myths about later-day extensions of darcy's law

Ž .Journal of Petroleum Science and Engineering 26 2000 187–198www.elsevier.nlrlocaterjpetscieng

Myths about later-day extensions of Darcy’s law q

Walter Rose)

Illini Technologists International, P.O. Box 2430, Station A, Champaign, IL 61825, USA

Received 12 December 1998; accepted 15 December 1999

Abstract

Low intensity mass and energy transport processes in porous solids historically have been treated as though they areŽadequately described by Darcy’s long-ago empirical relationship Darcy, H., 1856. Les Fountaines Publique de la Ville

. ŽDijon, Dalmont, Paris . Almost a hundred years had elapsed, however, before Yuster Yuster, S.T., 1951. Theoreticalconsideration of multiphase flow in idealized capillary systems. Proceedings of the Third World Petroleum Congress, 2,

.437–445 inferred that coupling effects might be involved during the ensuing pore-space fluid flows; therefore, Yuster inŽeffect hypothesized that the Onsager Onsager, L., 1931. Reciprocal relations in irreversible processes. Physical Reviews, 37,

.405–426; 38, 2265–2279 formulations probably should be considered. This is because of the way the latter take intoexplicit account the fact that entropy production rates for decaying irreversible processes inherently are positive-definitequantities. In this paper, it is specifically suggested that Yuster’s watershed ideas preferably should be employed for caseswhere there is a significant momentum transfer across the interstitial no-slip fluid–fluid interfaces of contact. And here, itwill be further suggested that these ideas can also be applied to describe the nature of other categories of porousmedia-coupled transport processes. Accordingly, the purpose of this paper is to call attention to what is believed to be agrowing need to clarify the ongoing dispute about what still remains as a fractious issue. The hope is, of course, that future

Ž .workers eventually will be able to anticipate if and when the so-called Onsager i.e. OrDogma models should be employedŽ .rather than the superficial empirical Darcian DrDogma ones. Thereby, perhaps the validity of simulation outputs can be

significantly enhanced. In other words, the intended goal here will be to establish a coherent way to examine if and whentwo or more contiguous massrenergy fluid particle streams will couple with each other in ways governed by the dictates of

Ž .the famous Onsager Reciprocity Relationships ORR . This approach was first proposed by Lars Onsager, and eventually, in1968, he was awarded for it the Nobel Prize in Physics. Accordingly, it is the intention in what follows to suggest thatwhenever traditional modeling algorithms do not seem to provide adequate forecasts for future process states, then othermore solidly-based means must be developed to replace what here are being referred to as somewhat suspect later-dayextensions of Darcy’s Law. q 2000 Elsevier Science B.V. All rights reserved.

Ž .Keywords: transport processes; porous media systems; coupling effects; Onsager ORR vs. Darcian formulations

q This Paper was a contribution to the Fundamental Session on Transport Phenomena at the International INRESC.98 Congress in Tehran,December 1998, that was organized and chaired by the author and his colleague, Professor Bahram Dabir. Refer to -

http:rrwww.staff.uiuc.edur;wdroserdarcy.html) .) Fax: q1-217-359-9289.

Ž .E-mail address: [email protected] W. Rose .

0920-4105r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0920-4105 00 00033-4

( )W. RoserJournal of Petroleum Science and Engineering 26 2000 187–198188

PROLOGUE — From Darcy to Onsager: ’TwasDarcy the first cat to do it: he sampled some sandjust to Õiew it, and confirmed that his dew hadseeped through it. Then Onsager agreed to pursueit! A Nobel Award, I will woo it! Though Darcycan’t claim that he knew it, the rest must agreethat he blew it! With good luck I’ll haÕe time toreÕiew it. For others who doubt, let them chew it.

1. Introduction

1.1. An oÕerÕiew of the subject

Some might say that it is a sacrilege to suggestthat fluid flow in natural porous solids quite often isnot well described by the famous empirical Law of

Ž .Darcy 1856 . In its modern form, this relationshipcan be written as:

J sK X where J s r q ,Ž .r r r r r r

also X s grad P qr g , 1Ž . Ž .r r r

k r kŽ .r r r[ K s or q s grad P qr gŽ .r r r rž /m mr r

Ž .Note that in Eq. 1 the subscript r is to identify theparticular fluid species that happens to be occupyingthe pore space of the porous medium under study.

As has been confirmed by the data consistentlyŽ .obtained over the years, clearly, Eq. 1 can be

accepted at least as a sensible way to describe lowintensity isothermal flows in isotropic media of ho-mogeneous Newtonian fluid under isothermal condi-tions at least when gas-slippage, electrokinetic andother categories of coupling effects are absent. Morethan that, evidence will be presented in what followsto indicate that transport involving coupling phenom-ena in general will often be more logically describedby invoking the so-called linear Onsager Reciprocity

Ž . Ž .Relationships ORR loc. cit. such as:

K Kr ,r r , sJ s K PX where K 'Ž .r r , s s r , s ž /K Ks ,r s , s

2Ž .

K Kr , s s ,rand K 'K , also -0Ž .r , s s ,r ž /K Kr ,r s , s

Ž . Ž .In Eqs. 1 and 2 the J terms are fluxes of somer

specified kind as associated with pore space fluidsŽand indeed, as also sometimes associated with theisotropy properties of the pore space network contin-

.uum ; the X terms are the conjugate energy gradi-rŽ . Žent per unit volume driving forces; and the K ,r

.K , k terms are the scalar elements of proportion-r ,s r

ality of the so-called material response transportcoefficients referred to a long time ago by Truesdell

Ž .and Toupin 1960 . Specific examples of transportprocess cases where claims in the literature have

Ž . Ž .been made that Eqs. 1 and 2 adequately describethem will be found in Sections 1.2.1–1.2.5 below.

1.2. EÕolution of ideas about porous media transportprocesses

( )1.2.1. What Darcy 1856 originally had to sayDarcy was an engineer who happened to be the

Ž .Mayor of a French provincial city Dijon in theŽ .1850s cf. Hubbert, 1956; Philip, 1995 . As such, he

became curious about the quantitative performanceof sand filters that were locally used for waterpurification purposes, so he arranged to undertakesome ingenious experimental measurements. The lin-

Ž .ear relationship given above in 1 that was observedbetween the flux of the downward seepage of waterwith the gravity driving force that caused the drainagebecame known as Darcy’s Law.

1.2.2. The elaborations of the obserÕant Richards( )1931

While even in the early years, the primacy ofŽDarcy’s Law already had been challenged e.g. by

Forscheimer, Dupuit, Klinkenberg, and many othersŽ .cited by authorities such as Bear 1972 and Schei-

Ž ..degger 1974 . Richards, a soil physicist with theUSDA, did not hesitate to speculate that an analog ofDarcy’s Law could also be utilized to describe whathe referred to as the so-called unsaturated flow ofwater in soils. This simplistic presumption based on

Ž .the earlier ideas of Buckingham 1907 is still con-Ž .sidered by many authorities but not all of them! to

be both plausible and useful. The problem was,however, that Richards erroneously had focused onlyon the case of where the pore space was partially

Ž .saturated by single viscous fluid e.g. water ascontained in pore space where no fluid–fluid inter-

( )W. RoserJournal of Petroleum Science and Engineering 26 2000 187–198 189

faces existed. In this early work, it was imaginedfurther that any leftover pore space remained emptyŽor at least, hypothetically filled by what would

Ž . .amount to an ideal i.e. zero viscosity fluid .

1.2.3. Early definitiÕe writings by Muskat and Hub-bert

Muskat, a classical physicist from CalTech whospent much of his professional life as Chief of thePhysics Division at Gulf Research and DevelopmentŽ .Pittsburgh , wrote two magnificent monographsŽ .1937, 1949 and hundreds of other key papers andreports dealing with fluid flow through porous mediasubject matter. While this coverage ignored some ofthe earlier ground-breaking ideas of Richards et al.introduced terms like relatiÕe permeability, by sup-posing that in multiphase saturated systems, the flow

Ž .of separate contiguous side-by-side streams wouldbe describable by analogous Darcian-like relation-ships. Another contemporary giant of those early

Ž .days was Hubbert 1940, 1956 , a physicist turnedgeophysicist, who also wrote with insight and under-standing on the same subject matter, but like Muskat,he overlooked practical ways to incorporate andutilize Richards’ earlier dynamic ideas. The polemi-cal dispute that erupted between these two giantsregretfully marked a low point in two otherwise

Ž Ž ..illustrious careers cf. Hubbert 1950 .

1.2.4. The paradoxical formulations of Buckley and( )LeÕerett 1942

Standing on the shoulders of the earlier trailŽ .blazers, Leverett a splendid thinker and doer and

Ž .Buckley his corporate ‘bean-counting’ manager di-rected attention to basing the description of dynamicmultiphase flow processes, where in effect, the ele-ments of immiscible fluids were to be thought of assuperimposed co-residents of the same macroscopicpore space domains. In other words, just as one canthink of pore space surrounding and simultaneouslybeing surrounded by a rigid and inert solid phasematrix when only one fluid is present, so as thissame pore space when occupied by two or moreimmiscible fluid phases gives rise to a situation thatsome fluid–fluid interfaces often will be found to

Žexist in the interstitial spaces as also will the fluid–

.solid pore wall junctions that also exist . Occasion-ally, whenever the Buckley–Leverett algorithmsparadoxically imply that multiple values of satura-tion as a function of time and position appear toco-exist, this can make one wonder if it is because of

Ž .explanations having to do with a conceptual misun-Ž . Ž .derstandings, b computing errors andror c labo-

ratory experimental design and instrumentation prob-lems.

( )1.2.5. Hassler’s 1944 ingenious patented labora-tory measurement method

Another contemporary giant of the Richards erawas Hassler, an occasional academician who heldvarious posts at Penn State and UCLA, and will beremembered as the one who pioneered in the devel-opment of modern understandings about the capillarynature of porous media. His patent for measuring theso-called relative permeability transport process co-

Žefficients has been scrutinized by Rose 1948, 1949,.1950a, 1980, 1987, 1991b and others. In fact, many

workers will at least partially agree with ScheideggerŽ .1974 who claimed that the Hassler methodology

Ž .A . . . is exceedingly accurate perhaps incorrect butŽ .very complicated perhaps correct B.

1.2.6. The last of the seÕeral watershed papers of( )professor Yuster 1951

Yuster had been the Chairman of the Penn StateUniversity Petroleum and Natural Gas Departmentfor many years, and he was soon to leave for atwilight posting at UCLA. And this was the settingfor his amazing Third World Petroleum Congresspaper. In it was an implied and unfamiliar rejectionof the classical Darcian points of view. Unmentionedby Yuster, however, was the fact later demonstrated

Ž .by Rose 1969a that Yuster’s formulations serendip-itously were more-or-less consistent with the earlierORR predictions of Onsager.

1.2.7. Framework ideas appearing from the 1950sonwards

Ž . Ž .Truesdell and Toupin 1960 , Wei 1960 , DeG-Ž . Ž . Ž .root and Mazur 1962 , Luikov 1966 , Bear 1972 ,

Ž . Ž . Ž .Jaynes 1983 , Whitaker 1986 and Dullien 1992


are just some of the trailblazers whose definitivemonographs treated with great insight the subject inthe context of how to model properly the macro-scopic transport phenomena, for example in porousmedia systems.

1.2.8. Acknowledged importance of Onsager’s workin the 1980s and 1990s

Because of the new interests and understandingsabout the thermodynamics of irreversible processesthat began to appear in the middle years of the 20thcentury, intrigued petroleum reservoir engineeringworkers more and more sought tentatively to modelthe results of the Yuster Gedanken experiment with

Žthe aid of Onsager’s ORR equations cf. for exam-Žple, the later papers of Rose 1965, 1966, 1969a,

1976, 1985, 1987, 1988a, b, 1989b, 1990a, b, c,1991a, 1993, 1995a, b, et. seq., . . . of KalaydjianŽ . Ž .1990 , of Bourbiaux and Kalaydjian 1990 , of

Ž . Ž .Bentsen and Manai 1993 , of Bentsen 1998, 2000 ,.and many others .

1.2.9. Current persisting disagreements Õs. emerging( )understandings 1990 to date

The reader is referred to the papers of the eminentPhilip of CSIROrAustralia for basing his objectionson arguments originally voiced by BuckinghamŽ . Ž .1907 . For example, in Philip and DeVries 1956

Ž .and Philip 1972 it is flatly stated that viscouscoupling phenomena are not to be observed duringmultiphase flow processes in natural porous media.Opposing views, however, are to be found described

Ž .in the recent publications of Ehrlich 1993 , LasseuxŽ . Ž .et al. 1996 , Dullien 1992 , Bourbiaux and Kalayd-Ž . Ž .jian 1990 , Liang and Lohrenz 1994 and again,

many others.

1.2.10. Examples of plausibly Õalid Õs. seeminglyinÕalid ORR applications

Many literature citations can be given to supportŽ . Ž .the use of Eq. 2 rather than Eq. 1 as the proper

starting point to obtain a description of transportprocesses in porous media. Three major categoriesmay be mentioned as follows:

Ž .i The first is when the transport processes in-volve one or more immiscible fluid separately con-fined in an inherently anisotropic pore space. Rose

Ž .1995a,b for example, suggests that one can betempted to write a complex 4th rank tensor form forthe flux–force relationship such as:

a ,a a ,a a ,b a ,bK K K Ki , i i , j i , i i , j

a ,a a ,a a ,b a ,bK K K Kj , i j , j j , i j , ja bJ s : Xi jb ,b b ,b b ,a b ,aK K K Ki , i i , j i , i i , j

b ,b b ,b b ,a b ,aK K K Kj , i j , j j , i j , j

a ,b a ,bw xand K s K P K 3Ž .i , j i , j

a,b Ž . Ž .where K refers to a,b fluids and i, j refers toi,j

flow directions.Ž .ii A second category is when one or more of the

immiscible fluids is involved in complex seriesandror parallel pore space networks. One then oftenmay have to deal with coupling between two or moredistinct energy species fluxes, as well as with theconjugate energy gradient driving forces as directedthrough the various immiscible pore space saturantseven as the latter independently may be in motion.This can be descriptively referred to as a ‘thermo-diffusion’ type of coupled transport process, which

Ž .can involve analogs of Eq. 2 . In other words, the� 4 � 4r,s subscripts of the J, X , terms respectivelydesignate energy fluxes and the conjugate energy

Ž .gradient driving forces. If the pore fluid s is notstationary, then, in applying the Onsager formalism,attention will have to be given to the to the so-calleddiffusive fluxes defined by:

J sJ qK where J sbulk fluid flow rate 4Ž .r b r , s b

.In other words, it is as though in a thermo-diffusionsystem, one may have to deal simultaneously with athermal energy transfer caused by a temperature

Ž .gradient Fourier’s Law , a mass flow caused by aŽ .concentration gradient Fick’s Law , coupling com-

ponents proportional to the K terms, and a bulkr ,sŽ .fluid flow rate defined by Eq. 4 due to the superim-

posed motion of fluid which carries thermal andchemical energy along with it.

Ž .iii When the transport process of interest in-volves only the flow of two or more immisciblefluids being driven by corresponding mechanical en-ergy gradients, allowance has to be given to thepossibility that the so-called viscous coupling phe-nomena will affect the process outcomes. This is thepossibility that first caught Yuster’s attention in 1951,


and that is why some workers today will model suchŽ . � 4processes by Eq. 2 where the subscripts r,s are to

be attached to the flux and force terms of the respec-tive fluid phases. Thus, for the two-phase case onewill be tempted to consider as applicable the rela-tionship:

J sK PX qK PX 5Ž .r r ,r r r , s s

J sK PX qK PXs s ,r r s , s s

Ž .And in Eq. 5 , it is to be noted that in explicit form,the driving forces measured by the X , X terms arer s

both conceptually similar entities as previously de-Ž .fined in Eq. 2 . Therefore, it can clearly be imag-

Ž .ined that that the viscous coupling effects if anyhas to do with the viscous drag that can occur acrossthe interstitial fluid–fluid interfaces.

2. The subject matter

ŽTo properly have a useful e.g. preferably an.analytical description of coupled transport processes

that are associated with various massrenergy fluxesthat occur within and through the interstices of poroussolids, there are two major requisite conditions thatideally should be taken into consideration. Firstly, aplausibly valid relationship between these fluxes andthe conjugate energy gradient driving forces needs tobe postulated, tested, modified, and eventuallyadopted in its best form. And secondly, these algo-rithm relationships must then be shown to lead tobelievable predictions of future dynamic processevents that in fact are in accord with what is ob-served to happen with respect to the behavior of theparticular prototype systems under study. Applyingthis two-step approach to various representativetransport process cases, it follows that a number of

Žimportant observations can be drawn includingthose, which sometimes, are the ones commonly

.overlooked .

2.1. The elementary cases

Using modern terminology, it can be said thatDarcy in 1856 was simply testing by experiment analready popular but simplistic hypothesis. This wasthat the gravity-driven and low Reynolds Number

seepage of a homogeneous fluid, completely occupy-ing the pore space of a uniform sand body, shouldgive rise to a downward flow that is more or lessdirectly proportional to the local magnitude of thehydraulic head acting on the mobile pore space fluid.This result now, as well as then, is not an unexpectedone, however, was given the dictates of the other

Žpioneer investigators such as Fourier who was in-. Žvestigating heat transfer in 1807 , Ohm who was

.interested in electrical conduction in 1827 , PoiseuilleŽwho was studying fluid flow in capillary tubes in

. Ž1841 and Fick whose focus was on diffusion in

. Ž Ž ..1870 cf. DeGroot and Mazur 1962 . Moreover,Darcy’s 1856 laboratory results, both then and now,have always been known to be largely reproducibleŽviz. especially for trivial cases of isothermal andlow intensity single-phase flows of homogeneousfluids of simple rheology fluids through the inter-stices of isotropic, rigid and chemically inert porous

.bodies .

2.2. More complex cases

Later investigators naturally have tried to justifythe utility of applying ‘generalized Darcian-like rela-tionships’ to other more special fluid flow cases thatare encountered during coupled massrenergy trans-port processes. These are described in detail by

Ž . Ž .Babchin and Yuan 1997 , Bear 1972 , DullienŽ . Ž .1992 , Philip and DeVries 1956 , Raats and KluteŽ . Ž .1968 , and Rose 1966, 1976, 1982, 2000a . It isnow well established, however, that the relationshipbetween fluxes and forces probably sometimes alsodepend in non-Darcian ways upon a host of sub-

Ž .sidiary factors, such as a having to do with isolatedŽ .fluid and solid phase properties and interactions, b

upon the wettability of the interstitial pore spaceŽ .boundaries, c upon whether such things as heat

flow and chemical diffusion simultaneously are oc-Ž .curring, andror d upon many other system condi-

tions which in the modern era are easy enough tovisualize but not always the ones that can be conve-niently quantified. In fact, in the most complexcases, the proper relationships may sometimes benonlinear, and therefore, they simply cannot be de-

Ž . Ž .scribed by algorithms based on Eqs. 1 and 2 .


2.3. Special case extensions

Many special transport processes cases that havebeen introduced from time to time by previous work-ers, however, in the end have proven to be withoutmerit. One reason has to do with the fact that thesubsequently made process observations have notalways been found to be in agreement with thepredictions of future states based on the earlier sim-ple-minded Darcy Law formulations. And then, thereis the fact that some of the other remaining doubtfulextensions are the ones that back and forth wouldbecome popular, but then fade away at least untilsomeone has the idea to have them reinvented again.The problem is that many of the bizarre extensionsŽ Ž .such as adapting Eq. 1 to describe the flow ofemulsions, foams andror other non-Newtonian flu-

Ž .ids mentioned by Scheidegger loc.cit often are leftŽnot scrutinized by those unaware of or indifferent

.to the practical impacts of trying to describe thewell-set natural phenomena with fallacious models.

2.4. The onset of the Yuster era

Almost a hundred years had elapsed between thetime of Darcy’s original work and the appearance ofYuster’s watershed paper at the 1951 Third WorldPetroleum Congress. Again, using the modern termi-nology, one now can say that Yuster, in effect, wasoffering a challenge against the indiscriminate use ofquasi-generalized linear modeling, especially for the

Ž .unsteady-states of coupled processes, say inanisotropic media. This contention inherently is inaccord with the idea that the description of irre-versible transport processes sometimes can best beaccomplished by making use of what prove to be thegoverning ORR in conjunction with an explicit state-ment of the Second Law of Thermodynamics along-side the various massrenergy balance equations.

2.5. The process irreÕersibility feature

Taking into consideration the senses of the obser-vations made in Sections 2.1–2.4 above, it followsfrom applying common sense that irreversible On-sager, rather than reversible Darcian, should proba-bly be employed for forecasting the nature of futuresystem states. Furthermore, it is being suggested herethat the difficulties to be faced in proving vs. dis-

proving the utility of the inherently complex Onsagermodeling is no greater than what would be required

Ž .to examine how much error if any is involvedwhen adopting formulations based on myth and tra-dition, rather than on logic and experience. Theseviews support those presented by the independentanalysis given in the recent paper of Ayub and

Ž .Bentsen 1999 . To summarize, it is perhaps mostprobable that no one knows how many times duringthe last 143 years that the phrase Darcy’s Law hasbeen cited in worldwide published literature. Thenumber undoubtedly is a very large one however,and for reasons that also explain why the structurallysimilar laws of Fourier, Ohm, Poiseuille and Fickhave been so frequently mentioned over similar spansof intervening time. Such, it may be added, is indeedthe appeal of simple-minded dogma to theCartesian-minded advocates of the primacy of linearrelationships!

3. Example of Darcian CmythsD

This paper is intended to serve as a critique ofŽwhat can be termed as Darcian AmythsB viz. those

generalized relationships for complex cases that tendto be accepted mostly because of simplistic butillogical Darcy Law conceptualizations seductively

.are embodied therein . It is a well-known fact, how-ever, that while useful empirical relationships some-times can be derived from a governing theoreticalframework, the converse will usually not be true.This is because of the rule of experience that indi-cates that particularities that need to follow fromŽ .rather than be precursors of the parent formula-tions.

Accordingly, in what follows, the phrase Darcy’sLaw is being employed here only to reference thelinear transport process behavior as observed byDarcy in 1856 for certain very simple systems. Andinstead of making dubious use of various later-day,over-generalized Darcian-like descriptions, the inten-tion in this paper is to recommend against the sus-pect use of any of the mythical and historical processdescriptions that so far have not been independentlyestablished. Instead, the proposal is made in whatfollows, to adopt the thermodynamically-based trans-port process theory of the Nobel Laureate, LarsOnsager, in a form which is inherently appropriate


for the description of both coupled and uncoupledirreversible processes. The latter is referenced hereinas the OrDogma Onsager Reciprocity Relations, orORR.

Ž .For example, it will be clear that Eq. 1 can beregarded as providing the limiting case values to Eq.Ž . Ž2 when no coupling of any sort is involved i.e.

.when K sK s0 . Furthermore, the rate of en-i , j j ,iŽ .tropy production say for a two-phase system is

Žgreatest when coupling is involved i.e. the.OrDogma cases , as will be indicated at least quali-

tatively by the idealized data shown in Fig. 1. Forexample, one can write:

w xu s sÝJ PX )0, but when˙ r r

X sX sXsConstant,r s

w x w xthen s 4 s 6Ž .˙ ˙OrDogma DrDogma

since when X sX , thenr s

K qK qK qK r K qK )1Ž . Ž .r ,r r , s s ,r s , s r s

Ž .In Eq. 6 , the left-hand side is the product ofthermodynamic temperature times the rate of entropy

Ž .production. Moreover, 6 , when applied to the caseŽof a two-phase flow through a horizontal crack Rose

.1990c, 1991a, 1993 gives results displayed in Fig. 2for the case where the viscosity ratio is unity. Inthese connections, note that for the inequality criteria

Ž .shown as the last of Eq. 2 to be established as anindicator of the positive-definite nature of the en-tropy production rate for irreversible processes, it isrequired that the discriminate be positive-definite for

Ž .the quadratic equations given in 6 defined by2 2K q2 K K qKŽ .r ,r r , s s ,r s , s

y4K K qd )0, 7Ž . Ž .r ,r s , s

s

where for X sX , then ds K PX susÝr s r , s sr

Ž .Thus, Eq. 7 guarantees that the roots of thequadratic equation will be real.

As regard the identification of what here are beingtermed as Darcian myths, an examination of thefollowing clarifications will be helpful.

Ž . Ž . Ž .a Eqs. 1 and 2 by themselves are essentiallyincomplete statements unless there are also availablecoherent and reliable experimental designs, for ex-ample, the ones already depicted in Fig. 1 as pro-

Ž .posed by Rose 1997 .

Fig. 1. The curves here are drawn to quantitatively describeviscous coupling phenomena in an idealized crack-shaped porespace. The abscissa is either the wetting phase saturation, S , or1

the nonwetting phase saturation, S , upon which values for the2

K and K depend according to the analytical formulations ofr ,s rŽ .Rose 1990a,b,c, 1993 . Since attention is being limited to cases

Žof equal viscosity in each fluid phase e.g. say m sm s11 2.centipoise as contained in a horizontally oriented uniform crack

of 1 mm thickness, the curves for the transport coefficients asfunctions of the respective saturations are symmetrical aroundS sS s0.5; and the same will be true for the so-called RATIO1 2

Ž .function as calculated by the last of Eq. 2 , and also, for theŽ .so-called DELTA function as calculated by the last of Eq. 2 .

Clearly demonstrated here is that at particular saturation andsaturation distribution conditions, the rate of entropy production isgreater for processes where viscous coupling is occurring than forthe corresponding purely Darcian flow cases. Moreover, the maxi-mum value for the RATIO function is 0.36 at S sS s0.5,1 2

confirming that entropy production in system where viscous cou-pling is occurring is the greatest at the intermediate saturation

Ž .levels. See Rose 2000b .

Ž .b Eventually, computerized simulations will beundertaken. This, in many cases, will be the mosttrustworthy way to see whether future process eventscan be reliably forecast.

Ž .c Workers should avoid letting themselves bemisled that plausible predictions are necessarily onlythe result of big causes.


Ž . Ž .Fig. 2. A schematic cartoon to give the sense of the coherent ways described by Rose 1976, 1997 and Dullien and Dong 1997 to measureŽ . Ž . Ž .coupled two-phase flow transport coefficients that appear in Eqs. 1 – 7 . In Case 1, the oil phase coded as Phase ‘s’ is syphoning upwards

Žthrough the core sample by a measured flow from the higher Reservoir A to the lower Reservoir B. At the same time, water coded as Phase.‘r’ that otherwise would be stationary since the C and D Reservoirs temporarily are being held at the same level will also be dragged

upwards at a measured rate along with the flowing oil because of viscous coupling, hence, values for K and K can be calculated withs ,s r ,sŽ .Eq. 5 by setting X s0. Similarly, in Case 2, the water phase is syphoning downwards from Reservoir D to C and dragging oil with it thatr

Ž .otherwise would be stationary because the A and B Reservoirs are now temporarily being held at the same level , hence, K and K canr ,r s ,rŽ .also be calculated with Eq. 5 by setting X s0. In other words, in two separate companion experiments performed at the same saturations

and saturation distribution conditions, with K and K extracted from the Case 1 data, while K and K are similarly extracted froms ,s r ,s r ,r s ,r

the Case 2 data, the wanted experimental information in principle can remarkably be obtained under the Hassler-like conditions of zerosaturation gradient in the directions of flow. Or to put the matter another way, it is being suggested specifically that in upwards or

Ž .downwards free gravity flow, conditions of zero capillary pressure gradient hence, zero saturation gradient can be attained, sustained andmaintained so that the macroscopically measured values for the four K transport coefficients can unambiguously be associated withr ,sŽ .hence, functions of the particular values of saturation and saturation distribution.

Ž .d Apparently, the greatest Darcian myth has todo with the claim about the utility and relevance ofthose antiquated effective and relative permeabilityideas on which DrDogma methodologies are based.Here, the uncertainties arise from the fact that inspite of contrary claims, no general agreement hasbeen reached about the proper way to make effectiveŽ .hence relative permeability measurements.

4. Particular research proposals

The literature citations given here can be thoughtof as representing the ‘tip of the iceberg’ about what

already has been said and published, and about whatfuture investigators eventually may reveal. For ex-

Ž .ample, Ayub and Bentsen loc. cit. alone list 171citations in their bibliography. From the listings hereand elsewhere, it is now possible to identify andprioritize which specific research tasks still need tobe undertaken. The major ones are as follows.

4.1. Selecting preferred algorithm statements

In the end, coherent mathematical models inclosed form for each coupled transport process ofinterest are wanted. According to OrDogma as ap-plied to low intensity processes, these in general will


often be reflective of linear relationships, where thetransport coefficients of proportionality can ideallybe fashioned from the data outputs of well-definedlaboratory procedures. For example, thermal diffu-sion types of cases, of course, the wanted algorithmsmight logically take on the form of the OnsagerRelations for coupled processes. Even so, more com-plex forms will occasionally have to be considered,for example, the ones where chemical reactionsandror phase changes occur along a deforming solid

Žmatrix and as well, along pore paths containing.non-Newtonian fluid in high Reynolds Number flow .

In any case, moreover, answers are still needed forthe provocative question why it has been so difficultheretofore to obtain full confirmation of an otherwisecompelling theory thoughtfully espoused by the pa-

Ž .per of Lasseux et al. 1996 .

4.2. Dealing with measurement methodologies forthe material response parameters

Many of the reference cited here in one way oranother address the need to develop and perfect waysto determine laboratory values for the transport coef-

Ž .ficients. Examples are: Bentsen and Manai 1993 ,Ž .Bourbiaux and Kalaydjian 1990 , Dullien and Dong

Ž . Ž .1996 , Goode and Ramakrishman 1993 , Liang andŽ . ŽLohrenz 1994 , and Rose 1997, 1990a, 1988b,

.1985, 1980, 1951b,c . . . to name a few. In particu-lar, Fig. 2 displays the sense of the ideas of RoseŽ .1997 for measuring the transport coefficients inexperimental arrangements where the fluid motionsare due entirely to gravity free-fall effects. If andwhen what the above investigators have been propos-ing is achieved, the remaining tasks for optimizingexperimental design will be greatly simplified. Andonly then will it be useful to have fully developedalgorithms available. Otherwise, it may be added,that investigators will be left with the ‘garbage in,garbage out’ dilemma.

4.3. Other computerization requirements

Two other important research tasks to considerŽ .investigating are: a developing approximate ways

Ž .to calculate rather than actually measure values forthe transport coefficient elements of the permeability

Ž . Ž .tensor Task 1 ; and b independently showing how

Ž .small the cross interaction coefficients have to bebefore excessive error is introduced by making use

Ž .of simplistic DrDogma modeling Task 2 . GivenŽ .the arguments of Lasseux et al. loc. cit. , success

with accomplishing the intent of Task 2 can indeedbe proven as a daunting one. This is because theavoidance of difficult laboratory measurements canonly be tolerated when qualitative modeling happensto be acceptable. Insights on this aspect of the prob-

Ž .lems can be found by reading Birkhoff 1960 , whopoints out in his Chapter on Paradoxes of ViscousFlow that very small causes sometimes can result invery large effects. A simple illustration of this easilyoverlooked proposition is given by the way a minus-cule degree of surface roughness in a capillary tubecan give rise to the premature onset of turbulence atsubstantial low Reynolds Number conditions.

5. Conclusions

To recapitulate, it is to be remembered that lowintensity massrenergy transport in porous solids his-torically has been treated as a reversible process, andas though, Darcy’s long-ago empirical observationsusually probably can be invoked. In 1951, however,Yuster suggested that at least, when viscous couplingis involved during multiphase fluid flows, Onsager-like formulations are worth considering. This isbecause the latter take into account the fact thatentropy production rates for decaying irreversibleprocesses inherently are positive-definite quantitiesin the senses implied by the ORR Reciprocity Rela-

Ž .tionships and as displayed in Eq. 6 .Accordingly, the purpose of this paper is clear. It

has been to call attention to the growing need for theŽcontinuing research to test if and when Onsager i.e.

.OrDogma models should be preferred over DarcianŽ .i.e. DrDogma models. In either case, both modelsdescribe those transport processes in terms of two ormore contiguous energy species that interact witheach other in response to the prevailing drivingforces.

By examining the contents of the representativepapers referenced herein, it will be seen that impor-tant research-based tasks still remain to be addressedbefore valid algorithms will be available to facilitatethe description and control of porous media-coupled


transport processes, for example, Darcy’s long-agowork has been cited mainly to underscore the propo-sition that most of the common place modern exten-sions of DrDogma modeling are not always consis-tent with laboratory observations. In fact, this is whatYuster’s 1951 paper purported to indicate as hadOnsager’s 1931 paper done 20 years earlier. Therecord shows, however, that those who offer com-mercial reservoir simulation services to the industryhave not been paying much attention to what theearly prophets have been saying! In fact, almost 150years of frequently flawed treatment of multiphaseflow processes based on antiquated Darcian notionshas been the legacy of lazy people in a hurry toaccept incorrect simulation forecasts just becausethey are easily and cheaply available.

To be noted, a proof can be made by mentioningin passing that the importance and relevance of theabove conclusions are underscored by the content of

Ž .the recent paper of Baggio et al. 1997 . Here, asgoverning equations, these authors invoke the self-

Ževident minded First Law rather than the more.complex Second Law of Thermodynamics and this,

as though irreversible processes can be adequatelyŽ .treated represented as reversible ones. What now

seems to be generally understood, however, is that itwill be as false economy to prefer employing asimplified methodology over a more difficult andexpensive one whenever it can be independentlyshown that such shortcuts inevitably can lead tofallacious and untrustworthy results!

As a final statement about the slowness withwhich old ideas are replaced by unfamiliar new ones,nothing is more telling than the following compar-isons. From 1856 to 1931, that is from Darcy toOnsager, 75 years were needed to make workerssuspect that useful analogies would be found to unifythe understandings about a wide variety of transport

Ž Žphenomena including in porous media cf. for ex-.ample, Luikov, 1966 . Still, from 1931 to 1951 to

1972, that is from Onsager to Yuster to Bear, 20years and then another 21 years were needed topersuade at least a few dedicated worker to considerthe quest further. And from 1972 to date, that ismore than a quarter century has elapsed withoutmuch progress having been achieved that has led to awide acceptance of the elementary theses espousedin a few of the contemporary papers of the 1990s.

And as one final comparison, the readers may noticeŽ .with Rose et al. 1999 some alarm that even with 41

years separating the otherwise elegant studies ofŽ . ŽPhilip and DeVries in 1956 and of Baggio et al. in

.1997 , no notice was taken then or now of the Soretand Deufour coupling effects that probably alwaysoccur during the thermo-diffusion processes notwith-standing that these already had been reported in

Žinternational publications since the 1870s! cf. DeG-.root and Mazur, 1962 .

EPILOGUE — From Darcy to Yuster, and Be-yond!: One seldom hears of engineers suspectingflows of Darcian Laws. Some laugh at Stokes andclaim as hoax his data plots of scattered dots.Some say G AbsurdH wheneÕer heard that Yusterknew what fluids do when sheared and stressedand decompressed, and caused to flow howeÕerslow. Oh how forlorn to hear this scorn by thosewho then weren’t eÕen born! Stokes was deceiÕedbecause he belieÕed that sharing fame withNaÕier’s name might then betray the careless wayhe missed tha Law that Darcy saw! But Yuster toocould not undo the jeers and sneers of pseudopeers! No matter what is known or not, thosetelling lies are most unwise when they misquotewhat giants wrote, for with them, dead Truthstays unsaid!

Acknowledgements

At the time of writing, the Author’s youngest son,Dean Michael Rose, a geologist and artisan, hasgrown to be exactly one-half of the Author’s presentage. It is timely, therefore, to publicly acknowledgemy colossal debt to him for teaching me so muchover the last 40 years about the fields of his special-izations which have ranged from butterflies, to theearth sciences, to computers, to Harz Berg Gemut-lichkeit, to good manners, and beyond. And in ex-change, I have thought to acquaint him with a fewthings I remember about the wisdom and generosityof the several deceased occasional Darcian boostersof yesteryear that once were mentors and closefriends, such as Morris Muskat, M. King Hubbert,L.A. Richards, John Philip, M.C. Leverett, GeraldHassler, A.I. Levorsen, Bill Hurst, Sam Yuster,


George Fancher, Parke Dickey, Al Bell, GeorgeCannon, Claude Hocott, M.R.J. Wyllie. Don Katz,and Alex Bruce to name a few.

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myths about later-day extensions of darcy's law

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