Calculation of the effective properties describing active dispersion in porous media: from simple to...

Calculation of the e�ective properties describing active dispersion inporous media: from simple to complex unit cells

A. Ahmadi a, A. Aigueperse b, M. Quintard c,*

a LEPT-ENSAM (UMR CNRS), Esplanade des Arts et M�etiers, 33405 Talence Cedex, Franceb ATI Services, 25 quai A. Sisley, B.P. 2, 92390 Villeneuve-La-Garenne, France

c Institut de M�ecanique des Fluides, All�ee du Prof. C. Soula, 31400 Toulouse, France

Received 29 November 1999; received in revised form 28 August 2000; accepted 31 August 2000

Abstract

Dissolution of a trapped non-aqueous phase liquid (NAPL) in soils and aquifers is a matter of great interest for the remediation

of contaminated geological structures. In this work, the Volume Averaging Method is used to upscale the ``active dispersion''

phenomenon, taking into account both dispersion and dissolution of the NAPL. The method provides a macroscopic equation

involving a dispersion tensor, additional convective terms and a linear form for the interfacial mass ¯ux. These ``e�ective properties''

are related to the pore-scale physics through closure problems. These closure problems are solved over periodic unit cells rep-

resentative of the porous structure. Two alternative approaches are considered. The ®rst involves a ®nite volume formulation of the

closure problems and therefore a detailed discretisation of the pore structure. The second is based on a ``network modeling'' of the

pore space and appears as a natural alternative for overcoming the limitations of the ®rst approach (simple unit cells containing a

small number of pores). The two approaches are presented and the in¯uence of NAPL volume fraction and the orientation of the

average velocity ®eld are studied in terms of the P�eclet number for simple unit cells and more complex ones containing a thousand

pores. Ó 2001 Elsevier Science Ltd. All rights reserved.

Keywords: NAPL aquifer contamination; Active dispersion; E�ective properties; Network models

1. Introduction

The fate of non-aqueous phase liquids (NAPLs) insoils and aquifers has received a lot of attention in thepast. E�orts have been developed to model the three-phase and two-phase ¯ows that lead to the developmentof the NAPL plume [18,25]. In this paper, NAPL dis-solution in water will be referred to as active dispersionas opposed to passive dispersion which corresponds tothe classical dispersion in porous media. The descriptionof NAPL active dispersion in water is very important asit determines the conditions under which the aquifer willbe contaminated beyond the NAPL plume.

This active dispersion mechanism can be described interms of local-equilibrium conditions, i.e., the averagedconcentrations are distributed following the thermody-namical equilibrium conditions at the interface betweenthe water and the NAPL phase [1±3,31]. However, ¯owconditions in the porous medium may be such that this

condition of local-equilibrium does not hold, and therate of mass exchange between water and the NAPLphases must be taken into account. For instance, in thecase of a binary system, macroscopic description of thisactive dispersion mechanism requires the knowledge ofan active dispersion tensor and a mass exchange coef-®cient [26±28,32,34]. These e�ective properties may beobtained from experiments or ®eld measurements.Several di�culties must be overcome, and if one con-siders the di�erent correlations available in the litera-ture (see for instance a discussion in [35]) they oftenspan over several orders of magnitude. While we shallnot discuss in this paper the comparative merits of allthe proposed correlations, it looks interesting to havesome quantitative predictions that would be associatedto a direct representation of the NAPL residual satu-ration and the water ¯ow. This would o�er, at least, aprecise understanding of the impact of the di�erentphysical parameters such as geometry, velocity, . . .However, the physics of dissolution in a real porousmedium is a highly intricate phenomenon involvingmany di�erent mechanisms, as discussed for instance in

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Advances in Water Resources 24 (2001) 423±438

* Corresponding author.

E-mail address: [email protected] (M. Quintard).

0309-1708/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 6 5 - 8

[27]. Indeed, dissolution is a�ected by the di�erentscales found in natural systems (pore-scale, variousheterogeneities), and the dissolution process itself maybe unstable leading to preferential channels that have a

tremendous impact on the dissolution kinetics [27]. Inaddition, macro-scale models involving pore-scalemoving boundaries pose a particular problem, and thetraditional linear exchange models represent an ap-

Nomenclature

Abc area of the b±c interface contained in theaveraging volume V; m2

Abr area of the b±r interface contained in theaveraging volume V; m2

av b±c interfacial area per unit volume, mÿ1

b vector ®eld that maps rCb onto theconcentration deviation cm for thenetwork model, m

bb vector ®eld that maps rCb onto theconcentration deviation ~cb, m

cb pore-scale contaminant concentration,kg mol/m3

Cb � hcbib average intrinsic contaminantconcentration in the b-phase, kg mol/m3

Ceqb equilibrium concentration, kg mol/m3

cm average concentration of the di�usingspecies over a pore-throat section, kgmol/m3

cm spatial deviation of the average sectionconcentration, kg mol/m3

hcmi Darcy-scale super®cial average of cm, kgmol/m3

hcmib Darcy-scale intrinsic average of cm, kgmol/m3

D molecular di�usion coe�cient, m2=sD� e�ective local scale dispersion tensor,

m2=sDT dispersion coe�cient for a cylindrical

pore-throat from Taylor and Aristheory, m2=s

D�xx longitudinal local scale dispersioncoe�cient, m2=s

D�yy transverse local scale dispersioncoe�cient, m2=s

I identity tensorl distance between two adjacent

pore-centers on the cubic lattice, mlc unit cell dimension, mlch characteristic length, mlb characteristic length for the b-phase at

the pore-scale, mli i � 1; 2; 3, lattice vectors used to describe

a unit cell, mn outward unit vector of the volume Vi

nbc unit vector normal to the b±cinterface

nbr unit vector normal to the b±r interfacePe P�eclet number

Petube P�eclet number associated to a tubecorresponding to a pore-throat

r position vector, mrb pore-body radius, m�r�b �rb=l, dimensionless average pore-body

radiusrij radius of a pore-throat connecting pores

i and j, mrt pore-throat radius, m�r�t �tt=l, dimensionless average pore-throat

radiuss a scalar that maps hcmib ÿ Ceq onto cm; ssb a scalar that maps Cb ÿ Ceq

b onto theconcentration deviation ~cb; s

Scr residual c-phase saturationSh Sherwood numberub a velocity like coe�cient in the volume

averaged transport equation, m/sV volume of the unit cell used for local

averaging, m3

Vb volume of the b-phase contained in V,m3

vb b-phase pore-scale velocity, m/s~vb b-phase velocity deviation, m/sVb � hvbi ®ltration velocity, m/svm norm of the average velocity over the

pore-throat section, m/svm average velocity over the pore-throat

section, m/svm velocity deviation for the network

model, m/shvmib Darcy-scale intrinsic average of the

velocity vm, m/shvbib Darcy-scale intrinsic average of the

velocity, m/s

Greek symbolsa mass exchange coe�cient, sÿ1

b subscript representative of the aqueousphase

e local scale porosityeb volume fraction of the b-phaseec volume fraction of the c-phasec subscript representative of the

contaminant phaser�rb rrb=l, dimensionless standard deviation

of the pore-body radiusr�rt rrt=l, dimensionless standard deviation

of the pore-throat radius

424 A. Ahmadi et al. / Advances in Water Resources 24 (2001) 423±438

proximation that may be inaccurate under some cir-cumstances.

This problem of determining the e�ective propertiesfrom a pore-scale description of the NAPL entrapmentin a porous medium was the motivation for thetheoretical work published by Quintard and Whitaker[35]. These authors obtained a macroscopic equationfor the concentration of NAPL constituent dissolved inthe water-phase which was coherent with the alreadyclassically used model. In order to obtain such a result,several assumptions were made. Some were reminiscentof assumptions classically made in deriving macro-scalemodels, i.e., separation of scale, others were speci®c toproblems involving pore-scale moving boundaries. Inparticular, it was assumed that the concentration ®eldat the pore-scale could be determined by assuming aquasi-stationary interface. This leads to a macro-scaleequation involving several ``e�ective properties'' thatcould explicitly be obtained from the solution of twopore-scale local problems, later referred to as closureproblems, one giving the e�ective active dispersiontensor, the other one the mass exchange coe�cient.These e�ective properties can be calculated for a givenmorphology, thus giving properties essentially de-pending on time, t, i.e., the history of the dissolutionprocess. While the closure problem could be used, asexplained in [36], to construct step by step this his-torical evolution of the interface in conjunction withthe historical evolution of the macro-scale concentra-tion ®eld, this represents a very complicated task. Inpractice, one replaces this direct time dependence bynon-linear relationships involving the NAPL satura-tion, and other parameters related to the velocity ®eld,such as the P�eclet number for instance. While theproposed theory has its limitations, it can be used tolook at the impact of several parameters, such as thegeometry of the pore-scale phase repartition, or thevelocity ®eld. For this reason, the closure problemswere solved in [35] for simple 2D unit cells, such asperiodic arrays of disks representing the solid andNAPL phases. Indeed, the results brought some inter-esting perspectives. The dependence of the e�ectiveproperties with the P�eclet number satis®ed the expectedgeneral behavior, i.e., the existence of a di�usive regimeand a dispersion regime. However, for these simpleunit cells, it was observed that:1. The di�usive regime is relatively important, which

would preclude the use of a correlation for the massexchange coe�cient vanishing with the P�eclet number(or the Sherwood number).

2. The active dispersion tensor may be di�erent from thepassive dispersion tensor calculated by replacing allpore-scale interfaces by passive interfaces (i.e., zeromass ¯ux). This would indicate that special correla-tions should be used for dispersion in the presenceof trapped NAPL.

3. The mass exchange coe�cient may not tend towardszero for vanishing NAPL saturation, depending onthe wettability conditions.

4. Dependence of the e�ective properties on saturationand the geometry of the pore-scale structure of thethree phases (solid, water and NAPL) may be verycomplicated.Practical implications are very important. However, a

question remains: would these complex features simplifyif one takes into account more complex pore-scale ge-ometries? There are already examples in the literatureshowing that some simpli®cations may arise if one re-places simple unit cells by more complex unit cells. Thisis the case for instance when calculating passive dis-persion tensors as illustrated by the work of Souto andMoyne [39]. For simple unit cells, the authors founddispersion tensors having very di�erent features fordi�erent orientations of the averaged velocity ®eld, whilethis complex behavior simpli®ed for more complex,randomized pore-scale geometry.

The present paper addresses these questions for thecase of active dispersion. First, the way e�ective prop-erties are calculated is brie¯y summarized to clarify theobjective and the notations. Examples of calculationsover simple unit cells are presented that emphasize thekind of complex behavior that may be observed. Asolution of the closure problems over network models isthen presented following the theoretical results pre-sented in [4], which allows to solve the closure problemsover unit cells involving thousands of pores. The resultsare ®nally compared to simple unit cells calculations.

2. Direct calculation of e�ective properties

In this paper, we consider the simple case of a binarysystem, in the porous medium represented in Fig. 1. Theb-phase corresponds to water, while r and c refer tothe solid and NAPL phases, respectively. Following theassumptions made in [35], we consider a binary system,where the NAPL phase is assumed to have a zero ve-locity. The associated macro-scale mass-conservationequation was obtained under the following form

oebCb

ot�r � �VbCb� � r � �Db � rCb�

ÿ a Cb

�ÿ Ceq

b

�� 1�

where eb is the b-phase volume fraction, Cb the averagedintrinsic contaminant concentration in the b-phase, Vb

the ®ltration velocity, Db the active dispersion tensor, athe mass exchange coe�cient, and Ceq

b is the equilibriumconcentration. Here, it must be noticed that a di�erentnomenclature is sometimes used in the literature for thedispersion tensor, which corresponds to Db=eb. Thisconservation equation will be completed with the

A. Ahmadi et al. / Advances in Water Resources 24 (2001) 423±438 425

appropriate macro-scale boundary conditions corre-sponding to the particular system studied. It is impor-tant to note that the knowledge of these boundaryconditions is not necessary for the developments in thispaper which focuses on the determination of the e�ectiveproperties appearing in this equation.

To be clear about the notations, we give the corre-sponding de®nitions of the macroscopic quantities interms of volume averages, as they are introduced in thecited literature. We have, for example, the macro-scaleaverage concentration, Cb, de®ned as a volume averageof the pore-scale concentration cb as follows:

Cb � 1

V

ZVb

cb dV � hcbib; �2�

where V and Vb are, respectively, the averaging volumeand the volume of the b-phase in V. The ®ltration ve-locity corresponds to

Vb � 1

V

ZV

vb dV � hvbi � ebhvbib; �3�

where vb is the pore-scale b-phase velocity.As usual in scaling-up theories, Eq. (1) is an ap-

proximate solution of the pore-scale to Darcy-scaleproblem. It is beyond the scope of this paper to recall all

the mathematical developments leading to this analysis,and we refer the reader to the cited literature [11,35,36]and to the introduction for a summary of the limita-tions. The dots in the right-hand side of the equation area reminder of the simpli®cations involved. In the above-mentioned papers, it is shown that the e�ective proper-ties are related to the pore-scale physics through twoclosure problems, which are listed below. The closureproblems involve two closure variables bb and sb whichappear in the description of the pore-scale concentrationas a function of the average concentration, i.e.,

cb � Cb � bb � rCb ÿ sb Cb

�ÿ Ceq

b

�: �4�

In this development, the porous medium is representedby a periodic system. The system is, therefore, com-pletely characterized by a single unit cell as large asnecessary taking into account all the complexity of thepore-scale geometry. The closure problems are thereforesolved over this representative unit cell using periodicboundary conditions. It must be noted that despiteperiodic boundary conditions, the use of this methodologyis not limited to periodic systems [5].

The ®rst closure problem giving bb allows to calculatethe active dispersion tensor. Over a periodic unit cellrepresentative of the NAPL entrapment, the followingboundary value problem has to be solved.

Problem I.

~vb � vb � rbb � r � �Drbb� � eÿ1b ub; �5�

B:C:1 bb � 0 at Abc; �6�

B:C:2 nbr � rbb � nbr � 0 at Abr; �7�

bb�r� li� � bb�r�; �8�

hbbi � 0; �9�

ub � 1

V

ZAbr�Abc

n � �Drbb� dAÿDreb: �10�

In this problem, the velocity deviation is given by

~vb � vb ÿ hvbib �11�and D is the molecular di�usion coe�cient. The twovectors nbr and n are the outward unit vectors normal tothe b±r interface and to the total b±r and b±c interface,respectively. The closure variable, bb, is then used toobtain the active dispersion tensor using the followingequation

Db � ebDI�D1

V

ZAbr�Abc

nbb dAÿ h~vbbbi: �12�

The mass exchange coe�cient, a, is obtained fromsolving closure Problem II, which is formulated asfollows.

Fig. 1. Sketch of NAPL repartition in groundwater: the b-phase cor-

responds to water, while r and c refer to the solid and NAPL phases,

respectively.


Problem II.

vb � rsb � r � �Drsb� ÿ eÿ1b a; �13�

B:C:1 sb � 1 at Abc; �14�

B:C:2 nbr � rsb � 0 at Abr; �15�

sb�r� li� � sb�r�; �16�

hsbi � 0; �17�

a � 1

V

ZAbr�Abc

n � �Drsb� dA: �18�

It must be noticed that the mass exchange coe�cient is apart of Problem II, through an integro-di�erential for-mulation. Special procedures were designed to handlesuch problems, taking into account periodicity con-ditions. Examples of solutions are available in [35] in thecase of 2D unit cells. The original numerical model(1994) has been extended to handle 3D cases, and resultsare presented in the next section.

3. Results for simple unit cells

The calculation of the e�ective properties follows thealgorithm below:1. de®ne geometry, both for the solid and NAPL phase,2. calculate the pore-scale velocity ®eld for a given

macroscopic velocity or pressure gradient,3. solve Problem I and compute the e�ective dispersion

tensor,4. solve Problem II, and obtain the mass exchange coef-

®cient.We refer the reader to [35] for a presentation of the

numerical schemes used in the actual numerical modelsdesigned for solving these closure problems. The phasedistribution is represented by assigning phase indicatorvalues on each block of a Cartesian grid, as illustrated inFig. 2.

The velocity ®eld is obtained by solving Stokesequations using an Uzawa algorithm. Quasi-second or-der accurate schemes are used to solve for the closureproblem equations at a given P�eclet number. Theproblem of the unit cell geometry is complex, as illus-trated by observation published by Lowry and Miller[24] and Mayer and Miller [25]. No experimental datawere used in the calculations presented in this paper.Our objective was rather to test for the impact of thedi�erent choices that can be made. Therefore, di�erenttypes of unit cells have been used, which are summarizedin Figs. 3±5. In addition, we did not try to obtain thehistorical evolution of the dissolved interface. We rather

Fig. 2. Example of phase discretised distribution. The scalar variables are estimated at the block center, while the components of the vectors (like the

velocity vector: vbx and vby) are calculated at the interface of the grid block.

Fig. 3. Simple 2D unit cell.

Fig. 4. Simple 3D unit cell.


calculated the e�ective properties for di�erent, arbitraryvalues of the saturation and P�eclet number.

A comparison between the results obtained fromsimple 2D and 3D unit cells is shown in Figs. 6 and 7 forthe dispersion coe�cient, and in Fig. 8 for the massexchange coe�cient. In the caption of these ®gures, theP�eclet number is de®ned as

Pe � hvbiblc

D; �19�

where lc is the unit cell dimension. The use of the P�ecletnumber is made possible because the velocity ®eld cor-responds to a laminar ¯ow, i.e., it is independent of theReynolds number. This is not a limitation of the theory,and a velocity ®eld involving inertia e�ects could be usedinstead without changing the numerical model solvingthe closure problem in which vb is only an input ®eld.

All three ®gures show the expected behavior of thee�ective parameters with respect to the P�eclet number.The di�usive regime, at low P�eclet number, is moreimportant for the transverse dispersion coe�cient thanfor the longitudinal dispersion coe�cient. It is also lessmarked, i.e., it appears at larger P�eclet number, for themass exchange coe�cient. However, one sees that thereis a dramatic impact of the geometry on the coe�cient

values. The in¯uence of saturation, for instance, cannotbe represented by simple correlations. This is moredramatic if one considers the in¯uence of the velocity®eld direction. This e�ect is illustrated in Fig. 9 for thedispersion coe�cient, and in Fig. 10 for the mass ex-change coe�cient, in the case of the simple 2D unit cellpresented in Fig. 3.

In the di�usive regime, our results show that themedium is macroscopically isotropic, as expected fromthe unit cell geometry. On the contrary, the dispersionmechanisms are very sensitive to the velocity orienta-tion, for these simple unit cells. Correlations extractedfrom these calculations may not be practical in the caseof real, natural systems. Following the results obtainedin the case of passive dispersion [39], we would expectthat a more complex, disordered unit cell would produceresults less sensitive to the pore-scale geometry.

In a ®rst attempt to check this problem, we havesolved the closure problems on ``disordered'' unit cells,like the one illustrated in Fig. 5. Results for the longi-tudinal dispersion coe�cient are shown in Fig. 11, andresults for the mass exchange coe�cient are shown inFig. 12.

There seems to be a smaller in¯uence of the velocityorientation in the case of the longitudinal dispersioncoe�cient, this is more clear for the mass exchange co-e�cient. This shows an interesting trend if one is in-terested in capturing the e�ect of real porous mediafeatures. However, there are computational limitationsthat prevent the use of such direct simulations for verycomplex systems. This called for a di�erent approach ofthe problem, and following the extensive literatureconcerning the use of network models in porous mediaphysics, we designed a speci®c numerical procedure tosolve the closure problems on network models as ex-plained in the next section.

Fig. 5. Simple disordered 2D unit cell.

Fig. 6. Longitudinal dispersion coe�cient: comparison between 2D and 3D unit cells for di�erent values of the c-phase volume fraction �ec�.


4. Network formulation

In order to capture the e�ects of real porous mediaand to obtain more signi®cant e�ective properties, it isnecessary to incorporate a larger number of pores and amore complex geometry in the averaging volume con-sidered. Network modeling provides the possibility ofachieving these two aims. The interest of networkmodels for active dispersion has already been demon-strated by the work of Lowry and Miller [24] or Grayet al. [17]. These considerations led us to formulate theupscaling problem on a network, and the theory is de-tailed in [4] To be clear: our contribution lies in thecalculation of the e�ective properties through a speci®cimplementation of the closure problems presented in theprevious section. It must be emphasized that all under-lying assumptions are kept. In addition, simplifying as-

sumptions speci®c to the treatment of networks will bemade, as we shall discuss later.

In this network model implementation, the porousstructure is idealized as a network of spherical porebodies connected to one another by cylindrical pore-throats. The pore-body-radius �rb� and the throat-radius�rt� are given by Gaussian distributions with user-speci®ed values of the mean and the standard deviation.Since our main objective has been to determine localscale transport properties on a network model, we havechosen a 3D network on a regular cubic lattice as a ®rstapproach. The methodology used can easily be extendedto more complex networks (with a variable number ofconnections to each pore-body for example).

It must be emphasized that the interest of networkmodels lies in the possibility of using a simple descrip-tion of the ¯ow (Poiseuille ¯ow, constant concentration

Fig. 7. Transverse dispersion coe�cient: comparison between 2D and 3D unit cells for di�erent values of the c-phase volume fraction �ec�.

Fig. 8. Dimensionless mass exchange coe�cient: comparison between 2D and 3D unit cells for di�erent values of the c-phase volume fraction �ec�.


Fig. 9. In¯uence of the average velocity orientation on the dispersion coe�cient for the simple 2D unit cell of Fig. 3.

Fig. 10. In¯uence of the average velocity orientation on the mass exchange coe�cient for the simple 2D unit cell of Fig. 3.

Fig. 11. In¯uence of the average velocity orientation on the dispersion coe�cient for the disordered 2D unit cell of Fig. 5.


in the sites, 1D ¯ows in the links, . . .). The impact ofthese simpli®cations may be checked by using the directsolution presented in the previous section. As a conse-quence, we believe that both approaches have their in-terest, and are complementary.

5. Preliminary steps: drainage, imbibition, velocity ®eld

approximation

The porous structure initially saturated by water is®rst penetrated by the contaminant. The contaminant isthen displaced by water leaving behind trapped con-taminant ganglia. The network must therefore undergosimilar physical phenomena. It must be noted thatmodeling of the drainage and imbibition allows to set upa NAPL saturation in the network model and will haveno consequence on the developments presented in thefollowing sections.

In this work the porous medium is assumed water wetand the capillary forces are assumed to dominatedrainage and imbibition mechanisms. For these steps,piston-displacement and ®lm ¯ow mechanisms are takeninto account. The piston-displacement in both drainageand imbibition are modeled using the Young±Laplaceequation [13,24] and are governed by the pore-scalegeometry. While ®lm displacement has not been con-sidered for the drainage due to the lack of a rigorouscriterion, it has been taken into account for the imbi-bition [20,24]. This ®lm ¯ow is responsible for a dis-placement mechanism called ``choke-o�'' or ``snap-o�'',in which interfaces in small pores become unstable andrupture. Once the two phases are distributed in the po-rous network, the single phase displacement of water inthe porous network containing ganglia of di�erent sizesand forms is studied. Network modeling associated witha number of simplifying assumptions (creeping ¯ow,

Newtonian, non-miscible, incompressible ¯uids, . . .)leads to a satisfactory approximation of the velocity®eld, while a detailed resolution of the ¯ow would havebeen impossible from the practical point of view. Ob-viously, with this simpli®ed treatment, details of the ¯owsuch as rotational ¯ow in dead end pore throats are nottaken into account and the velocity in these throats isconsidered to be zero. The phase-distribution as well asthe velocity ®eld are now considered known for thefurther study of NAPL transport.

6. Upscaling dispersion

The volume averaging methodology has been re-viewed for our special case of network geometry [4]. Thelocal equations and properties are obtained startingfrom a description of the transport in each pore-throatbased on the Taylor and Aris formulation of dispersionin a capillary tube [7,40]. These authors state that undersome limiting conditions listed below, the transport in acapillary tube is governed by a 1D classical convection±dispersion equation with the dispersion coe�cient givenby

DT � D� r2t v2

m

48D�20�

in which rt is the radius of the tube, D the moleculardi�usion coe�cient and vm is the mean velocity over thetube section. This result can also be found using generalupscaling theories [9,10,23,38]. Therefore, it is consistentwith the proposed averaging approach assuming suc-cessive upscaling are performed.

The Taylor and Aris formulation is valid under thefollowing limiting conditions [40]:

Fig. 12. In¯uence of the average velocity orientation on the mass exchange coe�cient for the disordered 2D unit cell of Fig. 5.


(a) The changes in concentration due to convectivetransport along the tube take place in a time whichis so short that the e�ect of molecular di�usion maybe neglected.(b) The time necessary for appreciable e�ects to ap-pear, owing to convective transport, is long com-pared with the time of decay during which radialvariations of concentration are reduced to a fractionof their initial value through the action of moleculardi�usion.

The condition (b) can be considered valid if [40]

lt

2vm

� r2t

3:82D; �21�

where lt is the length of the tube. This condition, whichmust be satis®ed for each cylindrical pore-throat in-cluded in the pore network, can also be written in termsof a P�eclet number related to each tube:

Petube � vmlt

D� 7:22

l2t

r2t

: �22�

There will be an attempt to take into account this con-dition in the presentation of the results in the followingsections.

Using the volume averaging procedure applied to thepore-scale equations, we obtain a local scale averagedequation similar to the one given by Eq. (1):

oebCb

ot�r � �VbCb� � r � �D� � rCb�

ÿ a Cb

�ÿ Ceq

b

�� 23�

The mass exchange coe�cient a and the local scaledispersion coe�cient D� are expressed as a function ofthe pore-scale properties and the two closure variables b

and s in the following manner:

a � 1

V

ZAbc

nbc �DT � rs dA; �24�

D� � ebhDTib ÿ ebhvmbib � ebhDT � rbib: �25�In this problem, vm is the average velocity over the pore-throat section and is written as the sum of the averagevelocity and a velocity deviation: the velocity deviationis given by

vm � vm ÿ hvmib: �26�In a manner similar to the development in [35], we ob-tain the following closure problems for the two closurevariables b and s:

Problem I.

vm � rb� vm � r � �DT � rb� ÿ eÿ1b ub; �27�

b � 0 at Abc; �28�

Fig. 13. The geometry of the NAPL blobs trapped in a network.


b�r� li� � b�r�; i � 1; 2; 3; �29�

hbib � 0: �30�Problem II.

vm � rs � r � �DT � rs� ÿ eÿ1b a; �31�

s � 1 at Abc; �32�

s�r� li� � s�r�; i � 1; 2; 3; �33�

hsib � 0: �34�The problems are similar to the ones described in

Section 2. Using a well chosen decomposition of theclosure variables, the integro-di�erential terms in theclosure problems can be eliminated. In this develop-ment, we will assume that the concentration is constantwithin the intersections or nodes. We recall that thepressure was also assumed to be constant at these in-tersections. A study similar to the one performed for thepressure ®eld by Koplik [21] has not been performed yetin order to estimate the error made. We note, however,that this assumption is consistent with classical treat-ment of networks. As a consequence, the closure vari-ables are considered constant on each nodes. Theclosure problems obtained are therefore solved analyti-cally over each tube (pore-throat) as a function of thevalues at the two pore-bodies occupying each end of thetube. Then a balance over each intersection of tubes willlead to a linear system. The resolution of the linearsystem leads to the values of the closure variables oneach pore, from which local properties are calculated.The details of the calculations are beyond the scope ofthis paper and are published elsewhere [4].

7. Results on the network model

It is obvious that in this case a similar algorithm aspresented in Section 3 is to be followed. The main dif-ference here is that now the NAPL distribution is givenby modeling physical processes such as drainage andimbibition on the network and is directly related to thenetwork geometry. An example of such a realization isshown in Fig. 13. The closure problems are then solvedover the network giving the dispersion tensor and theexchange coe�cient. Additional coe�cients interveningin the ®nal macro-scale equation can also be calculated.

The values found for the dispersion tensor and theexchange coe�cient are studied as a function of theP�eclet number given by

Pe � khvmibklD

�35�

in which l is the distance between two adjacent pore-centers on the cubic lattice and khvmibk is the norm ofthe local scale average velocity. The length l is also usedas a characteristic length for obtaining dimensionlesspore-body and pore-throat radii and their standard de-viations denoted �r�b, r�rb, �r�t , and r�rt, respectively. At thisstage of the problem Eq. (22) must be considered inorder to limit the results to their domain of validity.With our particular case of cubic lattice, since thevelocity in the tubes perpendicular to the direction of thepressure gradient is rather small, we can make the fol-lowing approximation to relate the local scale averagevelocity to the average tube velocity, �vm, of the tubesparallel to the pressure gradient:

hvmib � �vm

3: �36�

In addition, the length l is taken as an approximation forlt. As a ®rst approach condition (22) can be approxi-mated as

Pe� 2:4l�rt

� �2

: �37�

As a rough estimate, we will consider the followingrelation:

Pe <2:4

5

l�rt

� �2

: �38�

Results satisfying this condition are plotted in solidlines while the extrapolation of the results to greatervalues of Pe is plotted in dotted lines.

All calculations presented in this paper have beenperformed over unit cells containing 1000 pores. Resultspresented are the average values over ®ve realizations.Although, in the cases studied, the di�erence betweenthe results obtained from di�erent realizations is rathersmall, a larger number of realizations must be taken intoaccount in a systematic calculation procedure. In orderto study the in¯uence of the c-phase volume fraction, theresults for three cases listed in Table 1 are presented.The porosity, b-phase volume fraction and the c- phasesaturation are also given in this table. In Section 3, the

Table 1

Cases studied for active dispersion

Case �r�b r�rb �r�t r�rt e eb ec Scr

1 0.30 0.12 0.15 0.06 0.214 0.144 0.0704 0.329

2 0.20 0.08 0.10 0.04 0.095 0.066 0.0283 0.299

3 0.10 0.05 0.05 0.02 0.023 0.016 0.0061 0.271


dispersion tensors were studied as a function of ec forsimple unit cells by changing the size of the contaminantblob placed in the center of the cell. This means that theresults concern cases with varying b-phase and c-phasevolume fractions, while the porosity of the porous me-dium stays unchanged. For the network models of theporous medium, the volume fractions of the two phasesare intimately related to the geometry and vary with theporosity. In order to have a possible comparison be-tween the results presented in Section 3 and those ob-tained for cases listed in Table 1, we present thelongitudinal and transverse dispersion behavior inFigs. 14 and 15 in terms of the two coe�cients of thetensor D�xx=�De� and D�yy=�De� as a function of the P�ecletnumber. In this manner we overcome the problem ofvarying porosity for these cases.

The main features of the curves follow the expectedbehavior, i.e., di�usive and dispersive regimes. The

general tendency for the variations as a function of ec

seems identical as the ones observed for the simple unitcells with a much less amplitude. The variation of thedimensionless mass exchange coe�cient �al2=D� versusthe P�eclet number is plotted in Fig. 16. These curvesshow a di�usive regime at low P�eclet number and a re-gime more in¯uenced by advection for values of P�ecletnumber above 0.1. Similar behaviors were observed byQuintard and Whitaker [35], and in the new resultspresented in the previous sections. We notice, however,that this variable tends to an asymptotic advectiveregime at high P�eclet number.

In two recent network models [12,43], the roughnessand grooves of a real porous medium are represented bycorners in cubic pore-bodies and rectangular pore-throats. In this manner, the exchange between thetrapped water in the corners at the irreducible watersaturation and the trapped NAPL in these chambers ortubes is taken into account. In our work, this aspect ofthe problem has not been considered and the water isconsidered stagnant in pores containing NAPL. Thismay be considered as one possible explanation forconstant mass exchange coe�cient values for high P�ecletnumbers. Other factors can explain this behavior of themass exchange coe�cient for high P�eclet numbers.Consider dissolution in a tube (or heat transfer with aconstant wall temperature or any other similar InitialBoundary Value Problem), there will be an entranceregion with a development of a boundary layer, in whichthe mass exchange coe�cient will increase with the po-sition, and will have a strong dependence on the veloc-ity, among other factors. This dependence will also bevery sensitive to the ¯ow model, i.e., developed para-bolic ®eld or development of a boundary layer. Thisvariation of the exchange coe�cient with the positionmeans that non-local behavior is involved. It is wellknown that beyond the entrance region there is an as-ymptotic limit with a constant mass exchange coe�cient

Fig. 15. Transverse dispersion coe�cient as a function of the P�eclet

number for di�erent NAPL volume fractions for the networks.

Fig. 16. The mass exchange coe�cient as a function of the P�eclet

number for networks.

Fig. 14. Longitudinal dispersion coe�cient as a function of the P�eclet

number for di�erent NAPL volume fractions for the networks.


(or Nusselt number). Although for a real porous me-dium, the problem becomes much more delicate, webelieve that this behavior is not speci®c of the networkapproach. Indeed, it is mainly related to the fact that ourtheory, in which periodic boundary conditions areconsidered, corresponds to a fully developed exchangezone between the b-phase and the NAPL ganglia. Themodel developed therefore gives the asymptotic value ofthe exchanged coe�cient. The variation of the exchangecoe�cient as a function of ec seems to be much moreimportant for the network approach, but al2=D in-creases with increasing ec as observed for simple unitcells.

Let us now study the in¯uence of the orientation of thevelocity ®eld. Consider case 2 of Table 1. In this case, theaverage velocity is in the x-direction. Di�erent caseshave been considered with the average velocity rotatedat 30�, 45� and 60� about the z-axis. The results arepresented in Figs. 17±19. As expected when more com-plex (and therefore more realistic) unit cells are consid-ered, the orientation of the velocity has little in¯uenceon the dispersion tensor and the exchange coe�cient.

Finally we will study the impact of the speci®c surfaceon the exchange coe�cient. Indeed, many authors dis-cuss the mass exchange phenomena in terms of a Sher-wood number de®ned as

Sh � alch

Dav

�39�

in which lch is the characteristic length and av is theinterfacial area per unit volume. From the structure ofthe closure Problem II, a natural de®nition would be

Sh � al2ch

D: �40�

While there is certainly a relationship between lch andav, it is not necessarily simple. From our experience, wethink that the most important parameters are related to

length-scales characteristic of the distance betweenganglia or ganglia clusters. Indeed, simple examplesshow that there is not a priori a direct relationship be-tween a and av. For instance, the calculations performedon the simple unit cells represented in Fig. 20 byAigueperse [6] showed no di�erence for the values of a.This obvious result emphasizes that in the case of morecomplex clusters, some zones may be at a relativelyconstant concentration close to the equilibrium value,thus marginally contributing to the mass exchange whileincreasing the speci®c area. These unit cells may seemunrealistic, and one may think that some scaling be-tween lch and av exists that could make the introductionof the speci®c area useful. We check this idea below,using our network computations.

A number of correlations are presented in the litera-ture [8,14±16,19,22,28,30,32,41,42] in which the Sher-wood number is expressed in terms of the Reynoldsnumber, the Schmidt number, the P�eclet number and thevolume fraction of the b-phase. In all these correlations

Fig. 19. In¯uence of the average velocity orientation on the mass

exchange coe�cient for a network model.

Fig. 17. In¯uence of the average velocity orientation on the longi-

tudinal dispersion coe�cient.

Fig. 18. In¯uence of the average velocity orientation on the transverse

dispersion coe�cient.


the characteristic length considered is the longest blobdimension. Although many of these relationships arederived from the measurements of dissolution of solidorganic spheres by a uniform aqueous phase ¯ow ®eld ina packed bed or for di�usion-limited dissolution of ¯uidspheres suspended in a laminar ¯ow regime (see [32,34]for a discussion), as a ®rst approach they are comparedto our results. In all cases studied here, the Schmidtnumber is constant and equal to 1000. Since the productof the Schmidt number and the Reynolds number isequal to the P�eclet number, we can reduce the number ofparameters to two and study Sh as a function of Pe andeb. The characteristic length used for the calculation ofSh is the average of the dimensions of the largest con-taminant blob. In Fig. 21, the Sherwood number cal-culated using the expression given in Eq. (40) ispresented as a function of the P�eclet number for di�er-ent values of eb. In addition to the three cases listed inTable 1, three other cases for which the properties are

listed in Table 2 are considered. We must note that thevalues of the distance between two adjacent pore-centersis of 200 lm for cases 4 and 5 and of 150 lm for case 6.One can see that, for instance, for close values of eb theSh obtained for a large P�eclet number is rather di�erent.From these results we can conclude that simple corre-lations cannot characterize correctly the NAPL dis-solution process, and that, at least for the networkrealizations studied in this paper, the introduction of thespeci®c area does not produce simpli®ed correlations.As a consequence, we believe that the use of a de®nitionfor the Sherwood number like in Eq. (40) is more ap-propriate.

We can also compare the values of the mass exchangecoe�cient (Fig. 22) found by our work to the ones ob-tained experimentally and presented in the literature[16,19,28,29,32,33,37]. However, this comparison mustbe performed with great care and a number of pointsmust be discussed. Some of these experiments are per-formed under quasi-steady conditions, i.e., measure-ments are made before a signi®cant change in NAPLvolume or interfacial area occurs [32,37]. Others[19,28,29,33] take into account dynamic e�ects corre-sponding to the reduction of the NAPL saturation andthe shrinking of NAPL blobs during dissolution. Thecomparison of this second class of experiments with ourresults is inappropriate, although the results are plottedin Fig. 22 for completeness. Concerning the quasi-steadyexperiments, the comparison must still be done withcare. We believe that the macro-scale model involvingthe mass exchange coe�cient is an approximation of aproblem which has non-local properties. This meansthat this coe�cient is history and position dependent.Moreover, the experimental results depend clearly onthe way experiments are observed (cross-section aver-ages, ®nite-length tube averages, etc.).

The results are presented in terms of a Sherwoodnumber de®ned by Eq. (40). The characteristic lengthused for the experimental results presented is the averagegrain size. For our network results, the distance betweentwo pore-body centers which is taken to be equal to100 lm, seems to be a good candidate and comparableto the concept of the grain size. Di�erent authors [32,37]underline the importance of the experimental conditionson the values of the Sherwood number obtained. Inparticular, the procedure used for the NAPL emplace-ment seems to be of great importance and in¯uencesthe distribution of the trapped NAPL blobs. All

Fig. 21. The Sherwood number as a function of the P�eclet number.

Table 2

Additional cases studied for active dispersion

Case �r�b r�rb �r�t r�rt e eb ec Scr

4 0.10 0.04 0.05 0.02 0.022 0.017 0.004 0.19

5 0.15 0.075 0.075 0.03 0.051 0.035 0.016 0.31

6 0.20 0.08 0.10 0.04 0.095 0.069 0.025 0.33

Fig. 20. Simple unit cells with same mass exchange coe�cient and

di�erent speci®c area.


experimental results presented are conducted overporous media (beds of sand or glass beads) with theaverage grain sizes ranging from 70 lm up to 0.1 cm.Our results are in the same order of magnitude as theexperimental results. The behavior is, however, ratherdi�erent. The only experimental results showing a dif-fusive regime for low Reynold numbers is that of Ra-dilla [37] obtained for sand packing of average grain sizeof 70 lm. This di�usive regime is also observable in ourresults. However, one must point out here a major dif-®culty associated with the measurement of mass ex-change coe�cients at low P�eclet number, i.e., forconditions under which local mass equilibrium prevails.In this case, the mass exchange coe�cient is hardlyidenti®able. For instance, under local mass equilibriumobtained for su�ciently large values of the massexchange coe�cient, the averaged dissolution of aporous column will depend on the velocity whatever theexact value of the mass exchange (it may even have aconstant value). One may therefore infer a column scalemass exchange coe�cient, which will go to zero at zerovelocity, while the Darcy-scale mass exchange coe�cientwithin the porous column will keep a constant value (thedi�usive limit). It is beyond the scope of this paper toaddress the problem of the experimental determinationof mass exchange coe�cients, and we leave this discus-sion open.

8. Conclusions

Di�erent types of unit cell geometries have been usedto calculate active dispersion tensors and mass exchangecoe�cients. The ®rst series correspond to an accuratedescription of both the geometry and the pore-scalephysics. Computational limitations make di�cult toapproach with such unit cells the complexity of a real

disordered system. The correlations extracted from theseresults would incorporate saturation, the P�eclet number,in a highly non-linear complex fashion. In addition,velocity orientation e�ects may be important.

This called for a special treatment of unit cells in-volving thousands of pores. A speci®c, original treat-ment of the active dispersion case has been proposed inthe case of network models. The results presented in thispaper con®rm that scale e�ects may dampen the speci®cfeatures associated to simple unit cells, and that corre-lations involving a smaller number of well de®nedparameters may be expected.

However, such network model treatment requiresthat the pore-scale physics is represented by simplersolutions (1D, constant concentrations in some areas,etc.). The original full closure problems may be used tocheck the validity of such simple representations.Therefore, the two approaches are equally necessary.

For the unit cells studied in this paper, it does notseem that simple correlations involving the speci®c area,or even the saturation are available. More studies wouldbe needed for other geometries or network realizationsto check whether such simple correlations exist for someclasses of porous media and NAPL repartitions.

It must be emphasized that the study presented hereinvolves some length and time scale assumptions, as wellas other limitations associated with the e�ect of dis-solution. In particular, the history e�ects of the disso-lution process are not incorporated in the analysis. Thiscalls for further studies.

Acknowledgements

This work has been partially supported by CNRS/INSU/PNRH and Institut Francßais du P�etrole. Theauthors wish to thank Martin Blunt for his constructiveremarks on the paper.

Fig. 22. Comparison of our network results to experimental results in the literature.


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