Transport in chemically and mechanicallyheterogeneous porous media—III.

Large-scale mechanical equilibrium and theregional form of Darcy’s law

Michel Quintard a,* & Stephen Whitaker b

aL.E.P.T.-ENSAM (UA CNRS), Esplanade des Arts et Me´tiers, 33405 Talence Cedex, FrancebDepartment of Chemical Engineering, University of California at Davis, Davis, CA 95616, USA

(Received 6 July 1995; revised 30 July 1996; accepted 2 February 1997)

In this paper we develop the regional form of Darcy’s law when the condition oflarge-scale mechanical equilibrium is valid. This condition occurs for many steady orquasi-steady flows of practical importance, and analytical constraints are presentedwhich define the domain of validity for large-scale mechanical equilibrium. Given atwo-region model of a heterogeneous porous medium, the intrinsic region-averagedvelocities can be expressed in terms of the large-scale average velocity according to

Jhf〈vb〉hgh ¼ M pbh·f〈vb〉g, in the h¹ region

Jqf〈vb〉qgq ¼ M pbq·f〈Vb〉g, in the q ¹ region

Here the mapping tensorsM pbh andM p

bq, are specified by the same closure problemused to determine the effective thermal conductivity tensor for a two-phase system,and they are directly related to the regional permeability tensorsK p

bh andK pbq.

Regional permeability tensors are determined for stratified systems, for a nodularmodel of heterogeneous porous media, and for fractured porous media. These large-scale permeabilities are sensitive to the structure of the mechanical heterogeneitiesand in general they are not equal to the local permeability that is used to characterizethe mechanical heterogeneities.q 1998 Elsevier Science Limited. All rights reserved.

NOMENCLATURE

Ahq ¼ Aqh, area of the interface between theh andq-region contained in the averaging volume,V` (m2)

bhh vector field that maps=f〈Pb〉bhgh onto Pbh in the two-equation model (m)

bhq vector field that maps=f〈Pb〉bqgq onto Pbh in the two-equation model (m)

bqh vector field that maps=f〈Pb〉bhgh onto Pbq in the two-equation model (m)

bqq vector field that maps=f〈Pb〉bqgq onto Pbq in the two-equation model (m)

bbh ¼ bhh þ bhq, a vector field that maps=f〈Pb〉bg ontoPbh when the condition of large-scale mechanicalequilibrium is valid (m)

Advances in Water Resources, Vol. 21, pp. 617–629, 1998q 1998 Elsevier Science Limited

All rights reserved. Printed in Great Britain0309-1708/98/$19.00 + 0.00PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 00 1 5 - 8

617

*Corresponding author.

bbq ¼ bqh þ bqq, a vector field that maps=f〈Pb〉bg ontoPbq when the condition of large-scale mechanicalequilibrium is valid (m)

ch ¼ «bhcb þ (]«b/]pb)h, total compressibility in theh-region (Pa¹1)

cq ¼ «bqcb þ (]«b/]pb)q, total compressibility in theq-region (Pa¹1)

g gravitational acceleration (m2 s¹1)I unit tensori,j ,k unit base vectors parallel to thex, y, z coordinate

systemK bh Darcy-scale permeability tensor in theh-region

(m2)K bq Darcy-scale permeability tensor in theq-region (m2)K p

bhh h-region, large-scale permeability tensor in the two-equation model (m2)

1 INTRODUCTION

In Fig. 1 we have illustrated the two-region model of aheterogeneous porous medium that is under considerationin this series of papers. In Parts I and II of this work we dealtwith the flow of a slightly compressible fluid in such hetero-geneous porous media. Starting with the Darcy-scale equa-tions, we obtained a two-equation model that accuratelydescribes the flow when the condition of large-scalemechanical equilibrium is not valid. For a complete discus-sion of this problem, including a review of the literatureavailable on the subject, we refer the reader to these papers.

In this paper our objective is limited to the derivation of aregional form of Darcy’s law that provides the informationnecessary for the derivation of a large-scale nonequilibriummodel for mass transfer. If we consider the flow of a dilute

Fig. 1. Two-region model of a heterogeneous porous medium.

K pbhq ¼ K p

bqh, large-scale cross-effect permeability tensor inthe two-equation model (m2)

K pbqq q-region, large-scale permeability tensor in the two-

equation model (m2)K p

b ¼ K pbhh þ K p

bhq þ K pbqh þ K p

bqq, large-scale permeabil-ity tensor in the one-equation model (m2)

K pbh ¼ K p

bhh þ K pbhq, h-region permeability tensor for the

regional form of Darcy’s law (m2)K p

bq ¼ K pbqh þ K p

bqq, q-region permeability tensor for theregional form of Darcy’s law (m2)

l i i ¼ 1,2,3, lattice vectors used to describe a spatiallyperiodic system (m)

LP large-scale characteristic length associatedwith f〈Pb〉bhgh,f〈Pb〉bqgq andf〈Pb〉bg (m)

M pbh a second-order tensor that maps {〈vb〉} onto { 〈vb〉h}

M pbq a second-order tensor that maps {〈vb〉} onto { 〈vb〉} q}

nhq ¼ ¹ nqh , outwardly directed unit normal vectorpointing from theh-region toward theq-region

Pb pressure in theb-phase (Pa)P0

b reference pressure in theb-phase (Pa)Pb ¼ Pb ¹ P0

b þ r0bg:rb, a pressure associated with the

b-phase (Pa)〈pb〉b intrinsic average pressure in theb-phase (Pa)〈Pb〉b intrinsic average ofPb (Pa){ 〈Pb〉b} large-scale average of the intrinsic average pressure

associated with the one-equation model (Pa)f〈Pb〉bhgh intrinsic regional average pressure for theh-region (Pa)f〈Pb〉bhg Jhf〈Pb〉bhgh, superficial regional average pressure for

the h-region (Pa)f〈Pb〉bqgq intrinsic regional average pressure for theq-region

(Pa)f〈Pb〉bqg ¼ Jqf〈Pb〉bqgq, superficial regional average pressure

for the q-region (Pa)Pbh ¼ 〈Pb〉bh ¹ f〈Pb〉bhgh, large-scale pressure deviation

associated with theh-region (Pa)Pbq ¼ 〈Pb〉bq ¹f〈Pb〉bqgq, large-scale pressure deviation

associated with theq-region (Pa)Pbh ¼ f〈Pb〉bhgh ¹ f〈Pb〉bg, large-scale pressure deviation

associated with theh-region (Pa)Pbq ¼ f〈Pb〉bqgq ¹ f〈Pb〉bg, large-scale pressure deviation

associated with theq-region (Pa)r position vector (m)sh scalar field that mapsf〈Pb〉bqgq ¹ f〈Pb〉bhgh ontoPbh

sq scalar field that mapsf〈Pb〉bhgh ¹ f〈Pb〉bqgq onto Pbq

t time (s)t* characteristic process time (s)〈vb〉 ¼ «b〈vb〉b, superficial average, Darcy-scale velocity

(m s¹1){ 〈vb〉} large-scale average of the superficial velocity

associated with the one-equation model (m s¹1){ 〈vb〉h}

h intrinsic regional average velocity associated with theh-region (m s¹1)

{ 〈vb〉h} ¼ Jh{ 〈vb〉h}h, superficial regional average velocity

associated with theh-region (m s¹1){ 〈vb〉q}

q intrinsic regional average velocity associated with theq-region (m s¹1)

{ 〈vb〉q} ¼ Jq{ 〈vb〉q}q superficial regional average velocity

associated with theq-region (m s¹1)Vj small-scale averaging volume (m3)V Darcy-scale averaging volume (m3)V` large-scale averaging volume (m3)Vh volume of theh-region contained withinV` (m3)Vq volume of theq-region contained withinV` (m3)

Greek letters

a exchange coefficient

Jh volume fraction of theh-region,Vh/V`

Jq volume fraction of theq-region,Vq/V`

rb density in theb-phase (kg m¹3)r0

b reference density of theb-phase (kg m¹3)〈rb〉b intrinsic average density of theb-phase (kg m¹3)mb shear coefficient of viscosity (N-s m¹2)

618 M. Quintard, S. Whitaker

chemical species in the porous medium represented in Fig. 1,the velocity field will not be affected by the concentrationfield. However, the concentration field will certainly dependon the relative importance of advection and diffusion in thedifferent regions. If significantly different time and length-scales are involved, the condition of large-scale mass equi-librium may not be valid and a two-equation model for masstransfer may be necessary. This is the subject of Parts IV34

and V28 in which the reader will find a discussion of theproblem of convection, dispersion and adsorption in hetero-geneous porous media. Under normal circumstances, thelarge-scale nonequilibrium model takes the form of a two-equation model, and this is reminiscent of all models pro-posed in the literature for mobile and immobile zones (see areview by Brusseau and Rao1). These models require infor-mation about the regional velocities, i.e. informationdescribing the velocity field in each region. This is neces-sary, even if the flow can be described by a one-equationmodel based on the validity of large-scale mechanical equi-librium. It is the objective of this paper to develop a theoryfor the regional velocities and to identify the constraintsassociated with large-scale mechanical equilibrium.

When the condition of large-scale mechanical equi-librium is valid, and the flow can be treated as quasi-steadyand incompressible, the large-scale pressure and velocityare given by

=·f〈vb〉g ¼ 0 (1)

f〈vb〉g ¼ ¹1mb

K pb· =f〈pb〉bg¹ rbg� �

(2)

Here it is important to note that〈vb〉 represents a superficialvelocity at the Darcy scale, and that {〈vb〉} represents anintrinsic velocity at the large scale. Normally we would liketo identify intrinsic averages with a superscript; however,in the case of large-scale averages such as {〈vb〉} and{ 〈pb〉b} no suitable notation is available to us.

If the solute transport process is constrained by the con-dition of large-scale mass equilibrium2, a one-equationmodel is sufficient3,29 and only the large-scale velocity isrequired. This velocity is defined explicitly by

f〈vb〉g ¼1

V`

∫V `

〈vb〉 dV (3)

in which V ` is the large-scale averaging volume indicatedin Fig. 1. The large-scale velocity can also be expressed as

f〈vb〉g ¼1

V`

∫Vh

〈vb〉h dV þ1

V `

∫Vq

〈vb〉q dV

¼Jh

1Vh

∫Vh

〈vb〉h dV

� �þ Jq

1Vq

∫Vq

〈vb〉q dV

� �¼Jhf〈vb〉hgh þ Jqf〈vb〉qgq ð4Þ

Here {〈v〉h} h and {〈vb〉q}q are the intrinsic region-averaged

velocities as indicated by the superscriptsh andq. Thesevelocities are not directly available to us from eqns (1) and

(2) and under some circumstances they will be absolutelynecessary for a reliable description of the mass transportprocess. This occurs when large-scale mass equilibrium isnot valid and a two-equation model is required to describethe process of convection, dispersion and adsorption. Thisproblem is discussed in Parts IV and V.

In Fig. 2 we have illustrated a flow that is parallel to astratified system, and if theh-region represents a high per-meability region the convective transport in theq-regionmay be negligible. Under these circumstances, the regionalvelocities are related by

f〈vb〉qgq ,, f〈vb〉hgh, for Kbq , Kbh (5a)

and the transport process in theq-region will be diffusiverather than convective. If the flow illustrated in Fig. 2 isassociated with a remediation process4, the removal ofchemical contaminants from theq-region will be controlledby diffusion. For the condition indicated by eqn (5), a two-equation model for the mass transport process may be neces-sary even though the fluid-flow process can be accuratelydescribed by the one-equation model given by eqns (1) and(2). If theh-region represents a low permeability region, thevelocity in the two regions will be related by

f〈vb〉qgq < f〈vb〉hgh, for K bq q K bh (5b)

and the transport in theq-region will be convective ratherthan diffusive. Under these circumstances, a one-equationmodel may be satisfactory. However, we will see in Part IVthat a uniform velocity in the two regions is not a sufficientcondition for the use of a one-equation model. Whether atwo-equation model or a one-equation model is required isdetermined by the condition of large-scale mass equi-librium that is presented in Part IV.

If the flow is orthogonal to a stratified system, as illus-trated in Fig. 3, the regional velocities are related by

f〈vb〉qg < f〈vb〉hgh (6)

regardless of the values of the Darcy-scale permeabilitytensorsK bh andK bq. Under these circumstances, it seemsprobable that large-scale mass equilibrium will exist andthe mass transport process could be analyzed in terms of aone-equation model. Indeed, such a behavior has been

Fig. 2. Flow parallel to a stratified system.

Transport in chemically and mechanically heterogeneous porous media III 619

observed for purely conductive processes5. The processesillustrated in Figs 2 and 3 can be attacked directly at theDarcy scale4, and this eliminates the need to be concernedwith two or multi-region models. The analysis at the Darcyscale provides the most detailed description of the process,and it is used in our numerical experiments that aredescribed in Part V; however, the computational efficiencyassociated with the use of large-scale averaged equationscannot be overlooked.

One of our objectives in this series of papers is to deter-mine under what conditions the mass transfer process can beaccurately described by a large-scale, one-equation model.To accomplish this we need to be able to predict the region-averaged velocities, since they play a key role in the deter-mination of the condition of large-scale mass equilibriumthat is developed in Part IV. When the condition of large-scale mass equilibrium is not valid, the one-equation, equi-librium model must be discarded and systems of the typeillustrated in Figs 1–3 must be analyzed in terms of the two-equation model described in Part V. Under certain cir-cumstances, the two-equation model can be replaced by aone-equation, nonequilibrium model, and this problem isalso examined in Part V. The two-region model of a hetero-geneous porous medium may be a severe simplification formany mass transport processes that occur in aquifers or oilreservoirs, and multi-region models may be necessary. Whenmany large-scale averaged equations are required, the advan-tage of a complete solution at the Darcy scale is very appeal-ing4,6; however, this is currently impossible in terms ofpetroleum reservoir modeling where large-scale averagingis accomplished tacitly as part of the numerical simulation.

The impression given by Figs 1–3 is that we are dealing

with the Darcy scale and above; however, the general struc-ture of two-equation models has application at the Darcyscale and below. For example, thej-phase illustrated inFig. 1 may represent soil aggregates that can be character-ized as macropore–micropore systems, and Rappoldt7 hasstudied the problem of oxygen diffusion and reaction insuch systems using a two-region model. The two-regionmodel is illustrated in Fig. 4 where macropores are identi-fied by theg-phase and the micropores are imbedded in thek-region. The analysis of the micropore–macropore systemis simplified by the fact that the convective transport isnegligible, and it is the convective transport to which wedirect our attention in this study.

1.1 Regional forms

In order to develop the regional forms of Darcy’s law whenthe condition of large-scale mechanical equilibrium is valid,we need to make use of the developments presented in PartI8. There we derived the superficial average continuity equa-tions for the two-equation model which were given by

h-region

Jhch

]

]tf〈pb〉bhgh ¼ =·

1mb

K pbhh·=f〈pb〉bhgh þ

1mb

K pbhq

�·=f〈pb〉bqgq

�¹

a

mb

f〈pb〉bhgh ¹f〈pb〉bqgq� �

ð7Þ

Fig. 3. Flow parallel to a normal system.

Fig. 4. Two-region model of a macropore–micropore system.

620 M. Quintard, S. Whitaker

q-region

Jqcq

]

]tf〈pb〉bqgq ¼ =·

1mb

K pbqh·=f〈pb〉bhgh þ

1mb

K pbqq

�·=f〈pb〉bqgq

�¹

a

mb

f〈pb〉bqgq ¹ f〈pb〉bhgh� �

ð8Þ

Here we have described these two results as superficialaveraged equations, since the terms on the left-hand siderepresent accumulation per unit volume of both regions.

To develop the regional form of Darcy’s law associatedwith eqns (7) and (8), we recall the Darcy-scale momentumequation for theh-region:

〈vb〉h ¼ ¹1mb

Kbh· =〈pb〉bh ¹ 〈rb〉bg� �

(9)

which was expressed in the form

〈vb〉h ¼ ¹1mb

Kbh· =〈Pb〉b� �

(10)

Here the pressure

Pb

is defined by

Pb ¼ pb ¹ p0b þ r0

bg·rb (11)

and the simplification that takes place between eqns (9) and(10) is associated with the concept of a slightly compres-sible flow that was discussed in Part I. The region-averagedform of eqn (10) can be expressed as

f〈vb〉hgh ¼ ¹1mb

Kbh

· =f〈Pb〉bhgh þ1Vh

∫Ahq

nhqPbh dA

� �ð12Þ

To obtain this result we have followed the procedure out-lined in Section 2 of Part I, and we have made use of thefirst of the following two decompositions:

〈Pb〉bh ¼ f〈Pb〉bhgh þ Pbh (13a)

〈Pb〉bq ¼f〈Pb〉bqgq þ Pbq (13b)

Here we have used the convention that spatial deviationsassociated with quantities having different length scales areindicated by a tilde. Since〈Pb〉bh is a Darcy-scale variableandf〈Pb〉bhgh is a large-scale variable, we have usedPbh toindicate the difference between these two variables. FromSection 3 of Part I we have the following representationsfor the large-scale pressure deviations:

Pbh ¼ bhh·=f〈Pb〉bhgh þ bhq·=f〈Pb〉bqgq

¹ sh f〈Pb〉bhgh ¹ f〈Pb〉bqgq� � (14a)

Pbq ¼ bqh·=f〈Pb〉bhgh þ bqq·=f〈Pb〉bqgq

þ sq f〈Pb〉bqgq ¹ f〈Pb〉qhgh� � (14b)

and substitution of eqn (14) into eqn (12) provides us withthe regional form of Darcy’s law for theh-region:

h-region

Jhf〈vb〉hgh ¼ ¹1mb

K pbhh·=f〈Pb〉bhgh

¹1mb

K pbhq·=f〈Pb〉bqgq

þ1mb

Kbh·yh f〈Pb〉bhgh ¹f〈Pb〉bqgq� � (15)

Here the volume fraction,Jh, has been incorporated intoboth the permeability tensors,K p

bhh andK pbhq, and the

exchange vector,yh , in order to obtain a superficialregion-averaged momentum equation that is consistentwith the continuity equation given by eqn (7). The perme-ability tensors are defined in Part I and the exchange vectoryh is given by

yh ¼1

V `

∫Ahq

nhqsh dA (16)

This vector is similar to the velocity-like vectors that arecontained in eqns (128) and (129) of Part I, and one candetermineyh by means of Problem III given by eqn (125) inPart I. Quintard and Whitaker9 have shown that vectorshaving the form given by eqn (16) are zero when symmetricunit cells are used to complete the closure. While webelieve that the last term in eqn (15) is negligible for theslightly compressible flow studied in Part I, we will retainthis term for the present and complete our statement ofthe regional forms of Darcy’s law by listing the result forthe q-region:

q-region

Jqf〈vb〉qgq ¼ ¹1mb

K pbqh·=f〈Pb〉bhgh

¹1mb

K pbqq·=f〈Pb〉bqgq

þ1mb

K bq·yq f〈Pb〉bqgq ¹ f〈Pb〉bhgh� �

(17)

Here the exchange vectoryq has a form analogous to thatgiven by eqn (16).

In principle, one can determine the region-averaged velo-cities by first solving eqns (7) and (8) and then using thepressure fields to determine the velocities according to eqns(15) and (17). This procedure would be followed for thecase of transient, slightly compressible flows; however,there are many practical problems for which one can

Transport in chemically and mechanically heterogeneous porous media III 621

impose the approximation

f〈Pb〉bhgh ¼ f〈Pb〉bqgq (18)

and when this occurs the problem described by eqns (7),(8), (15) and (17) can be greatly simplified. The conditionindicated by eqn (18) will be referred to as large-scalemechanical equilibrium and the key to a successful attackon the two-region flow problem is to know under whatcircumstances eqn (18) is a valid approximation. For thepresent, eqn (18) represents an assumption, and associatedwith that assumption are restrictions that will be developedin the following paragraphs. Given the restrictions asso-ciated with eqn (18), we will make use of order of magni-tude estimates in order to obtain constraints10. Theconstraints allow one to determine when eqn (18) repre-sents an acceptable approximation, and these constraintsneed to be verified by detailed analysis or experiments.

2 LARGE-SCALE MECHANICAL EQUILIBRIUM

In order to understand the physical approximation associ-ated with the mathematical statement given by eqn (18), oneneeds to make a judgment about how small the differencebetweenf〈Pb〉bhgh and f〈Pb〉bqgq must be in order that eqn(18) be acceptable. This means that we must know howsmall the difference must be relative to some standard.One approach to this problem is to decompose the regionaverage pressures according to

f〈Pb〉bhgh ¼ f〈Pb〉bg þ Pbh (19a)

f〈Pb〉bqgq ¼ f〈Pb〉bg þ Pbq (19b)

and then search for conditions that allow us to neglectPbh

and Pbq. Here f〈Pb〉bg is the large-scale average pressuredefined by

f〈Pb〉bg ¼1

V`

∫V`

〈Pb〉b dV

¼ Jhf〈Pb〉bhgh þ Jqf〈Pb〉bqgq

(20)

and in eqns (19a) and (19b) we have used a circumflexaccent to identify the spatial deviations because theregion-averaged pressures and the large-scale average pres-sure have the same characteristic length scale.

In order to develop the governing equation forf〈Pb〉bg,we add eqns (7) and (8) and make use of eqns (19a) and(19b) to obtain

(Jhch þ Jqcq)]f〈Pb〉bg

]t¼ =·

1mb

K pb·=f〈Pb〉bg

� �

þ =·1mb

K pbhh·=Pbh þ K p

bhq·=Pbh þ K pbqh·=Pbq

ÿ�

þ K pbqq·=Pbq

��¹ Jhch

]Pbh

]tþ Jqcq

]Pbq

]t

!(21)

Here we have used

K pb ¼ K p

bhh þ K pbhq þ K p

bqh þ K pbqq (22)

in which K pb is the large-scale Darcy’s law permeability

tensor that appears in eqn (1). At this point we will requirethat Pbh and Pbq be small enough so that the followingrestrictions are satisfied:

Jhch

]Pbh

]tþJqcq

]Pbq

]t,, =·

1mb

K pb·=f〈Pb〉bg

� �(23)

K pbhh·=Pbh þ K p

bhq·=Pbh þ K pbqh·=Pbq þ K p

bqq·=Pbq

,, K pb·=f〈Pb〉bg ð24Þ

These restrictions, and the assumption that small causesgive rise to small effects11, represent the physical conditionassociated with the mathematical condition given by eqn(18).

In their current form, eqns (23) and (24) are of little valuesince they do not provide us witha priori knowledge of theconditions for which eqn (21) reduces to the one-equationmodel given by

Jhch þ Jqcq

ÿ � ]f〈Pb〉bg]t

¼ =·1mb

K pb·=f〈Pb〉bg

� �(25)

For an incompressible flow, this represents a combinationof eqns (1) and (2). If we make use of the definitions givenby eqns (19) and (20) to express the large-scale deviationsas

Pbh ¼Jq f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

(26a)

Pbq ¼ Jh f〈Pb〉bqgq ¹ f〈Pb〉bhgh� �

(26b)

we can write the restrictions given by eqns (23) and (24) ina form that is much more susceptible to physical interpreta-tion. This form of eqns (23) and (24) is given by

JqJh ch ¹ cq

ÿ � ]

]tf〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

,, =·1mb

K pb·=f〈Pb〉bg

� � (27)

JqK pbhh þ (Jq ¹Jh)K p

bhq ¹ JhKpbqq

� �·= f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

,, K pb·=f〈Pb〉bg

(28)

Here we have ignored both spatial and temporal variations

622 M. Quintard, S. Whitaker

in Jh andJq since the variation of these volume fractionswill have only a small influence on the condition of large-scale mechanical equilibrium. The transient restrictiongiven by eqn (27) is automatically satisfied when the twocompressibilities are equal, i.e.ch ¼ cq, and this has con-siderable intuitive appeal. When the two Darcy-scale per-meabilities are equal, i.e.K bh ¼ K bq , it seems plausiblethat the steady restriction given by eqn (28) should also beautomatically satisfied, and this is indeed true since one canprove that

JqK pbhh þ (Jq ¹ Jh)K

pbhq ¹JhK

pbqq

� �¼ 0,

whenK bh ¼ Kbq

(29)

While eqns (27) and (28) have considerable appeal as thephysical restrictions associated with the mathematical con-dition given by eqn (18), these restrictions are of no valueunless we can develop a reasonable estimate of the pressuredifference, f〈Pb〉bhgh ¹ f〈Pb〉bqgq. One approach to thisproblem12 is to derive the governing differential equationfor this pressure difference and then use that equation asthe basis for developing the desired estimate. In thisapproach, one begins with eqns (19) and (26) to developthe identities

f〈Pb〉bhgh ¼ f〈Pb〉bg þ Jq f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

(30a)

f〈Pb〉bqgq ¼ f〈Pb〉bg ¹ Jh f〈Pb〉bhgh ¹f〈Pb〉bqgq� �

(30b)

which are used in eqns (7) and (8) to obtain for theh-region

h-region

Jhch

]f〈Pb〉bg]t

þJqJhch

]

]tf〈Pb〉bhgh ¹f〈Pb〉bqgq� �

¼ =·1mb

K pbhh þ K p

bhq

ÿ �·=f〈Pb〉bg

� �þ =·

1mb

JqK pbhh ¹ JhK

pbhq

ÿ �·= f〈Pb〉bhgh��

¹f〈Pb〉bqgqÞÿ¹a

mb

f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

ð31aÞ

The analogous form for theq-region is given byq-region

Jqcq

]f〈Pb〉bg]t

¹ JhJqcq

]

]tf〈Pb〉bhgh¹f〈Pb〉bqgq� �

¼ =·1mb

K pbqh þ K p

bqq

ÿ �·=f〈Pb〉bg

� �þ =·

1mb

JqK pbqh ¹ JhK

pbqq

ÿ �·= f〈Pb〉bhgh��

¹ f〈Pb〉bqgq�i

þa

mb

f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

ð31bÞ

Here we can see accumulation and diffusive terms

involving the pressure difference,f〈Pb〉bhgh ¹ f〈Pb〉bqgq,and we can obtain a single equation for this quantity bysubtracting eqn (31b) from eqn (31a). It is convenient tocarry out this operation by first multiplying eqn (31a) byJq

and eqn (31b) byJh, and then subtracting the latter fromthe former to obtain

JhJq Jqch þ Jhcq

ÿ � ]

]tf〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

¹1mb

J2qK p

bhh ¹ 2JhJqK pbhq þ J2

hKpbqq

ÿ ��·= f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �i

þa

mb

f〈Pb〉bhgh�

¹f〈Pb〉bqgq�

¼ ¹JhJq ch ¹ cq

ÿ � ]f〈Pb〉bg]t

þ =·1mb

JqK pbhh þ Jq ¹ Jh

ÿ �K p

bhq ¹JhKpbqq

ÿ ��·=f〈Pb〉bg

ið32Þ

Here we have placed all the terms containing the dependentvariable on the left-hand side, while the source terms invol-ving the temporal and spatial derivatives of {〈Pb〉b}havebeen placed on the right-hand side. On the basis of eqn(29) we can see that the right-hand side of eqn (32) willbe zero when the porous medium is homogeneous, i.e.

ch ¼ cq, K bh ¼ K bq, for a homogeneous porous medium

(33)

Under these circumstancesf〈Pb〉bhgh should be equal tof〈Pb〉bqgqand this is certainly consistent with eqn (32).

In order to extract an estimate off〈Pb〉bhgh ¹ f〈Pb〉bqgq

from eqn (32), we need to estimate both the time andspace derivatives and also the pressure difference. Webegin with the right-hand side of eqn (32) and representthe derivatives according to the following equation (seeRef. 10, Section 2.9):

]

]tf〈Pb〉bg ¼ O

Df〈Pb〉bgtp

!(34)

==f〈Pb〉bg ¼ ODf〈Pb〉bg

L2P

!(35)

Here one must think ofD{ 〈Pb〉b} as some identifiablechange in the large-scale average pressure, with the char-acteristic timet* and the characteristic lengthLP beingdefined by the estimates represented by eqns (34) and(35). This means thatt* is infinite for steady processes.

In order to estimate the derivatives on the left-hand sideof eqn (32), we return to eqns (26a) and (26b) and note thatPbh andPbq represent large-scale spatial deviations, thus weexpect that the change in these quantities will be on theorder of the quantities themselves. This is comparable to

Transport in chemically and mechanically heterogeneous porous media III 623

assuming that the average of the large-scale deviation iszero, i.e.

Assumption :fPbhgh ¼ 0, fPbqgq ¼ 0 (36)

Under these circumstances, the time and space derivative ofthe pressure difference are estimated according to

]

]tf〈Pb〉bhgh ¹ f〈Pb〉bqgq� �¼ O

f〈Pb〉bhgh ¹ f〈Pb〉bqgq

tp

!ð37Þ

== f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �¼ O

f〈Pb〉bhgh ¹ f〈Pb〉bqgq

L2P

!ð38Þ

Here we have assumed thatf〈Pb〉bhgh, f〈Pb〉bqgq andf〈Pb〉bgall have the same characteristic time- and length-scalessince they are all averaged over the same length-scale.When the estimates given by eqns (34), (35), (37) and(38) are used in eqn (32), we can arrange that transportequation to obtain the following estimate:

f〈Pb〉bhgh ¹ f〈Pb〉bqgq� �

Df〈Pb〉bg¼ O(L) (39)

in which the parameterL is given by

In arriving at this result, we have ignored variations of thesystem parameters such asJh, K p

bhh, etc., as is suggested bythe form of the estimates given by eqns (35) and (38). Wecan now return to the two restrictions given by eqns (27)and (28) and make use of the estimate given in terms ofeqns (39) and (40) to obtain

JhJq ch ¹ cq

ÿ �mbL2

P

� �K p

btpL ,, 1 (41)

JqKpbhh þ Jq ¹ Jh

ÿ �K p

bhq ¹ JhKpbqq

� �K p

b

L ,, 1 (42)

For steady-state processes,t* → ` and the first of these isautomatically satisfied. If, in addition to being steady, the

process is one-dimensional, we have the condition that=f〈Pb〉bg is a constant, provided that variations ofJh,K p

bhh, etc., can be ignored. Under these circumstances,eqns (35) and (38) indicate thatLP → ` and the secondconstraint given by eqn (42) is automatically satisfied. Formulti-dimensional or transient flows, one needs to considerthe constraints given by eqns (41) and (42) carefully, and ifsignificant variations in the system parameters occur, oneneeds to take this into account in reformulating the esti-mates associated with the spatial derivatives in eqn (32).On the basis of a recent study of local thermal equi-librium13, we are confident that eqns (41) and (42) arereliable constraints for systems that are isotropic at theDarcy scale; however, detailed tests involving systemsthat are anisotropic at the Darcy-scale need to be carriedout. Estimates of the characteristic time,t*, and the char-acteristic length,LP, require some knowledge of the fieldsunder consideration. Examples of such estimates are givenelsewhere10,14.

When the constraints indicated by eqns (41) and (42) aresatisfied, we say that the condition of large-scale mechanicalequilibrium is valid and a single large-scale averagedpressure can be used to describe the pressure field in aheterogeneous porous medium. Under these circumstances,eqn (18) can be used to simplify eqns (15) and (17) to

h-region

Jhf〈vb〉hgh ¼ ¹1mb

K pbh· =f〈Pb〉bg ¹ rbg� �

(43)

q-region

Jhf〈vb〉qgq ¼ ¹1mb

K pbq· =f〈Pb〉bg ¹ rbg� �

(44)

Here we have made use of the ideas expressed by eqns (9)–(11), and for compressible flows it is understood that thedensityrb should be interpreted as〈rb〉b. The regional per-meabilities,K p

bh andK pbq, are crucial for the determination

of the region average velocities that appear in two-equationmodels of solute transport15, and these quantities aredefined by

K pbh ¼ K p

bhh þ K pbhq (45a)

K pbq ¼ K p

bqh þ K pbqq (45b)

The regional permeabilities are available to us through the

L ¼

JhJq ch ¹ cq

ÿ �mb

tpþ O

JqK pbhh þ Jq ¹Jh

ÿ �K p

bhq ¹ JhKpbqq

L2P

� �JhJq Jqch þ Jhcq

ÿ �mb

tpþ O

J2qK p

bhh ¹ 2JhJqK pbhq þ J2

hKpbqq

L2P

!þ O(a)

(40)

624 M. Quintard, S. Whitaker

closure problems given in Part I. To be explicit, we refer toeqns (124) and (125) of Part I and define two new mappingvectors:

bbh ¼ bhh þ bhq, bbq ¼ bqh þ bqq (46)

so that the regional permeabilities are given by

K pbh ¼ K bh· JhI þ

1V `

∫Ahq

nhqbbh dA

� �(47)

K pbq ¼ Kbq· JqI þ

1V`

∫Aqh

nqhbbq dA

� �(48)

The mapping vectors,bbh and bbq are determined by thesum of closure problems I and II in Part I and we list theresult as

Closure problem

Kbh : ==bhq ¼1Vh

∫Ahq

nhq·K bh·=bbh dA,

in the h¹ region(49a)

B:C:1 nhqK bh·=bbh ¼ nhq·Kbq·=bbq þ nhq· K bq ¹ Kbh

ÿ �,

at Ahq ð49bÞ

B:C:2 bbh ¼ bbq, at Ahq (49c)

Kbq : ==bbq ¼1

Vq

∫Aqh

nqh·Kbq·=bbq dA,

in the q ¹ region(49d)

Periodicity :bbh r þ l i

ÿ �¼ bbh(r ), bbq(r þ l i) ¼ bbq(r ),

i ¼ 1,2,3 eÞ

Average :fbbhgh ¼ 0, fbbqgq ¼ 0 (49f)

Concerning the integrals on the right-hand side of eqns(49a) and (49d), Nozadet al.16 have developed estimatesindicating that these terms make a negligible contributionto the bbh- and bbq-fields. In addition, Quintard andWhitaker9 have proved that these integrals are zero whensymmetric unit cells are used to solve the closure problemindicated by eqns (49). Under these circumstanceswe can simplify the computation associated with thenumerical solution of the closure problem by imposingthe condition

1Vh

∫Ahq

nhq·Kbh·=bbh dA¼ 0,

1Vq

∫Aqh

nqh·Kbq·=bbq dA¼ 0

(50)

The validity of this closure problem has been tested byNozadet al.16 for systems in whichK p

bh andK pbq are iso-

tropic. In addition, the two closure problems from whicheqns (49) was obtained have been successfully comparedwith numerical experiments in Part II17. Under these

circumstances, we feel confident that the regional perme-abilities in eqns (43) and (44) can be predicted on the basisof the closure problem forbbh and bbq; however, a directcomparison between eqns (43) and (44) and laboratoryexperiments needs to be carried out.

Eqns (43) and (44) can be added to obtain the large-scaleform of Darcy’s law18, which is given by

f〈vb〉g ¼ ¹1mb

K pb· =f〈Pb〉bg ¹ rbg� �

(51)

HereK pb was given earlier by eqn (22), and in terms of the

solution to the closure problem given above we have

K pb ¼ JhKbh þ JqK bq

ÿ �þ

Kbh ¹ Kbq

ÿ �V `

·∫

Ahq

nhqbbh dA

(52)

It is extremely important to understand that the condition ofperiodicity represents an approximation that is used inorder to compute the vector fields (bbh andbbq) that appearin eqns (47) and (48), and several remarks concerning thismatter are appropriate.

First we note that the use of periodic boundary conditionsis consistent with the assumption that a representative unitcell exists which accurately describes the geometry of theporous medium. For real systems this assumption requires athorough examination that is beyond the scope of this study;however, based on our experience, we can make the follow-ing comments concerning the use of spatially periodicmodels:

1. The derivation of the macroscopic equations is validboth for periodic unit cells and for disordered sys-tems, provided certain length-scale constraints aresatisfied. In particular, it is crucial to the theory thatthe length scales are conveniently separated. Thesematters have been examined in detail by Quintardand Whitaker19,30,31,32,33.

2. For obvious reasons, the choice of periodic boundaryconditions is appropriate for periodic systems. Thisis due to the fact that second-order gradients ofthe averaged quantities are discarded in the develop-ment at the closure level. As a consequence, devia-tions are periodic functions over the system, up to thissecond-order approximation. Other boundary condi-tions used in the closure problems would imposepreferential directions in the calculations, and wouldlead to less general results. If there are importantnonlinearities in the system, such as those encoun-tered in dynamic two-phase flow in heterogeneoussystems20, the periodicity must be removed from theanalysis.

3. The question of periodicity is more difficult in thecase of disordered systems. Let us assume that prop-erties are randomly distributed. If the system is suchthat there is a finite correlation length, it is intuitivelyappealing that results obtained with a periodic unit

Transport in chemically and mechanically heterogeneous porous media III 625

cell must be representative of the porous mediumitself, provided that characteristic length of the unitcell, ,j, is much larger than the correlation length. Ina series of examples (flow in heterogeneous systems,Ahmadi and Quintard21,22) it was found that the unitcell must be on the order of 10 times the correlationlength in order to obtain this behavior. It was alsofound that this estimate can vary dramaticallydepending on the statistical properties of themedium, and effects that appear as percolation thresh-olds are encountered.

4. The influence of geometrical variations depends onthe type of problem under investigation. Fabrieetal.23 found that geometrical variations are less impor-tant when dealing with diffusion effects than theyare when dispersion effects are dominant. Forexample, in the case of pure heat conduction, thecompilation of results by Nozad24 shows that thevolume fraction is the most important parameterfor unconsolidated, isotropic systems. Indeed, rela-tively accurate results can be obtained with a simpleunit cell involving a very small number of particles.However, more complex, anisotropic systems mayrequire the introduction of more sophisticated unitcells9,25–27. Here we have used the phrase isotropicsystems to mean porous media that are isotropic withrespect to a particular process, and we have used thephrase anisotropic systems in the same manner. Thisterminology is necessary because there are no isotro-pic porous media, but there are media that are isotro-pic with respect to certain processes. In the case ofimportant convective effects, it is well known thatdispersion is greatly affected by channeling paths,and in that case a representative unit cell wouldneed to incorporate all the geometrical features asso-ciated with these channeling effects. It is commonknowledge to think that this would mean a unit cellwith a larger size compare to the purely diffusivecases.

Throughout this discussion, we have assumed thatboundary effects at the exterior boundary of the entiredomain do not interfere with the homogenization problemwithin the macroscopic domain. In addition, it must beemphasized that the complexity of the representative unitcell is not a theoretical limitation, but only a computationallimitation.

For the purposes of computing the regional average velo-cities, it is convenient to make use of eqn (51) in eqns (43)and (44) to obtain

h-region

Jhf〈vb〉hgh ¼ M pbh·f〈vb〉g (53)

q-region

Jqf〈vb〉qgq ¼ M pbq·f〈vb〉g (54)

HereM pbh andM p

bq are the mapping tensors defined by

M pbh ¼ K p

bh· Kpb

ÿ �¹ 1 (55a)

M pbh ¼ K p

bh· K pb

ÿ �¹ 1 (55b)

and determined entirely by the closure problem given byeqns (49). A comparison of eqns (53) and (54) with eqn (4)indicates that the mapping tensors are related by

M pbh þ M p

bq ¼ I (56)

The regional velocities given by eqns (53) and (54) arenecessary for the solution of the solute transport equationsin chemically heterogeneous porous media, and for most ofthese problems the flow can be treated as incompressible.This means that one need only solve eqns (1) and (2) todetermine the large-scale velocity, {〈vb〉} which is thenused with eqns (53) and (54) to determine the superficial,region average velocities.

3 REGIONAL PERMEABILITIES AND MAPPINGTENSORS

When the condition of large-scale mechanical equilibrium isvalid, it seems clear that the procedure for determining theregion-averaged velocities will be based on eqns (53) and(54). This requires the solution of the closure problem givenby eqns (49), and results will be presented for stratifiedsystems, nodular systems and fractured porous media.

3.1 Stratified systems

For the stratified system illustrated in Fig. 5, we havederived the analytical solutions forK p

bh, K pbq andK p

b, andthese are given by

h-region

K pbh ¼ Kbh· JhI ¹

JhJq(jj )· K bh ¹ Kbq

ÿ �j · JqKbh þ JhK bq

ÿ �·j

" #(57)

q-region

K pbq ¼ Kbq· JqI ¹

JhJq(jj )· Kbq ¹ K bh

ÿ �j · JqKbh þ JhKbq

ÿ �·j

" #(58)

Large-scale

K pb ¼JhK bh þ JqKbq

¹JhJq K bh ¹ K bq

ÿ �·(jj )· Kbh ¹ K bq

ÿ �j · JqKbh þ JhKbq

ÿ �·j

(59)

In order to determine the regional velocities represented byeqns (53) and (54), we require the mapping tensors definedby eqns (55) and (55). Eqns (57)–(59) lead to very complexrepresentations forM p

bh andM pbq; however, when both the

h- andq-regions are isotropic at the Darcy scale we obtainthe following analytical representations:

626 M. Quintard, S. Whitaker

h-region

J¹ 1h M p

bh ¼Kbh

JhKbh þ JqKbq

� �I

þ 1¹Kbh

JhKbh þ JqKbq

� �jj

(60a)

q-region

J¹ 1q M p

bq ¼Kbq

JhKbh þJqKbq

� �I

þ 1¹Kbq

JhKbh þ JqKbq

� �jj

(60b)

These results provide the intuitively obvious relationsbetween the intrinsic region-averaged velocity in theh-region and the large-scale velocity given by

f〈vb〉hgh·i ¼Kbh

JhKbh þ JqKbq

� �f〈vb〉g·i (61a)

f〈vb〉hgh·k ¼Kbh

JhKbh þ JqKbq

� �f〈vb〉g·k (61b)

f〈vb〉hgh·j ¼ f〈vb〉g·j (61c)

It is of some importance to remark once again that is anintrinsic average velocity, thus we have compared thisquantity with the intrinsic form of the region-averagedvelocity,. Analogous relations are easily developed forthe q-region, and for completeness we list them as

f〈vb〉qgq·i ¼Kbq

JhKbh þ JqKbq

� �f〈vb〉g·i (62a)

f〈vb〉qgq·k ¼Kbq

JhKbh þ JqKbq

� �f〈vb〉g·k (62b)

f〈vb〉qgq·j ¼ f〈vb〉g·j (62c)

If either the h- or q-region are anisotropic at the Darcyscale, the components ofM p

bh andM pbq can be determined

numerically, and in that case one will not find the simple,intuitively appealing results that are given by eqns (61) and(62).

3.2 Nodular systems

The only results available for the nodular system illustratedin Fig. 6 are based on numerical solutions of eqns (49) forthe case of isotropic systems at the Darcy scale. This leadsto regional and large-scale permeabilities that are isotropicwith respect to rotations in thex–y plane, and thereforemapping tensors that are isotropic in thex–y plane. Thesingle distinct value ofJ¹ 1

h M pbh in the x–y plane is

Fig. 5. Stratified model of a heterogeneous porous medium.

Fig. 6. Nodular model of a heterogeneous porous medium.

Fig. 7. Mapping tensor for theh-region of a nodular system.

Fig. 8. Mapping tensor for theq-region of a nodular system.

Transport in chemically and mechanically heterogeneous porous media III 627

illustrated in Fig. 7 as a function ofKbq /Kbh , and the com-parable representation forJ¹ 1

q M pbq is given in Fig. 8.

3.3 Fractured systems

The fractured system shown in Fig. 9 has essentially thesame characteristics as the nodular system shown in Fig. 6;however, for fractured systems we are generally confined tosmall values ofKbq/Kbh and practical applications areassociated with values of that are small compared to one.For systems that are isotropic at the Darcy scale, theregional permeabilities and mapping tensors are isotropicand the single distinct values ofJ¹ 1

h M pbh andJ¹ 1

q M pbq are

shown in Figs 10 and 11.

4 CONCLUSIONS

In this work we have developed the regional form ofDarcy’s law that allows one to determine the region-averaged velocities that are required for the analysis ofsolute transport in a chemically heterogeneous porousmedium. The constraints associated with the condition oflarge-scale mechanical equilibrium have been identified,and when this condition is valid the region-averaged velo-cities can be determined easily in terms of the large-scaleaveraged velocity and the appropriate mapping tensors. Thelatter can be determined by the solution of a closure problem

that is normally used to calculate the effective thermal con-ductivity tensor for two-phase systems.

ACKNOWLEDGEMENTS

This work was completed while S. W. was a visitor at theLaboratoire Energe´tique et Phe´nomene de Transfert in 1994and 1996. The support of L.E.P.T. is very much appreciated.The partial support for M. Q. from CNRS/INSU/PNRH andInstitut Franc¸ais du Pe´trole is gratefully acknowledged.

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