s 
 
 14.GS: 
 CIR 242 
 
 (3&{A. Su-r^ 
 
 STATE OF ILLINOIS 
 
 WILLIAM G. STRATTON, Governor 
 
 DEPARTMENT OF REGISTRATION AND EDUCATION 
 
 VERA M. BINKS, Director 
 
 FLUID FLOW IN 
 PETROLEUM RESERVOIRS 
 
 II. - Predicted Effects of 
 Sand Consolidation 
 
 Walter Rose 
 
 ILLINOIS GEOLOGICAL 
 SURVEY LIBRARY 
 
 DIVISION OF THE 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 JOHN C. FRYE, Chief URBANA 
 
 CIRCULAR 242 
 
 1957 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://archive.org/details/fluidflowinpetro242rose 
 

 FLUID FLOW IN PETROLEUM RESERVOIRS 
 II. — PREDICTED EFFECTS OF SAND CONSOLIDATION 
 
 Walter Rose 
 
 ABSTRACT 
 
 There is much need for analytically developing valid descrip- 
 tions of the phenomena of fluid flow in porous media. The Kozeny 
 equation is frequently employed, and an important check on this ap- 
 plication can be made by considering the question of sand consoli- 
 dation. 
 
 Ordinarilyit would be assumed thata consolidated sand is less 
 permeable than the unconsolidated sand from which it was derived. 
 To provide an analytic basis for this expectation, I have applied 
 the Kozeny theory to the treatment of flow through various types of 
 regular packings of spheres. Uniform size sphere packings are 
 used frequently as models of unconsolidated sands, but a better 
 representation considers packings made up of more than one sphere 
 size. 
 
 Such models may be modified in three ways to represent the 
 physical consequences of lithification: (1) cement may be located 
 at points of particle contact as pendular rings; (2) cement may be 
 deposited as an even layer on all pore surfaces; or (3) consolida- 
 tion may be represented as a squeezing of all the particles together 
 so that the point contacts become fused surfaces. 
 
 The Kozeny theory predicts an unexpected increase in permea- 
 bility as pendular ring cement is deposited, but a sensible decrease 
 in permeability for other types of consolidation. Three perplexing 
 questions therefore arise: Is the Kozeny theory applicable to a 
 treatment of what happens during consolidation? Have good models 
 been chosen for study? Or ought we question the theory that con- 
 solidation always means a decrease in permeability? These ques- 
 tions are discussed, but none can be considered fully answered. 
 
 INTRODUCTION 
 
 In a previous paper (W. Rose, 1957a) the application of the Kozeny equation 
 to describe homogeneous fluid flow through consolidated sandstones was ques- 
 tioned. An analytic check was made by taking a regular packing (cubic and 
 rhombohedral) of uniform-sized spheres as the model of the unconsolidated sand, 
 and cementing materials were represented as pendular rings deposited at sphere 
 contacts. With such a representation, it was found that the specific surface 
 area decreased with increased consolidation, which meant that the permeability 
 predicted by the Kozeny equation would increase, according to: 
 
 K=f/tA 2 (1) 
 
 where K is the permeability, f is the porosity, t is a textural constant, and A is 
 the specific surface area per unit pore volume. 
 
 An increase in permeability with increase in cementation is contrary to ex- 
 pectations, but because the Kozeny equation has usually proved valid in appli- 
 cations of this sort, the result was reported as a paradox. 
 
 [1] 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 In an earlier work H. E. Rose (1950) also attempted to relate flow through 
 consolidated media to fundamental considerations, and he concluded that the 
 flow resistance was a statistical function of the quantity of cementing material 
 present, and so was subject to considerable variation depending upon the uni- 
 formity in distribution of the bonding material. In consequence, he asserted 
 that the permeability of consolidated sands cannot be uniquely calculated from 
 the physical characteristics of the bed. 
 
 Thus, two separate analyses indicate that sand consolidation has a special 
 effect on fluid flow phenomena. My own work, which has emphasized apparent 
 departure from the indications of the Kozeny equation, still awaits experimental 
 verification, but as H. E. Rose has stated, "It is also evident that the effect of 
 the bonding material can only be to increase the resistance of the bed. " Hence, 
 the purpose of this paper is to extend the application of my previous analysis to 
 consider a model composed of more than one sphere size. Although the influence 
 of consolidation on tortuosity cannot be treated at this time, certain conceptual 
 notions bring theoretical expectations and laboratory observations into closer 
 harmony. 
 
 First, I see no necessity for the premise made by H. E. Rose that uneven 
 distribution in bonding material is the proper explanation for the various anoma- 
 lies he observed. 
 
 Cementation may be thought of as usually having causes locally independent 
 of position, such as 1) flocculation and precipitation of clay minerals, micas, 
 and inorganic salts; 2) solution and fusion at grain contacts due to overburden 
 stress; and 3) entrapment of clay minerals and other suspended solids along 
 pore surfaces and at grain contacts. 
 
 In other words, the post-depositional variation in environmental conditions 
 throughout a large sedimentary sand body may give rise to variation in degree 
 of cementation between widely separated locations, just as there are observed 
 anisotropies in permeability, porosity, and other textural properties; but locally 
 the environmental factors which cause consolidation must be regarded as vir- 
 tually constant in space if not in time. Hence, the idea of uneven distribution 
 of bonding material appears to be unnecessary, at least for lightly consolidated 
 sands when small volume elements are being considered. 
 
 Actually the premise of uneven location of cement is not needed to explain 
 the anomaly reported by H, E. Rose, for pore shapes and size distribution can 
 not be represented by a single average particle diameter; moreover he admits 
 that his data are of doubtful value. 
 
 PROPOSED MODEL OF A CONSOLIDATED SAND 
 
 The random packing of random-sized spheres appears to be a satisfactory 
 model of an unconsolidated sand. True, quartz grains are double-domed elon- 
 gated pyramids , but they acquire high sphericity through natural weathering; 
 moreover, it can be shown that the sphere model is closely similar to a random 
 packing of variously sized oblate and prolate spheroids if the particles have no 
 preferential orientation. 
 
 Cementation may be equivalent to the result of compaction, or of an even 
 deposition of bonding material on all pore surfaces, or the cement may be de- 
 posited preferentially as pendular rings at points of grain contact. Hence, the 
 model to be adopted for consolidated sands is equivalent to the model proposed 
 for unconsolidated sands (that is, a random packing of random-sized spheres), 
 as modified by the change in porosity, specific surface area, and tortuosity 
 
PREDICTED EFFECTS OF SAND CONSOLIDATION 
 
 which accompanies compaction and/or the location of bonding material at points 
 of grain contact and/or as a layer on all pore surfaces. (In either case, the 
 point contacts between the sand grains of the original unconsolidated matrix 
 become altered as two- and perhaps three-dimensional surfaces of contact.) 
 
 The volumes and surface areas of pendular rings between spheres of unegual 
 size have been derived and presented in another publication (Rose, 1957?). In 
 regard to necessary accuracy for present purposes, these relationships can be 
 expressed as: „ 
 
 log (v/R ) = 3 log Q - 6, and 
 log (c/R 2 ) = 2.7 log 6 - 4 
 
 (2) 
 
 where v and a are the volume and area, respectively, of pendular rings regard- 
 less of the relative size of the sphere particles in contact, the angle Q is de- 
 fined by reference to figure 1, and R is the radius of each larger sphere in con- 
 tact with a smaller sphere. 
 
 Fig. 1. - Sphere packing models of porous media showing various types of cementation. 
 A - Pendular cement C - Uneven layered cement 
 
 B - Even layered cement D - Cementation by compaction 
 
 Figure 1 represents the types of consolidated sand models which will be used, 
 where diagram A depicts pendular ring bonding, diagram B shows cementation as 
 layers on all pore surfaces, diagram C allows for the sometime probable tendency 
 of layered cement to concentrate at points of grain contact, and diagram D refers 
 to consolidation by compaction with fusion at the contact surfaces. 
 
 These physical models should suffice to represent the porous (lightly con- 
 solidated) sandstones which occur in nature, whether the cementation comes about 
 through precipitation (or filtration) of dissolved or suspended minerals in the cir- 
 culating groundwater, or is the result of fusion at grain contacts in response to 
 the effects of overburden compaction. 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Fig. 2. - Plan view of rhombohedral packing and the geometry of the 
 filling process (after White and Walton, 1937). 
 
 A - The unfilled rhombohedral packing with center of spheres in 
 lower layer indicated by crosses and center of spheres in 
 upper layer indicated by circles. 
 
 B - The secondary sphere is shown in section X-Y. 
 
 C - The tertiary sphere is shown in section X-Y. 
 
 D - The quaternary sphere is shown in section X-Y. 
 
 E - The quinary sphere is shown in section X-Y. 
 
 That is , the environmental factors which give rise to cementation determine 
 just where the silicon dioxide, iron oxide and hydroxides, clays, and the cal- 
 cium and magnesium carbonates will be located, and in what quantity. Never- 
 theless, the local effect will be fairly uniform, and contiguous sand grains of 
 the original matrix can be expected to show similar lithification as depicted 
 schematically in figure 1 and also as shown by Spock (1953, fig. 22). 
 
 Figure 2 shows the specific sphere packing model which will be considered, 
 and it may be described as a "filled" rhombohedral packing. That is, secondary, 
 tertiary, quaternary, and quinary spheres of the sizes specified in table 1 are 
 inserted within (but without enlarging) the various void spaces of the rhombohe- 
 dral matrix to achieve a maximum sphere particle density (that is, minimum 
 porosity). This arrangement (White and Walton, 19 37) is used in the analyses 
 below in a way similar to the use previously made of the "filled" cubic packing 
 (Rose, 1957?). 
 
 As will be evident, the "filled" rhombohedral packing is characterized by a 
 high occurrence of large-to-small sphere contacts, more so than in the case of 
 the "filled" cubic packing, so that an advantage is gained of having a model of 
 the following description: 
 
PREDICTED EFFECTS OF SAND CONSOLIDATION 5 
 
 The model is regular, which simplifies the analytics; but it also possesses 
 wide particle size variation and sphere -to-sphere intermixing. 
 
 These features give the model a virtual character of randomness which, al- 
 though not real, provides excellent similitude with the prototype unconsolidated 
 sand sediments of nature. The latter is taken to be a locally isotropic assem- 
 blage of random-sized particles (of high sphericity) distributed randomly in 
 space. 
 
 INFLUENCE OF PENDULAR RING CEMENT 
 
 Table 1 specifies the way the secondary J sphere of radius 0.414R fits in the 
 largest void space left by the primary E spheres, which then leaves space to in- 
 sert two tertiary K spheres, eight quaternary L spheres, and eight quinary M 
 spheres. During the course of this filling, the fractional porosity is reduced 
 from 0.2595 to about 0.149. Table 1 also gives the resulting number of sphere- 
 to-sphere contacts. 
 
 As in my previous work (Rose, 1957?), it is next assumed that an angle, Q 
 equals 10 , defines a reasonable size for the pendular ring cement at sphere 
 contacts between the largest (primary E) spheres; hence the defining angles and 
 surface areas of all pendular rings are calculable from geometry as is discussed 
 in the cited references. The values given in table 1, in fact, have been obtained 
 by direct methods, rather than by making use of various available approximations 
 such as given by Equations 2. 
 
 Table 2 and figure 3 show the results obtained. If the rhombohedral packing 
 containing only E size spheres is filled with pendular ring cement occupying a 
 fractional pore space volume of only 4.09 x 10~3, the specific surface area of 
 the resultant system (A ) is reduced enough so the (A/ A ) is increased to 
 1. 15. Adding the smaller J, K, L, and M spheres continues this trend in a way 
 comparable to what was observed for the filled cubic packing. 
 
 If the Kozeny equation (Equation 1) is to be regarded as a valid relationship 
 between permeability, porosity, and specific surface area, and if cementation is 
 to be considered only as the deposition of pendular ring material, it is to be 
 expected that permeability increases with cementation to a degree somewhat 
 equivalent to the (A/A c )2 values as given in table 2. This is because the po- 
 rosity term is affected insignificantly by the addition of the cement material; 
 and apparently the same thing can be said about the effect of such minor ce- 
 mentation on the tortuosity term. 
 
 From my work to date on this subject (Rose, 1957a, 1957?, and herein), the 
 rule appears to be established for a large variety of sphere packings, that the 
 specific surface area bounding the flowing fluid (A c ) is always smaller when 
 pendular ring material is present than when it is absent (A). That is, the dome 
 area, Arj> (the influence of which is removed by the addition of pendular ring 
 material), is always greater than the added pendular ring surface area, Ap,- 
 moreover, this is true regardless of the sphere size ratio of the particles in 
 contact. 
 
 My original work (Rose, 1957a) showed how the (A/A c ) ratio could be 
 caused to increase as more and more pendular ring material was added. This 
 effect was demonstrated only for cases of regular packings of regular-sized 
 spheres, but it may be presumed to apply to filled as well as unfilled packings. 
 
ILLINOIS STATE GEOLOGICAL SURVEY 
 
 
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ILLINOIS STATE GEOLOGICAL SURVEY 
 
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 Fig. 3. - Fractional change in permeability, K, 
 as a function of the ratio of the volume of 
 cement added to the pore volume, S, for 
 various sphere packing examples. 
 
 Curve I shows the increase in permeability 
 resulting from the location of pendular ce- 
 ment at the particle contacts of various 
 filled-rhombohedral systems. Letters in- 
 dicate the particle size distribution of the 
 packing. 
 
 Curve II shows the effect of layered cement 
 in decreasing permeability for unfilled 
 rhombohedral packing (circles), unfilled 
 cubic packing (solid dots), and E, J-filled 
 rhombohedral packing (triangles). Numbers 
 indicate cement layer thickness. 
 
 In the current work (compare al- 
 so Rose, 1957'?), the (A/A c ) 2 ratio 
 is seen to be increased as more 
 pendular rings are added when pro- 
 ceeding with the sequence of adding 
 secondary and smaller spheres to 
 "fill" the basic regular patterns. 
 Figure 3 illustrates this latter ef- 
 fect. 
 
 It may be concluded, therefore, 
 that the indicated anomaly of pen- 
 dular ring cements causing an in- 
 crease in permeability is minimized 
 for the cases of well sorted sands 
 (minimum number of grain contacts) , 
 and of low degree of consolidation 
 (minimum amount of cement depo- 
 sition) . 
 
 INFLUENCE OF LAYERED CEMENTS 
 
 If a single sphere particle is 
 considered, it is apparent that add- 
 ing a layered cement would increase 
 the tortuosity and surface area but 
 would decrease the porosity by in- 
 creasing the grain volume. All three 
 effects would lead to a decrease in 
 permeability according to Equation 1, 
 and according to expectations. 
 
 With modification, this simple 
 argument can also be applied to a 
 treatment of complex sphere pack- 
 ings such as the filled cubic and 
 rhombohedral packings discussed 
 above. 
 
 Let K be the fractional change in 
 permeability due to consolidation, 
 and neglecting the influence of de- 
 posited cements on tortuosity, then 
 from Equation 1 we have: 
 
 K 
 
 (f c /f) 3 (a/a c )' 
 
 (3) 
 
 where f and a refer to the fractional 
 porosity and the surface area per 
 unit bulk volume, respectively, and 
 where the subscript "c" designates 
 the system after cement has been 
 added. 
 
PREDICTED EFFECTS OF SAND CONSOLIDATION 
 
 Then: 
 
 and: 
 
 A/A = 
 
 Z47TR N 
 
 V* 
 
 N [4^(R + r ) 2 - 2^(R + r)(r)(n) ] 
 
 V B -{Z(N [l.337r(R +r ) 3 -Z0.337T(n)(r 2 )(3R - r) ])} 
 
 V D - Zl.337TR n 
 B 
 
 (4) 
 
 (5) 
 
 where N is the number of R-sized spheres per unit cell volume; (n/2) is the 
 number of contacts between each R-sized sphere; V R is the bulk volume of the 
 unit cell; and r is the thickness of the layered cement. 
 
 For the special case of a rhombohedral packing (E spheres only), where 
 R = 1 , we may write : 
 
 (1.467 - I2.566r+ 25.133r 2 - 16.755r 3 ) 3 
 
 K= 0.317 
 If J-spheres of radius 0.414 are added, then we have 
 
 2 2 
 (1 - 4r - 5r ) 
 
 (6) 
 
 K= 0.854 
 
 (1.171 - 14.717r+ 64.084r 2 - 34.046r 3 ) 3 
 
 2,2 
 
 (1.171 - 6.685r - lOr ) 
 
 (7) 
 
 Calculations with equations 6 and 7 give the results tabulated in table 3 
 and shown in figure 3. Also shown is the influence on permeability where 
 layered cement is added to a cubic packing; these results also have been ob- 
 tained from calculations based on equations 3,4, and 5. 
 
 As in the case of adding pendular cement, layered cements also decrease 
 the surface area of the resulting system. This is because of the large number 
 of contacts between adjacent spheres which therefore limits the growth of any 
 one sphere. The effect is not large, however, and is more than offset by the 
 decrease in porosity which results when the cement is deposited, hence, K is 
 seen to decrease markedly in a way proportional to the degree of consolidation. 
 
 Table 3 and figure 3 show essentially the same trend, whether the packing 
 be cubic, rhombohedral, or filled-rhombohedral. It is true that adding a layer 
 of given thickness results in more pore space being filled in the rhombohedral 
 than in the cubic packing, but the decrease in permeability is also greater, as 
 would be expected, because of the greater packing density in the former case. 
 
 Regarding the latter, it should be apparent from equations 4 and 5 that still 
 further decreases in permeability result when the packing density is held constant 
 but the number of contacts is diminished. For example, calculation shows that 
 when n is decreased from 12 to 6 for the unfilled-rhombohedral packing, and 
 when r is taken as 0.1, then K is decreased from 0. 092 to 0. 008. 
 
 Of passing interest is the fact that all of the points in figure 3 showing the 
 decrease in permeability due to layered cements fall on approximately the same 
 curve regardless of the packing form. That is, for the systems studied, K seems 
 to be a unique function of the amount of layered cement added, whereas the 
 same is not true for the various reported cases of pendular cement which have so 
 far been investigated. 
 
10 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 Table 3. - Effect of Layered Cement on Permeability 
 
 
 Thickness 
 
 (A/A c )' 
 
 (f c /f)" 
 
 Fraction 
 
 of 
 
 K 
 
 
 of layer 
 
 r 
 
 pore spac 
 occupied 
 cement 
 S 
 
 :e 
 by 
 
 
 Unfilled 
 
 rhombohedral 
 
 packing 
 
 0.001 R 
 
 0.005 
 0.010 
 0.050 
 0.100 
 
 1.0040 
 1.0206 
 1.0422 
 1.2698 
 1.8182 
 
 0.9744 
 0.8781 
 0.7688 
 0.2319 
 0.0279 
 
 0.0086 
 0.0424 
 0.0839 
 0.3856 
 0.6967 
 
 
 0.9823 
 0.9145 
 0.8351 
 0.3739 
 0.0922 
 
 E, J-filled 
 
 rhombohedral 
 
 packing 
 
 0.001 R 
 0.005 
 0.010 
 0.050 
 
 1.0116 
 1.0595 
 1.1255 
 2.0790 
 
 0.9630 
 0.8266 
 0.6812 
 0.1314 
 
 0.0125 
 0.0615 
 0.1201 
 0.4916 
 
 
 0.9742 
 0.8758 
 0.7667 
 0.2732 
 
 Unfilled 
 
 cubic 
 
 packing 
 
 0.001 R 
 
 0.005 
 
 0.010 
 
 0.050 
 
 0.100 
 
 1.0020 
 1.0101 
 1.0207 
 1.1198 
 1.2913 
 
 0.9901 
 0.9515 
 0.9066 
 0.5906 
 0.3202 
 
 0.0033 
 0.0164 
 0.0328 
 0.1610 
 0.3168 
 
 
 0.9921 
 0.9611 
 0.9254 
 0.6614 
 0.4135 
 
 Further calculations for examples of filled porosity, where tertiary, quater- 
 nary, and quinary spheres are present, will show the same trend. This leads to 
 the conclusion that the Kozeny equation always predicts a decrease in perme- 
 ability when layered cements are added to the porous matrix. Evidently the ef- 
 fect is related to the amount of cement added, and to the nature of the packing 
 arrangement (that is, the number of contacts between adjacent spheres and the 
 particle size distribution). 
 
 Of some importance nevertheless is the similarity between the model which 
 has been used above to represent sand consolidated with layered cements, and 
 the model which can be selected to represent the consolidation of porous media 
 by sintering or compaction. For the simplest example, consider a cubic packing 
 of uniform size lead balls of radius R, and then apply a triaxial load so that the 
 original six point-contacts on the surface of each sphere become flattened to 
 circular contacts. 
 
 If now it is assumed that the density of the material is not changed, and 
 that, except for the flattened areas, the remaining surfaces of the compressed 
 lead pellets assume an approximate spherical shape of radius S, then: 
 
 a/a 
 
 S 2 - 3S(S + Z - R) 
 
 c 
 
 3 . _*: 
 
 V 
 
 f /f = 
 
 8(R - Z) - 1. 33^S + 2tt(Z + S - R) (4R - Z - S) 
 
 (8) 
 
 (9) 
 
 3.81 R 
 
 where Z/R is the fractional flattening of each sphere, and where Z, R, and § 
 are related by: 
 
 4.19 S 3 - 6.26 (Z+ S - R) 2 (4R - Z - S) - 4.19 = 
 
PREDICTED EFFECTS OF SAND CONSOLIDATION 11 
 
 Physically, this model and the layered cement model have an equivalent 
 appearance (fig. 1), which is that of spheres grown bigger during consolidation, 
 and which now have flat surfaces of contact with adjacent spheres. There is, 
 however, a basic difference in the two lithification processes. When layered 
 cements are deposited, the grain volume increases, decreasing pore volume 
 (and porosity) since bulk volume remains constant. In the sintering process, 
 on the other hand, the bulk volume decreases and the grain volume remains 
 constant, and consequently the pore volume (and porosity) decrease in what 
 proves to be a significantly different way. 
 
 Equations 8 and 9 show that permeability is decreased when a sediment is 
 squeezed together and compacted, a result which follows expectations. For 
 example, in the extreme case of compacting so that the sphere centers are 
 brought 10 percent closer in all directions (i.e. , Z/R = 0.1), the Kozeny equa- 
 tion predicts that the reduced permeability will be only 7 percent of the original 
 permeability (i.e. K= 0.0714). And if there is preferential orientation in the 
 compaction, the resultant permeability values will have a directional variation 
 not unlike that found in nature where the overburden loads a more or less later- 
 ally unconfined sedimentary stratum. 
 
 DISCUSSION 
 
 As it occurs in nature, cementation must result in uniform distributions (i.e. 
 layered cements) and/or localized depositions (i.e. pendular cements); or, per- 
 haps, compression and/or fusion occurs at points of grain contact due to the 
 compaction of the overburden sediments. 
 
 If various types of sphere packings are taken as the model of the originally 
 unconsolidated sand, the Kozeny equation predicts that permeability is decreased 
 as layered cements are added. This is because of the large decrease in free 
 volume (porosity), which more than offsets the effect of a slight decrease in the 
 pore surface area. 
 
 With these same models, the Kozeny equation predicts an increase in perme- 
 ability when the cements are located at points of grain contacts as pendular 
 rings. This is because a large decrease in pore surface area attends the loca- 
 tion of pendular cement, which offsets the rather negligible opposing effect of 
 decrease in porosity. 
 
 Sintering and compaction, it has been shown, produce a result similar to 
 the effect of adding layered cements; and since it is ordinarily supposed that 
 permeability should decrease with consolidation, one is left only with the need 
 to rationalize the paradox of finding intuition contradicted in the case of sands 
 cemented with pendular ring deposits. 
 
 At lease three explanations are possible. It has already been proposed (Rose, 
 1957a and 1957?) that the Kozeny equation itself might not accurately express 
 the phenomena being considered. Such a view is given much support by the 
 recent analysis of Scheidegger (1957), who points to various defects of the 
 Kozeny theory. On the other hand, we must admit that the Kozeny theory has 
 been applied with success to the treatment of fluid flow in both unconsolidated 
 and consolidated porous media (Carman, 19 5 6). 
 
12 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 One expects, therefore, to meet some success in transiting from an uncon- 
 solidated sand (where Kozeny theory certainly applies) to a consolidated sand 
 (where Kozeny theory sometimes applies). 
 
 For example, Carman (19 5 6) notes that the "two extreme cases of cubic and 
 close packing gave fairly close agreement. . . with the Kozeny equation. " 
 Elsewhere Carman implies "the failure of the Kozeny theory for most consolidated 
 media, " when this is observed, is related more to the difficulty of independently 
 measuring tortuosity and specific surface area, rather than to any fundamental 
 defect in the theory. And, as will be noted, the work discussed in this paper is 
 free from any uncertainty about tortuosity and specific surface area. 
 
 The second explanation of the anomoly has to do with the question as to 
 whether or not pendular ring cements actually are found in nature. If such con- 
 figurations do not exist, and if natural cementation is always of the layered and/ 
 or sintered and/or compacted type, then the Kozeny theory conforms with the ex- 
 pectation that consolidated media should have a lower permeability than the 
 original unconsolidated matrix. 
 
 The question of the prevalence of pendular ring cements in nature is involved. 
 Little information is to be found in the geologic literature. In essence, one must 
 know from the standpoint of the petrophysicist the structure of the original sand, 
 and then trace the history of diagenetic processes which have occurred over geo- 
 logic time to produce the resultant consolidated sand. And to find an explicit 
 solution, all of these factors must be known with some quantitative detail. 
 
 Perhaps as a compromise one might be tempted to postulate that natural 
 cementation is principally of the layered type, with only a minor tendency for 
 additional pendular cement to be deposited. Such a configuration is illustrated 
 in figure lc, which is a simplified depiction of the probable deposition of extra 
 cementing material at the junction point of adjacent sand particles as the latter 
 grow in size with layered cement. 
 
 From geometry, one can state that the ratio of the surface area added to that 
 subtracted at each contact can be approximated by: 
 
 (2Rr+ r 2 )K y (10) 
 
 (R+r) 
 
 where R is the sphere size, r is the thickness of the cement layer, and y is a 
 
 measure of the amount of pendular cement deposited on top of the layered cement. 
 
 It is easily seen that for all reasonable values of r and y, the subtracted area 
 
 greatly exceeds the added area, which means that the (A/A ) ratio thereby is 
 
 — c 
 
 markedly decreased; hence K tends to become unreasonably large. 
 
 A third idea to be considered is an interesting possible explanation as to why 
 
 the Kozeny theory predicts an increase in permeability for cases of pendular ring 
 
 deposition. The fact of the matter is that no one can say at this time what the 
 
 relationship is between the permeability of consolidated porous media and the 
 
 permeability of the parent matrix prior to diagenesis. Evidently no experiments 
 
 of this sort have been done except trivial ones which demonstrate that ultimately 
 
 a loose sand can be plugged (partially or completely) if enough cementation 
 
 occurrs. Available data, however, do not show whether permeability is increased 
 
 or decreased during the period of initial consolidation, and this is the question 
 
 being considered here. 
 
PREDICTED EFFECTS OF SAND CONSOLIDATION 13 
 
 Among other factors, it is clear that the Kozeny theoretical inclusion of the 
 influence of porosity and specific surface area on permeability is reasonable, 
 whether or not quantitative accuracy has been obtained. That is, porosity is a 
 measure of the total free volume, and hence for isotropic media, porosity is a 
 measure of the total cross -sectional area available for flow. The specific sur- 
 face area term bears a relationship to the mean hydraulic radius (cf. Scheidegger's 
 treatment of this point, 1957), and as such establishes the pore size equivalence 
 between the porous rock and a bundle -of -tubes model. 
 
 The surface area concept is also important in another way. Thus, in the final 
 analysis, the conductivity which porous media afford to the flow of fluids must 
 be related to the extent to which the pore wall boundaries exert a "drag" on the 
 moving fluids. Clearly, the greater the surface area (for a given porosity condi- 
 tion), the greater will be the influence of the zero-velocity boundary surfaces in 
 resisting fluid flow. 
 
 A simple case will illustrate the point. If we choose two capillary tubes of 
 unit length so that together they have the same internal volume as a single larger 
 tube of unit length, then the smaller tubes, when arranged in parallel, afford a 
 flow conductivity which is only one-half that afforded by the larger tube alone. 
 The total cross-sectional area (porosity) has not been changed, but the zero- 
 velocity boundary surfaces have increased by a factor of the square root of two. 
 Obviously, the decrease in flow conductivity which is indicated by Poiseuille's 
 law is directly attributable to the fact that the fluid must drag over more pore 
 wall surface area with no compensating increase in hydraulic radius or total 
 cross-sectional area. 
 
 In the case of packings of sphere particles, the first addition of pendular 
 cement at points of grain contact also does not change the porosity significantly; 
 but the surface area of the pore boundaries is greatly decreased, as has been 
 shown from geometry. Hence, perhaps it is not unreasonable that the permeability 
 should increase even though the surface area removed by the deposition of the 
 pendular cement is that portion most distant from the central pore spaces. 
 
 In order to study the magnitude of the effect, one needs experimental data 
 now nonexistent; for otherwise we have only the indications of the Kozeny theory 
 which are by no means conclusive. Presumably, there is little hope of applying 
 the "drag" theoretical approach to the permeability problem (Scheidegger , 1957). 
 because of the formidable obstacles of analysis which will be encountered. 
 
 CONCLUSIONS 
 
 1) Kozeny theory predicts an increase in permeability when various types of 
 sphere packings have cement deposited as pendular rings at points of grain con- 
 tact. 
 
 2) Kozeny theory predicts a decrease in permeability if the cement is depos- 
 ited as an even layer, or if the particles are pressed together by compaction. 
 This trend is diminished, and perhaps reversed, if pendular cement is subse- 
 quently deposited over the layered cement at certain preferential locations. 
 
 3) Since it is usual to assume that permeability should always decrease dur- 
 ing natural consolidation, one might question the validity of the Kozeny theory 
 in this application. 
 
 4) One also might question the reasonableness of the models which have been 
 selected, suggesting the need for further geologic work to classify the physical 
 characteristics of sedimentary rock before and after diagenesis. 
 
14 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 5) Finally, one might go to the laboratory and see if permeability can be in- 
 creased in certain cases, and thus reconcile what initially have been assumed 
 to be conflicting ideas. 
 
 REFERENCES 
 
 Carman, P. C. , 1956, Flow of gases through porous media: Academic Press Inc. 
 New York . 
 
 Spock, L. E. , 1953, Guide to the study of rocks: Harper and Bros. , New York. 
 
 Rose, H. E. , 19 50, The permeability of cemented beds and sandstones: Jour. 
 Inst. Water Engrs., v. 4, p. 535. 
 
 Rose, Walter, 1957a, Fluid flow in petroleum reservoirs. I - The Kozeny para- 
 dox: Illinois Geol. Survey Circ. 236. 
 
 Rose, Walter, 1957?, Volumes and surface areas of pendular rings: Jour. 
 Applied Physics. (Ms. accepted, 1957; publication date may be 1958.) 
 
 *Scheidegger, Adrian E. , 1957, The physics of flow through porous media: 
 The Macmillan Company, New York. 
 
 White, H. E. , and Walton, S. F. , 1937, Particle packing and particle shape: 
 Jour. Am. Ceramic Society, v. 20, p. 155. 
 
 * Compare with Scheidegger's personal communication to the author, dated 
 September 18, 1957: "I think the difficulties with the Kozeny equation orig- 
 inate in the fact that it assumes in essence that the resistance to flow re- 
 sults entirely from the sticking of the liquid to the walls of the pores and 
 thus automatically arrives at an equation where permeability is universally 
 proportional to the specific internal surface of the medium. All other effects 
 that might contribute to the flow resistance are summarily neglected. " 
 
Illinois State Geological Survey Circular 242 
 14 p. , 3 figs., 3 tables, 1957 
 
nnnnm 
 
 CIRCULAR 242 
 
 ILLINOIS STATE GEOLOGICAL SURVEY 
 
 URBANA "^'"