Wettability as Related to Capillary Action in Porous l\1edia
SOCONY MOBIL OIL CO. JAMES C. MELROSE DALLAS, TEX.
ABSTRACT interpreted with the aid of a model employing the concept of a cylindrical capillary tube. This
The contact angle is one of the boundary approach has enjoyed a certain degree of success Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 conditions for the differential equation specifying in correlating experimental results. 13 The general the configuration of fluid-fluid interfaces. Hence, ization of this model, however, to situations which applying knowledge concerning the wettability of a involve varying wettability, has not been established solid surface to problems of fluid distribution in and, in fact, is likely to be unsuccessful. porous solids, it is important to consider the In this paper another approach to this problem complexity of the geometrical shapes of the indi will be discussed. A considerable literature relating vidual, interconnected pores. to this approach exists in the field of soil science, As an approach to this problem, the ideal soil where it is referred to as the ideal soil model. model introduced by soil physicists is discussed Certain features of this model have also been in detail. This model predicts that the pore structure discussed by Purcell14 in relation to variable of typical porous solids will lead to hysteresis wettability. The application of this model, however, effects in capillary pressure, even if a zero value to studying the role of wettability in capillary of the contact angle is maintained. The model is phenomena has not previously been attempted in generalized to situations in which the contact angle detail. In the present paper, additional features of takes on values between zero and 40 degrees. For the model are introduced. These features are the imbibition branch of the capillary - pressure critical in determining the quantitative behavior of function, the model predicts a considerable departure the model. from the usually assumed cos e relationship. In fact, according to the model, it is possible that a GENERAL FEATURES OF displaced wetting phase will not be able to reimbibe, CAPILLARY HYDROSTATICS even when the contact angle does not exhibit BASIC PRINCIPLES hysteresis. When the interstices of a typical porous solid INTRODUCTION are occupied by two or more immiscible fluid phases, the fluids are microscopically commingled. The subject of capillarity in porous media has Hence, fluid-fluid interfaces are found within a long been of interest in many branches of engineer certain fraction of the pore openings. The funda ing and applied science. The earlier investigators mental equation of capillarity specifies the 1 s were those concerned with the physics of soils. - configuration of these fluid-fluid interfaces. This More recently, petroleum engineers and others is known as the Laplace equation, when derived dealing with the problems of petroleum production from mechanics, and as the Gibbs-Kelvin equation from reservoir rock have given much attention to when deri ved thermodynamically .15 6 10 the subject. - Also, important additions to the Given two fluid phases, a and {3, in hydrostatic literature of capillarity have been contributed from equilibrium, the Laplace equation states that the the field of chemical engineering. 11,12 These attest respective fluid pressures in regions close to the to the wide range of industrial applications in interface are related by which capillary phenomena playa role. The present paper is concerned with the role pO _ pf3 = (J"of3(_I_ + _1_) ..... (1) which wettability plays in capillary action in r I r 2 porous media. As is well known, capillary-rise (or Here aa{3 is the surface or interfacial tension and capillary pressure) phenomena have frequently been r1 and r2 are the principal radii of curvature of the Original manuscript received in Society of Petroleum Engi surface or interface. The pressure difference, pu_ neers office Nov. 27, 1964. Revised manuscript of SPE 1085 p {3, is the capillary pressure, Pc' As Buff 15 has received June 22, 1965. Paper presented at SPE-AIChE Joint Symposium on Wetting and Capillarity in Fluid Displacement shown, Eq. 1 states the condition for hydrostatic Processes held in Kansas City, May 17-20, 1959. equilibrium within the two-phase confluent region, lReferences given at end of paper. which is referred to as the interface. It thus can
SEPTEMBER, 1965 259 be regarded as a two - dimensional principle of related phenomena in porous media have been based hydrostatics. on experiments in which examples of non-zero con In the event that the surface of the solid phase tact angle have been largely excluded. y, within which a and (3 are enclosed, has a On the other hand, evidence has accumulated sufficiently symmetrical configuration, the interface which tends to show that in the case of the two a{3 becomes a surface of revolution. The radii of liquid phases encountered in oil reservoirs, i.e., curvature can then be written in well- known petroleum and formation water, the contact angle differential forms, and Eq. 1 becomes a differential measured through the water phase is often far from equation. However, this is also true in principle zero. 17 ,18 It seems highly probable that this even if the boundary surface lacks the required this situation can arise simply from the adsorption symmetry. The solution to Eq. 1 will always of polar constituents, which are present in describe the configuration of the interface, which petroleum in considerable quan tities, on the high is the central problem of capillarity. energy surface of the reservoir rock. As Zisman The boundary conditions for Eq. 1 are of two and co-workers have shown, when such adsorption types, both of which must be known if solutions to occurs on a high -energy surface, it becomes Eq. 1 are to be found. The first type, as suggested comparable to typical low - energy surfaces in above, specifies the spatial configuration of the hydrophobicity. 19,20 Furthermore, the adsorption solid surface, while the second is expressed in processes involved can clearly be expected to give Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 terms of the contact angle ea{3y, which the interface rise to considerable hysteresis in the water-oil a{3 subtends at the surface of the solid phase. contact angle. Experimental evidence for such Whereas aa{3 is a property of the two-phase surface hysteresis has in fact been presented by Benner, of contact, ea{3y is a thermodynamic property which Dodd, and Bartell. 21 It should be mentioned, depends on the three interfacial tensions, aa{3, a ay however, that to date very little experimental work and a{3y, in the immediate neighborhood of the has been carried out under conditions other than of the three-phase line of contact. room temperature and atmospheric pressure. This dependence is given by the classical Young equation, which actually corresponds to a one HYDROSTATIC HYSTERESIS dimensional principle of hydrostatics. 15 Since, The contact-angle hysteresis just referred to is however, no means exist for observing the quantities separate and distinct from the hysteresis in aaYand a{3y independently, Young's equation cannot capillary rise which is occasioned by the intercon be used to compute values of the contact angle. nected "ink - bottle" type of pore morphology The wettability of a solid surface, with respect to characteristic of porous media. More specifically, any two given fluid phases, can therefore be the latter effect is due to the fact that, within each specified only by values of the contact angle e individual pore space, a definite range of stable obtained from direct measurements. An interpreta fluid - fluid interface configurations exists. The tion, in terms of Young's equation, of contact-angle configurations having the greatest and least data taken from the literature has been presented possible values of the mean interface curvature elsewhere .16 correspond to the maximum and minimum values of The approach, by means of which wettability the capillary pressure sustained by the interface effects in porous media are to be studied in this and, hence, to the capillary rise. paper, can now be briefly stated. First, the intrinsic The conditions which establish the limits of wettability of the fluid-fluid-solid system (a{3y) is stability for interface configurations in porous defined by a specified value of the contact angle, media will be discussed in more detail below, in ea{3y. Second, the relationship of e to the capillary connection with the analysis of the ideal soil pressure is determined entirely within the context model. Because the relationship between interface of solutions to Eq. 1, taking the boundary conditions curvature and capillary pressure, Eq. 1, actually to be specified by (1) the contact angle and (2) the gives the condition for hydrostatic equilibrium in geometrical configurations of the surfaces of the the two-fluid phase system, the resulting hysteresis solid phase. The latter will in turn be specified by in capillary rise will be referred to as hydrostatic a detailed examination of the ideal soil model. For hysteresis. such a model, this approach appears to be the only The importance of this type of hysteresis has conceivable one which can claim tube well-founded, been particularly emphasized by soil physi in the sense that any such approach must rely cists, 1,2,4,5 who have unfortunately dealt entirely exclusively on the known physical principles which with cases of zero contact angle. Hydrostatic govern static capillary phenomena. hysteresis has been, however, in many instances completely ignored by those interested in contact CONTACT ANGLES AND ADSORPTION angle hysteresis. 22 Thus the important and In much of the experimental work reported in the necessary distinction between the two types of literature, one of the two fluid phases has been hysteresis has not always been maintained. gaseous. I- 6,8,10-13 Further, in all but a few of The effects of hydrostatic hysteresis have been these cases, the contact angle measured through most thoroughly explored in studies of capillary-rise the liquid phase has been zero. Thus, existing phenomena in unconsolidated beds or packs of theories and correlations for capillary - rise and small, dense, nonporous particles. Such studies
260 ~OCIETY OF PWI'IIOLEI M E'IGI'iEEHS JOIIIl."(.\1. are most easily interpreted if the grains do not limited range of fluid-fluid-interface curvatures depart radically from a spherical shape and if the which correspond to stable interface configurations grain - size distribution is neither too broad nor for pores of a given size. It is seen that the composed of more than one peak. Also, the average capillary-rise data directly reflect the phenomenon grain size should be small as compared with the of hydrostatic hysteresis. ratio of a to the density difference t1 p between the two phases. Under this condition the capillary LIMITATIONS OF PREVIOUS STUDIES pressure varies negligibly over the fluid - fluid In the interpretation of experimental capillary-rise interface within any particular pore space. Hence, data, it is necessary to adopt a model for the Eq. 1 can be written in terms of the capillary rise, description of the pore geometry, i.e., the configur h, ation of the solid surfaces which confine the fluid phases. For this purpose the model most frequently h= _a_ (_1- + _1_) ...... (2) employed is based on the cylindrical capillary tube. g6p r l r 2 A bundle of such tubes in parallel, having a highly curved, or tortuous, effective flow path, represents As was pointed out by the earliest investigators!, 6 the pore space of a packed bed of particles. Using
the capillary rise is a function of the fraction the concept of a hydraulic radius, 13 both permeability Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 of the total pore space occupied by one of and capillary rise can be discussed in terms of the two fluid phases. This fraction, when this model. Hence, as expressed by the Leverett referred to the fluid phase having a pressure j-function,6 capillary- rise data can be correlated deficiency with respect to the other (here, phase with permeability. Moreover, the model predicts (3), is called the "wetting-phase saturation", S. A that the magnitude of the j- function is primarily a typical functional dependence of capillary rise on measure of the tortuosity of the flow channels. saturation is given in Fig. 1. Due to the phenomenon For the case of drainage capillary-rise values, of hydrostatic hysteresis, the further specification the correlations of this type reported by different of the sign of saturation changes is required. Thus, investigators6,8, 12,13 are in good agreement. Also, Fig. 1 refers to the "desaturation" or "drainage" the observed values of the j-function correspond to branch of the capillary-rise function by means of a reasonable magnitude for the tortuosity. These arrows adjoining the upper curve, showing decreas studies, however, have been restricted to gas-liquid ing values of S. The lower curve, with increasing systems, and in only one case does the contact values of S, is called the "imbibition" branch of angle have a non-zero value. This case, represented the capillary- rise function. Fig. 1 also shows by the mercury injection data,8 appears to have a typical scanning loops, which lie between the value of the j-function which is 5 to 10 per cent drainage and imbibition curves. Thus, at any given greater than the other results. There does exist, in value of the saturation, there is a difference addition, a brief report 7 of measurements of drainage between the drainage and imbibition values of the capillary pressures, in which liquid-liquid systems capillary rise. As stated above, this is due to the were compared with a gas-liquid system in the same porous medium. In this work, the liquid-liquid-solid systems gave a zero value of the contact angle as A I R / WATER SYSTEM measured through the wetting phase. No appreciable GRAIN SIZ E: 25 TO 30 MESH differences in behavior from that of the gas-liquid 25 system were observed. It is significant that among the various studies E in which correlations with permeability were () 6 • 20 established, only that of Leverett included data ~ for the Imbibition branch of the capillary-rise curve. W en In this work, the j-function' for imbibition was about - 65 per cent of that for drainage. The capillary-tube 0:: 15 model, of course, completely fails to account for >- 0:: this discrepancy, since it provides no mechanism
SEPTEM6ER, 1965 261 the observed values of capillary rise was the of interconnected pores or cells, of which there are average grain size. Thus, a normalized interface three types. The first of these is the void space curvature can be defined as enclosed by eight spheres in open cubic packing. The other two arise from rhombohedral packing and are (1) the space enclosed by four spheres in J = Rgr (_1_ + _1_) = g,0,ph Rgr ••.• (3) tetrahedral arrangement, and (2) the space enclosed 2 r I r 2 2 contact angle. However, in a recent investigation Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 from cubic packing, there are six such openings by Harris, Jowett, and Morrow, 23 packed beds of for a given cell or element of void space. Similarly, galena were employed, and contact-angle variations for the two types of cells found in rhombohedral achieved by treatment with solutions of alkyl packing, the number of triangular array pore xanthates. Since adsorption is involved, it is openings will be four and eight, respectively. possible that some contact - angle hysteresis Also shown in Fig. 2 is a cross-section view in occurred. which are illustrated several fluid-fluid interfaces In summary, the experimen tal results which have in different positions. As indicated in the figure, been reported show that the capillary-tube model has an angle ¢ may be defined by the line of sphere a limited utility. Only when drainage conditions centers and the radius vector directed to the point prevail does the model provide a satisfactory basis of .contact between the interface and the sphere. for interpreting capillary- rise measurements in ThIs angle thus defines the position or location of unconsolidated beds. No insight is provided for the the interface relative to the pore opening. The consideration of the imbibition pressures which various interfaces in Fig. 2 are shown, for sim have been measured. Furthermore, only one case plicity, as having zero values of the contact angle involving a well established non-zero value of the made with the solid surface. The generalization contact angle has been reported, and the results e of the present discussion to include non-zero values obtained were restricted to the drainage branch of of the contact angle will be introduced later. the capillary-pressure curve. Thus, there is no The interface cross-sections illustrated in Fig. experimental evidence to suggest that the depend 2 are assumed to have a constant curvature, which ence of capillary rise or capillary pressure on contact angle is properly expressed by the factor cos e, as predicted by the capillary-tube model. In SQUARE ARRAY TRIANGULAR ARRAY view of this, since the model fails to account for the phenomenon of hydrostatic hysteresis, it is reasonable to expect a similar failure in the case of the contact-angle dependence which the model predicts. B DEVELOPMENT OF THE IDEAL SOIL MODEL B GEOMETRIC FEATURES OF THE MODEL ANGLE AOB=7]=45° ANGLE AOB =7]= 30· The problem of predicting values of capillary rise was studied by soil physicists quite early in the development of the subject. The "ideal soil model" was adopted to gain a theoretical under standing of unconsolidated porous media. This ANGLE DOC = cp mod~l was defined simply by regarding the porous ANGLE DOE =- cp med~um to be composed of perfectly spherical F o partIcles of identical size. The packing of the OF = { 262 ~()CIETY OF PETIIOLEI'M E:\GJ:\EEH~ JOIJRl'iAL is, of course, an approximation. It may be justified encountered for negative values of the angle ¢. by, first, recalling that the actual interface is Hence, for the drainage case, f is given by assumed to have a constant mean curvature, i.e., capillary pressure, over its surface. But the actual J = I ...... • . (5) interface, in the cross-section plane of Fig. 2, may max (cos 7] tl-I be considered to be approximately tangent to a synclastic surface of revolution. The only synclastic For the pore openings defined by the square and surface of revolution having a constant mean triangular arrays, the values of fare, respectively, curvature is a spherical surface; hence the assumed 2.414 and 6.464 (see Table 1). These values were interface configurations of Fig. 2. given by Hackett and Strettan3 and by Haines. 2 Under the approximation just discussed, which For imbibition processes, the situation is just admittedly is rather crude, the mean curvature can the reverse of the drainage process. In the region easily be derived. Since the curvature is inversely of stability, any advance of the wetting phase proportional to the grain radius Rgr, it is natural involved a decrease in interface curvature. At the to express the result in terms of the normalized point where such a decrease is no longer compatible curvature f, defined in Eq. 3. As a function of the with the advance of the interface, instability occurs. angles ¢ and Tf, where the latter specifies the type Again the interface is forced to jump very rapidly Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 of array defining the pore opening (see Fig. 2), to a stable position. Such a process was also the curvature is then given by observed by Haines. 1,2 The minimum interface curvature which is configurationally stable in the cos ~ sense described, clearly establishes the value of J (9= 0) .. (4) (cos 7] tl- cos ~ the imbibition capillary pressure. Because the conditions which determine this lower limit of INTERFACE CONFIGURATIONAL STABILITY stability are generally more complicated than the It is clear from Eq. 4 and from Fig. 2 that the drainage case, further discussion of imbibition will interface curvature depends on the position which be deferred to a later section. the interface occupies within the pore space. This COMPARISON WITH EXPERIMENTS ON DRAINAGE fact is the key to understanding hydrostatic Haines,2 and Hackett and Strettan 3 have reported hysteresis, and, hence, to the difference between drainage measurements using special beds composed imbibition and drainage capillary pressures. Thus, of a single layer of identical spheres, packed in there is a definite, limited range of positions which the two ideal arrays. The measured values of a stable interface may occupy. As the non-wetting f were about 5 and 12 per cent below those predicted phase begins to enter a given cell, it is clear that by Eq. 5 for the square and triangular arrays, this may be done in a reversible (quasi-static) respectively. It is interesting also to compare the manner only as long as any advance of the interface ideal soil values with the measurements obtained is accompanied by an increase in the capillary with various random packs, which were discussed pressure. But if a point is reached where the above. At a porosity of 0.4, the observed value of advancing interface undergoes a decrease in curvature and hence in capillary pressure, such an f is about 2.7. This is only 11 per cent greater interface is no longer configurationally stable. It than the value given for the ideal square array "jumps" very rapidly, displacing the wetting (porosity = 0.4764). phase from the cell. It is reasonable that the observed results should The conditions governing the stability of fluid more closely approximate the value for the square fluid interfaces were discussed in detail by Gibbs. 25 array than that for the triangular array. First, Unfortunately, this discussion has been ignored by assuming that the beds to which the experimental all later writers on capillarity, and hence little data refer are random mixtures of the two ideal attention has been paid to the principles underlying types of packing, a porosity of 0.4 corresponds to hydrostatic hysteresis. An exception to this neglect of an important feature of capillarity in porous media is the particularly clear discussion presented by TABLE 1 - NORMALIZED INTERFACE CURVATURES FOR Miller and Miller. 26 THE IDEAL SOIL * Pendular-Ring Interface jumps in displacement processes were Continuous Interface Interface Pore first observed and described by Haines,1,2 and Opening Position, Curvature, Position, Curvature, may be demonstrated very easily without any Displacement Mechanism Array p, degree __J__ 'P, degree J special apparatus. All that is required is a pack of Drainage square 2.414 29.9 2.414 coarse glass beads in a glass tube, through which Drainage triang. 0 6.464 20.3 6.464 Equil. imbibition square** 49.3 the interface jumps can be seen. 0.858 40.7 0.858 Equil. imbibition triang. 27.8 3.265 26.9 3.265 The maximum curvature of an interface passing Reinvosion, non-eq. square 60.1 0.541 29.9 2.414 through a pore opening represents the upper limit Reinvasion, non-eq. triong. 34.4 2.495 20.3 6.464 of configurational stability. The corresponding Primary displ., non-eq. square 90 0.000 0 capillary pressure is then the drainage pressure Primary displ., non-eq. triong. 54.7 1.000 0 * Contact angle::: O. for the given pore opening. From Eq. 4, as well as ** If ~ = 45°, thon SEPTEMBER, 1965 263 about 69 per cent of the void volume having pore pOSitIOn of a pendular-ring interface is specified, entrances of the square array type (open cubic as in Fig. 3, by the angle'll between the line of packing). Also, the experimental data are very sphere centers and the radius vector directed to probably based on capillary pressures measured at the interface-sphere point of contact. wetting-phase saturations of the order of 0.7. This, Because the pendular-ring interface is a surface of itself, provides a bias toward the pore openings of revolution, its configuration in the plane of Fig. having the smaller drainage curvatures. Finally, in 3 represents a complete specification of both considering the effect of slight distortions from the principal curvatures. The principal normal corre ideal pore - opening configurations, it maybe sponding to the curvature of larger magnitude is assumed that a given distortion produces only a that lying in the plane of Fig. 3, while that of the slight increase in the value of J in the case of the smaller curvature is always in a plane perpendicular square array, but a much larger decrease in J in to that of Fig. 3. The configuration as shown has the case of the triangular array. Evidence bearing a constant larger curvature. This is actually an on this point has been presented by Hackett and approximation, but the error involved is not large, Strettan. 3 as was shown by Fisher.27 Thus, the normalized A rather remarkable fact can now be pointed out curvature for the plane of Fig. 3 is - the ideal soil concept leads to predicted values Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 of drainage capillary pressures which are quite J = 1 . . • • • • . • • (6) comparable in magnitude to those predicted on the I (cos IJItl-1 basis of the capillary-tube model. It may be sur mised, then, that the application of this latter Under the approximation specified by Eq. 6, the model is somehow related to the particular config other principal curvature must vary from point to uration assumed by the interface for the drainage point along the curve shown in Fig. 3. At the situation. This configuration is defined by specifying interface-sphere contact point, this curvature is the location of the interface to be that for which ¢ given by = o. That is, the point of solid-interface contact lies exactly at the point of narrowest restriction in ...... (7a) the pore opening. Thus, it may be expected that unless this condition is satisfied, at least Moving along the curve, the curvature increases, approximately, the capillary-tube model will not be until it becomes, at the intersection with the line successful in predicting values of capillary rise or normal to line of centers at the point of intersphere capillary pressure. contact, ANALYSIS OF IMBIBITION PROCESSES J = -cos 1/1 ...... (7b) 2 sin IJI + cos IJI -I PENDULAR RING INTERFACES In addition to the interfaces of the type illustrated in Fig. 2, partially saturated porous media will CUBIC CELL commonly have a second type of fluid-fluid interface. 0 The interfaces discussed so far are considered to 'TJ ; 45 have continuity with many other interfaces of a 0 similar nature distributed through the porous medium. ANGLE DOH; t ; 90 Those of the second kind, however, are discontinu DOC; cP ous, enclosing what has been called a "pendular ring" of the wetting phase at points of intergrain GOH ;~ contact. In Fig. 3, a diagram is given for the ideal soil, showing a pendular ring of wetting-phase fluid situated at a point of contact of two grains, which o is immediately above an interface of the continuous type. Pendular-ring interfaces play a very important role in establishing the lower limit of interface TETRAHEDRAL CELL for stab,ility, i.e., in determining the magnitude of the imbibition capillary pressure. This conclusion 'TJ ; 30 0 is based on the work of W. o. Smith,S who apparently has been the only author to give any ANGLE DOH = t = 54.7 0 attention to this problem. ANGLE DOC; cP Under the assumption that the capillary pressure is the same at every point, pendular-ring interfaces ANGLE GOH = ~ are represented by anti clastic surfaces of revolution. Thus, the principal radii of curvature are opposite in sign. Just as the position of an interface lying o in a pore onpning is defined by the angle ¢, the FIG. 3 - PENDULAR-RING INTERFACES. 264 'iOCIETY OF Pl:THOLEI,M E"GIXEEHS JOI,H"AL Taking the average of the results given by Eq. 7, capillary pressure vs saturation curve which occurs the mean curvature of the pendular ring becomes at low saturations (see Fig. 1). Atkinson has attempted direct measurements of the angle 'P for J = 7 cos tjI - 5 i n tjI - 3 ...... (8) pendular rings formed by drainage processes. 11 He pr 8(I-cos tjI) found that for the square array, 'P was 30°. This agrees closely with the value obtained by solving Eq. 8, of course, is an approximate formula. Eqs. 5 and 8 simultaneously (see Table 1). Fisher27 and von Englehardt28 have presented Atkinson also measured the amount of wetting-phase numerical solutions to Eq. 1 for the pendular-ring fluid retained as residual saturation. This quantity interface. These solutions, although stated to be has received attention from several other investi exact deviate slightly from each other and from the gators. 12 ,28,29,31 solution represented by Eq. 8. Other authors have Under the condition that the non-wetting phase used approximate formulas based either on Eq. 7a is a gas, the formation of pendular rings also or 7b. 29 ,30 accompanies the process of imbibition. This occurs through the agency of the mass transfer processes ORIGIN OF PENDULAR RINGS of evaporation, diffusion and condensation. At Pendular rings of wetting-phase fluid, and hence equilibrium, the mean curvature of the pendular-ring Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 the corresponding interfaces, may have their origin interface must equal that of the continuous interface in either drainage (desaturation) or imbibition which corresponds to the imbibition capillary (resaturation) processes. In the case of drainage, pressure, as was pointed out by Smith.S Thus, the invasion of each individual pore or cell by the wetting-phase liquid is transferred by the processes non-wetting phase occurs through the mechanism noted from the continuous-liquid phase to the pen of an interface jump. But this process does not dular nngs, until the required equilibrium is result in complete occupancy of the void space by established. the non-wetting phase, since a pendular ring of When the non-wetting phase is itself a liquid, wetting-phase fluid is retained at each inter-grain the diffusional process is, of course, much slower. contact point. The curvature subsisting in a In this case it is possible to visualize an imbibition pendular-ring interface so formed has been a matter process carried out in such a way that the of some speculation. Several authors have suggested equilibrium between the advancing wetting phase that the curvature is equal to the maximum curvature and the non - continuous pendular rings is not established in the continuous interface, i.e., to the established. Hence, the curvature and capillary curvature corresponding to the drainage capillary pressure for the continuous interface remain smaller pressure. 9,11,29 in magnitude than the corresponding quantities for The basis for this conj ecture lies in the fact the pendular-ring interfaces, if such are present. that the interfaces which occupy pore openings Even for' the case of a gas phase as the non have continuity with other interfaces of the same wetting fluid, Morrow and Harris have raised the type through wetting-phase fluid configurations question of whether. equilibrium is established in which may be described as semipendular rings. It the imbibition process. 10 Certainly, as these is this fluid which is retained, having lost continuity, authors point out, the evidence is strong that the when the interface jumps to a new stable position. pendular rings formed by the drainage process do Simultaneously, the semipendular-ring interface, not tend to equilibrate rapidly with the continuous which has a curvature equal to the drainage value, fluid as changes in the drainage capillary pressure becomes a full pendular-ring interface. Hence, the occur. If this were to occur, it would be necessary supposition is justified that it has nearly the same for the vapor-phase mass transfer to take place over curvature as that corresponding to the drainage distances much greater than size of an individual capillary pressure. pore opening. On the other hand, for such an This conjecture has been tested recently by equilibration to be an important factor in imbibition, Morrow and Harris .10 The system studied was a the mass transfer required is local in nature, bed of glass beads in which water was displaced extending only over a distance comparable to an by air. Using a microcapillary probe, direct individual pore opening. Furthermore, the substantial measurements of the capillary pressure in pendular difference between drainage and imbibition capillary rings, relative to the capillary pressure in the pressure provides a somewhat greater driving force continuous wetting phase, were carried out. To an for such diffusional mass transfer. estimated accuracy of better than 5 per cent, it was MECHANISMS FOR IMBIBITION established that the pendular-ting capillary pres sures were the same as the drainage capillary It is now possible to discuss in detail the pressure. conditions under which a lower limit exists for the Pendular - ring fluid thus represents a major curvature of the continuous type of interface. Fig. contribution to "residual" wetting-phase saturation. 3 shows that the interface is stable for advancing When all, or nearly all, of the individual pore· spaces wetting- phase fluid only so long as ¢ < e - 'P. have been desaturated, the pendular-ring fluid is Here e is the angle which the line of centers of still retained throughout the porous medium. This the two spheres in successive layers of the packing is manifested by the sharp upward trend of the makes with its projection in the plane of the pore SEPTEMBER, 1965 265 opening. Thus, when the advancing interface comes Eqs. 4, 5, 8 and 10 to include the effect of contact in contact with the pendular -ring interface, the angle will be presented in a later section. position of which is denoted by the angle '¥, Numerical results for all of the above cases are instability ensues. The condition for the minimum given in Table 1. Both the square- and triangular curvature, then, is array models for the pore openings are included. Of particular interest are the values given for ¢=t-1jJ ...... (9) imbibition under equilibrium conditions, since this probably corresponds most closely to the available Substituting this relation in Eq. 4, the imbibition experimental results for imbibition capillary capillary pressure is given, in terms of the normal pressures. As pointed out, the observed ratio of ized curvature, by imbibition to drainage pressure is about 0.65. The results given in Table 1 lead to values for this ...(10) ratio of 0.36 and 0.51, for the square and triangular arrays, respectively. However, it is again reasonable to consider the Eq. 4, which expressed the curvature I as a effect of distortions from the ideal types of packing. function of the angle ¢, has been transformed to F or open cubic packing, such a distortion decreases give I as a function of the angle '¥. Recalling the the angle I; from its ideal value of 90°: According Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 discussion of the previous section, several possible to Eq s. 9 and 10, the value of the imbibition ways to specify the angle '¥ can be distinguished. curvature is quite sensitive to the magnitude of I;. These possibilities arise because of the slowness The minimum value which this angle could have is with which equilibrium may be established if the 45°, and, assuming this value, the curvature non-wetting phase is a liquid rather than a gas. 7r---,----,.--,----,---,----,---,----, Thus, in addition to the equilibrium case,S two non-equilibrium situations may be envisaged. In the 6 first of these it may be supposed that pendular W rings of wetting phase were formed by a drainage 0 « J(8,cf» process, as described previously. If the processes u.. 5 0::. leading to equilibrium then take place at a negligible W I-W- rate, the size of the pendular rings so formed will ZO:: 4 not change upon subsequent reinvasion of the -=> 01- DRAINAGE wetting phase. The second non-equilibrium case W« N> 3 J(8,cf> )max arises if pendular rings are entirely absent (,¥ = 0). -0:: ...J=> This situation occurs in the case of imbibition into «0 ~ 2 void space which is fully occupied by non-wetting 0:: phase. An example of this type of imbibition is 0 Z found in the displacement pressure studies of Bartell and Osterhof. 22 The various cases which have been discussed (including drainage) can be conveniently summar °O~---~2~---~4-----~6--~-8~---~IO~---~12~---IL4----~16 ized, and the various curvatures defined, as follows: INTERFACE POSITION.cf>.DEGREES 1. Drainage: FIG. 4 -- GRAPHICAL SOLUTION OF INTERFACE Idr = Imax (see Eq. 5) CURVATURE EQUATIONS, TRIANGULAR ARRAY. 2. Imbibition under equilibrium conditions: 7,,--~--,---,,---,----,---,----.---, I eq = Imin , with '¥ given by I pr = Imin (see Eqs. 8 and 10) 6 W 3. Imbibition by reinvasion, non-equilibrium con o « ditions: u.. 5 0::. Iri = Imin, with '¥ given by W • I-UJ I pr = I max (see Eqs. 5, 8 and 10) Za:4 -:::::> 4. Imbibition by primary displacement, non 01- UJ« equilibrium conditions: N>3 -a: Idp = Imin' with '¥ = 0 (see Eq. 10). ...I:::::> «0 Figs. 4 and 5 illustrate the manner in which the ~ 2 \ ri a: normalized interface curvature varies with the o J(8,'/t }min interface positions, ¢ and '¥. As seen, simultaneous Z solutions of Eqs. 4, 5, 8 and 10 establish the J interface curvatures appropriate to Cases 1, 2 and 0IL4-----=-'I6~--c':I a----:L----,L---:"--c----:L-----:-'-----' 3. The curves shown refer to the pore opening defined by a triangular array of spheres. Results PENDULAR RING for a contact angle of 40°, as well as for a zero FIG. 5 -- GRAPHICAL SOLUTION OF INTERFACE degree angle, are included. The generalizations of CURVATURE EQUATIONS, TETRAHEDRAL CELL IMBIBITION. 266 SOCIETY OF PETROLEUM E'iGI'iEERS JOUHNAL becomes 2.185. This gives a ratio of imbibition to pressures correspond to interface configurations drainage capillary pressure for the square-array pore having negative values of the angle rp. Hence, the opening of about 0.91. Taking the mean of this maximum value of the interface curvature for the value and that corresponding to ,; = 90° gives a advance of the non-wetting phase through a pore result, 0.63, which is quite close to the observed opening is reached when the point of interface-sphere ratio. contact has passed somewhat beyond the position These considerations of the magnitude of the of narrowest restriction. equilibrium imbibition capillary pressure are necessarily inexact. Yet, in a qualitative sense, CORRECTION FUNCTION FOR the model of the ideal soil serves to explain CAPILLARY-TUBE MODEL imbibition pressures and hydrostatic hysteresis in To express quantitatively the effect of the a manner which is entirely beyond the scope of the contact angle e on the drainage or imbibition capillary-tube model. curvature, it will be convenient to introduce a function which corrects for any deviation from the GENERALIZATION TO NON-ZERO dependence predicted by the capillary tube (or CONTACT ANGLES hydraulic radius) model. According to this model, the capillary pressure, or interface curvature, varies CONTACT-ANGLE RELATIONSHIPS Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 as cos e. Thus, the following function may be The discussion of the ideal soil presented so far defined, has assumed that the contact angle made between the solid surface and the fluid-fluid interface was J (8) l(8) =~-,---- ...... (13) zero. This was assumed for the sake of simplicity [J(8)]8=0 cos 8 in bringing out the differences in approach and in results between the development based on the z(e) is a factor suitable for correcting capillary capillary-tube model and that based on the ideal-soil pressure data obtained with systems for which the model. According to the former model, the normalized contact angle is zero. This follows from the interface curvature J is proportional to cos e. How substitution of Eq. 13 in Eq. 1 with the result that ever, no such simple relationship results from the adoption of the ideal soil model for porous media. The curvature for the conti'nuous type of inter ~=(!L) cos8 l(8)' ..... (14) face was given previously by Eq. 4. If the interface (J (J 8=0 makes an angle e with the solid surface, this equation becomes In Fig. 7, the function z(e) IS shown for the cos (cf; + 8) (11) J ( 8, cf; ) = -(-co=-=s:"':'TJ~)'-1 ---=---'-c -0s----:-cf;-· • • • r i The drainage value of the curvature is again specified by the maximum value which is reached en w SQUARE ARRAY by J (e), as the interface moves through the pore w opening. For non-zero values of e, however, this a::: eq maximum no longer corresponds to rp = O. Instead, ~ 50L=:~~c---JL------o ('-...... d P e.. TRIANGULAR ARRAY _ -sin(cp+8) Z 40 . (12) r i J (8, CP)mox - sin cP o I____ ~-}------I- en 30 Solving Eqs. 11 and 12 simultaneously then gives o L-______eq the dependence of both J(e)dr and rp on the contact a.. -L------angle e. That is, the drainage curvature is w specified by (,) TRIANGULAR ARRAY « 0 tL J(e)dr = J(e, rp), with rp given by a::: W J(e,rp)max = J(e,rp) (see Fig. 4). I- dr -10 A similar calculation of the effect of contact ~ angle on capillary pressure has been reported by SQUARE ARRAY dr PurcelI.14 Analyses of a closely related nature have been developed by several authors concerned -20L-----~-----L ______L- ____~ ____~ o with the water-repellency of porous materials com 10 20 30 40 50 posed of fibers. 32, 33 CON T ACT A N G L E, 9, DEGREES The dependence of the angle on contact angle rp FIG. 6 - EFFECT OF CONTACT ANGLE ON INTER is illustrated in Fig. 6, which shows that drainage F ACE POSITION. SEPTEMBER, 1965 267 case of drainage as well as for the several types Having introduced the angle W in the expression of imbibition. Calculations were not carried out for for interface curvature, it is necessary to consider contact angles greater than 40°, since the purpose the effect of contact angle on the curvature vs W in obtaining these results was to indicate the relation for pendular-ring interfaces. The generali general trend exhibited by the function z(e), zation of Eq.8 gives a rather cumbersome expression, bearing in mind the approximate nature of the although it is obtained in a straightforward way, various formulas adopted for the interface configur using the same argument as previously. The result ation. The values of Z(e) greater than unity, as IS shown in Fig. 7 for the drainage case, correspond to apparent contact angles, which are smaller in cos (lJI + 8l sin (lJI+8l magnitude than the true contact angles. Here, the J(B.lJIlpr- 2(I-cos lJIl 4sin lJI term "apparent contact angle" means the value of the contact angle which would be computed on the cos(lJI+B) basis of the capillary-tube model [Z(e) = 1]. - 4{ sin lJI cos (lJI+Bl- (1- cos lJIl[l-sin(ljItBl]) The magnitude of function Z(e) for drainage agrees closely with that computed from the mercury ...... (16) inj ection data reported by Brooks and Purcell. 8 For this, it would be necessary to assume that the Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 usual value for the Leverett function represents A numerical solution for Eq. 16 was obtained by the case of zero contact angle. However, in view machine computation. of the obvious limitations raised by the low precision The specification of the two imbibition situations of such measurements, the agreement obtained involving pendular rings can now be presented in a should not be given too great a significance. way entirely analogous to that used before. For imbibition under equilibrium conditions, the value DISCUSSION OF IMBIBITION SITUATIONS of the interface curvature is given by The values of z(e) given in Fig. 7 for the various ](e)eq = ](e,Wkin' with W given by imbibition situations were obtained by methods analogous to those used in the case of zero contact ](e,W)pr= ](e,Wkin (see Fig. 5). angle. Again, it is appropriate to specify the For the type of non-equilibrium imbibition which minimum interface curvature achieved by advancing was termed reinvasion, the specification of inter wetting-phase fluid by means of Eq. 9. This gIves, face curvature becomes with Eq. 11, J(e)ri = ](e,W)min' with W given by cos (t -lJI+ 8 1 ](e,W)pr = ](e,CP)max' and with cP given by . . (15) ](e,CP)max = ](e,cp) (see Fig. 5). The second type of non-equilibrium imbibition, primary displacement, has been defined as that dr which occurs when pendular rings are absent. The I1J dr previous method of specifying this situation, by Z ~"..----====::;:=.;T.~:-;:;-~ A R RAY setting W = 0 in Eq. 15, no longer applies. This is 0 ..... because another mechanism operates to cause U interface instability at a greater value of the Z curvature than that given by the condition W = ~ 0.8 o. u.. TRI ANGULAR ARRAY The new mechanism is due to the fact that, in the Z absence of pendular rings, some point on the 0 advancing interface will contact one of the spheres ..... 0.6 lying in the next layer of the packing. (In the U W previous mechanism the point of interface-sphere a:: a:: contact reached the point of contact of two spheres.) 0 In the case of the square - array type of pore u 0.4 opening, the expression for the interface curvature W a:: corresponding to the new mechanism becomes ~ quite complex. However, for the triangular array I- 0.2 SQUARE <[ the result is much simpler, > ARRAY a:: 1- sin(cp+8l ~ u 0 10 20 30 40 50 CONTACT ANGLE, 9, DEGREES .. (17) Hence, the specification for the interface curvature FIG. 7 - EFFECT OF CONTACT ANGLE ON INTER FACE CURVATURE. In this case can be written as 268 J(e)dp = J(e'¢)min' with ¢ given by Another aspect of the cooperative behavior of the ideal soil model should be briefly mentioned. J(e'¢)min = J(e,¢). This concerns recent theoretical studies of generalized hysteresis phenomena. The results of COOPERATIVE EFFECTS IN THE these studies have been referred to as the "domain IDEAL SOIL MODEL theory" of hysteresis. 40,41 Application of concepts derived from this work to experimental data have The discussion of the ideal soil model presented been reported by Poulovassilis,37 and the subject up to this point has emphasized the behavior of a has recently been further developed by Philip.42 single, isolated void space. Thus, attention has None of the several approaches to the cooperative been focused on the capillary pressures required behavior of the ideal soil model have so far been for the non-wetting phase to enter (drainage) or concerned with varying wettability. Such studies, leave (imbibition) an individual pore. It is now both experimental and theoretical, will certainly appropriate to extend the discussion somewhat to deserve attention in the future. take account of the manner in which the individual pores are mutually arranged within the total pore CONCLUSIONS space. This is necessary for the proper understand ing of all aspects of the relationships between In connection with the calculations leading to Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 capillary pressure and saturation. the values of the function Z(e) shown in Fig. 7, The key features are (1) the degree of intercon results were obtained for the location of the nectedness of the individual pores, and (2) the interface within the pore openings defined by the distribution in the size of the pores. For the ideal square and triangular arrays. These results, in soil, of course, the two types of packing give rise which the interface position is defined by the angle to elements of void space which are connected to ¢, are shown in Fig. 6. The effect of contact angle other similar elements by means of four, six or on interface position is not particularly strong. eight pore openings. Nevertheless, if only a moderate The curvature correction functions for drainage, size distribution is admitted, it seems clear that given in Fig. 7, show only a slight deviation from sufficient interconnectedness cannot be established the predictions of the capillary-tube model. This for all the pores of each particular size to model is implicitly assumed when pore-size distri communicate with at least one pore of a larger size butions are calculated from capillary-pressure data and one pore of a smaller size. Thus, in the course obtained by mercury injection. Since the contact of drainage or imbibition in an aggregate of pores, angle is of the order of 40° for this system, the certain pores will be bypassed. Trapped - fluid magnitude of the curvature - correction function structures occur, therefore, simply because of the suggests that the use of this model will introduce distribution of pore sizes and the finite degree of an error which is at most of the order of 10 per interconn ectedness. cent. Hence , the reliability of the usual interpretation This type of behavior, in which the accessibility of such data is confirmed, at least to within a of the individual units of the model becomes an tolerable error. This is true, however, only because important feature, has been extensively studied by the displacement mechanism is characteristic of Fatt.34 The model employed in this work consisted drainage, rather than imbibition. of a two-dimensional network of interconnecting In fact, the curves shown in Fig. 7 reveal a capillary tubes of varying size. The trapping or striking deviation of the dependence of the blocking of pores which occurs in this model was imbibition curvatures on contact angle from that reflected in the capillary pressure - saturation predicted by the capillary-tube model. This is the relationship. Everett has referred to such effects most significant result of the present investigation. as "cooperative behavior".35 Pre liminary theoreti The curves for the square array show the stronger cal studies of these cooperative effects in effect, and it is indicated that capillary pressure three-dimensional aggregates similar to the ideal may even change sign for values of the contact soil model have been reported by Barker.36 angle of the order of 40°. Thus a situation may Insight into the nature of these effects can also arise in which, after a non-wetting-phase fluid has be obtained from detailed experimental studies of invaded a porous medium, the displaced wetting the capillary pressure-fluid saturation relationships. phase cannot successfully re-imbibe. This is exemplified by the recent work of Morrow The curves presented in Fig. 7 for the various and Harris. 10 The scanning curves associated with types of imbibition show that the ideal soil concept the phenomenon of hydrostatic hysteresis were provides an entirely different dependence of studied in detail. 2, 37 The choice of the so-called capillary pressure on contact angle than does the branching point on the drainage curve appears to hydraulic radius or capillary - tube model. The control the amount of non-wetting fluid which is apparent contact angles appear to be of the order trapped upon imbibition. 10 It seems quite likely of three times the magnitude of the true contact that such studies will prove invaluable in inter angles. In view of the greater correspondence to preting the results of dynamic experiments such as the actual pore morphology found in porous media, have been discussed byNaar,Wygal,andHenderson38 it must be concluded that the ideal soil concept and by Raimondi and Torcaso. 39 provides a much better interpretation of the effect SEPTEMBER, 1965 269 of contact angle on capillary phenomena In such 4. Smith, W.O., Foote, P. D. and Busang, P. F.: systems. "Capillary Rise in Sands of Uniform Spherical Grains", Physics (1931) Vol. 1, 18. NOMENCLATURE 5. Smith, W. 0.: "Minimum Capillary Rise in an Ideal Uniform Soil", Physics (1933) Vol. 4, 184. g acceleration of gravity 6. Leverett, M. C.: "Capillary Behavior in Porous h capillary rise of wetting - phase fluid = Solids", Trans., AIME (1941) Vol. 142, 152. Pc /gt!.p 7. Calhoun, J. C., Lewis, M. and Newman, R. C.: "Experiments on the Capillary Properties of Porous J normalized interface curvature =~/l + l) Solids", Trans., AIME (1949) Vol. 186, 189. 2 \'1 r2 8. Brooks, C. S. and Purcell, W. R.: "Surface Area pressure in fluid phase Measurements on Sedimentary Rocks", Trans., AIME (1952) Vol. 195, 289. capillary pressure 9. Bethel, F. T. and Calhoun, J. C.: "Capillary De average radius of particle or grain saturation in Unconsolidated Beads", Trans., AIWIE radii of curvature of fluid-fluid interface (1953) Vol. 198, 197. saturation, fraction of void volume occupied 10. Morrow, N. R. and Harris, C. C.: "Capillary Equilib rium in Porous Materials", Soc. Pet. Eng. Jour. by wetting phase (June, 1965) 15. Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 Z curvature-correction function 11. Atkinson, D. I. W.: "The Mechanism of the Displace ment of Liquids from Porous Solids", Trans., Inst. Tf angle specifying type of pore opening array Chern. Engrs., London (1949) Vol. 27, 259. contact angle e 12. Dombrowski, H. S. and Brownell, L. E.: "Residual ~ angle specifying position of spheres neigh- Equilibrium Saturation of Porous Media", Ind. Eng. boring the pore opening Chern. (1954) Vol. 46, 1207. density difference for two fluid phases 13. Carman, P. C.: Flow of Gases through Porous Media, Academic Press, New York (1956) Chap. 11, 40. surface or interfacial tension a 14. Purcell, W. R.: "Interpretation of Capillary Pressure angle specifying position of interface within Data", Trans., AIME (1950) Vol. 189, 369. pore opening 15. Buff, F. B.: "The Theory of Capillarity", Handbuch 'P angle specifying position of pendular ring der Physik, Springer-Verlag, Berlin (1960) Vol. X, 281. interface 16. Melrose, J. C.: "Evidence for Solid-Fluid Interfacial Tensions from Contact Angles", Advan. Chern. Ser. SUBSCRIPTS (1963) No. 43, 158. dp primary displacement type of imbibition 17. Bobek, J. E., Mattax, C. C. and Denekas, M. 0.: "Reservoir Rock Wettability - Its Significance and dr drainage Evaluation", Trans., AIME (1958) Vol. 213, 155. eq equilibrium imbibition 18. Wagner, O. R. and Leach, R. 0.: "Improving Oil max maximum value of interface curvature Displacement Efficiency by Wettability Adjustment", Trans., AIME (1959) Vol. 216, 65. mIn minimum value of interface curvature 19. Bigelow, W. C., Glass, E. and Zisman, W. A.: pr pendular-ring interface "Oleophobic Monolayers II. Temperature Effects and Energy of Adsorption", Jour. Colloid Sci. (1947) Vol. Tl reinvasion type of imbibition 2, 563. a, f3 fluid phases 20. Baker, H. R., Shafrin, E. G. and Zisman, W. A.: "The y solid phase Adsorption of Hydrophobic Monolayers of Carboxylic Acids", Jour. Phys. Chern. (1952) Vol. 56, 405. ACKNOWLEDGMENT 21. Benner, F. C., Dodd, C. G. and Bartell, F. E.: "Evaluation of Effective Displacement Pressures The author wishes to acknowledge many valuable for Petroleum Oil-Water Silica Systems", Drill. and Prod. Prac., API (1942) 169. discussions with W. R. Foster and C. F. Brandner. 22. Bartell, F. E. and Osterhof, H. J.: "The Measurement Thanks are due to Mrs. Martha Nunn, who programmed of Adhesion Tension Solid Against Liquid", Colloid the solution to Eq. 16. Appreciation is also ex Symposium Monograph (1927) Vol. 5, 113. pressed to Socony Mobil Oil Co., Inc. for permitting 23. Harris, C. C., Jowett, A. and Morrow, N. R.: "'Effect publication of this work. of Contact Angle on the Capillary Properties of Porous Masses", Trans., Inst. Min. Metall. (1963-64) REFERENCES Vol. 73, 335. 24. Graton, L. C. and Fraser, H. J.: "Systematic Packing 1. Haines, W. B.: "Studies in the Physical Properties of Spheres-with Particular Relation to Porosity and of Soil. IV. A Further Contribution to the Theory of Permeability", Jour. Geology (1935) Vol. 43, 785. Capillary Phenomena in Soils", Jour. Agri. Sci. (1927) 25. Gibbs, J. W.: Scientific Papers, Dover Publications, Vol. 17, 264. New York (1961) Vol. 1, 242-252. 2. Haines, W. B.: "Studies in the Physical Properties 26. Miller, E. E. and Miller, R. D.: "Physical Theory for of Soil. V. The Hysteresis Effect in Capillary Prop Capillary Flow Phenomena", Jour. Appl. Phys. (1956) erties and the Modes of Moisture Distribution Vol. 27, 324. Associated Therewith", Jour. Agri. Sci. (1930) Vol. 27. Fisher, R. A.: "On the Capillary Forces in an Ideal 20, 97. Soil", Jour. Agri. Sci. (1926) Vol. 16, 492. 3. Hackett, F. E. and Strettan, J. S.: "The Capillary 28. Englehardt, W. V.: "Interstitial Water of Oil Bearing Pull of an Ideal Soil", Jour. Agri. Sci. (1928) Vol. Sands and Sandstones", Proc., 4th World Pet. Cong., 18, 671. 270 SOCIETY OF PETROLEUM ENGINEERS JOURNAL Rome (1955) Sec. IIC, 399. 36. Barker, J. A.: "Discussion" on Ref. 35, The Struc 29. Smith, W.O., Foote, P. D. and Busang, P. F.: ture and Properties of Porous Materials, Academic "Capillary Retention of Liquids in Assemblages of Press, New York (1958) 125. Homogeneous Spheres", Phys. Rev. (1930) Vol. 36, 37. Poulovassilis, A.: "Hysteresis of Pore Water, An 524. Application of the Concept of Independent Domains", 30. Rose, W.: "Volumes and Surface Areas of Pendular Soil Sci. (1962) Vol. 93, 405. Rings", Jour. Appl. Phys. (1958) Vol. 29, 687. 38. Naar, J., Wygal, R. J. and Henderson, J. H.: "Imbi 31. Harris, C. C. and Morrow, N. R.: "Pendular Moisture bition Relative Permeability in Unconsolidated in Packings of Equal Spheres", Nature (1964) Vol. Porous Media", Soc. Pet. Eng. Jour. (March, 1962) 203, 706. 13. 32. Baxter, S. and Cassie, A. B. D.: "The Water Repel 39. Raimondi, P. and Torcaso, M. R.: "Distribution of lency of Fabrics and a New Water Repellency Test", Oil Phase Obtained upon Imbibition of Water", Soc. Jour. Text. Inst. (1945) Vol. 36, T67. Pet. Eng. Jour. (March, 1964) 49. 33. Crisp, D. J. and Thorpe, W. H.: "The Water-Protecting 40. Everett, D. H.: "A General Approach to Hysteresis. Properties of Insect Hairs", Disc. Far. Soc. (1948) Part 3 - A Formal Treatment of the Independent Vol. 3, 210. Domain Model of Hysteresis", Trans., Far. Soc. 34. Fatt, I.: "The Network Model of Porous Media. I. (1954) Vol. 50, 1077. Capillary Pressure Characteristics", Trans., AIME 41. Enderby, J. A.: "The Domain Model of Hysteresis. (1956) Vol. 207, 144. Part 2 - Interacting Domains", Trans., Far. Soc. Downloaded from http://onepetro.org/spejournal/article-pdf/5/03/259/2153830/spe-1085-pa.pdf by guest on 26 September 2021 35. Everett, D. H.: "Some Problems in the Investigation (1956) Vol. 52, 106. of Porosity by Adsorption Methods", The Structure 42. Philip, J. R.: "Similarity Hypothesis for Capillary and Properties of Porous Materials, Academic Press, Hysteresis in Porous Materials", Jour. Geophys. New York (1958) 95. Res. (1964) Vol. 69, 1553. *** SEPTEMBER, 1965 271