Calculation of the eective properties describing active dispersion in
porous media: from simple to complex unit cells
A. Ahmadi
a, A. Aigueperse
b, M. Quintard
c,* aLEPT-ENSAM (UMR CNRS), Esplanade des Arts et Metiers, 33405 Talence Cedex, FrancebATI Services, 25 quai A. Sisley, B.P. 2, 92390 Villeneuve-La-Garenne, France cInstitut de Mecanique des Fluides, Allee 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 ``eective 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 Peclet 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; Eective properties; Network models
1. Introduction
The fate of non-aqueous phase liquids (NAPLs) in soils and aquifers has received a lot of attention in the past. Eorts have been developed to model the three-phase and two-three-phase ¯ows that lead to the development of the NAPL plume [18,25]. In this paper, NAPL dis-solution in water will be referred to asactivedispersion as opposed to passivedispersion which corresponds to the classical dispersion in porous media. The description of NAPLactivedispersion in water is very important as it determines the conditions under which the aquifer will be contaminated beyond the NAPL plume.
This active dispersion mechanism can be described in terms of local-equilibrium conditions, i.e., the averaged concentrations are distributed following the thermody-namical equilibrium conditions at the interface between the water and the NAPL phase [1±3,31]. However, ¯ow conditions in the porous medium may be such that this
condition of local-equilibrium does not hold, and the rate of mass exchange between water and the NAPL phases must be taken into account. For instance, in the case of a binary system, macroscopic description of this active dispersion mechanism requires the knowledge of an active dispersion tensor and a mass exchange coef-®cient [26±28,32,34]. These eective properties may be obtained from experiments or ®eld measurements. Several diculties must be overcome, and if one con-siders the dierent correlations available in the litera-ture (see for instance a discussion in [35]) they often span over several orders of magnitude. While we shall not discuss in this paper the comparative merits of all the proposed correlations, it looks interesting to have some quantitative predictions that would be associated to a direct representation of the NAPL residual satu-ration and the water ¯ow. This would oer, at least, a precise understanding of the impact of the dierent physical parameters such as geometry, velocity, . . .
However, the physics of dissolution in a real porous medium is a highly intricate phenomenon involving many dierent mechanisms, as discussed for instance in
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*Corresponding author.
E-mail address:[email protected] (M. Quintard).
[27]. Indeed, dissolution is aected by the dierent scales found in natural systems (pore-scale, various heterogeneities), and the dissolution process itself may be unstable leading to preferential channels that have a
tremendous impact on the dissolution kinetics [27]. In addition, macro-scale models involving pore-scale moving boundaries pose a particular problem, and the traditional linear exchange models represent an
ap-Nomenclature
Abc area of theb±cinterface contained in the
averaging volumeV; m2
Abr area of theb±rinterface contained in the
averaging volumeV; m2
av b±cinterfacial area per unit volume, mÿ1
b vector ®eld that maps rCb onto the
concentration deviationc^m for the network model, m
bb vector ®eld that maps rCb onto the
concentration deviationc~b, m
cb pore-scale contaminant concentration,
kg mol/m3
Cb hcbib average intrinsic contaminant
concentration in theb-phase, kg mol/m3
Ceq
b equilibrium concentration, kg mol/m3
cm average concentration of the diusing species over a pore-throat section, kg mol/m3
^
cm spatial deviation of the average section concentration, kg mol/m3
hcmi Darcy-scale super®cial average ofcm, kg mol/m3
hcmib Darcy-scale intrinsic average of cm, kg mol/m3
D molecular diusion coecient, m2=s
D eective local scale dispersion tensor, m2=s
DT dispersion coecient for a cylindrical pore-throat from Taylor and Aris theory, m2=s
D
xx longitudinal local scale dispersion coecient, m2=s
D
yy transverse local scale dispersion coecient, m2=s
I identity tensor
l distance between two adjacent pore-centers on the cubic lattice, m
lc unit cell dimension, m
lch characteristic length, m
lb characteristic length for the b-phase at
the pore-scale, m
li i1;2;3, lattice vectors used to describe a unit cell, m
n outward unit vector of the volumeVi
nbc unit vector normal to theb±c
interface
nbr unit vector normal to theb±rinterface
Pe Peclet number
Petube Peclet number associated to a tube corresponding to a pore-throat
r position vector, m
rb pore-body radius, m
r
b rb=l, dimensionless average pore-body radius
rij radius of a pore-throat connecting pores iandj, m
rt pore-throat radius, m
r
t tt=l, dimensionless average pore-throat radius
s a scalar that mapshcmibÿCeqonto^cm;s
sb a scalar that maps CbÿCeqb onto the
concentration deviationc~b;s
Scr residualc-phase saturation
Sh Sherwood number
ub a velocity like coecient in the volume
averaged transport equation, m/s
V 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/s
Vb hvbi ®ltration velocity, m/s
vm norm of the average velocity over the pore-throat section, m/s
vm average velocity over the pore-throat section, m/s
^
vm velocity deviation for the network model, m/s
hvmib Darcy-scale intrinsic average of the velocityvm, m/s
hvbib Darcy-scale intrinsic average of the
velocity, m/s
Greek symbols
a mass exchange coecient, sÿ1
b subscript representative of the aqueous phase
e local scale porosity
eb volume fraction of theb-phase
ec volume fraction of thec-phase
c subscript representative of the contaminant phase
r
rb rrb=l, dimensionless standard deviation of the pore-body radius
r
proximation that may be inaccurate under some cir-cumstances.
This problem of determining the eective properties from a pore-scale description of the NAPL entrapment in a porous medium was the motivation for the theoretical work published by Quintard and Whitaker [35]. These authors obtained a macroscopic equation for the concentration of NAPL constituent dissolved in the water-phase which was coherent with the already classically used model. In order to obtain such a result, several assumptions were made. Some were reminiscent of assumptions classically made in deriving macro-scale models, i.e., separation of scale, others were speci®c to problems involving pore-scale moving boundaries. In particular, it was assumed that the concentration ®eld at the pore-scale could be determined by assuming a quasi-stationary interface. This leads to a macro-scale equation involving several ``eective properties'' that could explicitly be obtained from the solution of two pore-scale local problems, later referred to as closure problems, one giving the eective active dispersion tensor, the other one the mass exchange coecient. These eective properties can be calculated for a given morphology, thus giving properties essentially de-pending on time, t, i.e., the history of the dissolution process. While the closure problem could be used, as explained in [36], to construct step by step this his-torical evolution of the interface in conjunction with the historical evolution of the macro-scale concentra-tion ®eld, this represents a very complicated task. In practice, one replaces this direct time dependence by non-linear relationships involving the NAPL satura-tion, and other parameters related to the velocity ®eld, such as the Peclet number for instance. While the proposed theory has its limitations, it can be used to look at the impact of several parameters, such as the geometry of the pore-scale phase repartition, or the velocity ®eld. For this reason, the closure problems were solved in [35] for simple 2D unit cells, such as periodic arrays of disks representing the solid and NAPL phases. Indeed, the results brought some inter-esting perspectives. The dependence of the eective properties with the Peclet number satis®ed the expected general behavior, i.e., the existence of a diusive regime and a dispersion regime. However, for these simple unit cells, it was observed that:
1. The diusive regime is relatively important, which would preclude the use of a correlation for the mass exchange coecient vanishing with the Peclet number (or the Sherwood number).
2. The active dispersion tensor may be dierent from the passive dispersion tensor calculated by replacing all pore-scale interfaces by passive interfaces (i.e., zero mass ¯ux). This would indicate that special correla-tions should be used for dispersion in the presence of trapped NAPL.
3. The mass exchange coecient may not tend towards zero for vanishing NAPL saturation, depending on the wettability conditions.
4. Dependence of the eective properties on saturation and the geometry of the pore-scale structure of the three phases (solid, water and NAPL) may be very complicated.
Practical implications are very important. However, a question remains: would these complex features simplify if one takes into account more complex pore-scale ge-ometries? There are already examples in the literature showing that some simpli®cations may arise if one re-places simple unit cells by more complex unit cells. This is the case for instance when calculating passive dis-persion tensors as illustrated by the work of Souto and Moyne [39]. For simple unit cells, the authors found dispersion tensors having very dierent features for dierent orientations of the averaged velocity ®eld, while this complex behavior simpli®ed for more complex, randomized pore-scale geometry.
The present paper addresses these questions for the case of active dispersion. First, the way eective prop-erties are calculated is brie¯y summarized to clarify the objective and the notations. Examples of calculations over simple unit cells are presented that emphasize the kind of complex behavior that may be observed. A solution of the closure problems over network models is then presented following the theoretical results pre-sented in [4], which allows to solve the closure problems over unit cells involving thousands of pores. The results are ®nally compared to simple unit cells calculations.
2. Direct calculation of eective properties
In this paper, we consider the simple case of a binary system, in the porous medium represented in Fig. 1. The
b-phase corresponds to water, while r and c refer to the solid and NAPL phases, respectively. Following the assumptions 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-conservation equation was obtained under the following form
oebCb
whereebis theb-phase volume fraction,Cbthe averaged
intrinsic contaminant concentration in the b-phase, Vb
the ®ltration velocity,Db the active dispersion tensor,a
the mass exchange coecient, andCeq
b is the equilibrium
concentration. Here, it must be noticed that a dierent nomenclature is sometimes used in the literature for the dispersion tensor, which corresponds to Db=eb. This
appropriate macro-scale boundary conditions corre-sponding to the particular system studied. It is impor-tant to note that the knowledge of these boundary conditions is not necessary for the developments in this paper which focuses on the determination of the eective properties appearing in this equation.
To be clear about the notations, we give the corre-sponding de®nitions of the macroscopic quantities in terms of volume averages, as they are introduced in the cited literature. We have, for example, the macro-scale average concentration,Cb, de®ned as a volume average
of the pore-scale concentrationcb as follows:
CbV1
Z
Vb
cbdV hcbib; 2
whereVandVb are, respectively, the averaging volume
and the volume of the b-phase inV. The ®ltration ve-locity corresponds to
VbV1
Z
VvbdV hvbi ebhvbi
b; 3
wherevb is the pore-scaleb-phase velocity.
As usual in scaling-up theories, Eq. (1) is an ap-proximate solution of the pore-scale to Darcy-scale problem. 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 are a reminder of the simpli®cations involved. In the above-mentioned papers, it is shown that the eective proper-ties are related to the pore-scale physics through two closure problems, which are listed below. The closure problems involve two closure variables bb andsb which
appear in the description of the pore-scale concentration as a function of the average concentration, i.e.,
cbCbbb rCbÿsb Cb
ÿCbeq: 4
In this development, the porous medium is represented by a periodic system. The system is, therefore, com-pletely characterized by a single unit cell as large as necessary taking into account all the complexity of the pore-scale geometry. The closure problems are therefore solved over this representative unit cell using periodic boundary conditions. It must be noted that despite periodic boundary conditions, the use of this methodology is not limited to periodic systems [5].
The ®rst closure problem givingbballows to calculate
the active dispersion tensor. Over a periodic unit cell representative of the NAPL entrapment, the following boundary value problem has to be solved.
Problem I.
~
vbvb rbb r Drbb eÿb1ub; 5
B:C:1 bb0 at Abc; 6
B:C:2 nbr rbbnbr0 at Abr; 7
bb rli bb r; 8
hbbi 0; 9
ubV1
Z
AbrAbc
n DrbbdAÿDreb: 10
In this problem, the velocity deviation is given by
~
vbvbÿ hvbib 11
and D is the molecular diusion coecient. The two vectorsnbrandnare the outward unit vectors normal to
theb±rinterface and to the totalb±randb±cinterface, respectively. The closure variable, bb, is then used to
obtain the active dispersion tensor using the following equation
DbebDIDV1
Z
AbrAbc
nbbdAÿ h~vbbbi: 12
The mass exchange coecient, a, is obtained from solving closure Problem II, which is formulated as follows.
Problem II.
vb rsb r Drsb ÿeÿb1a; 13
B:C:1 sb1 at Abc; 14
B:C:2 nbr rsb0 at Abr; 15
sb rli sb r; 16
hsbi 0; 17
aV1
Z
AbrAbc
n DrsbdA: 18
It must be noticed that the mass exchange coecient is a part of Problem II, through an integro-dierential for-mulation. Special procedures were designed to handle such problems, taking into account periodicity con-ditions. Examples of solutions are available in [35] in the case of 2D unit cells. The original numerical model (1994) has been extended to handle 3D cases, and results are presented in the next section.
3. Results for simple unit cells
The calculation of the eective properties follows the algorithm 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 eective 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 models designed for solving these closure problems. The phase distribution is represented by assigning phase indicator values on each block of a Cartesian grid, as illustrated in Fig. 2.
The velocity ®eld is obtained by solving Stokes equations using an Uzawa algorithm. Quasi-second or-der accurate schemes are used to solve for the closure problem equations at a given Peclet number. The problem 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 data were used in the calculations presented in this paper. Our objective was rather to test for the impact of the dierent choices that can be made. Therefore, dierent types of unit cells have been used, which are summarized in Figs. 3±5. In addition, we did not try to obtain the historical 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 andvby) are calculated at the interface of the grid block.
Fig. 3. Simple 2D unit cell.
calculated the eective properties for dierent, arbitrary values of the saturation and Peclet number.
A comparison between the results obtained from simple 2D and 3D unit cells is shown in Figs. 6 and 7 for the dispersion coecient, and in Fig. 8 for the mass exchange coecient. In the caption of these ®gures, the Peclet number is de®ned as
Pehvbiblc
D ; 19
wherelcis the unit cell dimension. The use of the Peclet number is made possible because the velocity ®eld cor-responds to a laminar ¯ow, i.e., it is independent of the Reynolds number. This is not a limitation of the theory, and a velocity ®eld involving inertia eects could be used instead without changing the numerical model solving the closure problem in whichvb is only an input ®eld.
All three ®gures show the expected behavior of the eective parameters with respect to the Peclet number. The diusive regime, at low Peclet number, is more important for the transverse dispersion coecient than for the longitudinal dispersion coecient. It is also less marked, i.e., it appears at larger Peclet number, for the mass exchange coecient. However, one sees that there is a dramatic impact of the geometry on the coecient
values. The in¯uence of saturation, for instance, cannot be represented by simple correlations. This is more dramatic if one considers the in¯uence of the velocity ®eld direction. This eect is illustrated in Fig. 9 for the dispersion coecient, and in Fig. 10 for the mass ex-change coecient, in the case of the simple 2D unit cell presented in Fig. 3.
In the diusive regime, our results show that the medium is macroscopically isotropic, as expected from the unit cell geometry. On the contrary, the dispersion mechanisms are very sensitive to the velocity orienta-tion, for these simple unit cells. Correlations extracted from these calculations may not be practical in the case of real, natural systems. Following the results obtained in the case of passive dispersion [39], we would expect that a more complex, disordered unit cell would produce results less sensitive to the pore-scale geometry.
In a ®rst attempt to check this problem, we have solved the closure problems on ``disordered'' unit cells, like the one illustrated in Fig. 5. Results for the longi-tudinal dispersion coecient are shown in Fig. 11, and results for the mass exchange coecient are shown in Fig. 12.
There seems to be a smaller in¯uence of the velocity orientation in the case of the longitudinal dispersion coecient, this is more clear for the mass exchange co-ecient. This shows an interesting trend if one is in-terested in capturing the eect of real porous media features. However, there are computational limitations that prevent the use of such direct simulations for very complex systems. This called for a dierent approach of the problem, and following the extensive literature concerning the use of network models in porous media physics, we designed a speci®c numerical procedure to solve the closure problems on network models as ex-plained in the next section.
Fig. 5. Simple disordered 2D unit cell.
4. Network formulation
In order to capture the eects of real porous media and to obtain more signi®cant eective properties, it is necessary to incorporate a larger number of pores and a more complex geometry in the averaging volume con-sidered. Network modeling provides the possibility of achieving these two aims. The interest of network models for active dispersion has already been demon-strated by the work of Lowry and Miller [24] or Gray et al. [17]. These considerations led us to formulate the upscaling problem on a network, and the theory is de-tailed in [4] To be clear: our contribution lies in the calculation of the eective properties through a speci®c implementation of the closure problems presented in the previous 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 be made, as we shall discuss later.
In this network model implementation, the porous structure is idealized as a network of spherical pore bodies connected to one another by cylindrical pore-throats. The pore-body-radius rband 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 local scale transport properties on a network model, we have chosen a 3D network on a regular cubic lattice as a ®rst approach. The methodology used can easily be extended to more complex networks (with a variable number of connections to each pore-body for example).
It must be emphasized that the interest of network models lies in the possibility of using a simple descrip-tion of the ¯ow (Poiseuille ¯ow, constant concentradescrip-tion
Fig. 7. Transverse dispersion coecient: comparison between 2D and 3D unit cells for dierent values of thec-phase volume fraction ec.
Fig. 9. In¯uence of the average velocity orientation on the dispersion coecient for the simple 2D unit cell of Fig. 3.
Fig. 10. In¯uence of the average velocity orientation on the mass exchange coecient for the simple 2D unit cell of Fig. 3.
in the sites, 1D ¯ows in the links, . . .). The impact of these simpli®cations may be checked by using the direct solution 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 is then displaced by water leaving behind trapped con-taminant ganglia. The network must therefore undergo similar physical phenomena. It must be noted that modeling of the drainage and imbibition allows to set up a NAPL saturation in the network model and will have no consequence on the developments presented in the following sections.
In this work the porous medium is assumed water wet and the capillary forces are assumed to dominate drainage and imbibition mechanisms. For these steps, piston-displacement and ®lm ¯ow mechanisms are taken into account. The piston-displacement in both drainage and imbibition are modeled using the Young±Laplace equation [13,24] and are governed by the pore-scale geometry. While ®lm displacement has not been con-sidered for the drainage due to the lack of a rigorous criterion, 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 and rupture. Once the two phases are distributed in the po-rous network, the single phase displacement of water in the porous network containing ganglia of dierent sizes and forms is studied. Network modeling associated with a 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 have been impossible from the practical point of view. Ob-viously, with this simpli®ed treatment, details of the ¯ow such as rotational ¯ow in dead end pore throats are not taken into account and the velocity in these throats is considered to be zero. The phase-distribution as well as the velocity ®eld are now considered known for the further study of NAPL transport.
6. Upscaling dispersion
The volume averaging methodology has been re-viewed for our special case of network geometry [4]. The local equations and properties are obtained starting from a description of the transport in each pore-throat based on the Taylor and Aris formulation of dispersion in a capillary tube [7,40]. These authors state that under some limiting conditions listed below, the transport in a capillary tube is governed by a 1D classical convection± dispersion equation with the dispersion coecient given by
DT Dr 2 tv2m
48D 20
in which rt is the radius of the tube, D the molecular diusion coecient andvmis the mean velocity over the tube section. This result can also be found using general upscaling theories [9,10,23,38]. Therefore, it is consistent with the proposed averaging approach assuming suc-cessive upscaling are performed.
The Taylor and Aris formulation is valid under the following limiting conditions [40]:
(a) The changes in concentration due to convective transport along the tube take place in a time which is so short that the eect of molecular diusion may be neglected.
(b) The time necessary for appreciable eects to ap-pear, owing to convective transport, is long com-pared with the time of decay during which radial variations of concentration are reduced to a fraction of their initial value through the action of molecular diusion.
The condition (b) can be considered valid if [40]
lt 2vm
r2 t
3:82D; 21 wherelt is the length of the tube. This condition, which must be satis®ed for each cylindrical pore-throat in-cluded in the pore network, can also be written in terms of a Peclet number related to each tube:
PetubevDmlt7:22l 2 t
r2 t
: 22
There will be an attempt to take into account this con-dition in the presentation of the results in the following sections.
Using the volume averaging procedure applied to the pore-scale equations, we obtain a local scale averaged equation similar to the one given by Eq. (1):
oebCb
ot r VbCb r D rCb ÿa Cb
ÿCbeq 23
The mass exchange coecient a and the local scale dispersion coecientD are expressed as a function of the pore-scale properties and the two closure variablesb
andsin the following manner:
a 1 V
Z
Abc
nbcDT rsdA; 24
De
bhDTibÿebh^vmbibebhDT rbib: 25 In this problem,vmis the average velocity over the pore-throat section and is written as the sum of the average velocity and a velocity deviation: the velocity deviation is given by
^
vmvmÿ hvmib: 26 In a manner similar to the development in [35], we ob-tain the following closure problems for the two closure variablesb ands:
Problem I.
vm rb^vm r DT rb ÿeÿb1ub; 27
b0 at Abc; 28
b rli b r; i1;2;3; 29
The problems are similar to the ones described in Section 2. Using a well chosen decomposition of the closure variables, the integro-dierential terms in the closure problems can be eliminated. In this develop-ment, we will assume that the concentration is constant within the intersections or nodes. We recall that the pressure was also assumed to be constant at these in-tersections. A study similar to the one performed for the pressure ®eld by Koplik [21] has not been performed yet in 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. The closure problems obtained are therefore solved analyti-cally over each tube (pore-throat) as a function of the values at the two pore-bodies occupying each end of the tube. Then a balance over each intersection of tubes will lead to a linear system. The resolution of the linear system leads to the values of the closure variables on each pore, from which local properties are calculated. The details of the calculations are beyond the scope of this paper and are published elsewhere [4].
7. Results on the network model
It is obvious that in this case a similar algorithm as presented in Section 3 is to be followed. The main dif-ference here is that now the NAPL distribution is given by modeling physical processes such as drainage and imbibition on the network and is directly related to the network geometry. An example of such a realization is shown in Fig. 13. The closure problems are then solved over the network giving the dispersion tensor and the exchange coecient. Additional coecients intervening in the ®nal macro-scale equation can also be calculated.
The values found for the dispersion tensor and the exchange coecient are studied as a function of the Peclet number given by
PekhvmDibkl 35
in which l is the distance between two adjacent pore-centers on the cubic lattice and khvmibk is the norm of the local scale average velocity. The lengthlis also used as a characteristic length for obtaining dimensionless pore-body and pore-throat radii and their standard de-viations denotedr
b,rrb,rt, andrrt, respectively. At this stage of the problem Eq. (22) must be considered in order to limit the results to their domain of validity. With our particular case of cubic lattice, since the velocity in the tubes perpendicular to the direction of the pressure gradient is rather small, we can make the fol-lowing approximation to relate the local scale average velocity to the average tube velocity, vm, of the tubes parallel to the pressure gradient:
hvmib
v3m: 36
In addition, the lengthlis taken as an approximation for
lt. As a ®rst approach condition (22) can be approxi-mated as
Pe2:4 rl t
2
: 37
As a rough estimate, we will consider the following relation:
Results satisfying this condition are plotted in solid lines while the extrapolation of the results to greater values ofPe is plotted in dotted lines.
All calculations presented in this paper have been performed over unit cells containing 1000 pores. Results presented are the average values over ®ve realizations. Although, in the cases studied, the dierence between the results obtained from dierent realizations is rather small, a larger number of realizations must be taken into account in a systematic calculation procedure. In order to study the in¯uence of thec-phase volume fraction, the results for three cases listed in Table 1 are presented. The porosity, b-phase volume fraction and thec- phase saturation are also given in this table. In Section 3, the
Table 1
Cases studied for active dispersion
Case r
b rrb rt rrt 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
dispersion tensors were studied as a function of ec for
simple unit cells by changing the size of the contaminant blob placed in the center of the cell. This means that the results concern cases with varyingb-phase andc-phase volume fractions, while the porosity of the porous me-dium stays unchanged. For the network models of the porous medium, the volume fractions of the two phases are intimately related to the geometry and vary with the porosity. 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 the longitudinal and transverse dispersion behavior in Figs. 14 and 15 in terms of the two coecients of the tensorD
xx= DeandDyy= Deas a function of the Peclet number. In this manner we overcome the problem of varying porosity for these cases.
The main features of the curves follow the expected behavior, i.e., diusive and dispersive regimes. The
general tendency for the variations as a function of ec
seems identical as the ones observed for the simple unit cells with a much less amplitude. The variation of the dimensionless mass exchange coecient al2=Dversus the Peclet number is plotted in Fig. 16. These curves show a diusive regime at low Peclet number and a re-gime more in¯uenced by advection for values of Peclet number above 0.1. Similar behaviors were observed by Quintard and Whitaker [35], and in the new results presented in the previous sections. We notice, however, that this variable tends to an asymptotic advective regime at high Peclet number.
In two recent network models [12,43], the roughness and grooves of a real porous medium are represented by corners in cubic bodies and rectangular pore-throats. In this manner, the exchange between the trapped water in the corners at the irreducible water saturation and the trapped NAPL in these chambers or tubes is taken into account. In our work, this aspect of the problem has not been considered and the water is considered stagnant in pores containing NAPL. This may be considered as one possible explanation for constant mass exchange coecient values for high Peclet numbers. Other factors can explain this behavior of the mass exchange coecient for high Peclet numbers. Consider dissolution in a tube (or heat transfer with a constant wall temperature or any other similar Initial Boundary Value Problem), there will be an entrance region with a development of a boundary layer, in which the mass exchange coecient will increase with the po-sition, and will have a strong dependence on the veloc-ity, among other factors. This dependence will also be very sensitive to the ¯ow model, i.e., developed para-bolic ®eld or development of a boundary layer. This variation of the exchange coecient with the position means that non-local behavior is involved. It is well known that beyond the entrance region there is an as-ymptotic limit with a constant mass exchange coecient
Fig. 15. Transverse dispersion coecient as a function of the Peclet number for dierent NAPL volume fractions for the networks.
Fig. 16. The mass exchange coecient as a function of the Peclet number for networks.
(or Nusselt number). Although for a real porous me-dium, the problem becomes much more delicate, we believe that this behavior is not speci®c of the network approach. Indeed, it is mainly related to the fact that our theory, in which periodic boundary conditions are considered, corresponds to a fully developed exchange zone between theb-phase and the NAPL ganglia. The model developed therefore gives the asymptotic value of the exchanged coecient. The variation of the exchange coecient as a function of ec seems to be much more
important for the network approach, but al2=D in-creases with increasing ec as observed for simple unit
cells.
Let us now study thein¯uence of the orientation of the velocity ®eld. Consider case 2 of Table 1. In this case, the average velocity is in the x-direction. Dierent cases have been considered with the average velocity rotated at 30, 45 and 60 about the z-axis. The results are presented 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¯uence on the dispersion tensor and the exchange coecient.
Finally we will study the impact of the speci®c surface on the exchange coecient. Indeed, many authors dis-cuss the mass exchange phenomena in terms of a Sher-wood number de®ned as
Shalch
Dav 39 in which lch is the characteristic length and av is the interfacial area per unit volume. From the structure of the closure Problem II, a natural de®nition would be
Shal2ch
D : 40
While there is certainly a relationship betweenlchand
av, it is not necessarily simple. From our experience, we think that the most important parameters are related to
length-scales characteristic of the distance between ganglia or ganglia clusters. Indeed, simple examples show that there is not a priori a direct relationship be-tweenaandav. For instance, the calculations performed on the simple unit cells represented in Fig. 20 by Aigueperse [6] showed no dierence for the values of a. This obvious result emphasizes that in the case of more complex clusters, some zones may be at a relatively constant concentration close to the equilibrium value, thus marginally contributing to the mass exchange while increasing the speci®c area. These unit cells may seem unrealistic, and one may think that some scaling be-tweenlchandavexists that could make the introduction of 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 Reynolds number, the Schmidt number, the Peclet number and the volume fraction of theb-phase. In all these correlations
Fig. 19. In¯uence of the average velocity orientation on the mass exchange coecient for a network model.
Fig. 17. In¯uence of the average velocity orientation on the longi-tudinal dispersion coecient.
the characteristic length considered is the longest blob dimension. Although many of these relationships are derived from the measurements of dissolution of solid organic spheres by a uniform aqueous phase ¯ow ®eld in a packed bed or for diusion-limited dissolution of ¯uid spheres suspended in a laminar ¯ow regime (see [32,34] for a discussion), as a ®rst approach they are compared to our results. In all cases studied here, the Schmidt number is constant and equal to 1000. Since the product of the Schmidt number and the Reynolds number is equal to the Peclet number, we can reduce the number of parameters to two and studyShas a function ofPeand
eb. The characteristic length used for the calculation of
Sh 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) is presented as a function of the Peclet number for dier-ent values of eb. In addition to the three cases listed in
Table 1, three other cases for which the properties are
listed in Table 2 are considered. We must note that the values of the distance between two adjacent pore-centers is of 200 lm for cases 4 and 5 and of 150lm for case 6. One can see that, for instance, for close values ofeb the
Shobtained for a large Peclet number is rather dierent. From these results we can conclude that simple corre-lations cannot characterize correctly the NAPL dis-solution process, and that, at least for the network realizations studied in this paper, the introduction of the speci®c area does not produce simpli®ed correlations. As a consequence, we believe that the use of a de®nition for the Sherwood number like in Eq. (40) is more ap-propriate.
We can also compare the values of the mass exchange coecient (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 must be performed with great care and a number of points must 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 NAPL volume or interfacial area occurs [32,37]. Others [19,28,29,33] take into account dynamic eects corre-sponding to the reduction of the NAPL saturation and the shrinking of NAPL blobs during dissolution. The comparison of this second class of experiments with our results is inappropriate, although the results are plotted in Fig. 22 for completeness. Concerning the quasi-steady experiments, the comparison must still be done with care. We believe that the macro-scale model involving the mass exchange coecient is an approximation of a problem which has non-local properties. This means that this coecient is history and position dependent. Moreover, the experimental results depend clearly on the way experiments are observed (cross-section aver-ages, ®nite-length tube averaver-ages, etc.).
The results are presented in terms of a Sherwood number de®ned by Eq. (40). The characteristic length used for the experimental results presented is the average grain size. For our network results, the distance between two pore-body centers which is taken to be equal to 100 lm, seems to be a good candidate and comparable to the concept of the grain size. Dierent authors [32,37] underline the importance of the experimental conditions on the values of the Sherwood number obtained. In particular, the procedure used for the NAPL emplace-ment seems to be of great importance and in¯uences the distribution of the trapped NAPL blobs. All
Fig. 21. The Sherwood number as a function of the Peclet number.
Table 2
Additional cases studied for active dispersion
Case r
b rrb rt rrt 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
experimental results presented are conducted over porous media (beds of sand or glass beads) with the average grain sizes ranging from 70 lm up to 0.1 cm. Our results are in the same order of magnitude as the experimental results. The behavior is, however, rather dierent. 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 size of 70 lm. This diusive regime is also observable in our results. However, one must point out here a major dif-®culty associated with the measurement of mass ex-change coecients at low Peclet number, i.e., for conditions under which local mass equilibrium prevails. In this case, the mass exchange coecient is hardly identi®able. For instance, under local mass equilibrium obtained for suciently large values of the mass exchange coecient, the averaged dissolution of a porous column will depend on the velocity whatever the exact value of the mass exchange (it may even have a constant value). One may therefore infer a column scale mass exchange coecient, which will go to zero at zero velocity, while the Darcy-scale mass exchange coecient within the porous column will keep a constant value (the diusive limit). It is beyond the scope of this paper to address the problem of the experimental determination of mass exchange coecients, and we leave this discus-sion open.
8. Conclusions
Dierent types of unit cell geometries have been used to calculate active dispersion tensors and mass exchange coecients. The ®rst series correspond to an accurate description of both the geometry and the pore-scale physics. Computational limitations make dicult to approach with such unit cells the complexity of a real
disordered system. The correlations extracted from these results would incorporate saturation, the Peclet number, in a highly non-linear complex fashion. In addition, velocity orientation eects 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 in the case of network models. The results presented in this paper con®rm that scale eects may dampen the speci®c features associated to simple unit cells, and that corre-lations involving a smaller number of well de®ned parameters may be expected.
However, such network model treatment requires that the pore-scale physics is represented by simpler solutions (1D, constant concentrations in some areas, etc.). The original full closure problems may be used to check the validity of such simple representations. Therefore, the two approaches are equally necessary.
For the unit cells studied in this paper, it does not seem that simple correlations involving the speci®c area, or even the saturation are available. More studies would be needed for other geometries or network realizations to check whether such simple correlations exist for some classes of porous media and NAPL repartitions.
It must be emphasized that the study presented here involves some length and time scale assumptions, as well as other limitations associated with the eect of dis-solution. In particular, the history eects of the disso-lution process are not incorporated in the analysis. This calls for further studies.
Acknowledgements
This work has been partially supported by CNRS/ INSU/PNRH and Institut Francßais du Petrole. The authors wish to thank Martin Blunt for his constructive remarks on the paper.
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