T.P.2563
DISCUSSION OF THIS AND ALL FOLLOWING TECHNICAL PAPERS IS INVITED Discussion in writing (3 copies) may be sent to the Editor, Journal of Petroleum Technology, 601 Continental Building, Dallas 1, Texas, and will be considered for publication in the Transactions volume Petroleum Development and Technology. Discussion will close December 15, 1949. Any discussion offered thereafter should be in the form of a new paper.
THEORETICAL GENERALIZATIONS LEADING TO THE EV ALlJATION OF RELATIVE PERMEABILITY
WALTER ROSE, JUNIOR MEMBER AI ME GULF RESEARCH & DEVELOPMENT CO., PITTSBURGH, PENNSYLVANIA Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021
ABSTRACT tain references can be cited"""-""·> where a qualitative aware ness of their existence and nature is indicated. This condition Theoretical expressions are presented to describe wetting is explainable in terms of the recognized complexity of the and non·wetting phase relative premeability relations. These problem, and in order to provide some basis for theoretical expressions have then been compared with existing published development, the general emphasis has been placed in the past data, the conformance noted being sufficiently good to satisfy on experimental procedures of evaluation, the validity of the requirements of some engineering use. As a consequence, which, in principle, could be confirmed by the analysis of well it may be supposed that relative permeability characteristics and reservoir performance. However, because of the experi of porous media now can be inferred from basic core analysis mental difficulties which have been encountered (notably, data, in a manner more convenient (although less direct) difficulties due to the so-called "end effects"), and because of than presently available methods of experimental evaluation. the general unreliability of field performance tests required to study the applicability of the data, relative permeability INTRODUCTION phenomena continue to be incompletely understood and This paper presents a new approach to the problem of described. t relative permeability evaluation. A classification of published In this paper is examined the possibility of predicting discussions of relative permeability concepts which have ap relative permeabilities entirely from fundamental considera peared in the petroleum literature will show that previously tions. In addition, attention is called to certain important emphasis has been placed on methods of experimental meas factors, previously unemphasized in the published literature, urement, on the interpretation of the data so obtained with which now can serve as a basis for the eventual experimental respect to the variable properties of the system, and on the confirmation of this and similar theoretical approaches, as oretical and practical considerations which relate relative well as the experimental solution of the problem in general. permeability to gross fluid behavior in petroleum reservoirs. Therefore, the analysis presented herein has the dual purpose Essentially, no detailed examination has appeared which both of orienting future experimental activity and also of pro treats the fundamental factors controlling the quantitative viding an immediate solution for the relative permeability features of the relative permeability relation*, although cer- porblem, sufficiently adequate for some engineering use. It will be seen that the conclusions reached are supported by theoreti • Intuitively, at least, we may suspect that the factors controlling the cal and intuitive considerations, and they are not contradicted quantitative features of relative permeability relations (where relative permeability of porous media to given fluid phases is plotted as function insofar as gross features are concerned by existing concepts of the saturation of these phases) will be describable in terms of basic rock texture and in terms of the character of the distribution of the and interpretations of published experimental data, except as immiscible fluids within the interstitial spaces. The term basic rock tex specifically noted. ture is used here to refer to the pore configurational ch~racters dependent on factors such as grain sizes and packing a& modified by cementation and other secondary processes. It will be realized that there are other factors Darcy's law, expressed simply as: of importance which in practical instances also will control the quantita tive features of the relative permeability relation, such as those related k dp to the various possibilities for variable interaction between the saturatinll q=- (1) fluids and the surfaces of the interstitial spaces. Thus, interstitial clays p. dx can be responsible for variable conditions of pore configuration obtaining in a given instance depending upon the nature of the saturating fluids. where q is linear rate of flow (assumed horizontal), k is the Variable conditions of preferential wettability also can obtain expressive of analogous possibilities in ways which will control heterogeneous fluid flow behavior. For the purposes of this paper, however, fixed pore con t The poor comparison usually obtained between laboratory and field figurational and wettability characters will be assumed. Moreover, no gas-oil ratios (c.f., Fig. 4 of paper by L. E. Elkins') is illustrative of attempt will be made to establish explicitly a dependence of relative conditions which discourage unqualified acceptance of the results of pre permeability on strictly fluid properties. vious studies except as presently useful approximations. In this connection, Manuscript received at Petroleum Branch office April 28, 1948. Pre Evinger and MuskatO have discussed in detail the discrepancy observed sented at Branch Fall Meeting, October 4-6, 1948. between field well produetivity values and values obtained from the References are given at end of paper. analysis of laboratory flow data.
May, 1949 PETROLEUM TRANSACTIONS, AIME 111 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY
permeability constant, f.' is fluid viscosity, and dp/ dx is the holds great promise, at least for engineering use, particularly pressure gradient, defines completely the viscous flow of if examination of newer data as they become available sub homogeneous fluids through porous media of uniform cro~s stantiates the general applicability of their correlations. section. The permeability constant, k, then is an expresi'ion of These attempts to explicitly relate permeability functions tu lhe properties of a particular porous body which collectively rock properties can be expressed by the following analytical require fluid flow to be controlled by fluid viscosity and pre; reasoning. Permeability can be defined functionally as: sure gradient in the manner indicated by Eq. (1). Now, in k = F(T) (2) media variously saturated with two or more immiscible con That is, there are a series of rock properties collectively rep tinuous fluid phases Darcy's law will still define the dynamic resented by the symbol T which control permeability accord character of flow for each phase separately, if effective perme ing to the prevailing physical requirements of the system. In ability constants are properly defined as functions of the general, the function, F (T), will be known in simple systems distributions of the various saturating fluids. This point of (e.g., capillary tube), whereas it will be only partially or iO view follows that accepted by Richards , who considered the imperfectly known in more complex systems (e.g., petroleum presence of one fluid saturating a porous body merely as a reservoir rock). In practice this is a serious limitation to the Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 factor which controlled (in the same manner as the structure extent that the approximations which are discovered to apply of the porous body itself) the geometry of the paths available nnder one set of conditions become invalid under slightly for flow to another immiscible fluid simultaneously saturating different conditions such that empirical generalization is im the porous body. The analysis to follow requires assuming, as possible. By analogy the effective permeability, k" can also be a simplification, steady state flow with the fluid phase satura considered as formally defined in terms of other functions of tion distribution uniformly constant during the time interval rock properties, where at least one function of particular of measurement; and in the study of reservoir behavior this importance involves the fluid phase saturation distribution, or: usually can be done rigorously, at least in principle, by select k, = Fe(T"p), (3) ing sufficiently small elements for consideration wherein the where p is employed here to denote the saturation distribution macroscopic flow character is nonetheless retained. Under of the fluid phases. The relative permeability, k" then can be such conditions, the observed effective permeabilities become expressed as: significant constants functionally related to the saturation distribution in addition to the other porous body properties ke F,(Te,p) F ( ) (4) which, as noted above, themselves control completely the kr =T= F(T) IV r p dynamic aspects of single-phase (homogeneous fluid) flow. Eq. (4) calls attention to the simplification employed in the In developing the basis for the analysis to follow where present treatment of relative permeability, which involves de relative permeability will be explicitly related to saturation scribing all of the properties of the system (responsible for distribution characters attention is called to certain previous the specific permeability in single phase flow) as implicit functions of saturation in polyphase flow. Therefore, the sat attempts which have been made to relate specific and effective uration (or more properly, the saturation distribution) ex permeability to various rock properties. Traxler and Baum" plicitly becomes the only variable controlling relative perme defined specific permeability in terms of average pore radius, ability in polyphase flow, and this concept is developed in the and Leverett' attempted to apply this concept to relative per paragraphs to follow. For example, the "effective" p.orosities* meability. Hassler'" and Botse!' also discussed this supposed which contribute to the separate elements of polyphase flow relationship, although none of these authors presented an can be related to the total porosity responsible for single phase exact definition of pore distribution, and Leverett" finally dis flow according to the character of the prevailing saturation carded the notion altogether because of the complicating fac distributions in the polyphase system. Assuming for the mo tors related to saturation hysteresis effects. In this connection, ment, for illustrative purposes, that effective porosities and Haines", and more recently, Leamer and Lutz14 and Ritter and total porosity are the only factors which determine effective
Drake15 showed the usefulness of relating pore distribution to permeability and specific permeability respectively, a ratio of the differential plot of the capillary pressure type relation,t these factors as defined in terms of a saturati'on function will implying the possibility of deriving relative permeability rela then serve uniquely as an expression for relative permeability. tions therefrom, although this has not yet been entirely accom Having now indicated a need for detailed examination of plished§. Carman" and others used the Kozeny equation in relative permeability concepts and having suggested a basis :I: That is, the plot of some function of the fluid phase distributions tions and three phase relations also can be inferred, at least versus some measure of the capillary pressure obtaining at the interfaces between phases (e,g. the interfacial curvature. or the height of the inter approximately, in a similar and related manner. face above some reference level of zero curvature, etc.). * Effective porosity is herein defined as the saturation of a phase under § c.f. discussion following Eq. (9). consideration per unit of bulk volume. 112 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 DEVELOPMENT OF THE GENERALIZED saturation range of interest.:f: These assumptions made for ex EXPRESSION FOR WETTING PHASE pediency, however, can be supported by the fact that a plaw; sible conclusiun is eventually obtained, as will be seen below. RELATIVE PERMEABILITY In any event, simply dividing Eq. (8) by Eq. (7) yields all expression for wetting phase relative permeability, k •. w, In a recent paper'" the authur investigated sume of the pos namely: sible applications of capillary pressure data, and arrived at the following equation fur wetting phase relative permeability. as: which leads to Eq. (10) since t = tew has been assumed, and PD)' krw == pw ( Po . (10) few since - = pw. fw Now, in order to contribute to a fuller interpretation of the theoretical considerations; to compensate for notational dif Eq. (10) can be differentiated to yield: Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 ferences occuring in this paper and the above mentioned Rose and Bruce paper (loc. cit.); and to allow the desired em- (ll) phasis of dynamical rather than static considerations of fluid behavior, the development of Eq. (10) will be reviewed as dpw Po dpw follows: where: D-=-=--- dr r dPe According to common usage the Kozeny equation for a is the pore radii distribution coefficient according to Ritter and single phase flow can be expressed as: Drake (loc. cit.), and r is the pore radius (i.e. a pore width) f which under a given condition of saturation (Pw) contains the k= (5) Nt interfaces between the non-wetting and wetting phases.§ Eq. (11) supports the view mentioned above that the relative per where f is the fractional porosity, A is the specific internal meability-saturation relation depends on the distribution of surface area of the pores (per unit of pore volume, and t is a pores in the system being considered. dimensionaless rock textural constant related to the shape and orientation (tortuosity) of the pores. Carman" later, in study Actually, Eq. (10) can be derived more fundamentally by ing the properties of unconsolidated sands, established em combining the Darcy and the Poiseuille equations for fluid pirically that: flow in the manner suggested by Traxler and Baum (loc. cit.) to yield: PD A= (6) f r2 (j k= const. where PD is the displacement pressure, and (j is the interfacial where r is presumed to be an average pore radius of the hypo tension. Eqs. (5) and (6) can be combined to yield: thetical system of capillary tubes equivalent to the porous f(j2 media actually being considered. By analogy, then, an expres k= (7) t sion for effective permeability in terms of an effective porosity Pn't and a similar effective average pore radius can be derived, Now, considering the wetting phase effective permeability which on combination with the above expression for specific (kew) in a polyphase flow system it can be argued by analogy permeability yields an expression for relative permeability. as: to Eq. (7) that: (lOa) £ewo-2 kew ==-- (8) Pc't",," where re refers to the average pore radius of that portion of where the effective porosity, few, is defined as noted above; the system containing the phase under consideration having a and where Pc, the capillary pressure, is accepted as being the saturation, p. To the extent that the ratio (re/r)' is related to effective displacement pressure characterizing the partially the distribution of fluids within the interstitial spaces, Eq. saturated system. The effective rock textural constant, tm, has (lOa) shows how relative permeability depends both on phase not been evaluated as yet, and therefore, for the purposes of saturation and distribution. In order to illustrate the signifi this analysis it will be assumed that tow = t throughout the cance of Eq. (lOa) Curves E and F (as applied to non-wetting t The constant, t, is defined fundamentalIy19 as a product of dimension t Although according to the derivations of Eq. (5), (6). (7) •. less pore shape and orientation factors. This could suggest that tew char these relations apply strictly to unconsolidated systems [c.f. Sullivan and acterizing a polyphase system is approximately equal to the t character HerteP9 point out. for instance, that "where the media are consolidated, izing the equiva~ent single-phase system in the sense that the incidental or where the media contain bridging, agglomeration or considerab!e chan presence of a non-wetting phase does not appreciably alter the config neling" there is some Question as to the applicability of Eq. (5)], it is urational properties of the pores containing the wetting phase~ although assumed that suitable adjustment of the rock textural constant, t, will its presence does reduce the porosity available for wetting phase saturation. make these relations applicable to a consideration of consolidated systems. Carman has recently discussed the applicability of Eq. (5) to consolidated § Note, a plot of r versus p,v will exhibit the same hysteresis effects porous media.33 observed in the capillary pressure-saturation relation. May, 1949 PETROLEUM TRANSACTIONS, AIME 113 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY and wetting phase relative permeability respectively) are is a reasonable definition for this effective surface area,§ anal presented in Fig. 6, as calculated for a hypothetical case ogous to Carman's definition of specific surface area, (cf. where it is assumed that a normal probability distribution of Eq. (6)). pores obtains and that at any given phase saturation condition Eq. (13) also suggests the importance of the function Pv=F(o/r) and Po = F(o/r.). 1 In order to solve Eq. (10) it is necessary to have available Pc dpw -+ J Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 pressure versus saturation data such as are obtained in capil pw lary pressure (static) experiments or displacement (dynamic) is the interfacial surface area previously defined by Leverett." experiments.* It will be shown now, however, that a more gen eral solution of the wetting phase relative permeability versus It will be noted from Eq. (12) that (pw) = 1, and lim Pc = PD . (15) (12) p;t-l Also: versus the wetting phase saturation, pw:r The dimensionless group, 114 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 view, as have numerous others previously). It is further sup- above, and which serves as a basis for deriving the desired posed that all porous systems of the type being considered expression for relative permeability, is: can be classed, according to their fluid behavior properties, as intermediate between the limiting systems described as uni (18) [(3Pw,-2) pw + (4-5 P"I)J' form sized, uniformly oriented capillary tubes on the one hand, and random sized, randomly oriented capillary tubes on the Eqs. (12) and (18) can now be combined to yield: other. These two limiting systems, abbreviated U.S.O. and Po 2pw'(2-3pWi) +3pw Pw,(3Pw,-2) +pwi(4-5pWi) R.S.O. respectively, can be analyzed in detail as follows: --p;; = 3 (19) 4YPw(l-pwi) (PW-PWi) The uniform sized, uniformly oriented system of capillary It will be observed that the conditions: tubes (which is actually equivalent to a single capillary tube lim Pc = Pn and lim Po = 00 oriented in the direction of flow) has this character as defined p,..-H PW+PWI in terms of capillary pressure concepts. The capillary pressure are satisfied by Eq. (19). Combining Eq. (19) with Eq. (10), equals the displacement pressure for all values of saturation,t we obtain finally: and the irreducible wetting phase saturation in zero. It follows k".=, 16pw' (PW-PWi)' (1-pwl) 2 Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 then that: [2p". (2 - 3pw.) + 3pw pwi (3pWi - 2) + pw.( 4 - 5pw.) ] l (20) Po dpw = Pn (1 - pw), where Eq. (20) evidently reduces to the limiting values 1 and J pw 0, as pw approaches 1 and pWi respectively. Or, by Eq. (12): While Eq. (19) has been developed primarily as an inter p... mediate relationship for the purpose of deriving Eq. (20), it .p (pw ) u.s.o. = ---.,- (16) (2 - Pw)' has the same type of validity as the empirical representation Eq. (16) satisfies the limiting requirements of Eq.(15). of capillary pressure data by the function .p (pw) plotted in Fig. 1. It is not suggested that Eq. (19) gives a universal The random sized and oriented system of capillary tubes description of capillary pressure curves, although, in contrast 2 to other proposed empirical equations,t it satisfies all the basic (R.S.O.) simply is characterized by the fact that pwi =-, 3 1.0 since it is supposed that the random orientation reduces two j thirds of the pore volume effectively to cul-de-sac porosity (i.e .. porosity which cannot be invaded by the non-wetting phase since the zero permeability conditions prevent the escape of 0.8 II the wetting phase therefrom).* Now, it is observed from / [I Eq. (15) that the intercept on the pw axis of the tangent to 0.6 II d 2 lim .p (pw ) is also - such that the curve / / pw~1 dpw 3 .p (pw) "".0. = (3pw - 2) !;I / is taken as the most convenient definition for .p (Pw h.s.o .. It is ~ // readily vertified that Eq. (17) also satisfies the requirements ~ 7/ / of Eq. (15). 0.2 / / The limiting functions .p (pw) u.s.o. and .p (pw) R.S.O. are shown d~~/ II in Fig. 1, and, as assumed above, data obtained on any porous P system should yield .p (pw) versus pw plots belonging to the ~ 8" eY J/ / family of curves intermediate to these limts and having the ----- 0.2 0.4 0.6 0.8 1.0 properties indicated by Eqs. (15). A suitable.generalized ex· Pw -Wetting Phase Saturation pression for .p (Pw) which fits the required conditions stated FIG. 1 - VARIATION OF THE RHO FUNCTION WITH WETTING PHASE SATURATION. t Starting with a completely saturated capillary tube a displacing pres Curve I - Theoretical Curve applying to a U.S.O. system. sure which is infinitesimally g-reater than P D is required before the non .. wetting phase can enter the capillary and produce a desaturation of the Curve II - Theoretical Curve applying to a R.S.O. system. wetting phase. However, at any saturation condition between 1 and 0, Curve A - Experimental Curve for Leverett's Sand III". the pressure difference between the phases arbitrarily can be set exactly Curve B - Experimental Curve for an Ohio Sandstone." equal to P , thereby establishing condition of statis equilibrium. so that D Curve C - Experimental Curve for a Magnolia Limestone." d po Curve D - Experimental Curve for Thornton and Marshall "Well -=0. A".t! dpw Curve E - Experimental Curve for a California Sand.' * Klinkenberg23 in his analysis of homogeneous flow through an R.S.O. system makes a similar sort of an assumption. t c.f. for example, the equation of Hassler et "I (loc. cit.). May, 1949 PETROLEUM TRANSACTIONS, AIME 115 T.P. 2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY physical requirements as expressed by Eq. (15). It is Eq. (20) systems will not be accepted simply because available data or which represents the immediate goal of the above outlined intuitive argument does not require unqualified acceptance. theoretical considerations. Its validity and usefulness will be discussed below. The data of Leverett and Lewis suggest that the following qualitative generalizations, expressed symbolically,§ apply inso IOO%Gas far as gross features are concerned to a consideration of the properties of a given porous body: (21-1) b·g system boo system b-o-g system . That is, the relative permeability to brine in a given porous body is independent of the fluid properties of the other sat urating fluid (s) in those cases where brine is considered to be the wetting phase. Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 Also: keg (pg) k"g (pg) --"-----'-'-~ > --"-----'-'-~ (21-2) gob system goo system g-o-b system That is, the relative permeability to gas in two phase systems is independent of the nature of the wetting phase, and in three phase systems it is always less at a given saturation than in a FIG. 2 - THREE-PHASE RELATIVE PERMEABILITY DIAGRAMS. two phase system. Solid Lines, Data of Leverett and Lewis." Dashed Lines, Theoretical Gas and Oil Isoperms. Finally: (Water Isoperms not shown.) keo (po) < k,·o (Po) ~ keo (po) > keo (po) o-g system o-b system »o-g-b system o-g system (21-3) DISCUSSION OF RELATIVE PERMEABILITY That is, at a given oil saturation, the relative permeability to CONCEPTS SUGGESTED BY AVAILABLE oil is less in an oil-gas system than in an oil-brine system; EXPERIMENTAL DATA the latter itself being greater than, equal to, or less than tne oil relative permeability in a three-phase system; the oil rela First, however, -in order to develop the range of usefulness tive permeability in the three-phase system in turn being in which relationships of the type indicated by Eq. (20) can always greater than that in a two-phase oil-gas system. be applied to evaluate relative permeability, some considera Now, to the extent that gas and oil can both be regarded as tion will be given to the published empirical results of pre non-wetting phases in brine-oil or brine-gas two-phase systems, vious workers. This is required, since the validity of Eq. (20) is not to be considered as rigorously proved by the above argu and to the extent that appropriate compensation is made for ment, so that conformance between theory and experiment is gas slippage (Klinkenberg) eifects,* it is regarded that: the only direct proof presently available. One danger in this approach to be noted and avoided stems from the uncertainty keg (pg) keg (p,,) k,o (po) (21-4) regarding the detail which justifiably can be employed in the gob system goo system o-b system examination of the data, since it is desired only to generalize on significant features independent of the experimental pro § The folIowing symbo}s are employed as subscripts: cedures followed and the experimental materials employed.t b refers to a brine (aquous) phase g refers to a gas phase The data of Leverett and Lewis" which have been replotted o refers to an oil phase n refers to a non-wetting phase (unspecified) in Fig. 2 for convenience provide an indication of the type of w refers to a wetting phase (unspecified) relative permeability relations which prevail in two and three Thus, the notation: phase systems. Although this work pertains specifically to the keb (pb) bog system properties of unconsolidated media, the conclusions applying to two-fluid-phase systems are confirmed more or less by later implies we are concerned with the relative permeability of brine plotted studies of consolidated sand bodies. The conclusions applying as a function of brine saturation, in a brine-gas (two-phase) system. to three-fluid-phase systems, however, have not yet been shown * Detailed consideration has been given to the problem of slippage effects in gas relative permeability phenomena, and it has been concluded valid when applied to consolidated sands, and in the discus that these effecis are often ncgligib!e as an approximation even in in sion to follow this fact is recognized to the extent that certain stances where appreciable s!ippage corrections must be applied to the of the Leverett and Lewis statements about three-fluid-pha"c values of effective and specific permeability used to calculate the relative permeability relation. The theory and data supporting this risu1t have been reported elsewhere,:~2 together with a statement of the slippage term t It is significant to note that all the cited references on permeability measurement emphasize the exploratory nature of the work as a precau which must be employed when it is desired to derive exactly gas relative tion against the tendency to over-generalize on the experimental results. permeability as a function of gas mean free path. 116 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 this being more or less supported by experimental data.t Also, relative permeability data of Botset (loc. cit.) , Martinelli since oil can be regarded as a wetting phase in a two· phase oil et al", Morse et al {loco cit.) , and Bulnes and Fitting". A cer gas system Eq. (21.1) can be expanded as: tain conformance is noted immediately between the shape and character of the experimental and theoretical curves, this con k,·o (Po) k,b (Pb) k,b (pb) k,b (Pb) (21-5) formance being of particular interest because the cited data o-g system b-g system b-o system - b-o-g system were obtained variously in studies of several dissimilar porous systems (i.e. capillary tube, unconsolidated sands, consolidated On combining the equalities (21A) and (21-5) with the sands and dolomites). inequalities (21-2) (and 21-3) we obtain: Now it has been mentioned that Eq. (20) was developed specifically as an expression for wetting phase relative perme ken (Pn) k,·,. (pw) k,o (p,,) ~ > < ability. In this connection it should be observed that Eq. (20) 2-phase system 2 or 3-phase system o-g-b system? conforms to the experimental conclusion first reached by k,n (Pn) k," (Pg) Wyckoff and Botset and later by Leverett and Lewis, that wet --~~--->.---~~~-- (21-6)+ 2-phase system g-o-b system ting phase relative permeability in a given porous body is a function of saturation alone (independent of fluid properties) Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 Eq. (21-6) shows how in a given porous body the relative per as expressed by the relation (21-5). Therefore, by insertion of meability to the three phases, brine, oil and gas, compare with suitable saturation terms in Eq. (20) it may be assumed thaI each other under conditions of two and three-phase saturation. the brine relative permeability in 2 and 3-phase systems and Specifically, it is shown that of the nine possible systems of the oil relative permeability in oil-gas systems will thereby be interest those pertaining to wetting phase relative permeability established. t in 2 or 3-phase systems can be classed together; as can those pertaining to non-wetting phase relative permeability in 2- At this point it becomes necessary to consider an apparent phase systems.§ The relative permeabilities to gas and to oil in 3-phase systems according to presently available data pro 1.0 vide an anomolous situation insofar as their description by the inequality (21-6) is not supported by theory nor entirely ~ confirmed by experiment, * and this will receive discussion I/;VI below_ However, first it will be shown how the various ele .2" 0.8 - --- :.0 ments of Eq. (21-6) can be evaluated, at least approximately, o Q) f---- c-- WV 1/ from a consideration of various modified forms of Eq. (20). E v'/ / r; rf 0.6 f------1/1 APPLICATION OF THE GENERALIZED Q) V/ V :g> Vj / I ----f-- / !f I EXPRESSIONS FOR RELATIVE PERMEABILITY Q) 0:: 1// Q) Xlj Fig_ 3 shows a plot of Eq. (20) where various values for the 8 0.4 f---- V; II I term, pw;, are assumed arbitrarily in the interval 0 to 2/3. Also L (L l/VA C III U shown for comparison are the representative wetting phase 0' / c V ~>Il Ei 5 Qj :// t Actually. this question is not entirely settled since the recent data: ,:!G / 'IDA I in some instances suggest a dependence of the relative permeability on 3:: 0 U ~ / fluid properties. This paper does not consider systems where there is I • 2 ill interaction between solid and liquid phases (e.g. clay hydration). it being ~ C// / / assumed that in the absence of such interaction the flow of fluids will be /[/ ~ controlled entirely by preferential wettability characteristics rather than by incidental fluid properties such as viscosity, etc. /V ~~ ~ :I: That is, krn at a given saturation, pn' in 2-phase systems will be o v:: ~ 0.2 0.4 0.6 0.8 1_0 greater than k rw at an equivalent saturation, pw' in 2 or 3-phase systems; o which in turn is less than k ro at an equivalent saturation, po' in 3-phase ~ - Wetting Phose Saturation systems; which in turn is greater than, less than or equal to above men tioned krn in two-phase systems; which in turn is greater than krg at pg FIG, 3 - EXAMPLES OF WETTING PHASE RELATIVE PERMEABILITY. in a g-o-b system. Theoretical Curves (Eq. 20). § These are: (1) k,.b in b-o system I. pw; assumed to be zero. (2) k,.b in b-g system II. pw; assumed to be 0.0]. (3) k,b in b-g-o system III. pw; assumedr to be 0.1. (4) k,.o in o-b system (5) k,o in o-g system IV. pw; assumed to be 0.3. (6) k '0 in o-g-b system V. pw; assumed to be 0.5. (7) k,g in g-b system V I. pw; assumed to be 0.67 (%) (8) k,.g in g-o system (9) keg in g-o-b system Experimental Curves: (1), (2), (3) and (5) may be expressed as k,w in 2 or 3-phase systems. A. Data obtained on single capillary tube". (4), (7) and (8) may be expressed as krn in 2-phase systems. B. Data obtained on unconsolidated sand'. * In this connection, it is observed that the Leverett and Lewis data. c. Data obtained on a synthetic rorc". (representing all that is presently known about relative permeability in 3 fluid phase systems) merely support but in no way confirm or establish n. Data obtained on I! dolomite". Fcq. (21-6) above, even for the unconsolidated systems they specifically studied, as can be seen simply by replotting the ternary diagrams they F. Data obtained 011 II ('ol1solidlltet! slInd'. reported. This observation, of course, does not detract from the pioneering accomplishment of their experimental work, or from the qualitative im portance of their conclusions except where a theoretical contradiction can t Here and elsewhere in this paper hydrophilic interstitial sut'{aces have be established. been assumed merely to simplify the presentation. May, 1949 PETROLEUM TRANSACTIONS, AIME 117 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY discrepancy between existing data and the generally accepted Fig. 4 is presented to illustrate the significance of the va theoretical concept symbollically stated as: rious saturation parameters employed in this paper. It is evi lim krw = 0 lim pw = pWi dent that whether pwm is used in Eq. (22) or whether pwl is where pw+PW! Po+OO used in Eq. (20) to derive relative permeability, this will de In the development of Eq. (20) the validity of this concept pend upon the nature of the flow process being considered. was assumed and the difficulty arises from the fact that appar Further, it is believed that these limiting saturation parameters ently in the general case: can be approximated from existing data correlations* or ex lim knv = 0 perimentally obtained in a manner less involved than present where pwm> pw! (usually) methods of relative permeability measurement.t This is illus For example, an examination of Leverett's capillary drainage trated with the aid of Fig. 5 where an assumption is made experimental data" gives values of pw! between 5 and 10 per (based on Botset's3 plot of gas-oil ratios) that 0.5 is a suitable cent, and these are to be compared with pwm values of 20-30 value to use for pwm when it is desired to fit Eq. (22) to Bot per cent elsewhere reported on similar sands'··'''t. This appar set's data as obtained under his stated flow conditions. This ent discrepancy is normally explained by assuming that the gives curve "A" in Fig. 5, and for comparison Botset's original relative permeability at saturation pwm is so small that it is Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 Nichols Buff data points are included. Also shown in Fig. 5 mistaken for zero, and that actually conditions of pendular is a plot of Eq. (20) where pw! is chosen to be 0.35 (curve configuration must be attained (i.e. pw = pw,) before flow will "B"). Since this value (pWi = 0.35) is a reasonable "irre cease and relative permeability will become zero. ducible" saturation for the Nichols Buff sand, it is concluded Although this explanation is plausible, it is of interest to that curve "B" shows a limiting position of the relative per consider another point of view. Fig. 3 in the recent paper by meability relation, which would characterize this sand under Holmgren" calls attention to the fact frequently observed in conditions quite different from those experimentally imposed the laboratory, that the production of a wetting phase from a porous system, effected by the displacing action of an invading by Botset.t non-wetting phase under pressure, depends upon various fac In concluding this discussion on the applicability of Eqs. tors associated with the conditions of flow, and that maximum (20) and (22) to the evaluation of k •. w, it is of interest to note production will be attained generally in the static type capil that these equations reduce to: lary pressure experiment rather than in a dynamic flow type dkrw experiment. It seems reasonable that pwm can be accepted as knv = pw or -- = 1 (22a) the dynamic equivalent of pw!, which itself is an "irreducible" dpw saturation attained only under conditions of static equilibrium. when the condition: pwm = pw! = 0 is satisfied, as identified In this sense pwm is not a fixed and unique saturation, but with a system of uniformly sized capillary tubes oriented in rather it is determined by the prevailing conditions of flow, the direction of flow. Eq. (22a) is of particular interest in and it becomes equal to pw i only in the limiting case. Therefore, connection with Leverett's argument' where he concludes that it will be assumed that pwm is an appropriate value to substitute relative permeability curves characterizing "a system of non in Eq. (20) when the conditions of polyphase flow are such 0 interconnected, parallel capillary tubes" will be simply 45 that pwm » pw!, and indeed this is required if Eq. (20) is to lines when plotted conventionally (i.e. as in Fig. 3) according be accepted as a suitable generalized expression for wetting phase relative permeability. Thus, Eq. (20) is rewritten as: * In thi! connection. Brownell and Katz5 develop the relation: 1 ( k dp ) ·0.26< krw pwm = 86.3 G"cos {J dx [2p,,' (2 - 3pwm) + 3Pwpwm (3pwm - 2) + pwm (4 - 5pwm)]' (22) § where G is the acceleration due to gravity and 8 is the angle of contact. Now. thi3 implies that the term pwm is determined by the pressure gradi t Similar experimental discrepancies have been elsewhere noted. Re ents in the flowing phases. requiring (acording to a strict interpretation cent capillary pressure experiments on NicHols Buff sandstone indicate of the equations appearing in this paper) that in some instances at least pwi values considerably lower than pwm values obtained by the extrapola the entire relative permeability curve will be shifted upon variation in tion of Botset's3 relative permeability data. Fig. 16 of the paper by the pressure gradients. Experimental evidence of such possibilities as Bulnes and Fitting (loc. cit.) illustrates this discrepancy as applied to applied to non-wetting phase relative permeability cap be found by exam limestones. ining Fig. 4 of the recent paper by Henderson and Yuster (loc. cit.). § According to these views, Eq. (10) must also be rewritten as: Actually. more recently at the 1948 Annual Conference on Secondary Recovery at the Pennsylvania State College these authors presented addi tional indications of such hysteretic shifts as applied to either wetting k TW == Pw ( -"'-0-"'D)' where 1T'D and 1T'c are the dymanic equivalents of the or non-wetting phases relative permeability. these shifts being entirely in accord with the theoretical interpretation prese1lted in the legend fOl static quantities, P D and Pc respectively. In general, it will be found "'0 7rD , Fig. 4 of this paper (Private Communication, October 16. 1948). The that for a given saturation > Po and that > P D which simply means that an interface in motion in a pore of a given size will have recent paper of Childs and George gives further evidence of these a larger specific surface area than the same interface in the same pore hysteretic shifts.34 under conditions of static equilibrium. (In this connection, note that Eq. (10) was developed to apply to a draining system, i.e. one where pw t For instance, pw i is simply the irreducible saturation observed in a is always constant or decreasing in magnitude. Similar argument. how drainage type capillary pressure experiment; that is. the maximum sat ever. can be applied also to an imbibing system. suggesting the possibility uration where the wetting phase configurationl is still entirely penduIar. of extremely complex hysteresis effects). Eq.(19) then can be rewritten as: t Curve "D" illustrates what might be expected if Nichols Buff capillary ~ = 2pw'(2 - 3pwm ) + 3pwPwm (3pwm - 2) + pwm (4 - 5pwm ) pressure data were used in Eq. (10) to describe wetting phase relative ~ permeability. The deviation between curves Band D supports the view "'D 4Ypw(1 pwm) (pw-pwm)3 that dynamic flow data are not directly derivable from static capillary in accordance with these views. pressure data. 118 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 to the assumed requirement that in such a system the sum of the effective permeabilities for the separate phases should total the specific permeability. Although as Henderson and 00 Yuster" have recently pointed out the experimental data of Martinelli et al" do not confirm Leverett's prediction,§ at least the equations developed in this paper are in accord with the 11 dp concept that in the absence of capillary interaction between c Pc Q) dx immiscible fluid phases (as would prevail, for instance, in a I E II hypothetical U.S.O. system of the type above mentioned) Q) u --- phase relative permeability is linearly related to phase satura· o 0.. II tion (c.£. Eq. (22a)). If) (:5 0 LI-+---+--~I------:::::::j004.--!1 0 Having indicated the usefulness of Eq. (20) and (22) to l'fw IPwi I~m 0 I I I evaluate wetting phase relative permeability under the condi oI '-w .11 I 1 I Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 tions described by Eq. (21-5) when reasonable values for pWi 1 ~r--'----r------~~I 1 (or pwm) can be assumed or otherwise obtained, it is of inter est now to see what further extensions can be made in the manner of evaluating specific non-wetting phase relations un der the 2-phase saturation conditions described by Eq. (21-4). 1 It has been observed that if we define 'l' w as the limiting wetting phase saturation associated with k,n (1); and then if we arbitrarily define the "effective" non-wetting phase satura tion range of interest to be within the interval, pw (1) and pw ('l'w); and then if we substitute in Eq. (22) the effective non-wetting saturation terms so defined for the comparable wetting phase saturation terms; an expression is evolved which apparently can serve as an equation for non-wetting phase relative permeability according to the conformance with experimental data indicated in Fig. 5 (Curve "C") and Fig. 6.* Thus, o ~---L--~~------~-H~I 0 1 pw == '.I-'w (Definition) : 1 " II 10 I I PnmllPnil Also, 11· Pn ---~o pn 1-pw pen =:--=-- (Definition) l--¥w l--¥w FIG. 4 - DEFINITIONS OF SATURATION PARAMETERS. pnrn penm ==-- Curves A, B, and C refer to the plots of a dependent variable 1-'1' w versus some function of fluid saturation as the independent variable. where pnm will be identified with equilibrium non-wetting phase In the top diagram the dependent variable is either the capillary saturation as commonly defined. Then by analogy to Eq. (22) : pressure difference measured across the interfaces of contact be k .. = tween the immiscible fluids saturating the interstitial spaces, or some 16p,,' (pn - pnm) 3 (1- -¥w - Pnm) function of the pressure gradients existing in these phases when [2pn' (2 - 2'l'w - 3Pnm) + 3pnpnm( 3pn~ - 2 + 2-¥w) + pnon (1 - 'l'w) they are flowing. In the bottom diagram the dependent variable is (4--¥w-5Pnm)J' (23) either the wetting phase or the non·weting phase relative permeabil ity. In particular, Curves A refer to fluid distributions obtaining where, under conditions of dynamic equilibrium (that is. under conditions lim krn lim k,n 1 of steady-state heterogeneous flow of immiscible fluids). Curves B pw+-¥w Pn+(l-'l'w) and C refer to fluid distributions obtaining under conditions of and static equilibrium which have been approached respectively by o. drainage or by imbibition capillary pressure processes. Thus, any given group of Curves A, B, and C show possible limits of hysteresis in the relation associated u'ith these curves. Note, for instance, that § Nor do these data deny Leverett's prediction since it may be supposed that the conditions of the Martinelli et al experiment are not entirely the fluid distribution whlch yields the largest ulI!lIe ior non·wetting applicable to a consideration of phenomena occurring in Leverett's ideal~ ized system. phase relative permeability (at a given non·wetting phase satura • Attention is called to the recent gas relative permeability data reported tion) invariably yields the lowest value for wetting phase relative by the author32 where conformance between the results of experimentation and the theoretical analysis as presented above also has been obtained. permeability, and vice versa. May, 1949 PETROlEUM TRANSACTIONS, AI ME 119 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY Thus, in the derivation of Eq. (23) the existence of limiting pore volume from which the portion occupied by the immobile non-wetting phase saturation terms, pn! and pnrn, analogous to wetting phase has been eliminated. t the wetting phase terms, pw! and pwrn, has been implied, as This discussion suggests that a character of symmetry is a shown graphically by Fig. 4 and as discussed in greater detail common feature associating wetting and non-wetting phase below. relative permeability to each other, for it is evident that a comparison of Eqs. (22) and (23) will yield: The manner in which eXIstmg experimental data are de scribed by Eq. (23), of course, can be fortuitous, to the ex krw (Pw) = krn (Pn) tent that insufficient data are available for examination. How when: ever, the applicability of Eq. (23) itself can be developed in pw = pn a reasonable manner from more or less plausible concepts in a 2-phase system where: 'lrw = 0 and where: pwrn = pw!. as follows: Physically, a condition of symmetry probably requires assum In the first place, Brownell in his discussion of the Martinelli ing that relative permeability is a function of saturation et al paper (loc. cit.), and later Brownell and Katz (loc. cit.) alone, and that "effectively" the pore configuration may be develop the view that the pore space saturated wtih capillary represented as an assemblage of equivalent capillary tubes. Thus, when a little non-wetting phase is introduced into an bound (immobile) wetting phase should be eliminated from Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 definitions of saturation functions employed in flow equations. initially non-wetting phase saturated system it occupies sev It is evident that if we identify the term 'lrw defined above with eral of the smaller pores, so that it is not unreasonable that the immobile wetting phase saturation, then the terms, a decrease in the non-wetting phase relative permeability pn pnm should reSillt approximately equivalent to the above referred -- and -- to decrease in wetting phase relative permeability. (Note, how- l-'lrw l-'lrw which were substituted in Eq. (22) to yield Eq. (23) actually 1.0 express non-wetting phase saturation parameters per unit of \ '" r--... 1 ~ "-.... ~ 1.0 .0 0.8 ~l \. ~ / ~ ° 0 \ \ i' (]) IF "- § \ \\ \A I~ >. '* ~ol~ ·W (]) .8 >. 0.. ~0.8 ' /J (]) o ° \1 B\\ ~ (l) .~ / '\\0 :D 0.6 E i..[l o "5 \' \ (l) Q) \ II ~ \ /.0:. E 0:: \ "\ 0.6 ~ 120 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 ever, this concept of symmetry has not been derived directly In connection with the application of the developments pre through analysis of the properties of any particular system). sented in this paper to the three-phase flow problem some Presumably, this assumed character of symmetry has not been quantitative statements now can be made, again based on the emphasized in the past simply because the end-points of the concept of inherent symmetry. In the first place the evaluation wetting and non-wetting phase relative permeability curves of the relative permeability of a wetting phase in the presence of one or more non-wetting phases has already been accom do not in general originate from com parables positions on plished to the extent that the equivalence of wetting phase the conventional data plot (cf. Fig. 4). Thus, whereas k,w (1) relations for a given porous body is established by existing occurs at pw(l), km(l) does not occur at pn(l) = pw(O); but data to be independent of fluid properties (c.£. Eq. 21-5). The rather it occurs at pn (1 - >¥ w = pw (>¥ w) where pw (>¥ w) only relative permeability to oil in a o-b-g 3-phase system has been occasionally equals pw (0). In a like manner, knv (0) occurs described by Eqs. (21-3) and (21-6). In order to explain these at pw(pwrn) although k,n(O) occurs at pn(Pnrn), where pwrn experimental observations, one is inclined to observe with usually is considerably larger than Pnrn.* Leverett and Lewis" that starting with a gas-oil system and In this connection it is observed that pwrn and pnrn are both holding po constant, the oil changes its distribution by migrat ing from smaller to larger pores as some of the non-wetting limiting saturation parameters associated with saturation dis gas is replaced by water_ However, to say that this in itself Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 tribution conditions of the respective phases such that an tends to increase the oil relative permeability (as apparently infinitesimal decrease in either pw or pn beyond pwrn or pnrn observed) invalidates the argument presented above which would result in a phase discontinuity. pwrn then represents the lead to the concept of symmetry. That is to say, granted the end point of a displacement experiment where the non-wetting average pore radius of the oil saturated portion of the system phase reduces the wetting phase to a minimum. (In a capil is increased as some of the gas is replaced by water, still the lary pressure experiment pwrn = pw i). Likewise, pnrn represents number of pores containing the oil is decreased simultaneously, the end point when the non-wetting phase is reduced to a such that it is assumed for the present that the resultant of minimum. Referring to the generalized pressure-saturation these effects does not entirely (if at all) account for the ob diagrams in Fig. 4 it is noted pnrn is approached in a more served facts expressed by the inequalities (21-3) and (21-6). Instead, the problem is simply solved by applying Eq. (22) abrupt manner than is pwrn. That is to say a funicular chord when the oil acts as a wetting phase, to give the relative per of non-wetting phase in the central pore portion of an assem meability of oil in an o-g system; and applying Eq. (23) when blage of sand grains and surrounded by wetting phase will be the oil acts as a non-wetting phase to give the relative perme abruptly pinched-off as globules by the wetting phase and ability of oil in an oil-brine system. This has been done for a thereby rendered immobile as pn is decreased to pnrn. On the typical case as illustrated by the dotted lines in Fig. 2, and it other hand, as pw is decreased to pwrn (or pw i) the wetting is seen that the form of the inequality (21-3) should be phase adhering to the pore wells is more gradually forced changed by this argument to: into a strictly pendular configuration, and, assuming 0 0 con k,o(po) > k,o (Po) >_k_'o(:c-Po.:....-) _ tact angle, actually never does become entirely discontinuous o-b system o-b-g system o-g system although it may so behave from a flow standpoint. In. any since pwrn was chosen greater than pnrn according to the general event, there is reason to expect in the region of low phase sat case. Although indicating the relative permeability to oil in uration a departure from wetting and non-wetting phase sym the three-phase system as immediate between that obtaining metry in the respective relative permeability curves (that is, in oil-brine and oil-gas two-phase systems represents a depart assuming as has been done above that a general character of ure from the Leverett and Lewis data (loc. cit.) intuitively it symmetry does exist at other levels of saturation) which mani is a necessity since there has not appeared an argument which fests itself in the form of a sharp non-wetting phase relative requires permeability cut-off in contrast to a more gradual wetting phase relative permeability cut-off. This then introduces a k,o (Po) § --:--"-'--- to sometimes be greater than minor limitation into the unqualified use of Eq. (23) for non o-b-g system o-b system wetting phase relative permeability, the quantitative magni Thus, combining Eq. (22) and (23) in a manner suggested tude which is suggested by observing the conformance be by the above discussion we obtain as an expression for oil tween theory and experiment as presented in Fig. 5c and § In a like manner Fig. 2 has been prepared to indicate that Eq. (21-2) Fig. 6. Presumably, the theoretical relationship for wetting should be modified as: phase relative perIl1eability (Eq. 22) does not invoke this k,g (pg) k,g (pg) limitation since it was developed from basic concepts to apply ~g--~O~S-y~st~e-m-=~g~-b~s-y~st~e-m- g-o-b system exactly to wetting phase flow in contrast to the intuitive devel Again, although this departs from the usual interpretation of the Lev erett and Lewis data no argument is apparent which supports the observa opment of the non-wetting phase relat'ion (Eq. 23). tion that the gas non-wetting phase relative permeability relation should be different in three-phase systems than in two-phase systems. This latter t It will be noted that Eq. (lOa) appears to contradict the intuitively would imply the gas knows the nature of the wetting phase(s) simul developed concept of inherent symmetry. as suggested by Curves E and F taneously saturating the porous system, which according to the concepts of Fig. 6. Indeed, all attempts at theoretical analysis of non-wetting phase assumed or established in this paper would be true only in unsteady state relative permeability have lead to monotonic curves concave downward (as flow. These arguments suggest that Eq. (21-6) should be modified as: opposed to the concave upward wetting phase relative permeability curves) when plotted as in the conventional diagram (c.r. Fig. 6). In k,n(pn) > k"n(pn) > k".(pw) view of the conformance obtained between non-wetting phase relative 2-8 phase syst.ems --o-g-b system-- -- 2-3 phase sysi~ permeability curves of this type and the" experimental data of Botset, as iEustrated by Curve E of Fig. 5, the concept of inherent symmetry should which is herein accepted as a qualitative description of the manner in be accepted for the present only to the extent that it is usefuL for ap which wetting and non-wetting phase relatvie permeabilities compare with proximation purposes. For it is believed that in some respects at least each other in a given porous medium under the indicated conditions of more exacting experimentation and more rigorous theoretical development equivalent phase saturation. It is apparent that this modified form of are yet to be accomplished before non-wetting phase relative permeability phenomena are completely understood. Eq. (21-6) will hold only when pwrn > pnm and/or when >¥w> o. May, 1949 PETROLEUM TRANSACTIONS, AIME 121 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY relative permeability in the presence of both brine and gas: x (24) * o-g-b system y'z' x=256po' (Po-Porn)' (I-Pbl-pom) (I-Pbl)' (I-Pbl-pgm)' (I-'l>'w-pgm) y = 2po' (2 - 2Pbl - 3pom) + 3popom (3pom + 2pbi - 2) + porn (1- pbi) (4 - 4Pb' - 5pom) z = 2 (1- Pb'), (2 - 2>f;w - 3Pim) + 3Pim (1- Pb;) (3pgm + 2>f;w - 2) + pgm (I-;fw) (4 - 4>f;w - 5Pirn) and where we may specifically identify Pbi as the irreducible Fig. 8 shows a plot Eq. (25) as it compares with cited data,t connate water, Pim as the equilibrium gas saturation, and pom and again fair conformance is noted, suggesting the applic- as the minimum oil saturation resulting from a gas drive. Oil Saturation in an Oil-Brine System Fig. 7 shows how oil relative permeability varies with connate 10 0.8 06 04 02 o 1.0 water saturation as predicted by Eq. (24), this plot being somewhat of the form suggested by the cited data of Leverett \ J Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 and Lewis (c.f. Fig. 9 of their paper, loco cit.) It is con 0.8 \ J cluded then that Eq. (24) can be useful to calculate oil relative permeability in three phase systems, for instance, \ /1 / when only two-phase laboratory data are available for study ~ or when assumptions can be made regarding the magnitude of =g0.6 \. II / krg = [(1- po - Pbl - pgm) (1- Pbl - porn) (1- >f;w - pgm)]' Curve A - Oil relative permeability in an oil-gas system containing kro (1- Pbi - pgm) (po - porn) (1- pgm) no connate water (viz. Eq. (22)). (25)t Curves B, C, D and E - Oil relative permeability in oil-gas-brine where: systems where Pbi is assumed to be 0.1, 0.2, 0.3 and 0.5 respectively and where ;fw is assumed to be 0.1 in each case. (viz. Eq. (24)). po + pg + Pb i = 1 Curve F - Oil relative permeability in an oil-brine system. (viz. Eq. (23)). Here again, Pbi represents the connate water, pgm represents the equilibrium gas saturation, and porn is the non-producible oil saturation by gas drive as was obtained for instance in the :I' For example, in setting up Eq. (25) to describe a typical three-phase case the following assumptions were made: experiments of Holmgren (loc. cit.). >f;w == 0.1; pgm == 0.1; porn == 0.3, and, * Thus, Eq. (24) reduces to Eq. (22) when Pbi = 0 and the oil acts Pbi == 0, 0.1, 0.2 and 0.4 as a wetting phase; and Eq. (24) reduces to Eq. (23) when pg = 0 and the oil acts as a non-weting phase. When all three phases (oil, gas and for the four examples given, yielding curves which can be compared connate water) are present Eq. (24) implies that the oil is acting as a directly to the relations illustrated in Fig. 7. Eq. (25) is rewritten as: wetting phase with respect to gas in the pore spaces left when the con nate water is considered as a part of the confining matrix along with . . (25a) the solid phase. This would seem to indicate that Eq. (24) should not be employed to predict the oil relative permeability when unreasonably when it is desired to describe approximately two~phase experimental data. high values are assumed for the connate water term~ Pbi' Therefore, in setting up Eq. (25a) to fit Botset's' data t More exact evaluation of this ratio entails appropriately solving Eqs. (20) , (22) , (23)' and/or (24) as required by the conditions under >f;w == 0.35, pgm == 0.1 and pwm == 0.5 consideration. were assumed as suggested by Fig. 5. 122 PETROlEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 ability of the concepts presented in this paper at least for saturation configurations as well as determine the exact oc. engineering use. Thus, it is seen that by Eq. (25): curence of the phase discontinuities. Therefore, hysteresis l' k,g 0 effects analogous to those observed in static capillarity studies 1m k,o = are to be anticipated as observable phenomena in relative p,,~-p.m permeability work. In this connection, the generalized expres. and, sions for relative permeability presented in this paper pre. 1Im-=· k,g 00 k,o sumabily can be used to calculate the various limits of po+pom hysteresis through suitable selection of the saturation paramo eters, although experimental proof is nonetheless desired to in accord with accepted concepts. Further, it is noted increas· further establish the validity of the assumptions which have ing Pbl, or increasing porn, or decreasing pgm, in each case hold· ing the other limiting saturation parameters constant, will lead to these expressions. Finally, it is concluded that insofar tend to increase the k,./k,o ratio at a giv8n total liquid as confirmation is offered by available data a character of saturation. symmetry seems to interrelate wetting and non.wetting phase Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 relative permeability relations in a manner requiring only the SUMMARY AND CONCLUSIONS knowledge of the above mentioned limiting saturation paramo To summarize, t.his paper has treated the subject of calcu Total Liquid Saturation lated relative permeability according to the point of view that 09 08 07 06 05 04 03 the configuration of the saturating fluid phases alone deter· mines the relationships which will prevail in steady state flow. 40 \ ..li Specifically an expression for wetting phase relative perme· \ ability has been derived from fundamental considerations 20 1'\ IAI based on modified forms of the Kozeny equation for flow and 'j\ \\ 1\ 11/\ Leverett's expression for capillary retention. Although the 10 -'>. derivation presented is neither rigorous nor unique it is sup· AI\ ~ ...l o \\ / ported through the indicated conformance of predicted reo o 4 {3 0:: \ '\ suIts with existing experimental data. Then, postulating the '\ 2 \\ IX' ~ concept of inherent symmetry as a character interrelating wet. '\~ \c ting and non· wetting phase relative permeability relationships ..D >-: o 1.0 1 1 the limiting saturation parameters (]) " y. ~ E ~ II ~ U. lim P and lim P 0.4 I "- k,+l ~ D / / [\'\ k,+O (]) <\ 1\ > o. 2 1 have been defined. This in turn has led to an expression for o o.1 I V t,\ t1 \ non-wetting phase relative permeability which again is sup· & 1 -" ported by conformance to existing data. Finally, general con. "0 I~ lTII~N sideration has been given to the problems of three· phase ·s 0.04 / /:L CT 1/ relative permeability phenomena and gas.oil ratios. :.=i rr\ ,\\ ,\ I 0.0 2 VJ In view of the considerations which have been examined in o o 0.0 I ~\ this paper it is concluded that the pore widths of the inter. II stitial spaces in contact with the immiscible fluid phase inter. 0.004 ~\ faces saturating these spaces is the factor of importance in ~ ~ h controlling the effective permeability of the porous body to 0.00 2 these phases. Essentially, capillary pressure data identifys the t distribution of pore widths with a range of fluid phase satura. 0.00.I 3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 tion configurations under conditions of static equilibrium, and Total Liquid Saturation it is shown herein how this fact can be employed to derive a generalized expression for the relative permeability relation FIG. 8 - RELATION OF GAS·LlQUID RELATIVE PERMEABILITY RATIO explicitly defined in terms of saturation parameters. In sug. TO TOTAL LIQUID SATURATION. gesting that the effective permeability characteristics in poly. phase systems are controlled by the magnitude of the pore Curves A, Band C gas-oil ratio data 0/ Leverett and Lewis" where spaces available for flow under the prevailing conditions of Pbl = 0, 0.2 and 0.4 respectively. saturation distribution, it is recognized that the limiting sat. Curves I, II, III and IV - Eq. (25) with limiting saturation param urations where phase discontinuities occur also have an im. eters chosen to approximately describe the Leverett and Lewis data, portance in defining the detailed features of the observed and where Pbl = 0, 0.1, 0.2 and 0-4 respectively. relations. Moreover, it has been suggested that in a given Curve D - Data of Botset'. porous body the conditions of flow determine the effective distribution of pore widths associated with the prevailing Curve V - Eq. (25a) with limiting saturation parameters chosen to approximately describe Curve D. May, 1949 PETROLEUM TRANSACTIONS, AIME 123 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY eters in order to treat the various problems of steady state REFERENCES polyphase flow. 1. Gerald L. Hassler: The Measurement of the Permeability of Reservoir Rock and Its Application. Science of Petru ACKNOWLEDGMENT leum, 1, 198-208 (1938). The author has apprecIated the opportunity to discuss this 2. M. C. Leverett: Flow of Oil-Water Mixtures Through Un paper with Morris Muskat and to benefit from his critical consolidated Sands. Trans. AIME, 132, 149 (1939). comments. Acknowledgment is given to Paul D. Foote, execu 3. Holbrook G. Botset: Flow of Gas-Liquid Mixtures Through tive vice-president of the Gulf Research and Development Co., Consolidated Sand. Trans. AIME, 136, 91 (1940). for permission to publish this paper. 4. G. L. Hassler, E. Brunner, and T. J. Deahl: The Role of Capillarity in Oil Production, Trans. AIME, 160, 155-174 NOMENCLATURE (1945). (1) Permeability Terms 5. Lloyd E. Brownell and Donald L. Katz: Flow of Fluids k -specific permeability Through Porous Media. Chem. Engl'. Progress, 43, 601- Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 kc - effective permeability 612 (1947). kr - relative permeability (fractional) (2) Saturation Terms (all per unit of pore volume expressed 6. R. D. Wyckoff and H. G. Botset: The Flow of Gas-Liquid as fractions) Mixtures Through Unconsolidated Sands. Physics, 7, 325 (1936:r. p - fluid sat~ration -lop, -limiting minimum saturation attained in static 7. Morris Muskat and Milan W. Meres: The Flow of Hetero 0 ••-, capillary pressure experiment geneous Fluids Through Porous Media. Physics, 7, 346 pm -limiting minimum saturation attained· in dynamic (1936) ; d. Morris Muskat, The Flow of Fluids Through flow experiment Porous Media, J. App. Phys., 8, 274-282 (1937); Morris if;w - immobile wetting phase saturation Muskat, Fluid Flow Through Porous Media, Proc. Sec. (3) Rock and Fluid Property Terms Hydraulic Conference, Bull. 27, University of Iowa Studies ", in Engineering. (1943). A - specific surface area of pores per unit of pore volume 8. L. E. Elkins: The Importance of Injected Gas as a Driving f - porosity (fractional) Medium in Limestone Reservoirs. API Drill. and Prod. t - Kozeny Rock textural constant Practice, 164, (1946). T - collective term for Rock Textural Properties 9. H. H. Evin~er and M. Muskat: Productivity Factor Calcu I" - fluid phase viscosity lations for Oil-Gas Water Systems in the Steady State. ()' - interfacial tension Trans. AIME, 146, 194 (1942). Po - capillary pressure PD --: displacement pressure 10. L. A. Richards: Capillary Conduction of Liquids Through 11'0 - dynamic equivalent of Po Porous Mediums. Physics 1, 318-333 (1931). 1I'D - dynamic equivalent of PD 11. 'R. N. Traxler and L. A. H. Baum: Permeability of Com j (Pw) - capillary pressure function (wetting phase) dp/ dx - pressure gradient in the x-direction pacted Powders-Determination of Average Pore Size. q -linear rate of fluid flow Physics, 7, 9-14 (1936). D;:- - distribution coefficient of pore radii 12. M. C. Leverett: Capillary Behavior in Porous Solids. r -pore radius Trans. AIME, 142, 152-169 (1941). o- contact angle 13. William B. Haines: Studies in the Physical Properties of (4) Miscellaneous Symbols Soils. J. Agr. Sci., 17, 264-290 (1927); ibid, 20, 97-116 F - denotes functional relationship (1930). G - acceleration due to gravity q, (Pw) - Rho Function 14. Ross W. Leamer and J. F. Lutz: Determination of Pore Size Distribution in Soils. Soil Sci., 49, 347-360 (1940) (5) Subscripts w -refers to a wetting phase (unspecified) 15. H; L. Ritter and L. C. Drake: Pore-Size Distribution in n-refers to a non-wetting phase (unspecified) Porous Materials. Ind. Eng. Chem. Anal. Ed. 17, 782 b - refers to a brine (aquous) phase (1945). o - refers to an oil phase 16. P. C. Carman: Fluid Elow Through Granular Beds. Trans. g - refers to a gas phase Inst. Chem. Eng., 15, 150-166 (1937). e - denotes effective 17. W. C. Krumbein and G. D. Monk: Permeability as a r - denotes reiative m - refers to minimum saturation value attained under Function of the Size Parameters of Unconsolidated Sands dynamic flow conditions Trans. AIME, 151, 153-163 (1943). i-refers to minimum saturation value attained in a 18. P. C. Carman: Capillary Rise and Capillary Movement of static capillary pressure experiment. Moisture in Fine Sands. Soil Sci., 52, 1-14 (1941). 124 PETROLEUM TRANSACTIONS, AIME May, 1949 WALTER ROSE T.P.2563 19. R. R. Sullivan and K. L. Hertel: The Permeability Method DISCUSSION For Determining Specific Surface. Adv. Coll. Sci., 1, 36-80 By A. C. Bulnes, Shell Oil Co., Midland, Texas (1942) . Walter Rose is to be congratulated on an excellent paper 20. J. J. McCullough, F. W. Albaugh and P. H. Jone~: Deter and one which constitutes, in my opinion, a most important mination of the Interstitial Water Content of Oil and Gas contribution to the subject of polyphase flow in porous media. Sands. Am. Pet. Inst. Drill. and Prod. Practice, 1944 (1945) . Particularly gratifying is the fact that it has been found possible to develop a comparatively simple expression for 21. W_ A. Bruce and H. J. Welge: The Restored State Method relative permeability from the fundamentals of capillary for Determination of Oil in Place and Connate Water. Pre phenomena and the quantitative measures of rock properties, sented Amarillo Meeting of the Division of Production, together with the aid of certain rather generous assumptions, A.P.I. (May, 1947). instead of being forced to fall back upon some unobvious 22. O. F. Thornton and P. L. Marshall: Estimating Interstitial and complicated empirical result. Moreover, in the process Water by the Capillary Pressure Method. AIME Petroleum of this derivation, the relationship existing between the meas Technology, T.P. 2126 (Jan. 1947). ured data of relative permeability, capillarity, and the void Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 structure of the medium is also brought out in a clear and 23. L. J. Klinkenberg: The Permeability of Porous Media to simple fashion and should be of interest to many readers. Liquids and Gases. A.P.l. Drill. and Prod. Pract. (1941), 200-213. The relative permeability of the wetting phase is shown to be a function of pw and of an irreducible saturation pwm or 24. M. C. Leverett and W. B. Lewis: Steady Flow of Gas-Oil pwl. The determination of pwl is relatively simple, or at any Water Mixtures Through Unconsolidated Sands. Trans. rate, it is easy to describe an experiment whereby it can be AIME 142, 107-116 (1941). measured. This does not appear to be the case for pwm; in fact, 25. R. A. Morse, P. L. Terwilliger and S. T. Yuster: Relative it is not clear to me just how this quantity can be determined Permeability Measurements on Small Core Samples. Oil except in terms of the relative permeability; that is, by solving & Gas J., 46, 109-125 (August 23, 1947). it in Eq. (22), a procedure that would defeat the purpose for which this equation was developed. Moreover, although it 26. J. H. Henderson and S. T. Yuster: Relative Permeability appears reasonable to conclude that pwl is a lithologic prop Studies. Prod. Monthly, 12, 13-20 (January, 1948). erty independent of the fluid properties,. it is not evident that 27. R. C. Martinelli, J. A. Putnam and R. W. Lockhart: Two the same statement can be made as regards pwm. I would Phase, Two-Component Flow in the Viscous Region. Trans. appreciate the comments of the author on the foregoing obser Am. Inst. Chem. Engr., 42, 681, 705 (1946). vations. 28. A. C. Bulnes and R. V. Fitting: An Introductory Discus The assumption that tew = t is of considerable importance sion of the Reservoir Performance of Limestone Forma in the mathematical developments and may be accepted as a tions. Trans. AIME, 160, 179 (1945). reasonable first approximation. Rose indicates that if not equal to t, tew is some function of the saturation. I would like to ask 29. C. R. Holmgren: Some Results of Gas and Water Drives Rose if he considers that tew may be also a function of the on a Long Core. Paper presented AIME Pet. Tech, July, interfacial tension, in which case we would expect the relative 1948, T.P. 2403. permeability to display a dependence upon this quantity. Granting the inequality of tew and t, I wonder whether Rose 30. R. G. Russell, F. Morgan, and M. Muskat: Some Experi would care to express an opinion as to the range of variation ments on the Mobility of Interstitial Waters. Trans. AIME to be anticipated in t/tew for normal variation in pw and/or (T, 170, 51 (1947). and also whether abandonment of the assumption of unity as 31. Walter Rose and W. A. Bruce, Evaluation of Capillary the value of the textural constant ratio would lead to funda Character in Petroleum Reservoir Rock", J. Eet. Tech., mental changes in the shapes of the theoretical curves in Fig. 3. (May, 1949). 32. Walter Rose: Permeability and Gas Slippage Phenomena. Presented before the Steering Committee of the Topical Committee on Petroleum Technology of the Division of Author's reply to A. C. Bulnes Production, A.P.I., at Chicago during the 28th Annual Upon reflection I find that my paper is an admixture of Meeting, November, 1948. cf. A.P.l. Drill. and Prod. Prac., theoretical probabilities and theoretical possibilities. The lat in Press, (1948). ter are not to be too seriously considered, although perhaps 33. P. C. Carman: Some Physical Aspects of Water Flow in it might be said they stimulate interesting argument. On the Porous Media, DisCltssions of the Faraday Society, 3, 72-77 other hand the probabilities are believed to be of some im (1948) . portance, and Bulnes has cleverly focused attention on this importance by questioning a major assumption upon which 34. E. C. Childs and N. C. George: Interaction of Water and the theoretical developments are based. In view of the fact Porous Materials, Discussions of the Faraday Society, 3, that no one heretofore has dealt with the textural characteriza 78-85 (1948). tion of polyphase saturated porous media it is appropriate May, 1949 PETROlEUM TRANSACTIONS, AIME 125 T.P.2563THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY therefore to set forth the rationale of the concept that the Curve D of Fig. 5 in the paper and the associated experimental ratio, t/t.w, is reasonably constant throughout the saturation data follows if the above described variation of the t/t.w ratio interval of interest. This is best done by recalling that the is employed in the solution of Equation 10. Actually, however, textural factor is defined as the product of a pore shape factor it appears that the development followed in the paper (based and a pore tortuosity factor. There appears to be no reason to on the Rho function concepts) obviates the necessity of giving attempt to describe the variation of shape factors with fluid explicit treatment to the rock textural factors, for the analyti saturation since it is well established in the literature that cal expressions finally developed to describe relative perme these generally fall within the narrow range of 2.0 to 2.5, ability phenomena give good conformance between theory and as applied to homogeneous fluid saturated media, and by experiment. In any event, it will be clear from the definition of implication at least as also applied to polyphase saturated the texture factor that the term, t.w, does not depend directly porous media.§ On the other hand, the tortuosity factor cannot on fluid properties such as interfacial tension. In this respect, be so easily disposed of as an universal constant, for values perhaps Bulnes implies that the dynamics of a given system are found to range between 2 and over 200 for consolidated might result in a dependence of the fluid distribution on such porous media. (A tortuosity value close to 2 is usually quoted fluid properties, which conceivably could result in a depen for unconsolidated porous media of various particle size, dence of tortuosity (and therefore texture) factors on fluid shape and surface characters). Therefore, perhaps we should properties since tortuosity certainly must depend on fluid dis Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 expect that as the "equilibrium" saturation of the non-wetting tribution character. Such dependence, however, will not be fluid is developed, the tourtuosity factor will increase since easily evaluated. Indeed, it may be regarded that an object a given element of flowing wetting liquid will have to pass of the writer's paper has been to present the view that the around additional barriers of non-wetting fluid in the central characters of polyphase flow through porous media are (a) in pore spaces. However, it is suspected that once the equilibrium adequately understood, and (b) difficult to evaluate either saturation of the non-wetting fluid is established, the tortuosity experimentally or theoretically. This condition, sadly enough, factor thereafter will remain sensibly constant at all other permits theorists to say almost anything and empiricists to wetting liquid saturation levels above the minimum where the report almost any data without immediate fear of serious wetting liquid itself becomes discontinuous. This is because contradiction. the tortuosity factor is a dimensionless ratio of actual length To reply to the comments of Bulnes about the difficulty of of fluid streamline to apparent length (i.e. actual bed length). evaluating the term, pwm, appearing in Eq. (22) of the paper, By representing the non-wetting fluid saturation as intercon further reference is made to the correlation of Brownell and nected spheres located in the central pore spaces it is seen Katz as cited in the paper footnote following Eq. (22).* The that the ratio will be independent of the distribution of spheres qualitative correctness of the form of this correlation has been (i.e. the non-wetting fluid saturation) as an approximation. supported recently by the examination of certain unpublished Thus, whereas t/t.w has been assumed equal to unity in the data obtained at this laboratory. It is believed therefore that paper, it would appear that the ratio actually tends to de controlled flooding experimentation should serve adequately crease initially as continuity in the non-wetting fluid is being for the determination of the limiting saturation parameters established, thereafter remaining nearly constant and less such as pwm and pnm. In any event, the feeling expressed by than unity throughout the wetting liquid saturation interval Bulnes that these limiting parameters will be dependent on of interest. Although this conclusion is based on highly specu fluid properties is born out by the cited result due to Brownell lative argument it is noted that better conformance between and Katz. * * * § See for example the recent Carman paper (Discussions of the Faraday * In the preprint version of the paper the Brownell-Katz equation for Society, No.3, 72-77. 1948), discussing these considerations. pw m was incorrectly given. 126 PETROLEUM TRANSACTIONS, AIME May, 1949