modeling forced versus spontaneous capillary imbibition processes commonly occurring in porous...

Ž .Journal of Petroleum Science and Engineering 30 2001 155–166www.elsevier.comrlocaterjpetscieng

Modeling forced versusspontaneous capillary imbibitionprocesses commonly occurring in porous sediments

Walter Rose)

Illini Technologists International, P.O. Box 2424, Champaign, IL 61825-2424, USA

Received 3 April 2000; accepted 16 May 2001

Abstract

The extraction of ganglia of non-wetting petroleum fluid from underground reservoirs can occur whenever they areŽ .displaced and replaced by an invading wetting e.g. aqueous fluid phase. The problem is how to best specify, quantify,

observe, and beneficially control such petroleum recovery processes. Cases that are considered are those that result form thefact that there may be two separate but superimposed imbibition mechanisms involved that inherently can reinforce andsupplement each other. For example, when the displacing fluid phase more-or-less flows colinearly with the one beingreplaced, this will happen whenever it is only the imposition of mechanical energy gradients that causes it to happen. Thesekinds of processes can be thought of as being due to aforced imbibition mechanism. At the same time, however, a parallelspontaneous imbibition mechanism also can come into play whenever the local free interfacial surface energy of the systemsimultaneously is being diminished with time. The latter happens, of course, whenever the higher energy oil–rock interfacialtensions spontaneously become replaced by lower energy water–rock interfacial tensions. In effect, it is as though theentering wetting fluids push forward in ways somewhat akin to why a ball will roll down an unobstructed hill path by itself!

ŽTo obtain a clear picture of what is involved, consideration should be given to the following three self-evident but easily.overlooked propositions. First, it is easy to show that petroleum recovery due to the spontaneous imbibition effects can

occur even when externally imposed forcing effects are totally absent. Second, it turns out to be clearly indicated that evenin cases when superficially the magnitudes of thespontaneous imbibition effects appear to be minuscule compared to thosecaused by prevailing superimposedforced imbibition mechanisms, the influence of the former effects may not themselves

Žprove to be necessarily inconsequential. Third, wheneverrationally based algorithms are employed viz. rather than.simplified ones which are intended to mitigate and alleviate some of the computational problems , this will be the best way

to guarantee that quantitatively believable reservoir performance simulation outcomes will be achieved. This latterw Ž . xproposition implicitly is affirmed by the work of Rose Math. Geol. 22 1990 641 where reference was made to the

apparent way simulation outputs for idealized systems were different depending on either or not spontaneous imbibitioneffects were incorporated in the analyses.q2001 Elsevier Science B.V. All rights reserved.

Keywords: Porous media, multiphase flows; Imbibition processes; Simulation algorithms, surface energy effects

) Corresponding author. Fax:q1-217r359-9289.Ž .E-mail address: [email protected] W. Rose .

1. Background ideas

Ž .In Rose 1949 , as well as in the Rose and BruceŽ . Ž .1949 , early attempts were made: 1 to characterize

0920-4105r01r$ - see front matterq2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 01 00111-5

( )W. RoserJournal of Petroleum Science and Engineering 30 2001 155–166156

the capillary properties of hydrocarbon-bearingŽ .porous sediments; 2 to account for prevailing initial

Ž .static equilibrium distribution conditions, and 3 tofacilitate an accurate simulation of the ensuing dy-namic fluid motions as well as of the final end-pointstates of the fluidrfluid saturation levels and inter-

Ž .pore distributions. Muskat 1949 remarked, how-ever, that the cited papers merely confirmed the needto undertake complicated capillary pressure and rela-tive permeability experimental measurements if theuse of empirically over-simplified relationships wereto be avoided. In this paper it is shown, however,that simulation algorithms must be well-founded intheory if their outputs are to be trusted. Even then,however, the algorithms for such processes will stillrequire the input of complex and sometimes diffi-cult-to-obtain observational field andror laboratoryinformation. The latter are required so that the simu-lation outputs can be fully validated and logicallyapplied for the many complex two-phase processcases that need to be studied. Facilitating that, ofcourse, requires that a discriminating examination be

made of the several applicable pore space fluid phasedistributions schematically depicted in Fig. 1.

Accordingly, it may be supposed that the overviewdescriptions of imbibition processes that appear inthis paper, only require the consideration of a limitednumber of the many replete pore space configurationcases that otherwise are of practical importance. Forexample, the configurations shown as 2 and 3 in Fig.1 cover those cases where the wetting and non-wet-

Ž .ting pore fluids may or may not share mutualfluid–fluid interfaces of contact as viewed at thepore space level, and that in general the immiscible

Ž .fluid phases may or may not be similarly boundedby the fluid–solid interfaces in ways that dependupon how the local pore space happens to be parti-tioned. Also worth considering are the limiting con-figuration cases such as where either the wetting ornon-wetting phase pore space saturations are so lowthat flow through them cannot occur. These are thecases illustrated by pore configurations 4 and 7 ofFig. 1. Since it is the intention only to offer prospec-tive ideas about the nature of imbibition processes in

Fig. 1. In CASES 1 and 2, the interstitial fluids each are everywhere bounded by pore walls except at tube entry source and sink locations. Ifthe fluids are homogeneous, and if the flow occurs at low Reynolds numbers, the transport can be described by Darcy’s Law unless there areopportunities for coupling. In CASES 3 and 4, the immiscible interstitial fluids are shown to be bounded in part by fluid–fluid interfacejunctions so that viscous coupling may have to be taken into account. CASES 5 and 6 are special ones that occasionally may be encounteredwhen slug flow of ganglia or of emulsions occur, or when non-Newtonian fluids are involved, but such processes are not that can belogically described by the algorithms discussed in this paper. CASE 7 itself depict conditions showing stationary wetting and non-wettingfluid phases in the reservoir pore space such as will be the case before fluid production is first commenced, and also after an irreduciblesaturation end-state for the displaced fluid has been reached.

( )W. RoserJournal of Petroleum Science and Engineering 30 2001 155–166 157

general, overlooked details must be left for futureworkers with clearer minds consider.

This paper is based in part on previous work by:Ž . Ž .Adam 1930, 1949 , Champion and Davy 1936 ,

Ž .Morel-Setoux 1969 , Wooding and Morel-SeytouxŽ . Ž . Ž .1976 , Kistler 1993 , Bartley and Ruth 1999 , and

Ž .Rose and Heins 1962 . The cited papers deal insignificant part with the commonly overlooked ideathat capillary imbibition processes inherently can

Žoccur spontaneously i.e. so that displacement workis performed without an externally imposed energy

.supply being added to the system . In addition, theŽ .publications by Berg 1993 and Neumann and Spelt

Ž .1996 have served to enrich the understandings ofŽmodern workers. On the other hand and surpris-

.ingly , the monographs of giants such as MuskatŽ . Ž . Ž .1949 , Von Engelhardt 1960 , Collins 1961 ,

Ž . Ž . Ž .Luikoiv 1966 , Bear 1972 , Scheidegger 1974 ,Ž . Ž . Ž .Marle 1981 , Scholz 1981 , Allen 1988 , Dullien

Ž . Ž .1992 , Kaviany 1995 , etc., all have essentiallyignored the possibility that thespontaneous imbibi-tion effects are based an important concept that needto be considered.

Apparently, the authors of the cited monographswere supposing that it would be enough only toreference the earlier evolving ideas dating back 60

Ž .years to Buckley and Leverett 1942 and to RichardsŽ .1931 . What had made sense in the early pre-com-puter days was pragmatically based on simplistic andsomewhat untrustworthy empirical evidence bothabout the dynamics of capillary-affected flows, andabout the nature of the initial and final static equi-

Žlibrium end-states viz. where the capillary forceswould be exactly balanced by other prevailing body

.forces .Ž .Rose 2000a argued that even today the most

frequently employed algorithms for simulation workare those that are habitually based on the 1942outmoded ideas of Buckley and Leverett. A startlingindicator of the truth of this contention is illustrated

Ž .by the content of recent papers such as: 1 SiddiquiŽ .and Lake 1992 , which deals with the migration of

petroleum from organic-rich source beds into adja-Ž .cent aquifers back in geologic time; and 2 Hadad et

Ž .al. 1996 , which deals with mathematical modelingof imbibition at the capillary fringe of oil-con-taminated aquifers. Both of these otherwise interest-ing papers deal with the simulation of reservoirprocesses by employing over-simplified algorithmsthat explicitly ignore the very underlying governingspontaneous capillary imbibition effects that shouldbe taken into consideration.

2. Forced versus spontaneous imbibition algorithms

The objective of this paper is to focus attention on the theory of how to formulate algorithms forforcedŽ . Ž .imbibition processes in Section 2.1 , then forspontaneous imbibition processes in Section 2.2 , and finally for

Ž .combined forced and spontaneous imbition processes in Section 2.3 .

2.1. The Buckley–LeÕerett model quasi-description of forced imbibition

Ž .Eq. 1 , which follows below, is the particular form of the modified Buckley–Leverett relationship employedŽ .by Rose 1988, 1990 to model two-phase displacement processes for the cases where viscous coupling effects

Ž . Ž .are or are not occurring in ways that were predicted earlier by Yuster 1951 . For example:

w x w xletting i , j s 1,2 ; S S '1; S J 'const.Ž .i i

E SŽ . 1Ž .1 2[ f sq=PJ sy=PJ s J qJ P=AqB = P q =P qDG P =BŽ . Ž . Ž .Ž .1 2 1 2 c c� 0ž /Et

K K K1 1 2whereAs ; also Bs with K s L qL ; and K s L qL ,Ž . Ž .1 11 12 2 21 22ž / ž /K qK K qK1 2 1 2

with L 'L G0.12 21


Ž .Eq. 1 may be thought of essentially as a way to formulate conservation-of-mass ideas necessary for thedescription of displacement processes driven by gravity and other mechanical energy gradient terms for cases

Ž .where these are being opposed by viscous resisting forces. In them it should be noted that: 1 theJ and Si iŽrespectively refer to fluxes and the fractional pore space saturations of theis1, 2 i.e. the wettingrnon-wet-

. Ž .ting fluids; 2 =P and DG respectively refer to the local difference of the dynamic pressure gradients of thecŽ .two contiguous phases i.e. sometimes referred to as the dynamic capillary pressure gradient , and to the gravity

force that is proportional to the fixed density difference between what are taken to be two immiscible andŽ .incompressible fluid phases; and 3 theK and L coefficients of proportionality respectively refer to thei i j

ambiguous effective permeability parameters which are assumed to be associated with Darcian-like idealizeduncoupled multiphase flow processes, andror to the measurable material response transport coefficients which

Ž .according to Truesdell and Toupin 1960 should characterize naturally occurringcoupled transport processes.Ž .As written, Eq. 1 also is intended to provide descriptions of how local values of saturation change with time in

dynamic transport systems under conditions of low Reynolds Number, and those where the porosity,f, and thetotal flow rate sum are being taken as fixed and independently known constants.

Ž .By itself, Eq. 1 at most can only provide for a full description of those imbibition flow processes that arecaused to occur solely by the action of the prevailing mechanical energy gradient driving forces as imbeddedtherein. When the action of surface energy gradient driving forces needs to be considered, however, the otherprocess algorithms given below in Sections 2.2 and 2.3 will, of course, have to be taken into explicit account.

2.2. Modified YoungrDupre-based model description of spontaneous imbibition´

Ž .Statements such as those given in Eq. 2 can be taken for the local free surface energy of the two-phasesaturated porous medium,F , and for the prevailing surface energy gradient driving force,X s=F below, or:c c c

F sg A qg A qg Ac sn sn sw sw nw nw

A qA 'A'constantŽ .sn sw

g yg (g cosuŽ . Ž .sn sw mw2Ž .

X s=F fg =A q cosu =AŽ .c c nw nw sn

EA ES EF .sw w cwhere )0 if )0;[ -0� 0ž / ž / ž /Et Et Etx x xi i i

Ž . Ž .In Eq. 2 , which to some extent had been offered earlier by Rose 1963 , theg terms are the interfacialŽ .tensions, theA terms are the prevailing specific interfacial surface area per unit pore volume terms, thea ,b

� 4 Ž .s,w,n subscripts refer respectively to the solid pore wall phase, the wetting fluid phase, and the non-wettingfluid phase, andF refers to the free surface energy associated with a representative macroscopic volumec

element of interest which is being taken to represent local conditions at selected points and times in the reservoirŽ .continuum. Specifically, the first of Eq. 2 is the famous DuprerYoung relationship that was shown by Adam´

Ž . Ž .1930, 1949 and Champion and Davy 1936 to be only exact for systems that have reached static equilibriumŽ .states. Therefore, the last of Eq. 2 itself can only be employed to make rough estimates of what the local

spontaneous imibition driving force values will be at particular locations and times. On this central question,however, the reader is advised to carefully study the arguments presented in the recent monograph of KavianyŽ .1995 .

Ž .Two major problems arise because of the temptation to accept the validity and utility of the last of Eq. 2 .Basically, there is the problem of finding a coherent laboratory methodology to obtain values for changingmagnitudes of the fluid–solid and fluid–fluid interfacial surface areas located within the interstitial pore space.

Ž .Here, as noted by Rose 1959 , the creation of these interfaces causes subtle heats-of-wetting effects thatperhaps can be measured by calorimetric procedures. As an ambitious way to bypass the impediment, this sameauthor showed how qualitative estimates of the magnitude of the effect could perhaps be approximated by the


results of geometric surface area magnitudes calculated for idealized regular packings of sphere particles.During an isothermal increase of interfacial surface area, however, the amount of heat added to a system, for

Ž . Ž .example, is given by Tribus 1961 and Scheidegger 1971 , as:

EgabdQ 'yT . 3Ž .ab ž /ET A , p ,m

Ž . Ž .In practical terms, however, Eq. 3 probably cannot be employed for the reason given by Morrow 1970 ,namely that the heat capacity of systems of interest often will be so large that the temperature changesaccompanying loss of surface energy will be miniscule. Until that question is resolved, modern workers have the

Ž .opportunity to address the problem in two ways, namely: 1 Whether or not it will be practical to haveindependently measured values for fluid–solid interfacial tensions, the capillary driving force for spontaneous

Ž .imbibition processes can be obtained directly from the energy gradient term defined in the last of Eq. 2Ž .without utilizing the suspect third of Eq. 2 as discussed by May in Chapter 7 of the Neumann and Spelt edited

Ž . Ž .volume 1996 ; and 2 whether the above equation at least can be employed for giving useful approximateŽ .information for low intensity i.e. quasi-static capillary imbibiton processes.

Accordingly, there remains the conceptual problem presented by those who persist in using the definitionŽ .given in the last of Eq. 2 for a spontaneous capillary imbibition processes driving force. Thereby this might

serve to justify the use of the standard definition of the approximate magnitude of the equilibrium contact anglerelationships when considering dynamic phenomena during the course of an irreversible process withoutviolating the sense of the Second Law of Thermodynamics. Thus, it is being accepted that the entropyproduction in the irreversible transport processes under discussion is a positive-definite quantity as given by:

EsJ PX s T 4Ž .c c ž /Et

wheres is the local entropy density, andT is the local thermodynamic temperature of the system.Ž . ŽTribus 1961 has called attention to the distinction between processes where only thermodynamical say,. Žmechanical forces are involved as against those which involve the creation of new surfaces e.g. as occurs in

.many capillarity cases, and those that involve gravity as well . Clearly, such forces can be considered to beŽ .essentially of a statistical i.e. thermostatic rather than thermodynamic nature. In these cases, energy andror

Žmaterial particles will be caused to flow through the system in ways by which transport properties e.g. thermal.conductivity, diffusivity, viscosity can be defined in terms of measured values for Onsager-like forces and

fluxes. On the other hand, the thermostatic measurements are concerned withequilibrium forces on the oneside, and withrates of displacement caused by them on the other. Thus, a distinction is being made between:Ž . Ž .1 how moving fluid particles sometimes can have transported with them fixed say unit amounts of suchentities as mechanical, thermal and chemical energy as the Lagrangian particles flow from source to sink

Ž .regions; versus 2 how no moving fluid particle will be transporting such entities as location-dependent freesurface energy, or as gravitational energy. This is because the latter have fixed or changing energy values withtime at particular spatial locations, hence they have nothing to do with the fluid particles as they innocentlyhappen to be moving by. After all, theA -terms are properties of the interfacial composite regions that referab

both to the physical–chemical nature of the fluid cover and to that of the solid phase substrate, while similarlyŽthe gravitational energy implicitly also depends upon fixed spatial locations e.g. the distance between fluid

.particles moving in a gravity field, and the center of the Earth .Ž .In any case, clearly the total free interfacial surface energy defined in Eq. 2 undergoes local displacements

and replacements. Rather than giving rise to energy fluxes in response to the action of the energy gradientŽ .driving forces defined by the last of Eq. 2 , this behavior is distinctly unlike that where finite amounts of other

Žinternal energy species e.g. thermal, chemical, electrical as respectively described by theAlawsB of Fourier,. Ž .Fick and Ohm become attached to i.e. associated with Lagrangian fluid particles that flow around from source

Ž .to sink regions in ways expressed by the Onsager Reciprocity Relationships such as Eqs. 5a,b below.


2.3. Combined forced and spontaneous imbibition process algorithm

J L L L Xw ww wn wc w

J L L L Xs P [ J sL PX with L sL 5aŽ .n nw nn nc n i i j j i j ji� 0 � 0 � 0J L L L Xc cw cn cc c

L sL ;L sL ;L sLwn nw wc cw nc cn

5bŽ .also s T s J PXŽ . Ž .˙ Ý i i� 0isw,n,c

° ¶withy X s =p q r gŽ . Ž . Ž .w w w

y X s =p q r qgŽ . Ž . Ž .n n n

yX sq=F cf. Eq. 2cŽ . Ž .c c 5cŽ .(= g =A cosuŽ .wn sn

DYNAMIC INTERFACIAL ² :=P s =p y=p /=P s g cosu r RŽ . Ž .c n w c wn¢ ßhenceJ s0 wheneverrwherever=PJ s0c w

ES ESw nf syf sy=PJ sq=PJ . 5dŽ .w nž / ž /Et Et

Clearly, in cases where bothspontaneous and forced imbibition processes simultaneously are occurring,Ž .then the above Eqs. 5a,b,c,d apply. Here it is to be noted that there are some 17 independent and dependent

Ž . Ž .parameters, namely: three flux termsJ , J , J . plus three force termsX , X , X , plus nine transportw o c w n cŽ . Ž .coefficients L , plus two saturation termsS , S . The required 17 relationships are to be found imbeddedab w o

Ž .in Eqs. 5a,b,c,d since there arethree transport equations,two continuity equations,one statement that the sumŽ .of the saturation terms equals unity,two independent definitions forJ in Eqs. 6a,b to follow, and forX inc c

Ž .Eq. 2 , two input values for the imposed mechanical energy gradient terms,X and X , three Onsagerw n

Reciprocity Relationship between three sets of the transport coefficients,two empirical relationships givingJw

and J as a function of laboratory time, andtwo effective permeability relationships for theL as a functionn 11,22

of saturation.Ž .Thus, in Eqs. 5a,b,c,d ,J and J in the first two equations refer to steady-state fluxes of bulk fluidw n

Ž w x.dimensions: mrs caused both by the capillary imbibition forces,X , and by the conjugate prevailingc

mechanical energy gradient forces externally imparted by imposed boundary condition values of given pressuredrop in the two phases, where:

L 'L )0 implies viscous coupling occurs;wn nw

but L 'L '0 implies no such couplingwn nw6aŽ .w x[q s k rm D p qX with as w,nŽ . Ž .a a a a c� 0

where usuallyX <D pc a

w xE A cosusnq s sc ž /lEt

[ . 6bŽ .w xL X qL X qL Xcw w cn n cc c� 0with L s J rXŽ .cc c c


Ž . ² : Ž .In Eq. 6b , R is the average pore radius in the local REV volume elements,J s d x rdt is the–c interface

average rate of fluid–fluid interface advance, andl has the sense of being a petrophysical measure of theaverage pore perimeter seen in the cross-section plane perpendicular to the fluid flow direction for thecontiguous REVs. cross-section divided by the pore volume of the REV. finite difference volume element.

w x w xAccordingly, J is a vector quantity with mrs dimensions of the fluxes of the w,n immiscible fluid phases. Inc

other words, the above equations apply to the case where in addition to the capillary imbibition process, thetwo-phase flow processes under consideration can be caused by imposed mechanical forces and then furtheraffected by viscous coupling phenomena. Presumably, the measurement of the petrophysical parameters,Aab

Ž .and l, can be approximated by applying the scanning methodology of Chalkley et al. 1949 .

3. Discussion

Ž . Ž .Applications of Eqs. 5a,b,c,d and 6a,b apply toŽ .particular limiting cases such as: 1 where neither

external mechanical forces are being imposed nor areviscous coupling phenomena involved, as described

Ž . Ž .by Eq. 7 below, and 2 where viscous couplingphenomena and mechanical energy induced capillaryimbibition processes simultaneously but indepen-dently occur, butspontaneous imbibition processes

Ž .are absent, as described by Eq. 8 . Thus:

J L Xw wc c

J L Xs 7Ž .n nc c� 0 � 0J L Xc cc c

and

[given L 'L 'L '0wc cw nc

J s L =X qL =XŽ .w ww w wn nthen 8Ž .ž /J s L =X qL =XŽ .n nw w nn n

L m k Xww n w wwhere s if .ž / ž / ž /L m k Xnn w n n

Ž . Ž .Clearly, comparing Eqs. 7 and 8 with Eqs.Ž .6a,b , the fluxes of the wetting and non-wettingfluids variously can be modified and enhanced byviscous coupling effects as measured byL sLwn nw

terms, and by theL sL and L sL terms.wc cw nc cn

Notice from the fact that the dependent variableF-terms like saturation, specific interfacial surfaceareas, and local interfacial surface energy, inherently

are functions both of space and time according todefinitions such as:

EF EFi idF ' dtq d x wherei , jsn,w;i ž /Et Ex

hence dS 'ydS , also dA 'yd A . 9Ž .w n sn sw

Finally, to illustrate how experimental work canŽ .be undertaken so that Eqs. 5a,b,c,d can be em-

ployed to generate algorithms for modeling com-bined spontaneous and forced imbibition processes,

Ž .it will simplify analyses by rewriting Eq. 5a in thenotational form where 1 and 2, respectively, are thewetting and non-wetting fluids, and whereJ3, X 3and L33 designate the free surface energy ‘flux’,‘driving force’, and associated transport coefficientsthat cause spontaneous imbibition, and the associated‘transport coefficient’ of proportionality. In general,the set of equations is given by:

J1 L11 L12 L13 X1s P 10aŽ .J2 L21 L22 L23 X 2ž / ž / ž /J3 L31 L32 L33 X 3

where when transport process experiments are per-formed, arbitrary but measured values for the me-chanical energy gradient driving forces,X1 and X 2,can be freely selected, and theJ1,J2 conjugatefluxes can be experimentally measured. Assumingthat the Onsager Reciprocity Relationships can beinvoked, there will be sixL transport coefficientsi j

to be measured. Additional relationships will be


needed to obtain values forX 3 and J3 as a func-tions of saturation as obtained during the experimen-tal work that is undertaken under conditions whereinputs of X1 and X 2 are those specified for thefollowing four cases.

Case I. Set I X1sa and I X 2s0; then observeI J1andI J2 as a function of saturation.

Case II. Set II X1s0 and II X 2sb ; then observeII J1 andII J2 as a function of saturation.

Case III. Set III X1sa andIII X 2sb ; then observeIII J1 andIII J2 as a function of saturation.

Case IV. Set IV X1s0 and IV X 2s0; then observeIVJ1 andIVJ2 as a function of saturation.

Ž .For these four cases, Eq. 10a takes on the forms:For Case I:

I J1 aL11 L12 L13I 0s P . 10bŽ .L21 L22 L23J2 ž / � 0I� 0I L31 L32 L33 X 3J3

For Case II:

II 0J1 L11 L12 L13II bs P . 10cŽ .L21 L22 L23J2 ž / � 0II� 0II L31 L32 L33 X 3J3

For Case III:

III aJ1 L11 L12 L13III bs P . 10dŽ .L21 L22 L23J2 ž / � 0III� 0III L31 L32 L33 X 3J3

For Case IV:

IVJ1 0L11 L12 L13IV 0s P . 10eŽ .L21 L22 L23J2 ž / � 0IV� 0IV L31 L32 L33 X 3J3

Ž .Note that considering cases wherea)b, Eqs. 2Ž .and 6b combine to yield:

with X 3sg =A cosuŽ .wm sn

I III IV II[ X 3 ) X 3 ) X 3 ) X 3Ž . Ž . Ž . Ž .

E A cosuŽ . . 11Ž .snalso withJ3s ž /lEt

I III IV II[ J3 ) J3 ) J3 ) J3Ž . Ž . Ž . Ž .

Ž . wIV xEq. 11 is a reflection of the fact that X 3 is thedriving force term when onlyspontaneous imbibi-tion is occurring, hence, combinedspontaneous and

Ž .induced driving forces clearly are greatest and leastŽ .when bs0 and whenb™a .

Ž .Combining Eqs. 10b,c,d, and e , it follows there-fore that values for the six initially unknown trans-port coefficients as functions of saturation given by

Ž .the following Eq. 12 matrix<

Ž .12


where the first, third and fifth rows of the matrixw xhave been obtained by summingJ , X values byi i

considering values such as that which appear in thew x w x w x w xqI , the yII , the qIII and the yIV sM cases;and where the second, fourth and sixth rows of the

w xmatrix have been obtained by summingJ , X val-i i

ues by considering values such as that which appearw x w x w x w xin the qI , the yII , the yIII , and the qIV sN

Cases. Notice, however, that the elements of thematrix that appear in each of the columns providemagnitude values for the Lij transport coefficientsthat are shown in the last row that underlies thebracketed matrix.

4. Conclusions

Given below is an enumeration of circumstanceswhich explain why the author of this paper has feltcompelled to revisit one more time a subject thatalready has been considered on and off by so manyof the pioneer authorities both before and after the

Ž .papers of Richards 1931 and Buckley and LeverettŽ .1942 first appeared. Following that will be a listingof conclusions that now can be drawn after reconsid-ering the validity of what historically have beentaken to be inviolate ‘common wisdom’ understand-ings.

For much more than a half a century, manyworkers have continued to employ what were only

Ž .assumed but not proven to be empirically justifiedideas. The aim was to provide useable descriptionssuitable for the simulation of many kinds of Darcianscale multiphase flow processes. Here, and for rea-sons which are already partially explained in a series

Žof recent papers such as Rose 1999, 2000a,b,c, in.press , the decision was reached to investigate those

newer methodology approaches that seem to besolidly based on the tenets of non-equilibrium ther-modynamics, and as well as on indisputable empiri-cal evidence. To illustrate this possibility, it is con-cluded that new speculations now can and should beexamined about the likely nature of commonplaceimbibition processes that only involve mechanicalandror free surface energy gradient terms to directlyforce andror to spontaneously cause displacementand replacement of immiscible Newtonian fluidscontained in rigid and inert porous media.

Some of the important and useful conclusions thatcan be drawn from considering the nature of ideal-ized imbibition processes of the sorts referenced inthis paper include the following.

Ž .1 The paradigm description of an imbibitionprocess is where one Newtonian fluid phase slowlyflows through the intersticies of a porous medium,and thereby displaces and replaces a previously resi-dent immiscible fluid.

Ž .2 If the driving forces that solely cause theimbibition effects to occur have the physical natureof being prevailing space–time gradients in the me-chanical energy to be associated with contiguouspore space fluid particles, a so-calledforced imbibi-tion process can be said to be occurring. This, forexample, would be the case when the free surfaceenergy of the fluid–solid interfaces for both theentering and replaced immiscible fluids are identi-cally equal to each other. Therefore, to initiate andsustain increases for the local upstream, mechanicalenergy values will somehow have to be forciblymade to occur.

Ž .3 If the driving forces that solely cause theimbibition effects to occur have the nature of beingprevailing space–time gradients in the free interfa-cial surface area energy to be associated with con-tiguous pore space fluid–fluid and fluid–solid inter-faces, a so-calledspontaneous imbibition processcan be said to be occurring. This, for example,would be the case when local values for the mechan-ical energy levels were held everywhere to be thesame. Therefore, the local capillary driving forces,X , would continue to be finite in magnitude untilc

eventually local values for the saturation gradientsbecame identically equal to zero.

Ž .4 Since many ordinary imbibition processes ofinterest are ones where both mechanical and localfree surface energy gradients are finite in magnitude,hence the resulting imbibition processes would beseen to continue on as a hybrid process.

Ž .5 Imbibition processes that are those to which( )coclusion 2 applies are the ones that historically

Ž .have been based on Eq. 1 at least to the extent thatthe K are empirically measured rather than calcu-a

lated as though Darcy’s Law can be applied. On theother hand, imbibition processes to whichconclu-

( ) ( )sions 3 and 4 apply are ones best modeled by thealgorithms given by the equations of Section 3 herein.


Ž .6 Of all of the research papers and monographsŽ . Ž .that have been cited in this paper, Eqs. 2 and 12

herein have not been previously mentioned exceptŽ .superficially by Rose 1963 , nor much less have

they ever been tested. To support the contentions thathave been given in the body of this paper, one needsonly to generalize from the propositions about thenature of coupled transport processes as discussed

Ž .previously by Bear 1972 , to Dullein, to Lasseux etŽ . Ž .al. 1996 , to Ayub and Bentsen 1999 , to Rose

Ž .2000a,b,c , and to the many other cited writings ofthese authors that deals with the Onsager theory ofnon-equilibrium coupled transport processes whichin fact continues to be under development up to thepresent day.

Nomenclatureg interfacial tension between fluid–fluid,

w 2 xfluid–solid interfaces, MrTf dimensionless fractional porosityA,B Ž .coefficients in Eq. 1Aa ,b specific interfacial surface area per unit pore

w y1 xvolume, wherea ,bsw,n, LLab

w 3 xtransport coefficients, L TrMDG w 2 2xgravity driving force, MrL Tg acceleration due to gravity parameter,

w 2 xLrTJ Jw,n c mechanical and interfacial surface energy

w xfluxes, LrTKa classical empirical effective permeability

w 2 xterm, LDab

w 3 xOnsager transport coefficients, L TrMF any space and time varying parameterQ w 2 2xheat of wetting term, MLrT=P ,=p dynamic capillary pressure gradient, fluidc a

w 2 2xpressure gradient, MrL T² :R local average pore radius, mean hydraulic

w xradius, LSa dimensionless fractional fluid phase satura-

tionT w xthermodynamic temperature, degt, x, y, z w x w xtime and spatial coordinates, T and LX Xa c mechanical and surface energy gradient

w 2 2xdriving forces, MrL Ta ,b ;ij subscriptsrsuperscripts designating phase

species, coordinate directionsAab interfacial surface energies per unit area,

w xMrT

u advancing contact angle, or inclination fromthe horizontal plane

m w xfluid phase viscosities, MrLTs w 2 2 xentropy, MLrT rdegF w 2 2xenergy ofa-species, MLrTra w 3xfluid density, MrLg Ž . Ž .average pore perimeterr pore volume in

w 2 xlocal R.V.Es, L

Added in proof

Ž . Ž .Eq. 13 below shows how the matrix of Eq. 12is to be calculated, namely by taking a summation ofcertain elements as indicated by considering the vari-

� 4 � 4ous superscripted A, B, C, D andror M, N ,� 4 �andror a , b, g . Here the superscriptedA, B, C,

4D elements identify the driving forces that are em-Ž .ployed for the four Eqs. 10b,c,d, and e cases, and

� 4where the a , b, g designate the magnitudes ofthose driving forces.

J M q J A y J B q J C y J D1 1 1 1 1 q2aL11N A B C DJ q J y J y J q J y2bL1 1 1 1 1 12M A B C DJ q J y J q J y J q2aL2 2 2 2 2 21

' 'N A B C D y2bLJ q J y J y J q J 222 2 2 2 2

M A B C D � 0q2aL31� 0 � 0J q J y J q J y J3 3 3 3 3y2bLN A B C D 32J q J y J y J q J3 3 3 3 3

13Ž .Ž .Accordingly, and to facilitate rendering Eq. 12

solveable and determinate by combining it with Eq.Ž .13 , notice that:

w A x a w g x AJ ' J q J a 0 gXB b g Bw x w x 0 b gJ ' J q J X

since ' 14Ž .CC a b g a b gw x w x XJ ' J q J q J � 0 � 0DD g 0 0 gXw x w xJ ' J

� 4where the five superscriptedA, B, C, D, g termsturn out to be ones that cannot be directly measured

Ž .by the experimenter if in Eq. 14 they are shown toŽ .be bracketed. In these cases, Eq. 11 must be em-

w g x w g xployed to obtain numerical values forJ and X .

Acknowledgements

Special thanks are due to Emeritus Scientist AlexBabchin and to his colleague J-Y. Yuan, both of the


Alberta Research Council, and also to Emeritus Pro-fessor Ramon Bentsen of the University of Alberta,for their generous sharing of insights as this paperwas being composed, and also to Professor HansOlaf Pfannkuch of the University of Minnesota, forhis enduring friendship enjoyed in times of turmoiland triumph while traveling together throughout theAmericas, France, Germany, Israel, Nigeria, Mexico,Canada, and beyond . . .Aaber immer von Haus zuHausB. And not to be forgotten, of course, are somany other enrichments provided over the years bythe likes of long-ago mentors such as Morris Muskat,King Hubbert, Alex Bruce, Parke Dickey, A.I. Le-vorsen, Sam Yuster, L.A. Richards, Jacob Bear, etc.

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modeling forced versus spontaneous capillary imbibition processes commonly occurring in porous...

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