Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

Advances in Water Resources 24 (2001) 423±438
www.elsevier.com/locate/advwatres

Calculation of the e€ective properties describing active dispersion in
porous 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, France
b
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 representative 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; E€ective 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. E€orts have been developed to model the threephase and two-phase ¯ows that lead to the development
of the NAPL plume [18,25]. In this paper, NAPL dissolution in water will be referred to as active dispersion
as opposed to passive dispersion which corresponds to
the classical dispersion in porous media. The description
of NAPL active dispersion 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 thermodynamical 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
*

Corresponding author.
E-mail address: quintard@imft.fr (M. Quintard).

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 e€ective properties may be
obtained from experiments or ®eld measurements.
Several diculties must be overcome, and if one considers the di€erent correlations available in the literature (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 saturation and the water ¯ow. This would o€er, at least, a
precise understanding of the impact of the di€erent
physical parameters such as geometry, velocity, . . .
However, the physics of dissolution in a real porous
medium is a highly intricate phenomenon involving
many di€erent mechanisms, as discussed for instance in

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

424

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

Nomenclature
area of the b±c interface contained in the
Abc

averaging volume V; m2
Abr
area of the b±r interface contained in the
averaging volume V; m2
av
b±c interfacial area per unit volume, mÿ1
b
vector ®eld that maps rCb onto the
concentration deviation c^m for the
network model, m
vector ®eld that maps rCb onto the
bb
concentration deviation c~b , m
cb
pore-scale contaminant concentration,
kg mol/m3
b
Cb ˆ hcb i
average intrinsic contaminant
concentration in the b-phase, kg mol/m3

eq
Cb
equilibrium concentration, kg mol/m3
cm
average concentration of the di€using
species over a pore-throat section, kg
mol/m3
c^m
spatial deviation of the average section
concentration, kg mol/m3
hcm i
Darcy-scale super®cial average of cm , kg
mol/m3
b
hcm i
Darcy-scale intrinsic average of cm , kg
mol/m3
D
molecular di€usion coecient, m2 =s


e€ective local scale dispersion tensor,
D
m2 =s
DT
dispersion coecient for a cylindrical
pore-throat from Taylor and Aris
theory, m2 =s

longitudinal local scale dispersion
Dxx
coecient, m2 =s

Dyy
transverse local scale dispersion
coecient, m2 =s
I
identity tensor
l
distance between two adjacent
pore-centers on the cubic lattice, m

unit cell dimension, m
lc
characteristic length, m
lch
characteristic length for the b-phase at
lb
the pore-scale, m
li
i ˆ 1; 2; 3, lattice vectors used to describe
a unit cell, m
n
outward unit vector of the volume Vi
nbc
unit vector normal to the b±c
interface
unit vector normal to the b±r interface
nbr
Pe
Peclet number

[27]. Indeed, dissolution is a€ected by the di€erent
scales found in natural systems (pore-scale, various
heterogeneities), and the dissolution process itself may
be unstable leading to preferential channels that have a

Petube
r
rb
rb
rij
rt
rt
s
sb
Scr
Sh
ub
V
Vb
vb

~vb
Vb ˆ hvb i
vm
vm
^vm
hvm i
hvb i

b

b

Peclet number associated to a tube
corresponding to a pore-throat
position vector, m
pore-body radius, m
rb =l, dimensionless average pore-body
radius
radius of a pore-throat connecting pores
i and j, m

pore-throat radius, m
tt =l, dimensionless average pore-throat
radius
a scalar that maps hcm ib ÿ Ceq onto c^m ; s
a scalar that maps Cb ÿ Cbeq onto the
concentration deviation c~b ; s
residual c-phase saturation
Sherwood number
a velocity like coecient in the volume
averaged transport equation, m/s
volume of the unit cell used for local
averaging, m3
volume of the b-phase contained in V,
m3
b-phase pore-scale velocity, m/s
b-phase velocity deviation, m/s
®ltration velocity, m/s
norm of the average velocity over the
pore-throat section, m/s
average velocity over the pore-throat

section, m/s
velocity deviation for the network
model, m/s
Darcy-scale intrinsic average of the
velocity vm , m/s
Darcy-scale intrinsic average of the
velocity, m/s

Greek symbols
a
mass exchange coecient, sÿ1
b
subscript representative of the aqueous
phase
e
local scale porosity
volume fraction of the b-phase
eb
volume fraction of the c-phase
ec
c
subscript representative of the
contaminant phase
rrb =l, dimensionless standard deviation
rrb
of the pore-body radius
rrt =l, dimensionless standard deviation
rrt
of the pore-throat radius

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-

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

proximation that may be inaccurate under some circumstances.
This problem of determining the e€ective 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 ``e€ective properties'' that
could explicitly be obtained from the solution of two
pore-scale local problems, later referred to as closure
problems, one giving the e€ective active dispersion
tensor, the other one the mass exchange coecient.
These e€ective properties can be calculated for a given
morphology, thus giving properties essentially depending 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 historical evolution of the interface in conjunction with
the historical evolution of the macro-scale concentration ®eld, this represents a very complicated task. In
practice, one replaces this direct time dependence by
non-linear relationships involving the NAPL saturation, 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 interesting perspectives. The dependence of the e€ective
properties with the Peclet number satis®ed the expected
general behavior, i.e., the existence of a di€usive regime
and a dispersion regime. However, for these simple
unit cells, it was observed that:
1. The di€usive regime is relatively important, which
would preclude the use of a correlation for the mass
exchange coecient vanishing with the Peclet number
(or the Sherwood number).
2. The active dispersion tensor may be di€erent 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 correlations should be used for dispersion in the presence
of trapped NAPL.

425

3. The mass exchange coecient may not tend towards
zero for vanishing NAPL saturation, depending on
the wettability conditions.
4. Dependence of the e€ective 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 geometries? There are already examples in the literature
showing that some simpli®cations may arise if one replaces simple unit cells by more complex unit cells. This
is the case for instance when calculating passive dispersion tensors as illustrated by the work of Souto and
Moyne [39]. For simple unit cells, the authors found
dispersion tensors having very di€erent features for
di€erent 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 e€ective properties 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 presented 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 e€ective 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 velocity. The associated macro-scale mass-conservation
equation was obtained under the following form
oeb Cb
‡ r
…Vb Cb † ˆ r
…Db
rCb †
ot 

ÿ a Cb ÿ Cbeq ‡




…1†

where eb is the b-phase volume fraction, Cb the averaged
intrinsic contaminant concentration in the b-phase, Vb
the ®ltration velocity, Db the active dispersion tensor, a
the mass exchange coecient, and Cbeq is the equilibrium
concentration. Here, it must be noticed that a di€erent
nomenclature is sometimes used in the literature for the
dispersion tensor, which corresponds to Db =eb . This
conservation equation will be completed with the

426

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

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 limitations. The dots in the right-hand side of the equation are
a reminder of the simpli®cations involved. In the abovementioned papers, it is shown that the e€ective properties are related to the pore-scale physics through two
closure problems, which are listed below. The closure
problems involve two closure variables bb and sb which
appear in the description of the pore-scale concentration
as a function of the average concentration, i.e.,


…4†
cb ˆ Cb ‡ bb
rCb ÿ sb Cb ÿ Cbeq :
In this development, the porous medium is represented
by a periodic system. The system is, therefore, completely 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 giving bb allows 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.
Fig. 1. Sketch of NAPL repartition in groundwater: the b-phase corresponds to water, while r and c refer to the solid and NAPL phases,
respectively.

appropriate macro-scale boundary conditions corresponding to the particular system studied. It is important 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 e€ective
properties appearing in this equation.
To be clear about the notations, we give the corresponding 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 concentration cb as follows:
Z
1
b
cb dV ˆ hcb i ;
…2†
Cb ˆ
V Vb
where V and Vb are, respectively, the averaging volume
and the volume of the b-phase in V. The ®ltration velocity corresponds to
Z
1
vb dV ˆ hvb i ˆ eb hvb ib ;
…3†
Vb ˆ
V V
where vb is the pore-scale b-phase velocity.
As usual in scaling-up theories, Eq. (1) is an approximate solution of the pore-scale to Darcy-scale
problem. It is beyond the scope of this paper to recall all

Problem I.
~vb ‡ vb
rbb ˆ r
…Drbb † ˆ eÿ1
b ub ;
B:C:1

bb ˆ 0

B:C:2

nbr
rbb ‡ nbr ˆ 0

…5†
…6†

at Abc ;
at Abr ;

…7†

bb …r ‡ li † ˆ bb …r†;

…8†

hbb i ˆ 0;

…9†

ub ˆ

1
V

Z

n
‰Drbb Š dA ÿ Dreb :

…10†

Abr ‡Abc

In this problem, the velocity deviation is given by
~vb ˆ vb ÿ hvb ib

…11†

and D is the molecular di€usion coecient. The two
vectors nbr and n are the outward unit vectors normal to
the b±r interface and to the total b±r and b±c interface,
respectively. The closure variable, bb , is then used to
obtain the active dispersion tensor using the following
equation
Z
1
nbb dA ÿ h~vb bb i:
…12†
Db ˆ eb DI ‡ D
V Abr ‡Abc
The mass exchange coecient, a, is obtained from
solving closure Problem II, which is formulated as
follows.

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

Problem II.
vb
rsb ˆ r
…Drsb † ÿ
B:C:1

sb ˆ 1

B:C:2

nbr
rsb ˆ 0

eÿ1
b a;

at Abc ;
at Abr ;

…13†
…14†
…15†

sb …r ‡ li † ˆ sb …r†;

…16†

hsb i ˆ 0;

…17†

aˆ

1
V

Z

n
‰Drsb Š dA:

…18†

427

The velocity ®eld is obtained by solving Stokes
equations using an Uzawa algorithm. Quasi-second order 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 illustrated 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
di€erent choices that can be made. Therefore, di€erent
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

Abr ‡Abc

It must be noticed that the mass exchange coecient is a
part of Problem II, through an integro-di€erential formulation. Special procedures were designed to handle
such problems, taking into account periodicity conditions. 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 e€ective 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 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 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.

Fig. 3. Simple 2D unit cell.

Fig. 4. Simple 3D unit cell.

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.

428

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

Fig. 5. Simple disordered 2D unit cell.

calculated the e€ective properties for di€erent, 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 coecient, and in Fig. 8 for the mass
exchange coecient. In the caption of these ®gures, the
Peclet number is de®ned as
b

Pe ˆ

hvb i lc
;
D

…19†

where lc is the unit cell dimension. The use of the Peclet
number is made possible because the velocity ®eld corresponds 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 e€ects could be used
instead without changing the numerical model solving
the closure problem in which vb is only an input ®eld.
All three ®gures show the expected behavior of the
e€ective parameters with respect to the Peclet number.
The di€usive regime, at low Peclet number, is more
important for the transverse dispersion coecient than
for the longitudinal dispersion coecient. It is also less
marked, i.e., it appears at larger Peclet number, for the
mass exchange coecient. However, one sees that there
is a dramatic impact of the geometry on the coecient

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 e€ect is illustrated in Fig. 9 for the
dispersion coecient, and in Fig. 10 for the mass exchange coecient, in the case of the simple 2D unit cell
presented in Fig. 3.
In the di€usive 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 orientation, 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 longitudinal dispersion coecient are shown in Fig. 11, and
results for the mass exchange coecient are shown in
Fig. 12.
There seems to be a smaller in¯uence of the velocity
orientation in the case of the longitudinal dispersion
coecient, this is more clear for the mass exchange coecient. This shows an interesting trend if one is interested in capturing the e€ect 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 di€erent 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 explained in the next section.

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

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

429

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 †.

4. Network formulation
In order to capture the e€ects of real porous media
and to obtain more signi®cant e€ective properties, it is
necessary to incorporate a larger number of pores and a
more complex geometry in the averaging volume considered. Network modeling provides the possibility of
achieving these two aims. The interest of network
models for active dispersion has already been demonstrated 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 detailed in [4] To be clear: our contribution lies in the
calculation of the e€ective properties through a speci®c
implementation of the closure problems presented in the
previous section. It must be emphasized that all underlying 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 porethroats. The pore-body-radius …rb † and the throat-radius
…rt † are given by Gaussian distributions with userspeci®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 description of the ¯ow (Poiseuille ¯ow, constant concentration

430

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

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.

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

431

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

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 consequence, we believe that both approaches have their interest, 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 contaminant 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 considered for the drainage due to the lack of a rigorous
criterion, it has been taken into account for the imbibition [20,24]. This ®lm ¯ow is responsible for a displacement 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 porous network, the single phase displacement of water in
the porous network containing ganglia of di€erent 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. Obviously, 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 reviewed 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 coecient given
by
DT ˆ D ‡

rt2 v2m
48D

…20†

in which rt is the radius of the tube, D the molecular
di€usion coecient and vm is 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 successive upscaling are performed.
The Taylor and Aris formulation is valid under the
following limiting conditions [40]:

432

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

(a) The changes in concentration due to convective
transport along the tube take place in a time which
is so short that the e€ect of molecular di€usion may
be neglected.
(b) The time necessary for appreciable e€ects to appear, owing to convective transport, is long compared 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
di€usion.
The condition (b) can be considered valid if [40]
lt
rt2
;

2vm
3:82 D

Petube

ÿ a Cb ÿ Cbeq ‡




l2
7:22 t2
rt

:

…22†

There will be an attempt to take into account this condition 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):

…23†

The mass exchange coecient a and the local scale
dispersion coecient D are expressed as a function of
the pore-scale properties and the two closure variables b
and s in the following manner:
Z
1
nbc
DT
rs dA;
…24†
aˆ
V Abc
b

…21†

where lt is the length of the tube. This condition, which
must be satis®ed for each cylindrical pore-throat included in the pore network, can also be written in terms
of a Peclet number related to each tube:
v m lt

ˆ
D

oeb Cb
‡ r
…Vb Cb † ˆ r
…D
rCb †
ot 


b

b

D ˆ eb hDT i ÿ eb h^vm bi ‡ eb hDT
rbi :

…25†

In this problem, vm is the average velocity over the porethroat section and is written as the sum of the average
velocity and a velocity deviation: the velocity deviation
is given by
^vm ˆ vm ÿ hvm ib :

…26†

In a manner similar to the development in [35], we obtain the following closure problems for the two closure
variables b and s:
Problem I.
vm
rb ‡ ^vm ˆ r
‰DT
rbŠ ÿ eÿ1
b ub ;

…27†

bˆ0

…28†

at Abc ;

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

433

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

b…r ‡ li † ˆ b…r†;

i ˆ 1; 2; 3;

…29†

b

hbi ˆ 0:

The values found for the dispersion tensor and the
exchange coecient are studied as a function of the
Peclet number given by

…30†

b

Problem II.
vm
rs ˆ r
‰DT
rsŠ ÿ eÿ1
b a;

…31†

s ˆ 1 at Abc ;

…32†

s…r ‡ li † ˆ s…r†;

i ˆ 1; 2; 3;

Pe ˆ

b

…35†

in which l is the distance between two adjacent poreb
centers on the cubic lattice and khvm i k is the norm of
the local scale average velocity. The length l is also used
as a characteristic length for obtaining dimensionless
pore-body and pore-throat radii and their standard deviations denoted rb , rrb , rt , and rrt , 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 following approximation to relate the local scale average
velocity to the average tube velocity, vm , of the tubes
parallel to the pressure gradient:
v


m
b
…36†
hvm i
ˆ :
3

…33†

hsi ˆ 0:

khvm i kl
D

…34†

The problems are similar to the ones described in
Section 2. Using a well chosen decomposition of the
closure variables, the integro-di€erential terms in the
closure problems can be eliminated. In this development, 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 intersections. 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 treatment of networks. As a consequence, the closure variables are considered constant on each nodes. The
closure problems obtained are therefore solved analytically 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].

In addition, the length l is taken as an approximation for
lt . As a ®rst approach condition (22) can be approximated as
 2
l
:
…37†
Pe  2:4
rt

As a rough estimate, we will consider the following
relation:
 2
2:4 l
:
…38†
Pe <
5 rt
Results satisfying this condition are plotted in solid
lines while the extrapolation of the results to greater
values of Pe 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 di€erence between
the results obtained from di€erent 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 the c-phase volume fraction, the
results for three cases listed in Table 1 are presented.
The porosity, b-phase volume fraction and the c- phase
saturation are also given in this table. In Section 3, the

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 difference 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 coecient. Additional coecients intervening
in the ®nal macro-scale equation can also be calculated.
Table 1
Cases studied for active dispersion
Case

rb

rrb

rt

rrt

e

eb

ec

Scr

1
2
3

0.30
0.20
0.10

0.12
0.08
0.05

0.15
0.10
0.05

0.06
0.04
0.02

0.214
0.095
0.023

0.144
0.066
0.016

0.0704
0.0283
0.0061

0.329
0.299
0.271

434

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

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 varying b-phase and c-phase
volume fractions, while the porosity of the porous medium 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 between the results presented in Section 3 and those obtained for cases listed in Table 1, we present the
longitudinal and transverse dispersion behavior in
Figs. 14 and 15 in terms of the two coecients of the
tensor Dxx =…De† and Dyy =…De† as 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., di€usive and dispersive regimes. The

Fig. 14. Longitudinal dispersion coecient as a function of the Peclet
number for di€erent NAPL volume fractions for the networks.

Fig. 15. Transverse dispersion coecient as a function of the Peclet
number for di€erent NAPL volume fractions for the networks.

Fig. 16. The mass exchange coecient as a function of the Peclet
number for networks.

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 coecient …al2 =D† versus
the Peclet number is plotted in Fig. 16. These curves
show a di€usive regime at low Peclet number and a regime 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 pore-bodies and rectangular porethroats. 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 coecient values for high Peclet
numbers. Other factors can explain this behavior of the
mass exchange coecient 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 coecient will increase with the position, and will have a strong dependence on the velocity, among other factors. This dependence will also be
very sensitive to the ¯ow model, i.e., developed parabolic ®eld or development of a boundary layer. This
variation of the exchange coecient with the position
means that non-local behavior is involved. It is well
known that beyond the entrance region there is an asymptotic limit with a constant mass exchange coecient

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

(or Nusselt number). Although for a real porous medium, 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 the b-phase and the NAPL ganglia. The
model developed therefore gives the asymptotic value of
the exchanged coecient. The variation of the exchange
coecient as a function of ec seems to be much more
important for the network approach, but al2 =D increases with increasing ec as observed for simple unit
cells.
Let us now study the in¯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. Di€erent 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 complex (and therefore more realistic) unit cells are considered, the orientation of the velocity has little in¯uence
on the dispersion tensor and the exchange coecient.
Finally we will study the impact of the speci®c surface
on the exchange coecient. Indeed, many authors discuss the mass exchange phenomena in terms of a Sherwood number de®ned as
Sh ˆ

alch
Dav

435

Fig. 18. In¯uence of the average velocity orientation on the transverse
dispersion coecient.

…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
Sh ˆ

al2ch
:
D

…40†

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

Fig. 17. In¯uence of the average velocity orientation on the longitudinal dispersion coecient.

Fig. 19. In¯uence of the average velocity orientation on the mass
exchange coecient for a network model.

length-scales characteristic of the distance between
ganglia or ganglia clusters. Indeed, simple examples
show that there is not a priori a direct relationship between a and av . For instance, the calculations performed
on the simple unit cells represented in Fig. 20 by
Aigueperse [6] showed no di€erence 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 between lch and av exists 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 literature [8,14±16,19,22,28,30,32,41,42] in which the Sherwood number is expressed in terms of the Reynolds
number, the Schmidt number, the Peclet number and the
volume fraction of the b-phase. In all these correlations

436

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

Fig. 20. Simple unit cells with same mass exchange coecient and
di€erent speci®c area.

Fig. 21. The Sherwood number as a function of the Peclet number.

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 di€usion-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 study Sh as a function of Pe and
eb . The characteristic length used for the calculation of
Sh is the average of the dimensions of the largest contaminant blob. In Fig. 21, the Sherwood number calculated using the expression given in Eq. (40) is
presented as a function of the Peclet number for di€erent 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 150 lm for case 6.
One can see that, for instance, for close values of eb the
Sh obtained for a large Peclet number is rather di€erent.
From these results we can conclude that simple correlations cannot characterize correctly the NAPL dissolution 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 appropriate.
We can also compare the values of the mass exchange
coecient (Fig. 22) found by our work to the ones obtained 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 performed under quasi-steady conditions, i.e., measurements 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 e€ects corresponding 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 coecient is an approximation of a
problem which has non-local properties. This means
that this coecient is history and position dependent.
Moreover, the experimental results depend clearly on
the way experiments are observed (cross-section averages, ®nite-length tube averages, 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. Di€erent 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 emplacement seems to be of great importance and in¯uences
the distribution of the trapped NAPL blobs. All

Table 2
Additional cases studied for active dispersion
Case

rb

rrb

rt

rrt

e

eb

ec

Scr

4
5
6

0.10
0.15
0.20

0.04
0.075
0.08

0.05
0.075
0.10

0.02
0.03
0.04

0.022
0.051
0.095

0.017
0.035
0.069

0.004
0.016
0.025

0.19
0.31
0.33

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

437

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

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
di€erent. The only experimental results showing a diffusive regime for low Reynold numbers is that of Radilla [37] obtained for sand packing of average grain size
of 70 lm. This di€usive regime is also observable in our
results. However, one must point out here a major dif®culty associated with the measurement of mass exchange coecients at low Peclet number, i.e., for
conditions under which local mass equilibrium prevails.
In this case, the mass exchange coecient is hardly
identi®able. For instance, under local mass equilibrium
obtained for suciently large values of the mass
exchange coecient, 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 coecient, which will go to zero at zero
velocity, while the Darcy-scale mass exchange coecient
within the porous column will keep a constant value (the
di€usive limit). It is beyond the scope of this paper to
address the problem of the experimental determination
of mass exchange coecients, and we leave this discussion open.

8. Conclusions
Di€erent types of unit cell geometries have been used
to calculate active dispersion tensors and mass exchange
coecients. The ®rst series correspond to an accurate
description of both the geometry and the pore-scale
physics. Computational limitations make dicult 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 e€ects may be important.
This called for a special treatment of unit cells involving thousands of pores. A speci®c, original treatment of the active dispersion case has been proposed in
the case of network models. The results presented in this
paper con®rm that scale e€ects may dampen the speci®c
features associated to simple unit cells, and that correlations 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 e€ect of dissolution. In particular, the history e€ects of the dissolution 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.

438

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

References
[1] Abriola LM. Modeling multiphase migration of organic chemicals
in groundwater systems ± a review and assessment. Environ
Health Perspectives 1989;83:17±143.
[2] Abriola L, Pinder GF. A multiphase approach to the modeling of
porous media contaminated by organic compounds ± 1: equation
development. Water Resour Res 1985a;21:19±26.
[3] Abriola L, Pinder GF. A multiphase approach to the modeling of
porous media contaminated by organic compounds ± 2: numerical
simulation. Water Resour Res 1985b;21(1):19±26.
[4] Ahmadi A, Aigueperse A, Quintard M. Upscaling of nonwetting
phase residual transport in porous med