Interfacial viscous coupling: a myth or reality?

Ž .Journal of Petroleum Science and Engineering 23 1999 13–26

Interfacial viscous coupling: a myth or reality?

Muhammad Ayub ), Ramon G. Bentsen 1

UniÕersity of Alberta, School of Mining and Petroleum Engineering, Department of CiÕil and EnÕironmental Engineering,606-Chemical–Mineral Engineering Building, Edmonton, Alberta Canada T6G 2G6

Received 30 July 1998; accepted 31 December 1998

Abstract

This paper is a review of interfacial viscous coupling in multiphase porous media flow which has been a matter of debatefor the last two decades. Several researchers have viewed the occurrence of interfacial viscous coupling phenomena as areality and strongly recommended its incorporation into the existing Darcy formulation for multiphase flow through porous

Ž .media. Other investigators, however, have argued that the effect s of mutual transfer of momentum between flowing fluidsis extremely small; hence, it can be ignored and there is no real need to modify the conventional Darcy formulation. Others,however, disagree, even with the existence of such phenomena. This paper is an attempt to resolve this controversy. Toaccomplish this task, and to present a fair judgment about the controversy, three theoretical approaches are reviewed. These

Ž . Ž . Ž .are 1 the volume averaging methods, 2 the irreversible thermodynamic methods and 3 the use of analogous models,which were used to develop a theoretical understanding of viscous coupling phenomena. On the basis of this review, it wasfound that the phenomena of interfacial viscous coupling is closer to a reality rather than a myth. Therefore, it is stronglyrecommended that the conventional two-phase flow formulation of Darcy’s law should be modified. Some of the importantproblems, related to validation of interfacial viscous coupling theory, and possible means for resolving them, are identified.q 1999 Elsevier Science B.V. All rights reserved.

Keywords: Multiphase flow; Coupled processes; Flow mechanism; Permeability coefficients; Permeability

1. Introduction

Researchers have studied problems in fluid flowthrough porous media for many years. These include

Ž . Ž .Muskat 1949, 1982 , Scheidegger 1960 , CollinsŽ . Ž .1961 and Bear 1972 . The well known empirical

Ž .law of Darcy 1856 is fundamental to a macroscopicdescription of single-phase fluid flow through porousmedia. This law has been used extensively by

) Corresponding author. Fax: q1-780-492-3409; E-mail:[email protected]

1 E-mail: [email protected].

petroleum engineers for single-phase flow problems.However, the practical problems confrontingpetroleum engineers very frequently involve mix-tures of immiscible fluids, such as water and oil orgas and oil. This type of multiphase flow throughporous media is not well understood, and it is ex-tremely difficult to describe analytically because ofseveral factors such as the interaction between fluidsand rock and the complexity of the pore structureŽ .Islam and Bentsen, 1987 .

Conventionally, it has been assumed that theDarcy law, which was developed to describe single-phase fluid flow through porous media, is equally

0920-4105r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 99 00003-0

( )M. Ayub, R.G. BentsenrJournal of Petroleum Science and Engineering 23 1999 13–2614

good for two-phase flow through porous media. Be-cause of the extremely complex nature of the fluid–fluid and the fluid–rock interactions, this assumptionis highly questionable. To test the applicability ofDarcy’s law to two-phase flow problems, severalexperimental studies were performed using two im-

Žmiscible phases Cloud, 1930; Plummer et al., 1937;.Fletcher, 1949 . These experimental studies found

that the presence of a second phase can cause areduction in the permeability to the first phase. How-ever, the investigators were unable to provide asatisfactory explanation.

Several other investigators have attempted toovercome the difficulties involved in extending

Ž .Dacry’s law to two-phase fluid flow. Leverett 1941 ,for example, introduced the capillary pressure termin a formulation which was suggested initially by

Ž . Ž .Muskat and Meres 1936 and Muskat et al. 1937as an extension of Darcy’s law to two-phase flow. Inthis formulation, it was assumed that Darcy’s law is

Ž .independently applicable, without any cross effect s ,to both of the fluids during two-phase flow through

Ž .porous media. Fatt 1953 highlighted the importanceof rock compressibility during multiphase flow andemphasized its inclusion into the equations of multi-phase flow through porous media. Moreover, MooreŽ . Ž . Ž . Ž .1938 , Child 1945 , Muskat 1949 , Hubbert 1950

Ž .and Rose 1954 discussed several useful aspectsrelated to the principles of the multiphase flowthrough porous media.

Another questionable assumption inherited by theconventional formulation for two-phase flow throughporous media is the sole dependence of relativepermeability on saturation, and that it is independentof the pressure and velocity of the fluids. To provethis assumption valid, many investigators such as

Ž . Ž .Hessler et al. 1936 , Wyckoff and Botset 1936 andŽ .Richardson et al. 1952 conducted experimental

studies; however, this assumption is only an approxi-Ž .mation Scheidegger, 1960 .

Ž .The suspected cross effect s between flowingfluids, and other non-Darcian behaviors discussed by

Ž . Ž .Scheidegger 1960 and Bear 1972 during two-phase flow through porous media, were also investi-

Ž . Ž .gated by Yuster 1951 , Scott and Rose 1953 ,Ž . Ž .Klute 1967 and Bolt and Groenevelt 1969 . These

investigators indicated that the cause of non-Darcianbehavior during multiphase flow could be due to the

presence of a relatively new phenomenon, which inthe recent literature is commonly known as ‘viscouscoupling’.

1.1. Viscous coupling

ŽConventionally, in immiscible two-phase oil and.water flow problems, it is common practice to

assume that only one driving force, i.e., the potentialgradient acting across each phase, is responsible forfluid flow in porous media. Such an approach ne-glects the possibility that momentum transfer be-tween the two flowing phases may act also as adriving force. This additional driving force, whichmay occur at the fluid interfaces, is known as vis-cous coupling. In other words, as described by RoseŽ .1991a , ‘‘the term ‘viscous coupling’ connotes theidea that adjacent immiscible fluids flowing side-by-side in a porous medium will exert a reciprocalviscous drag on each other that will significantlyaffect the ensuing motions of the contiguous fluidparticles. This is the predicted consequence of theno-slip boundary condition, which allows the forceof momentum to be transferred across the fluid–fluidinterfaces.’’

Coupled processes such as thermo-electric, elec-tro-osmosis and chemical kinetics are well recog-

Žnized. Similarly, many researchers Rose, 1988b,1989; Bourbiaux and Kalaydjian, 1990; Kalaydjian,

.1990; Mannseth, 1991; Bentsen and Manai, 1993have suggested a considerable effect from viscouscoupling during two-phase flow through porous me-dia. The role of viscous coupling in multiphase flow,however, is controversial, as some of the studiesŽ .Zarcone and Lenormand, 1994 suggested either a

Ž .negligible effect or no effect at all Philip, 1972 .Moreover, many of the text book writers on

Ž .petroleum engineering such as Scheidegger 1960 ,Ž . Ž . Ž .Collins 1961 , Muskat 1982 and Greenkorn 1983

have ignored the idea of viscous coupling.For a complete understanding of the viscous cou-

pling phenomenon, and its possible or probable af-fects on the behavior of multiphase flow throughporous media, an extensive review of research con-ducted by various researchers is helpful. Therefore,in this paper, it is intended to describe the previouswork done related to viscous coupling. Specifically,

Ž .three theoretical approaches, that is, 1 the volume

( )M. Ayub, R.G. BentsenrJournal of Petroleum Science and Engineering 23 1999 13–26 15

Ž .averaging methods, 2 the principles of irreversibleŽ .thermodynamics and 3 the use of analogous models

to understand the novel formulations for two-phaseflow through porous media are reviewed. In addition,various modes for further theoretical and experimen-tal development are discussed briefly. In order tosimplify the task at hand of presenting a fair judg-ment, which may help in resolving the controversialissue of viscous coupling, only the stable, colinearflow of two immiscible, incompressible fluidsthrough a water-wet isotropic and homogeneousporous medium is considered.

2. Theoretical developments

Ž .Several papers referenced in Bear 1972 indicatethat the effect of viscous coupling has been known

Ž .from the early 1950’s. Yuster 1951 was the firstŽ .person to mention this phenomenon Yuster’s effect .

Ž .Scott and Rose 1953 tried to explain the Yustereffect by introducing the concept of viscous cou-

Ž .pling. Bear 1972 discussed the non-Darcian behav-ior of fluid-flow at very low hydraulic gradients,

Ž .which serve as the driving force. Klute 1967 at-tributes this non-Darcian behavior to electro-osmosiseffects, particle movement and quasicrystalline water

Ž .structure, and so forth. Bolt and Groenevelt 1969 ,however, held coupling phenomena responsible forthe non-Darcian behavior of fluid flow throughporous media. To understand the mechanism of mul-tiphase flow through porous media, several theoreti-cal approaches, such as the theory of mixturesŽDrumheller, 1978; Bowen, 1980, 1982; Bedford andDrumheller, 1983; Murdoch and Kowalski, 1992;Wang and Beckermann, 1993; Cheng and Wang,

.1996; Wang and Cheng, 1996; Wang, 1997 , theŽvolume averaging methods Whitaker, 1967, 1973,

1986; Slattery, 1969, 1970; Bear, 1972; Gray, 1975;Gray and Lee, 1977; Hassanizadeh and Gray,1979a,b, 1980; Narasimhan, 1980; de la Cruz andSpanos, 1983; Bachmat and Bear, 1986; Bear and

.Bachmat, 1986 and the principles of irreversibleŽthermodynamics Onsager, 1931a,b; de Groot, 1963;

de Groot and Mazur, 1963; Katchalsky and Curran,1967; Bear, 1972; Kalaydjian, 1987, 1990; Kalayd-jian and Marle, 1987; Longeron, 1987; Del Riop and

.de Harro, 1992 have been adopted by various inves-tigators. Moreover, the use of analogous modelsŽ .Yuster, 1951; Bacri et al., 1990; Rose, 1990a, 1993is also a common practice among researchers. It

Ž .should be noted that Trapp 1976 has presented aone-to-one correspondence between the various pa-rameters of the continuum theory of mixtures and thevolume averaging methods.

2.1. Volume aÕeraging methods

In an effort to extend Darcy’s law for single-phaseflow to multiphase flow through porous media, de la

Ž .Cruz and Spanos 1983 constructed a theoreticalversion of Darcy’s empirical law by using volume

Žaveraging theorems Hassanizadeh and Gray,1979a,b, 1980; Bachmat and Bear, 1986; Bear and

.Bachmat, 1986 . For a horizontal, one-dimensional,homogeneous, porous medium, the flow equationsdeveloped by these investigators are:

1 1 D p1™ ™m q y q sy 1Ž .1 1 2ž /K K L11 12

and

1 1 D p2™ ™m q y q sy , 2Ž .2 2 1ž /K K L22 21

where K and K are the generalized phase per-11 22

meabilities for phase 1 and 2, respectively, andwhere K and K are the viscous drag coefficients12 21

which represent the viscous drag that exists betweenphases 1 and 2. Note that the conventional Darcy lawapproach involves only a single permeability coeffi-cient for each phase due to its neglect of interfacialviscous drag.

Ž .Another effort Whitaker, 1986 involving the useof the volume averaging method to develop thegoverning equations for two immiscible fluids hasresulted in a similar generalized theoretical versionof Darcy’s law which involves a system of twoequations with four unknown parameters. The finaloutcomes of these two formulations are similar toeach other; however, some of the parameters such asq , and K are defined differently in the two ap-i i j

proaches. Nevertheless, the main point worth notingis that both of the theoretical formulations indicate


the presence of fluid–fluid drag or viscous couplingphenomena.

2.2. Principles of irreÕersible thermodynamics

Several attempts, using the ideas of irreversiblethermodynamics, have been made to achieve a satis-factory explanation of the cross-effect of one fluidon the other in a multiphase flow system. BearŽ .1972 has provided a summary of Onsager’s theorywhich is the basis of the thermodynamics of irre-versible processes. In irreversible processes, by usingthe methods of continuum mechanics, it is assumedthat the relationship between the fluxes and forces islinear. The assumption of linearity between fluxesand forces is common in other types of irreversibleprocesses such as Fourier’s law which relates heatflow to thermal gradient; Fick’s law for a multicom-ponent system which relates the flow of each compo-nent to its concentration gradient; Ohm’s law whichrelates the flow of electrons to the potential gradient;

Ž .and Newton’s law for Newtonian fluids whichrelates the shearing force to the velocity gradient. Inall these cases, however, the simple relationship oflinearity between forces and fluxes does not alwaysexist. The main cause of the overall non-linearity ina multicomponent system is thought to be the non-

Ž .linear or linear coupling between one type of forceŽ .to another type of flux Bear, 1972 . For further

details of Onsager’s theory and other concepts ofirreversible thermodynamics, such as entropy andentropy production, the reader is referred to OnsagerŽ . Ž .1931a,b ; de Groot 1963 , de Groot and MazurŽ . Ž .1963 and Katchalsky and Curran 1967 .

Unfortunately, a sufficient understanding of non-linear processes in non-equilibrium thermodynamics

Ž .is not available. Hassanizadeh and Gray 1987 ,however, were able to develop a non-linear relation-ship between the pressure gradient and the flowvelocity at the macroscopic level by employing thegeneral continuum approach to the description ofthermodynamic processes in porous media. Berge

Ž .and Bolt 1988 investigated the phenomena of heatflow and liquid flow through porous media, andestablished a connection between the two classicalapproaches. More recently, Del Riop and de HarroŽ .1992 used a thermodynamics approach and devel-

oped approximate time evolution equations for fluxesin a porous system.

More importantly, an excellent theoretical devel-Ž .opment was reported by Kalaydjian 1987, 1990 ,

who used the concepts of irreversible thermodynam-ics to develop a macroscopic understanding of two-phase flow through porous media. The flow equa-

Ž .tions developed by Kalaydjian 1990 are:

kPk 1 kPk 2E P E Pr r1 21 1q s y q y , 3Ž .1 ž / ž /m E x m E x1 2

and

kPk 1 kPk 2E P E Pr r1 22 2q s y q y . 4Ž .2 ž / ž /m E x m E x1 2

The diagonal terms k 1 and k 2 are the permeabil-r r1 2

ities of individual phases and the nondiagonal termsk 2 and k 1 represent the interfacial viscous couplingr r1 2

between the fluid phases. An experimental method toevaluate these nondiagonal terms has also been re-

Ž .ported by the author, Kalaydjian 1990 . As far asŽ .the cross-effect s between various species of two-

phase flow through porous media is concerned, theseequations are analogous to those developed by de la

Ž .Cruz and Spanos 1983 , and also to those developedŽ .by Whitaker 1986 .

2.2.1. Applicability of the Onsager’s relationsAccording to Onsager’s fundamental theorem

Ž .Onsager, 1931a,b , the cross coefficients, which inthis case are l and l , are symmetric, provided12 21

that a proper choice of fluxes and forces is made.This principle of equality between the cross coeffi-cients is the central point of the Onsager’s theory ofirreversible processes at the microscopic level.

Ž .Kalaydjian 1990 tentatively assumed that On-sager’s reciprocity relations of irreversible thermody-namics for microscopic systems may also be validfor macroscopic systems. In other words, non-diago-nal terms in the equations above are equal to eachother. This assumption, though tentatively adopted

Ž .by Kalaydjian 1990 , is questionable to many inves-tigators who argue that the complex nature of theporous medium does not allow for such an assump-

Ž .tion. Bentsen 1994 , for example, on the basis of theŽ .results presented in Mannseth 1991 , showed that

the non-diagonal transport coefficients responsible


for viscous coupling, l and l , are not equal12 21

because of the non-linear relationship between thefluxes and the forces. More recently, BentsenŽ .1998b , by using the fractional flow theory, inconjunction with Kalaydjian’s transport equationsand Liang and Lohrenz’s method for generalizedmobilities, has constructed new equations for calcu-lating effective mobilities. The author’s investiga-tions led to the conclusion of inequality between thecross coefficients, l and l , which control mo-12 21

mentum transfer between fluid phases flowing inporous media; this implies non-applicability of On-sager’s theory at the macroscopic level.

There are clearly two schools of thought regard-ing the applicability of Onsager’s relationships to thegeneralized formulation involving four generalizedpermeability coefficients developed and advocated

Ž .by Rose 1951, 1988a,b, 1990a,b, 1995, 1997 ;Ž . Ž .Whitaker 1967, 1986 ; Slattery 1969, 1970 ; de la

Ž . Ž .Cruz and Spanos 1983 , Kalaydjian 1987, 1990and others. Among those who recently developedtheir theoretical formulations and consistently arguedin favor of the applicability of Onsager’s theory to

Žcoupled flows in porous media are Rose 1969,. Ž .1990a,b, 1991a,b, 1995 , Kalaydjian 1987, 1990 ,Ž .Auriault and Lebaigue 1989 and Auriault and

Ž .Lewandowska 1994 . Those who do not favor fullandror partial implementation of Onsager’s relationsto coupled flow phenomena include Bentsen and

Ž . Ž .Manai 1993 , Goode and Ramakrishnan 1993 ,Ž . Ž .Bentsen 1994 and Avraam and Payatakes 1995 .

Some other studies such as Rakotomalala and SalinŽ .1995 conditionally recommended the applicabilityof the above mentioned theory of irreversible ther-modynamics to coupling processes. They found thatOnsager’s relations hold only for unit mobility ratioand that a very small amount of viscous couplingwas observed for other mobility ratios. Bentsen and

Ž .Manai 1993 , however, clearly observed unequalnondiagonal permeability coefficients, even for thecase of unit mobility ratio, which implies non-appli-cability of Onsager’s relations.

Keeping in view the diversity in opinions regard-ing the applicability of Onsager’s relations to theviscous coupling phenomena, the situation seems tobe very complicated. Therefore, it is desirable andlogical to validate viscous coupling theory by furtherexperimental research.

2.3. Analogous models

ŽA number of investigators Yuster, 1951; Bacri et.al., 1990; Rose, 1990a, 1993; Ehrlich, 1993 have

used simple analogous models to gain insight intohow two immiscible phases flow through a porousmedium. In the porous media literature, two types of

Žanalogous models tubular flow and Hele–Shaw.flow are frequently used. In these simple models,

where possible interaction between the wetting fluidand the rock surface is commonly ignored, it is usualto assume that the boundary separating the wettingphase from the nonwetting phase is, in its entirety, a

Žfluid–fluid interface Rapoport and Leas, 1951;.Chatenever and Calhoun, 1952; Bear, 1972 . Ne-

glect, in the simple analogous models, of the rockfluid surfaces in the interfacial boundary gives rise totwo problems. First, the simple analogous modelsoverestimate the amount of momentum transferredacross the interfacial boundary separating the twophases. Second, the relative permeabilities take onunrealistic physical values, in certain cases, at limit-ing values of the viscosity ratio. To overcome these

Ž .difficulties, Scott and Rose 1953 adopted a morerealistic approach by assuming that the wetting phaseis separated from the nonwetting phase in part by a

Ž .solid-phase boundary of infinitesimal thickness andin part by a fluid–fluid boundary. The flow equa-tions developed by employing two types of analo-

Ž .gous models tubular flow and Hele–Shaw flow aregiven next.

2.3.1. Tubular flowThe analogous model described here is that used

Ž . Ž .by Yuster 1951 and Bacri et al. 1990 . The systemanalyzed is a simple tube of small diameter with

Ž .phase 2 nonwetting flowing in the central portionŽ .of the tube, and phase 1 wetting flowing concentri-

cally in the annular space between phase 2 and thewall of the tube. It is supposed that the radius of theinterface, R , is a function of x, the coordinate ofi

Ž .the direction of flow, and that d R x rd x<1.i

Under these conditions, the velocities in the directionof flow are dependent upon the radial coordinateonly, and are defined by:

1 E pi 2u s r qa ln rqb ; is1,2 5Ž .i i i4m E xi


The four constants of integration, a ,b can be deter-i i

mined by applying the four boundary conditions: now Ž . xslip condition at the wall of the tube u R s0 ;1

finite shear stress at the center of the tube, rs0;continuity of velocity at the fluid–fluid interfacew Ž . Ž .xu R su R and continuity of shear stress at1 i 2 i

w Ž . Ž .xthe fluid–fluid interface t R st R . Ther x ,1 i r x ,2 i

boundary separating the two fluids is supposed to bein part a fluid–solid surface and in part a fluid–fluidinterface. The latter two boundary conditions areassumed to apply only on the fluid–fluid part of theboundary. That is, continuity of velocity and shearstress is not invoked on the fluid–solid part of theboundary. Rather, boundary conditions appropriatefor a fluid–solid surface are utilized. Moreover, thelubrication approximation for film flow is employedso that:

E p E p1 2s s0 6Ž .

E r E r

Finally, the fluxes are assumed to be defined by

2 Rn s ru d r 7Ž .H1 12R Ri

and

2 Rin s ru d r 8Ž .H2 22R 0

where R is the radius of the tube. By applying theŽ .four boundary conditions to Eq. 5 , and by making

Ž . Ž .use of Eqs. 7 and 8 , it may be shown that

E p E p1 2n syl yl 9Ž .1 11 12E x E x

and

E p E p1 2n syl yl 10Ž .2 21 22E x E x

where, the phenomenological coefficients, l , arei j

generalized conductances or mobilities.Ž . Ž .It is to be noted that, Eqs. 9 and 10 are the

tubular analogues to the transport equations proposedŽ . Ž . Ž .by Kalaydjian 1987, 1990 , that is, Eqs. 3 and 4 .

Moreover, it is possible to derive tubular analoguesto the transport equations proposed by de la Cruz

Ž . Ž .and Spanos 1983 and Whitaker 1986 . In theinterest of brevity, however, these results are notreported here.

2.3.2. Hele–Shaw flowThe analogous model is that employed by Rose

Ž .1990a . That is, the system which is analyzed is aHele–Shaw cell wherein two fluids are flowing, withthe less dense fluid located between the interface andthe top of the cell, and the more dense fluid locatedbetween the interface and the bottom of the cell.Utilizing the same approach as before, it may beshown that:

E p E p1 2n syL yL 11Ž .1 11 12E x E x

and

E p E p1 2n syL yL . 12Ž .2 21 22E x E x

Ž . Ž .Eqs. 11 and 12 are the Hele–Shaw analogues tothe flow equations obtained by utilizing the othertheoretical approaches, as described in the precedingsections.

3. A common message

Although, for the sake of brevity, the final formu-lations obtained by utilizing only three theoreticalapproaches are described in the preceding sections,the other theoretical approach mentioned in Section

Ž2, that is, the continuum theory of mixtures Murdoch.and Kowalski, 1992; Wang, 1997 has arrived also at

a more or less similar final formulation. Because ofthe various definitions of different parameters in-volved in the theoretical development, however, theappearance of the final formulations obtained bymeans of the various theories may differ slightly

Ž .from each other. For example, Wang 1997 , usingthe theory of mixtures, has developed a formulationfor immiscible fluids flowing through a porousmedium, where he has introduced the concept of‘mixture pressure’, just like the concept of ‘globalpressure’ which earlier was introduced by ChaventŽ . Ž .1976 and Yortsos 1987 for immiscible displace-ment in porous media. The other porous media theo-

Žries, such as the one adopted by Kalaydjian 1987,.1990 , use separate pressure gradients, rather than a

mixture pressure or global pressure, for each fluidduring multiphase flow through porous media. Note


Ž .that, Wang 1997 , although taking a different ap-proach to investigate the viscous coupling phe-nomenon, has indicated a considerable effect ofviscous coupling on the relative mobility and thecross-interaction mobility terms used in his formula-tion.

A common message is conveyed by all of theresearch reviewed above; that is, the interfacial vis-cous coupling phenomenon in multiphase flowthrough porous media is an important phenomenon,which must be addressed by more vigorous research

Žefforts. In 1996, Walter Rose a prominent re-.searcher of the viscous coupling phenomenon , while

addressing a seminar held by the Alberta ResearchCouncil of Canada, emphasized the need of suchvigorous efforts and said, ‘‘if the viscous couplingmodel is selected, a correct result will be obtainedwhether or not viscous coupling is actually occur-ring. On the other hand, if the conventional model isselected, a correct result will be obtained only in thecase that viscous coupling is not present’’. Obvi-ously, a genuine question can be raised; why then,despite all this theoretical evidence, has the viscouscoupling phenomenon been unable to achieve its duerecognition within the scientific community? Oneanswer to this question is: because it is extremelydifficult to validate the theoretical indicators by ex-perimental means! In the following section some ofthe efforts to validate the theoretical findings arereviewed.

4. Verification of the theory

In an applied scientific discipline, such as multi-phase flow through porous media, no theoretical

Ž .claim s is acceptable unless andror until supportedby the physical evidence. Several investigators haveadopted andror adapted various physical ap-proaches to validate the viscous coupling phe-nomenon. Three main approaches to validate thenewly postulated viscous coupling theory are de-scribed below.

4.1. Laboratory experiments

Various theoretical approaches as described abovehave resulted in a set of large-scale flow equationsfor two immiscible phases in a porous medium with

four generalized permeability coefficients. Spanos etŽ .al. 1986 further extended the formulations by re-

interpreting the parameters of the Buckley and Lev-Ž .erett 1942 theory. They were able to incorporate

the four generalized permeability coefficients intothe Buckley–Leverett conventional formulation. In

Ž .another similar study, Rose 1988a has shown thatthe Buckley–Leverett equation can be modified andapplied to linear displacement processes involvingviscous coupling.

ŽHaving pointed out that a generalized theory Eqs.Ž . Ž ..3 and 4 is available, and that Buckley–Leveretttheory may be modified to include the effects ofviscous coupling, attention is now focused on experi-mental procedures to measure the four newly devel-oped generalized permeability coefficients. These arethe coefficients which are needed to validate the newgeneralized theory of immiscible displacement.

To obtain the values of the generalized permeabil-ity coefficients, various experimental schemes havebeen reported in the literature. At least two types ofexperiments are needed to quantify the matrix of

Ž .four generalized transport coefficients Rose, 1988b .Ž .Rose 1988b suggested conducting two types of

experiments, namely, horizontal flow and verticalflow experiments to obtain the required parameters.However, in this scheme two major difficulties areinvolved which are associated with measuring thegravitational effect, and the difference in the verticaland horizontal velocities. In their experimental re-

Ž .search, Bentsen and Manai 1991, 1993 and ManaiŽ .1991 presented a scheme involving cocurrent flowand countercurrent flow to overcome the difficultiesassociated with Rose’s approach which used horizon-

Ž . Ž .tal and vertical flows Rose, 1988b . Goode 1991suggested indirect measurement methods involving

Ž .two types of experiments. Liang 1993 and LiangŽ .and Lohrenz 1994 proposed a combination of

steady-state and unsteady-state experiments, onceagain, overcoming the above mentioned difficultiesof Rose’s method.

Ž .Recently, Dullien and Dong 1996 presented anexperimental approach to determine the four general-ized permeability coefficients by setting the pressuregradient in one of the two flowing fluids equal tozero. The values of the cross coefficients with re-spect to the effective permeability to water and oilwere found to be significant. Their approach of using


one flowing fluid, while keeping the other stationary,is a limiting case of two-phase flow phenomena; thatis, it does not represent the overall mechanism oftwo-phase flow through a porous system. Moreover,they were unable to determine the net contribution ofviscous coupling to the water or oil flow, when therespective pressure gradient across the water or oilphase was set equal to zero.

In the author’s opinion, however, to overcomethese difficulties and to include the effects of allpossible modes of fluid flow, various combinationsof steady-state and unsteady-state displacement ex-periments with cocurrent and countercurrent flowscenarios should be adopted.

4.2. ConÕentional numerical methods

Commonly, the laboratory experimental approachis considered as the best option to validate any noveltheoretical formulation. Perhaps, this is the reason

Žthat conventional numerical approaches finite-dif-.ference methods and finite-element methods are very

scarce in the literature related to the validation of theviscous coupling theory. By using a finite-differencealgorithm based on a Lagrangian modeling of two-

Ž .phase porous media flow, Rose 1990b , presentedsome results such as the importance of including theeffects of viscous coupling, capillarity and gravityinto the conventional numerical algorithm.

Ž .For an idealized porous medium, Goode 1991developed a percolation algorithm by using a finite-difference scheme to demonstrate the momentumtransfer during two-phase porous media flow. In

Ž .another study, Goode and Ramakrishnan 1993 useda network model to simulate the invasion of nonwet-ting phase using their percolation-like algorithm. Inthis study, however, by making use of the triangulargrids a finite-element scheme rather than a finite-dif-ference scheme was employed. They observed asignificant effect of the off-diagonal terms on thebehavior of fluid flow. Moreover, the cross coeffi-cients were found to be a strong function of satura-tion and viscosity ratio.

Investigative studies to validate the viscous cou-pling theory by utilizing the conventional numericalmethods are extremely limited. Therefore, based on afew available articles, it is very difficult to concludesomething definite.

4.3. Lattice gas automata

Lattice gas automata, a branch of cellular au-tomata, is a relatively new method for computationalfluid dynamics and is essentially a discrete model intime and space, where on a regular lattice, particlesof identical mass move from one site to another witha unit speed. Upon collision at a site or vertex, theparticles conserve their mass and momentum accord-ing to pre-specified simple mathematical rules. Re-markably, at an extended level, this simplicity is ableto handle extremely complex systems, such as porousmedia flow, at a macroscopic scale. For furtherdetails and relevant studies the reader is referred to

Ž . Ž .Wolfram 1986a,b , Rothman 1988, 1990 , Roth-Ž . Ž .man and Keller 1988 , Kadanoff et al. 1989 ,

Ž . Ž . Ž .Monaco 1989 , Zanetti 1989 , Doolen et al. 1990 ,Ž . Ž .Gutman 1990 , Gao and Sharma 1994a,b , Ferreol

Ž .and Rothman 1995 , van Genabeek and RothmanŽ . Ž .1996 and Olson and Rothman 1997 .

Ž .Frisch et al. 1986 suggested a lattice gas au-tomata technique for incompressible Navier–Stokesequations and its behavior at a macroscopic scale.Later, this work was extended by other investigators

Ž . Ž .such as Kadanoff et al. 1989 and Zanetti 1989 .Ž . Ž .Rothman and Keller 1988 and Rothman 1990

showed that this can be used to simulate two-phaseflow through a porous medium at the macroscopic

Ž .scale. Rothman 1990 conducted a numerical simu-Žlation study to validate the macroscopic laws earlier

Ž . .developed by Kalaydjian 1987, 1990 and othersfor immiscible two-phase flow in porous media.Under conditions of negligible capillarity, the macro-scopic laws were validated, and a linear relationshipbetween forces and fluxes, at a sufficiently high levelof forcing, was observed. Moreover, Onsager’s rela-tions were found to be valid only in the abovementioned linear regime. At high flow rates, how-

Žever, where inertial effects become significant i.e.,.Reynolds number41 , the symmetry in Onsager’s

reciprocal relations was unobservable.Ž .In a recent study, Olson and Rothman 1997

attempted to compute both coupling coefficients kwnŽ .and k same as K and K , respectively in anw 12 21

complex digitized rock geometry which was deter-Žmined by X-ray micorotomography Kinny et al.,

.1993 . The coupling coefficients were computed in-dependently and found to be equal, thus maintaining


the reciprocity of Onsager’s relations; however, theauthors found that, while Onsager’s relationship ap-peared to hold for equilibrium flows, it did not holdfor non-equilibrium flows because such flows areoutside the domain of applicability of the Onsagertheory. Moreover, the author’s claim that every pre-vious experimental determination of viscous cou-pling was conducted by utilizing an artificial porousmedium is not, strictly speaking, true. For example,

Ž . Ž .Bentsen and Manai 1991, 1993 , Manai 1991 andŽ . ŽBentsen 1992a,b used unconsolidated cores Ot-.tawa sand , which are likely to be a better representa-

tion of a porous medium than most other types ofŽporous medium with the possible exception of ac-.tual reservoir cores used in other experimental, sim-

Ž .ulation or lattice gas automata technique s .Although lattice gas automata and conventional

numerical simulation methods for validation of thetheoretical hypothesis provide certain limited advan-

Žtages over laboratory experiments Rothman, 1990;.Olson and Rothman, 1997 , the requirements of ad-

vanced computational techniques and the use of ficti-tious particles to represent the fluids make theseapproaches more difficult to adopt on a routine basis.Therefore, in our opinion, for the sake of a basicunderstanding of the viscous coupling theory, morecommonly used methods, that is, laboratory experi-mentation should be adopted first to validate thetheory; and then, to gain more insight and to authen-ticate the theoretical and the experimental results,conventional numerical methods and lattice gas au-tomata should be used.

5. Quantitative impact of ignorance

From the discussions in the preceding sections, amajor observation is that most of the available litera-ture on the subject at hand is either in full or partialsupport of the existence of viscous coupling phe-nomenon during two-phase flow through porous me-dia. Some of the experts, however, argued rightly orwrongly, that the transfer of momentum is small;hence, it can be ignored. Some investigators, such as

Ž .Philip 1972 , have rejected the idea of viscouscoupling completely. The main issue, however, seemsto be the question; how much?

Ž .Kalaydjian 1990 described an experimental ap-proach to quantify the effect of viscous coupling onboth microscopic and macroscopic levels in a squarecross-section capillary tube and in porous media,respectively. His investigation showed nonnegligibleviscous coupling terms in the matrix of generalizedpermeability coefficients, even at the macroscopiclevel.

Ž .In a similar effort, Rose 1991a compared twomodels: one which included viscous coupling, the‘C-model’, and one which did not, the ‘D-model’.

ŽEquality of the reciprocal coupling coefficients K12.'K was assumed. He showed that, when conven-21

Ž .tional methods viscous coupling not included wereused to determine the effective mobilities, a signifi-cant amount of error was incurred. As discussed inthe preceding sections, however, the assumption ofequality of the reciprocal coupling coefficient iscontroversial. In a previous study, to demonstrate the

Ž .effect of ignorance of viscous coupling, Rose 1990bpresented a Lagrangian algorithm for the simulationof coupled two-phase flow in porous sediments. Heobserved a considerable effect of viscous coupling

Fig. 1. Comparison of effective mobilities with and withoutviscous coupling between fluid phases.


by predicting various positions of the flood-frontagainst different saturations for two cases wherecoupling was and was not involved.

Ž .Moreover, Bentsen 1998b explored the impactof neglecting viscous coupling between fluid phaseson the effective mobility curves. He demonstratedthat the effective mobility curves that include theeffect of viscous coupling between fluid phases dif-fer significantly from those that exclude such cou-pling, Fig. 1. Moreover, he showed that the conven-tional effective mobilities that pertain to steady-state,cocurrent flow, steady-state, countercurrent flow andpure countercurrent imbibition differ significantly,Fig. 2. Thus, it appears that traditional effectivemobilities are not true parameters; rather, they areinfinitely nonunique.

Ž .Recently, Bentsen 1997 attempted to quantifyŽ .the effect s of neglecting viscous coupling by esti-

Ž .mating the relative error incurred when this effect sis neglected. By investigating several one-dimen-sional problems, he concluded, that relative errors ofabout 40% are introduced into the analysis when

Fig. 2. Comparison of effective mobility curves for cocurrent,countercurrent and pure countercurrent flow.

viscous coupling across fluid–fluid interfaces is ne-glected. This is important in pure countercurrentimbibition problems, vertical flow problems andwhen using quasi-analytical methods for estimatingrelative permeability.

In another recent investigation, as mentioned inŽ .Section 2.2.1, Bentsen 1998a , in addition to sug-

gesting the non-applicability of Onsager’s relation totwo-phase flow problems for the sand–fluid systemused in the study, also shows that the neglect ofviscous coupling between the two flowing phasescan result in the introduction of relative error aslarge as 30% into the calculated values of the effec-tive mobilities.

Obviously, the work done to quantify the effect ofviscous coupling and the ultimate consequences ofits ignorance is relatively limited. In certain cases,

Ž .however, as shown by Bentsen 1997, 1998a,b , anerror of about 30 to 40% is significant and cannot beoverlooked for one reason or another. At least thesefindings should be enough to alert the scientificcommunity involved in the study of multiphase flowthrough porous media. Particularly, the oil industry,where reservoir simulation is commonly used torepresent the behavior of multiphase flow, must takeextra precautions by introducing the newly suggestedmodifications in the conventional Darcian approach.

6. Concluding remarks

The discussions above have revealed that an ade-quate theoretical framework related to viscous cou-pling in two-phase flow through porous media isavailable in the literature. Moreover, many re-searchers realize that the phenomenon of viscouscoupling is probably not a myth; rather it is a reality,as indicated in many theoretical, numerical and ex-perimental studies. Hence, efforts should be made to

Ž .incorporate the effect s of viscous coupling into theexisting formulation of the multi-phase flow throughporous media. No theoretical work can establish itscredibility unless and until validated by laboratoryexperimental work, especially in applied scientificdisciplines such as the fluid flow through porousmedia. Some of the experimental approaches men-tioned in Section 4.1 can be considered as an encour-


aging start, but not enough to settle the controversialissue of viscous coupling. To further authenticate theviscous coupling theory, and to demonstrate the prac-tical implications of its ignorance, some of the prob-lematic areas which need to be investigated experi-mentally andror theoretically are:Ø to develop a more comprehensive theoretical

model which must be able to handle more com-plex cases such as the one where chemical reac-tions andror phase changes may occur;

Ø to devise economical experimental equipment, en-abling the measurement of the four generalizedmobility coefficients with relative ease;

Ø to develop techniques for quantification of vis-cous coupling so that it can be incorporated intoexisting formulations of fluid flow through porousmedia;

Ø and, for the time being, to develop approximatemeans for estimating the desired parameters untilmore authentic means, that is, the experimentalmeans become available.

7. Nomenclature

k absolute permeabilityk relative permeability to phase i; is1,2i

L length of porous mediap pressure of phase i; is1,2i

q flow rate of phase i; is1,2i

S normalized saturation of phase i; is1,2i

x distance in direction of flow™Õ velocity to phase i; is1,2i

l k rm , effective mobility of phase i;i i i

is1,2l 8 k rm , effective mobility of phase i;i i i

i s 1,2; obtained from steady-statecocurrent flow experiments

lU k rm , effective mobility of phase i;i i i

i s 1,2; obtained from steady-statecountercurrent flow experiments

lX k rm , effective mobility of phase i;i i i

is1,2; obtained from steady-state purecountercurrent flow experiments

l , L k rm , generalized mobility of phase i;i j i j i i

i,js1,2m viscosity of phase i; is1,2i

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