovery, which has become of major importance in Illinois. Because a study of the flow of fluids through porous rock involves complex problems, it may be helpful to choose a simpler system for an analytical treat- ment. This paper, for example, discusses what will happen in an idealized sys- tem of capillary tubes called a "pore doublet." This approach may seem to have nothing to do with oil recovery in the field, but we believe that these in- direct methods of analysis will ultimately enable us to understand the basic problems so as to develop better methods for predicting oil recovery, and de- termine the most efficient rate of water injection to get maximum recovery of oil. PREVIOUS WORK Recently Moore and Slobod (1956) have given an interesting and informa- tive, but not altogether accurate, discussion of the VISCAP concept of oil re- [3 ] 4 ILLINOIS STATE GEOLOGICAL SURVEY covery. Their work attempts to summarize the notion that the efficiency of a given process of oil recovery depends to a large extent on the interplay between viscous (shear) forces and capillary forces. The viscous forces act as resis- tance that must be overcome by the driving force before oil can be displaced and moved towards the production well; but the capillary forces either oppose or add to the driving force in effecting oil recovery. The purpose of this type of approach is to provide a basis for explaining why, for example, waterflooding efficiency should be rate -sensitive. Thus it has been reasoned that, with op- posing forces, there might be an optimum intermediate condition that would mark the point at which the maximum amount of oil would be recovered. We do not object to these proposed principles as the statement of a possi- bility. Perhaps in time we will know enough about the pore structure, fluid properties, and capillarity of reservoir systems so that optimum operating conditions can be assigned at the beginning of a secondary recovery operation. We would, however, question the application of the pore-doublet model that Moore and Slobod utilized to illustrate the contention that an intermediate rate of flooding will result in maximum oil recovery. The pore -doublet model has been used widely to evaluate oil recovery. For example, Bartell and co-workers (Benner, Riches, and Bartell, 1943) con- cluded that the phenomenon of counterflow would be observed in pore doublets where the water -oil interface would be advancing in the smaller pore and re- ceding in the larger. Figure 1 illustrates what they thought would happen in reservoir situations where doublet pore configurations were abundant. As has been discussed elsewhere, their analysis is not relevant to a description of what is likely to occur in the reservoir during waterflooding (Rose, in press). Other authors (for example, Yuster, 1940) also have made use of the pore- doublet model to evaluate the importance of the various factors that determine waterflooding efficiences. We believe, however, that the mechanics of oil dis- placement in a pore doublet have been incorrectly evaluated in some cases. ANALYSIS OF THE PORE-DOUBLET MODEL This paper simply analyzes what will happen in the pore -doublet model, and comes to conclusions different from those presented by other authors. For example, our analysis does not predict the frequent occurrence of Bartell's counter -flow phenomena, nor does it suggest that there is an intermediate rate of waterflooding that gives greater recoveries than either higher or lower rates. On the contrary, our analysis suggests that recovery is greatest when the in- jection rate (that is, the rate of flood-front advance) is least, rather than at some intermediate value. We consider it premature, however, to imply that any conclusions drawn from an analysis of the pore -doublet model (including ours) has any direct bearing on what will happen in actual reservoir situations. For example, we predict from the pore -doublet model that recovery is increased by employing low flooding velocities in water -wet sands, which may seem quite contradictory to certain field indications (namely at Bradford). Figure 2 illustrates schematically what we feel to be the case. The pore doublet is indicated by two pores of different size, joined both at the inflow and outflow ends by other pores of arbitrary size. Barrer (1948) correctly gives the rate of interface travel in circular pores as: TRAPPING OIL IN A PORE DOUBLET water oil Fig. 1. - The pore-doublet model as originally depicted by Benner et al. (1943). direction of water flooding / s- *\-^^r \ \ i <- .i\ \* \ / x / / 2 dx dv »r 4 Up-g{d x-d (l.-x))sin$l + 2irr 3 acose irr — — _h i w o { J 8ij x + 8^ (1-k) dt dt *= AF » 2? cos© /*■ ^S t oc constant \ N * = ° 2tcos( AP« — s / ■ t a constant "" — n — N \ /_ 0/ \ \ /_ o/ / KEY a oil-water m interfacial tension AP imposed pressure drop across tube g gravitational constant d fluid densities w designates water designates oil x designates interfoce position 1 pore length 9 contact angle dv volumetric flow rate dt r pore radius * angle of inclination V fluid viscosity Fig. 2. - The correct representation of the pore-doublet model. A - The water -oil interface in the inflow tube, advancing as described by Barrer (1948). B - Oil displacement in the doublet effected by pressure- gradient drive. C - Oil displacement in the doublet effected by capillary imbibition of water. ILLINOIS STATE GEOLOGICAL SURVEY 2, r [AP - g(d x - d H-x)\ sin<£] + 2rcrcos 8 L w o J _ dWdt = 8 V x + 8l7 ((-x) (1) w o where r is the pore radius AP is the difference between the pressures as measured at each end of the pore g is the gravitational constant d and d are the densities of the water and oil, respectively w o x denotes the interface position, between zero and £ where i is the pore length

= 0), the time required for the water -oil interface to move the entire length of the tube is: 2 Lim t _ 877 I r (AP + 2crcos 8 It) Assuming a water-wet system with 8=0, two limiting cases can now be considered. The first is one in which the capillary forces are negligible com- pared to the viscous forces, (AP»2cr/r), and in this case the time required for the water -oil interface to move the entire length of the tube is proportional to 1/r^ according to: „2 Lim t _ 877I x+{ = 2 [i} On the other hand, if capillary forces predominate (2 left in the pore doublet at break-through versus R, for various limiting values of V and F. TRAPPING OIL IN A PORE DOUBLET 9 Lim t (4t 2 ) (77 W + Vo ) j, = (°) x_ * t r 2 [AP + 2cr/r] Thus it is seen, as before, that the large tube always empties more quick- ly than the smaller, in a way inversely proportional to the ratio of radii (when capillary driving forces predominate), and inversely proportional to the square of the ratio of radii (when capillary driving forces can be neglected). Combining Equations 1 and 6 gives the more complete expression for re- sidual oil as : 2 F + 1 , 2 , v 1 1/2, 2 r f, 2 F + 1 , 2 ,.") 1/2, S = — (7) ° (R +1) (V- 1) where: R is the ratio of rj to r,, V is the water to oil viscosity ratio, and F is the ratio of AP to 2cr/r 1 Figure 3 gives various families of curves showing the relationship between S Q and R for various conditions of V and F. It is seen that minimum values of residual oil result when R is zero or unity, or when capillary forces control the displacement process (F =0), or when the water-oil viscosity ratio is ex- tremely high (that is, approaching infinity). This last consequence is especially interesting in that it provides a basis for understanding the relationship be- tween mobility ratio, or "fingering" phenomenon, and oil recovery as discussed by Aronofsky and Ramey (1956). Clearly, if the water viscosity is considerably greater than the oil viscos- ity, the movement of the flood-front in the smaller tube more closely approaches the faster rate in the larger tube (than if V were smaller) because there is al- ways more of the lower viscosity oil in the smaller tube to increase its rela- tive conductivity. Likewise, it should not be unexpected that the dominance of capillary forces would favor recovery, or that, when R is zero or unity, the re- sidual oil saturation will be zero. These, in fact, are the intuitively expected results. More to the point, the question may be appropriately asked: Will the trap- ped oil stay permanently in the smaller pore (of the pore doublet) in water-wet systems ? The answer is clearly no, unless the interfacial curvature (and hence the capillary pressure) at both ends of the trapped oil-leg are strictly identical, and there is no net effective pressure gradient acting across the smaller tube. This rather improbable situation is depicted schematically in figure 4A, and depends on postulating: 1) that flow of water has ceased in the larger tube (this could happen, for example, if,after the flood-front had moved through the larger pore of the pore doublet, subsequent trapping occurred at the downstream end); and 2) that the advancing and receding contact angles are equal. Or, if a AP is allowed across the pore doublet, then the opposing capillary forces must provide an exact balance, either because of difference between advancing and receding contact angle, or slight variation in pore radius of the smaller tube occurring exactly where the ends of the trapped oil-leg happen to be, or varia- tion in the two contact angles resulting from difference in wettability conditions 10 ILLINOIS STATE GEOLOGICAL SURVEY Fig. 4. - Situations in the pore -doublet model after break-through A - There is no net force acting on the trapped oil so it remains immobile. There is counterflow of the oil trapped in the smaller pore so it moves to a more stable configuration in the larger pore. C - During counterflow, globules of the trapped oil are en- trained in the water moving through. the larger pore. • • at different portions of the smaller tube. Such an exact balance of forces might not be encountered, however. In a more likely case (namely, where advancing and receding contact angles are not equal, and/or a finite pressure difference continues across the pore doublet after breakthrough, and/or there is slight variation in pore size and/or wettability of the smaller tube along its length) it would appear that the oil ini- tially trapped in the smaller pore would ultimately tend to move to some other position. For example, a variation of Bartell's counterflow concept might be responsible for movement of oil from the smaller tube to the larger tube, as depicted schematically in figure 4B. This would occur in response to the ten- dency of such systems to approach a practical minimum in free energy, for clearly, trapped oil in the smaller tube of the pore doublet has greater inter- facial surface area of contact with the water and pore walls than if the same volume of oil were moved (via counterflow) into the larger tube. The latter, of course, would happen only if the water motion in r^ had ceased, and if there were some net driving force to start this movement. An interesting observation is that the dimension of the effluent connecting tube that joins the down-stream end of the pore doublet assumes importance, as depicted in figure 4C. For if there is no bottle-neck constriction there (as evidently was assumed in the Benner et al. analysis) to prevent the free entry of oil, it would appear possible that oil globules could be entrained in the mov- ing water stream and be moved (via slug flow) at least until new barriers were met. In such a case it would seem possible that zero residual oil would event- ually be left in the subject pore doublet, without reference to the effect of vis- cosity ratio (V) pore radius ratio (R), or the A P versus capillary force ratio (F). Residual oil, of course, is invariably left in field operations, which prob- ably reflects the fact that the pore doublet itself is too idealized a model of real reservoir situations. That is to say, it seems reasonable that pore doub- lets occur in nature, but the surrounding environment has more than a little to do with how they function during the recovery process. In this sense, one TRAPPING OIL IN A PORE DOUBLET 11 perhaps can take the values for residual oil, predicted by pore -doublet theory, as maximum values of oil that cannot be produced at the point in the recovery process where oil phase continuity is broken. Observed recoveries may be greater in nature (that is, residual values may be lower) because initial trap- ping does not mean unalterable trapping; but inasmuch as the practical (econom- ic) end-point of the depletion process always occurs before all the oil that can be moved is produced, concepts regarding pore doublets may prove a useful in- dex of anticipated recoveries. Another compensating factor results from the fact that pore -doublet theory is speaking of displacement efficiency, whereas actually observed recoveries are always lessened because of sweep efficiency considerations. Thus we con- sider it not entirely unreasonable as a future possibility that lithologic exam- ination of core samples and knowledge of operating conditions, fluid properties, etc., will permit a reasonable assigning of the R, V, and F terms, so that equal- ly reasonable predictions of recovery can be made by use of Equation 7. A still more general formulation to use instead of Equation 7, which con- siders the possibility of more than two parallel paths (but still neglects gravity and non-uniform wettability influences) is: Xr. [i - l.] s =— *-. L_ (8) v 2 1 + 2. R i where: { , + R 2 FM ^ _ 1)} M _ l L. = F + R i V - 1 The above analysis admittedly is highly simplified, where, among other things, the flow of water due to a pressure gradient in the larger tube of the doublet has been neglected. Surely, the latter would determine the extent of counterflow that would result or the amount of transfer via slug flow that might be observed; likewise, an exact analysis should consider differences between advancing and receding contact angles, and effects of gravity. But the specu- lations as presented apparently suggest how to account for part or all of the so- called "subordinate phase of production" discussed in the theory of Buckley and Leverett (1942). The speculations also imply that cases may exist where intermittent water injection might benefit recovery by allowing time for oil trapped in small pores to move (via counterflow) to adjacent larger pores, so that later water injection will carry the oil on further towards the production point (via slug flow). Perhaps the most interesting observation to be made is that the above con- siderations present an explanation for the success of the so-called "imbibition" waterflood process in water-wet reservoirs (Brownscombe and Dyes, 1952). This recovery method depends entirely on capillary forces to bring water into the sand for oil displacement, which (in accordance with the theory presented above) gives the oil more time to be displaced from trapped positions in small- er pores so that smaller residuals result. The foregoing discussion defines (in our view) the usefulness of pore- doublet theory. For example, it provides a qualitative picture that explains the ef- 12 ILLINOIS STATE GEOLOGICAL SURVEY ficiency of imbibition waterfloods, and suggests ideas that help explain subor- dinate phase oil production and fingering phenomena. We reject, however, many of the conclusions of other authors, even though they may have been based on a model and analytic formulations entirely equivalent to those presented here. For example, Moore and Slobod, in citing an example in which oil is trap- ped in the larger (instead of the smaller) tube of the pore doublet, chose such a low rate of flow that they were in effect considering a negative (back-pres- sure) pressure gradient. These authors cited the example of r^ and r-> being one and two microns, {being 5 microns , 0" being 30 dyne/cm., oil and water viscosity being one centi- poise, and total flow rate through both tubes being 1.6 x lO"-* ccs./ second. From this the pressure gradient, A P, can be calculated as having a value of -2 x 1CP dynes per square centimeter, which is sufficient to so retard the ad- vance of the water-oil interface in r^ that displacement only occurs in r , . Analysis shows that when AP equals -Za/r^ + r^, the flood front moves at equal velocity through both tubes of the pore doublet so that the residual oil saturation as calculated by Equation 5 [or by Equation 7, letting F equal -R/R + l] is zero. However, if the back-pressure is increased to -Zcr/r^, forward mo- tion of the water-oil interface (namely, displacement in r^) ceases, and as AP is further increased in the negative sense, oil first enters r^ and then finally enters the smaller tube rj so that waterflooding displacement ceases altogether. Discussion of the exact analytics of these situations, however, is beyond the scope of our paper. We simply note in passing that evidently previous conten- tions that oil recovery is optimum only when there is an ideal balance between capillary and other driving forces appear to have been based on the unusual condition of a back-pressure existing at the downstream end of the pore doub- let. CONCLUSIONS In conclusion, we state the following as the principal conditions that favor maximum recovery from pore doublets (where again for simplicity gravity ef- fects are neglected, and zero advancing and receding contact angles are as- sumed): (1) If both pores of the pore doublet are nearly the same size, or if they are of considerably different sizes, then near-zero residual oil will result (that is, perfect sorting or poor sorting are better than intermediate degrees of sorting). (2) Recovery is always favored by a high water-oil viscosity ratio. (3) Examination of Equations (5) and (7) shows that imposing a back-pres- sure equal to 2cr/r i + r^ makes both pores of the doublet empty simultaneously so that zero oil-residuals result. (4) If imposing a back-pressure is impractical, then maximum recovery in water-wet systems results when capillary forces alone bring in the displac- ing water phase (namely, a low AP driving force). (5) The tendency of oil trapped in the smaller pores to seek a lower energy state in the contiguous larger pores would favor additional recovery via slug flow. Another conclusion which we sense is apropos (although it is not fully dem- onstrated by the content of this paper) is that one does not expect unlimited TRAPPING OIL IN A PORE DOUBLET 13 success with the pore-doublet model in predicting the performance of actual reservoir systems. This is suggested by the observation that even the more elegant network models of Fatt (1956) leave much to be desired in achieving exact representation of the prototype system. Thus, although the pore doublet microscopically may be found here and there in nature, its occurrence is not relevant unless and until the surrounding environment is taken into consideration. Our current inquiries, based on the use of complicated network model systems of the Fatt type, demonstrate this as will be shown in a later publication (Rose et al., in preparation). We admit, therefore, the temptation is to disregard what others say about such simple things as pore-doublet models, especially when more powerful methods of anal- ysis are now available; but we must recognize that the projected work with the more complex network models rests to an extent on a correct understanding of what happens in the pore-doublet unit. REFERENCES Aronofsky, J. S., and Ramey, H. J., Jr., Mobility ratio - its influence on in- jection or production histories in five -spot waterflood: paper presented at AIME Petroleum Branch Fall Meeting, Los Angeles, Calif., October, 1956. AIME Trans., v. 207, p. 205. Barrer, R. M., 1948, Fluid flow in porous media: Faraday Society Discussions No. 3, p. 61. Benner, F. C, Riches, W. W., and Bartell, F. E., 1943, Nature and importance of surface forces in production of petroleum: in Fundamental research on occurrence and recovery of petroleum, p. 74: American Petroleum Insti- tute. Brownscombe, E. R., and Dyes, A. B., 1952, Water -imbibition - a possibility for the Spraberry: API Drilling and Production Practice. Buckley, S. E., and Leverett, M. C, 1942, Mechanism of fluid displacement in sands: AIME Trans., v. 146, p. 107. Fatt, I., 1956, The network model of porous media: AIME Trans., v. 209, p. 144. Moore, T. F., and Slobod, R. L., 1956, The effect of viscosity and capillarity on the displacement of oil by water: Producers Monthly, v. 20, no. 10, p. 20; also presented at the New Orleans meeting of the Petroleum Branch AIME, Oct. 1955, under the title, "Displacement of Oil by Water - Effect of Wettability, Rate, and Viscosity on Recovery." Rose, Walter, Studies of waterflood performance. I. Causes and character of residual oil: Illinois Geol. Survey Bull. 80, in press. (Abstract in Produc- ers Monthly, v. 20, no. 10, Sept. 1956.) Rose, Walter, et al., Studies of waterflood performance. III. The network mod- el approach: Illinois Geol. Survey publication in preparation. Yuster, S. T., 1940, Fundamental forces in petroleum production: paper pre- sented at AIME New York Meeting, Feb. 15, 1940. (Abstract in Oil Weekly, v. 96, no. 11, p. 43.) Illinois State Geological Survey Circular 224 13 p., 4 figs., 1956 mmnai CIRCULAR 224 ILLINOIS STATE GEOLOGICAL SURVEY URBAN A