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EFFECTIVE MACRODIFFUSION IN SOLUTE TRANSPORT

THROUGH HETEROGENEOUS POROUS MEDIA

BRAHIM AMAZIANE∗, ALAIN BOURGEAT† , AND MLADEN JURAK‡

Abstract. Homogenization method is used to analyse the equivalent behavior of transientflow of a passive solute through highly heterogeneous porous media. The flow is governed by acoupled system which includes an elliptic equation and a linear convection-diffusion concentrationequation with a diffusion term small with respect to the convection, i.e. a high Peclet number isconsidered. Asymptotic expansions lead to the definition of a macroscale transport model. Numericalcomputations to obtain the effective hydraulic conductivity and the macrodiffusivity tensor have beencarried out via finite volume methods. Numerical experiments related to the simulations of solutetransport have been performed comparing the heterogeneous medium to the corresponding effectivemedium. The results of the simulations are compared in terms of spatial moments, L2 errors andconcentration contours. Results obtained from the simulations using the model obtained by thehomogenization method show good agreement with the heterogeneous simulations.

Key words. dispersion, heterogeneous porous media, homogenization, solute transport

AMS subject classifications. 74Q15, 74S10, 76M50, 76R99, 76S05

1. Introduction. The focus of the present paper is on the prediction of the flowand transport through heterogeneous porous media where mechanical dispersion couldappear. The understanding and prediction of fluid flow through porous media is ofgreat importance in various areas of research and industry. Petroleum engineers needto model multiphase and multicomponent flow for production of hydrocarbons frompetroleum reservoirs. Hydrologists and soil scientists are concerned with undergroundwater flow in connection with applications to civil and agricultural engineering, and,of course, the design and evaluation of remediation technologies in water qualitycontrol rely on the properties of underground fluid flow. More recently, modeling flowand transport of contaminant received an increasing attention in connection with thebehavior of an underground repository of radioactive waste (see, e. g., [12]).

The physical situation we address here corresponds to the transport of solute ina highly heterogeneous aquifer by groundwater flow due to the hydraulic head andby diffusion coming from the dilution of the solute in the water. The governing equa-tions arise from the laws of conservation of mass of the fluid, along with a constitutiverelation relating the fluid velocity which appears in the conservation law to the hy-draulic head. Traditionally, the standard Darcy equation provides this relation. Thisprocess can be formulated as a coupled system of partial differential equations whichincludes an elliptic hydraulic head-velocity equation and a linear convection-diffusionconcentration equation, subject to appropriate boundary and initial conditions. In aprevious paper, [11] we gave a mathematical model for explaining how the upscalingof a diffusive but dominant convective transport model at a mesoscale, in an aquiferwith highly contrasted geology, may produce a global model including, at a macroscaleconvection enhanced effective diffusivity tensors. For this it was necessary to makeprecise assumption on the Peclet number at the aquifer scale, or equivalently, on the

∗Laboratoire de Mathematiques Appliquees, CNRS-FRE2570, Universite de Pau, av. del’Universite, 64000 Pau, France, ([email protected]).

†Universite Lyon I - ISTIL, Equipe MCS, 15 bld. Latarjet, 69622 Villeurbanne Cedex, France,([email protected]).

‡Department of Mathematics, University of Zagreb, Bijenicka c. 30, 10000, Zagreb, Croatia,([email protected]).

1

http://arxiv.org/abs/math-ph/0503004v1

2 B. AMAZIANE, A. BOURGEAT AND M. JURAK

averaged Darcy’s velocity, assuming it was of order 1/ε, where ε is a small parameter.

It is known that periodicity could be seen as an example of stationary randomfield, see [24], [33], and we assume herein periodicity of the heterogeneities in or-der to simplify both the mathematical derivation of the macroscopic model and thecomputation of the macroscopic hydrogeological parameters. In fact, with generalstochastic assumption, the computation of local problems are done on a sufficientlybig representative elementary volume R.E.V. instead of a single periodic cell, andinstead of usual asymptotic expansion we should use either two-scale convergence forpartial differential equatons in probability space like in [13] or convergence of processsolution of a stochastic differential equation like in [16]. However as seen in [22] theresults with random flows could be different from periodic flows.

The goal of the present paper is to present numerical simulations based on periodichomogenization to prove numerically the accuracy of our mathematical modeling andto compare to classical models used in geohydrology literature like for instance [5],[9], [19], [34], [18], [36].

The outline of the remainder of the paper is as follows. In Section 2 we give ashort description of the mathematical and physical model used in this study and themain assumptions. In Section 3 and Section 4, the homogenization method is brieflypresented by recalling the mesoscale and the macroscopic equations and the localproblems. Starting from the geometry of the heterogeneities and from the spatial lo-calization of these heterogeneities we show how to compute the corresponding convec-tion enhanced effective diffusivity tensor and the effective hydraulic conductivity fordifferent heterogeneity shapes and different hydrogeological properties. Finally fromthe results of these computations, we derive the corresponding effective macroscopicmodel including the effects of mechanical dispersion in its effective macrodiffusivitytensor. The determination of the effective hydraulic conductivity requires numericalapproximation of elliptic equations solutions in a periodic cell. To compute the ef-fective macrodiffusivity tensor we have to solve local convection-diffusion problems ina periodic cell. We use finite volume methods to solve these local problems to com-pute the effective hydraulic conductivity and the effective macrodiffusivity tensor. InSection 4 we discuss the importance of different symmetries for the heterogeneitiesin the aquifer and show how they modify the calculation of the effective hydrogeo-logical properties. Section 5 is devoted to the presentation of the numerical results.We illustrate the performance of our upscaling by presenting two cases derived fromtypical 2D examples of pollutant spreading in an aquifer, like in [5]. We simulate thetransport of a solute in a porous domain using both the mesoscopic model and themacroscopic dispersive model, and we compare the heterogeneous and homogeneousscaled up results by computing the corresponding spatial moments, L2 errors andconcentration contours.

2. Formulation of the problem. The miscible displacement of an incompress-ible fluid with a dissolved solute in a heterogeneous confined aquifer Ω ⊂ Rd d = 1, 2or 3 over a time period (0, T ), is given by (see, e.g., [9]):

φ(x)∂c

∂t+ ~q(x) · ∇c = div(D(x)∇c) in Ω× (0, T ), (2.1)

where ~q(x) is the Darcy velocity, given by the hydraulic gradient ∇H :

~q = −K∇H, div~q = 0 in Ω, (2.2)

EFFECTIVE DIFFUSION IN SOLUTE TRANSPORT 3

subject to appropriate boundary and initial conditions. Here c(x, t) is the transportedsolute concentration, φ and K are the porosity and the hydraulic conductivity tensorof the heterogeneous medium, and D is the diffusivity tensor at the Darcy scale.

We assume, as explained in the introduction, that the heterogeneous porousmedium in the aquifer has a network of uniformly spaced heterogeneities with a blocksize l which is small compared to the size L of the reservoir. Namely, we assume theseparation of scales which can be expressed as follows: 0 < l ≪ L, and if ε denotesthe ratio of the period size to the size of the whole region, ε is then a small parameter0 < ε ≪ 1. First we express this local description (2.1)-(2.2) in a dimensionless formand we analyse the orders of magnitude of the dimensionless parameters appearingin this new formulation. The dimensionless equations are obtained by introducingdimensionless variables for all physical variables. For this, we denote by q0, D0 andφ0 the characteristic Darcy’s velocity, diffusivity and rock porosity. We assume more-over that the porosity, the hydraulic conductivity tensor, and the diffusivity tensorin the heterogeneous medium are rapidly oscillating functions depending on y = x/εthe mesoscale.

Introducing in (2.1)–(2.2) the dimensionless space variable x 7→ x/L and thedimensionless characteristic convection time t 7→ t/τc, where τc = φ0L/q0, we obtainthe dimensionless governing system of equations:

φ(x

ε)∂c

∂t+ ~q ε(x) · ∇c =

ε

Pediv(D(

x

ε)∇c) in Ω× (0, T ), (2.3)

~qε = −K(x

ε)∇H, div~q ε = 0 in Ω, (2.4)

where

ε =l

L, Pe =

q0l

D0.

In the above equations (2.3)-(2.4), although all the quantities are now dimensionless,we have kept the same previous notations.

The constant Pe is the local Peclet number and it represents the ratio of con-vective and diffusive terms at the scale of heterogeneities (mesoscale). As usual forhomogenization of convection-diffusion problems (see for instance [37], [31], [24]), ourmain assumptions are the separation of scales, 0 < ε ≪ 1 and Pe = O(1). The di-mensionless parameters φ, D and K are assumed to be Y -periodic functions, wherethe periodicity cell is Y = (0, l1) × · · · × (0, ld). The domain Ω, and the time T , arerescaled dimensionless domain and time.

Using multiscales asymptotic expansions of (2.3)-(2.4), we could derive the macro-scopic behavior of the system (see, for instance, [7, 32, 11]). Furthermore, in [11]boundary layers correctors were constructed for Dirichlet boundary conditions in aninfinite strip, and the convergence of the expansion, at any order, was rigorouslyjustified.

Throughout the paper we will use the convention of summation over repeatedindices.

3. Asymptotic analysis of the model. In this Section we recall the notationsand the main results of [11], and for simplicity, we will proceed with the expansion,up to the second order, in the interior of the domain, but neglecting the boundarylayers influence.


3.1. Asymptotic expansion in the Darcy law. The method and convergenceresults for the asymptotic expansion at any order of the Darcy velocity, including theboundary layer correctors, were presented in details in Section 3 of [11]. Here wepresent only the first two terms in the asymptotic expansion of the Darcy velocity,without any boundary layer correctors. All these results are standard and we willonly write the results and notations necessary for understanding the next section. Westart by defining the expansion:

Hε(x) ≈ H0(x) + εH1(x,

x

ε) + ε2H2(x,

x

ε) + · · · (3.1)

H1(x, y) = χ1i (y)∂iH

0(x) +H

1(x), (3.2)

H2(x, y) = χ2i,j(y)∂

2i,jH

0(x) + χ1

i (y)∂iH1(x) +H

2(x).

Then the Darcy velocity ~q ε in (2.4), is approximated by:

~q ε(x) ≈ ~Q0(x, x/ε) + ε ~Q1(x, x/ε) + · · ·

with

~Q0(x, y) = −K(y)(∇yχ1i (y) + ~ei)∂iH

0(x),

~Q1(x, y) = −K(y)[

(∇yχ2i,j(y) + χ1

i (y)~ej)∂2i,jH

0(x) + (∇yχ

1i (y) + ~ei)∂iH

1(x)

]

,

where (~ei)1≤i≤d is the standard basis of Rd. The functions χ1i and χ2

i,j are Y -periodicand defined by the two sequences of local problems defined in Y :

divy(

K(y)(∇yχ1i + ~ei)

)

= 0 in Y (3.3)

and

−divy

(

K(∇yχ2i,j + ~ejχ

1i ))

=(

K(∇yχ1i + ~ei)− 〈K(∇yχ

1i + ~ei)〉

)

· ~ej in Y,(3.4)

for i, j = 1, . . . , d.We use in (3.4), and through out the paper, the standard notation 〈·〉 for the

mean value over the periodic cell Y . From the vectors

~w i(y) = −K(y)(∇χ1i (y) + ~ei), i = 1, . . . , d

we build the matrix

Q(y) =[

~w 1(y), · · · , ~w d(y)]

, (3.5)

and we define effective hydraulic conductivity

Kh = −〈Q〉 = −1

|Y |

∫

Y

Q(y) dy. (3.6)

Then, the first term in the expansion of ~q ε is now denoted:

~Q0(x, y) = Q(y)∇H0(x), with div

(

Kh∇H0)

= 0 in Ω. (3.7)


Similarly, to compute the second term in the expansion, we should solve theproblem

−div(

Kh∇H1)

= div

2∑

i,j=1

N2i,j∂i∂jH

0

in Ω; (3.8)

where

N2i,j = 〈K(∇yχ

2i,j + χ1

i~ej)〉. (3.9)

But using variational formulation of the problem (3.4) we remark that:

~N2i,j · ~el = 〈K(∇χ1

l + ~el)χ1i −K(∇χ1

i + ~ei)χ1l 〉 · ~ej ; (3.10)

namely, the vectors ~N2i,j are skew-symmetric: ~N2

i,j · ~el = 0 for i = l, and ~N2i,j · ~el =

−~N2l,j · ~ei for i 6= l. With this last property, equation (3.8) reduces then to

div(

Kh∇H1)

= 0 in Ω. (3.11)

Remark 3.1. In (3.1) we impose the first term H0to satisfy the same bound-

ary conditions as the heterogeneous head Hε. In order to correct oscillations on theboundary produced by the following periodic terms in the expansion (3.1), boundarylayer correctors should be added. For this, in [11] Section 3, we split each χ1 and χ2

in Y -periodic term χk,# and two boundary layer corrector terms χk,±. But, in turn,these boundary layer correctors produce a small error, of order ε, in the whole domain

which is then corrected at the next order by the term H1. Since we consider herein a

simplified expansion without considering the boundary layers, we will have H1= 0 in

(3.2).Finally, discarding boundary layers and oscillating higher order terms, we approx-

imate ~q ε by:

~q ε(x) ≈ ~q 0(x) + ε~q 1(x), (3.12)

with

~q 0 = 〈 ~Q 0〉 = −Kh∇H0, ~q 1 = 〈 ~Q 1〉 = −~N2

i,j∂2i,jH

0. (3.13)

3.2. Asymptotic expansion for concentration. Like for the hydraulic head,we seek an asymptotic expansion for the concentration equation in the interior of thedomain, outside the boundary layers, in the form:

cε(x, t) ≈ c0(x, x/ε, t) + εc1(x, x/ε, t) + · · · . (3.14)

First the term, c0 = c0(x, t), is the solution of the first order hyperbolic equation

〈φ〉∂c0

∂t+ 〈 ~Q0〉 · ∇xc

0 = 0 in Ω× (0, T ), (3.15)

and the second one, c1 has the form

c1 = ψk(x, y)∂c0

∂xk(x, t) + c1(x, t). (3.16)


In (3.16) ψk(x, ·) is a Y -periodic solution of the local problem:

−1

Pedivy (D(y)(∇yψk + ~ek)) + ~Q0 · ∇yψk =

[

φ(y)

〈φ〉〈 ~Q0〉 − ~Q0

]

· ~ek in Y, (3.17)

and c1 is a solution of:

〈φ〉∂c1

∂t+ 〈 ~Q0〉 · ∇c1 + 〈 ~Q1〉 · ∇c0 =

1

Pediv(Dh

∗(x)∇c0) in Ω× (0, T ), (3.18)

with

Dh∗(x)~ek = 〈D(∇yψk + ~ek)〉+ Pe〈(

φ(y)

〈φ〉〈 ~Q0〉 − ~Q0)ψk〉. (3.19)

Definition of the macroscale concentration c1,ε(x, t):Following Section 6 in [11] we define a macroscale concentration, c1,ε(x, t), from

the non-oscillatory terms in the expansion (3.14); precisely, we denote

c1,ε(x, t) = c0(x, t) + εc1(x, t).

Summing up, equations (3.15) and (3.18), leads to the equation satisfied by c1,ε(x, t):

〈φ〉∂c1,ε

∂t+ ~q 0 · ∇xc

1,ε + ε~q 1 · ∇xc0 =

ε

Pedivx(D

h∗(x)∇xc

0) in Ω× (0, T ),

and we conclude (see for details Section 6 in [11]) that the macroscale concentrationobey, with precision O(ε2), the effective macroscale equation:

〈φ〉∂c1,ε

∂t+ (~q 0 + ε~q 1) · ∇xc

1,ε =ε

Pedivx(D

h∗(x)∇xc

1,ε) in Ω× (0, T ). (3.20)

It is interesting to notice that equation (3.20) could be seen as a viscous approximationof the first order hyperbolic equation (3.15).Definition of the effective macrodiffusivity tensor Dh:

Macroscopic effective hydraulic conductivity Kh is obtained by standard homog-enization procedure as recalled in (3.3), (3.5) and (3.6). The determination of Kh

requires the numerical approximation of the solution of d elliptic equations (3.3) inthe periodic cell Y .

According to (3.5)-(3.7) and (3.17), in order to compute the effective macrod-

iffusivity Dh we have first to find the Y -periodic solutions y 7→ ψk(y;~λ) of the dconvection-diffusion equations:

−divy (D(y)(∇yψk + ~ek)) +Q(y)~λ · ∇yψk =

[

φ(y)

〈φ〉〈Q〉~λ−Q(y)~λ

]

· ~ek in Y, (3.21)

k = 1, . . . , d, for every ~λ = Pe∇H0(x), and secondly to compute, according to (3.19),

the tensor

Dh(~λ)~ek = 〈D(∇yψk + ~ek)〉+ 〈(φ(y)

〈φ〉〈Q〉~λ−Q~λ)ψk〉. (3.22)

Finally, taking into account (3.24), we obtain the effective macrodiffusivity

Dh(x) = Dh(Pe∇H0(x)) +M(Pe∇H

0(x)). (3.23)


The first order correction ~q 1 = 〈 ~Q1〉 to the effective Darcy’s velocity ~q 0 = 〈 ~Q0〉

is given by (3.13), and using the skew-symmetry of the vectors ~N2i,j (see (3.10)), we

denote in (3.20)

−~q 1 · ∇c1,ε = ~N2i,j∂

2i,jH

0· ∇c1,ε = div(M(∇H

0)∇c1,ε), (3.24)

with the skew-symmetric matrix M(∇H0) given by

M(∇H0)i,l = (~N2

i,j · ~el)∂jH0, (3.25)

and we can write dimensionless effective macroscale equations in the form:

〈φ〉∂c1,ε

∂t+ ~q 0 · ∇xc

1,ε =ε

Pedivx(D

h(x)∇xc1,ε) in Ω× (0, T ), (3.26)

~q 0 = −Kh∇H0, div~q 0 = 0 in Ω. (3.27)

Remark 3.2. Note that ψk(y;~λ) depends on the global variable x through the

vector ~λ = Pe∇H0(x) and moreover the dependence on ~λ is nonlinear.

Although Dh(x) is not symmetric in general, it is not difficult to show that it ispositive definite, and therefore problem (3.26), (3.27), with suitable boundary condi-tions, is well posed. Namely, from the variational formulation of equation (3.21), for

any ~ξ ∈ Rd, it follows

Dh(~λ)~ξ · ~ξ = 〈D(∇yψ~ξ+ ~ξ) · (∇yψ~ξ

+ ~ξ)〉, (3.28)

where ψ~ξ= ξkψk. Since ψ~ξ

is a periodic function, the right hand side is strictly

positive for all ~ξ 6= 0.It is easy to see, as for instance when D is constant and diagonal, after developing

formula (3.28) that our effective macrodiffusivity Dh corresponds to a convectionenhanced diffusion like in [24] or [21].

4. Material symmetries. In general, the effective macrodiffusivity tensor (3.23)

is not symmetric; it also contains the skew symmetric term M(Pe∇H0(x)) coming

from the convective part. Inspired by [25] and [21], we will show that with suitablesymmetry of the aquifer heterogeneities the additional skew symmetric term in (3.23)disappears and the effective macrodiffusivity tensor becomes symmetric.

Let us consider a group of orthogonal matrices that preserve our periodicity lat-tice. For simplicity we consider only the reflections with respect to the coordinateaxes and we denote Ci = diag(1, . . . ,−1, . . . , 1), the diagonal matrix that has -1 ini-th row, and all other diagonal entries equal to 1, with C

Ti denoting the transpose of

the matrix Ci.Firstly, we formulate some well known results [24] concerning the periodic solu-

tions χ1i ∈ H1

#(Y ) of the problem (3.3), where H1#(Y ) is the space of the functions of

H1(Y ) that are Y -periodic. Below we will use the uniqueness of the solutions of thecell problems, and for this we have normalized them by requiring the mean value tobe equal to zero.

Proposition 4.1. Assume that for some i ∈ 1, . . . , d

∀z ∈ Rd, CTi K(Ciz)Ci = K(z). (4.1)


Then for all z ∈ Rd it holds

χ1i (Ciz) = −χ1

i (z), χ1j(Ciz) = χ1

j(z), for j 6= i

Q(Ciz) = CiQ(z)CTi , Kh = CiK

hCTi ,

where the matrix Q is given by (3.5), and Kh by (3.6).Proof. By simple change of variables y = Ciz change of function v(z) = χ1

j(Ciz)and orthogonality of the matrix Ci we find

divy(K(∇yχ1j + ~ej))(Ciz) = divz

(

CTi K(Ciz)Ci(∇zv(z) + C

Ti ~ej)

)

.

Since 〈v〉 = 〈χ1j 〉 = 0, from uniqueness of the solution of (3.3) we have v(z) = χ1

j(z)

for j 6= i and v(z) = −χ1i (z) for j = i. Other formulas follow from (3.5) and (3.6).

Proposition 4.2. Let K be diagonal and let (4.1) be satisfied for all i ∈1, . . . , d. Then

~N2i,j = 0 for all i, j ∈ 1, . . . , d.

Proof. If we assume that the tensor K(y) is diagonal and that (4.1) is fulfilled forindices i and l, then from Proposition 4.1 we have

〈K(∇χ1l + ~el)χ

1i 〉 = 〈K(∇χ1

i + ~ei)χ1l 〉 = 0,

and therefore from (3.10) ~N2i,j · ~el = 0, for all j.

Moreover it is clear that under the assumptions of Proposition 4.2, from (3.25)

we have M(Pe∇H0(x)) = 0.

We now turn to the symmetry properties of the solution ψk ∈ H1#(Y ) of the

problem (3.21), and the effective macrodiffusivity tensor Dh(~λ), given by (3.22).

Proposition 4.3. For all ~λ ∈ Rd it holds Dh(−~λ) = Dh(~λ)T .Proof. Following [25], we denote by ψi ∈ H1

#(Y ) the solution of equation (3.21),

corresponding to a vector ~λ, and ψi ∈ H1#(Y ) the solution corresponding to −~λ. Then

using the variational formulation of (3.21), and taking two different indices i, j we get

〈D(∇ψi + ~ei) · ∇ψj〉+ 〈Q~λ · ∇ψi ψj〉 =〈φψj〉

〈φ〉〈Q〉~λ · ~ei − 〈Q~λψj〉 · ~ei

and

〈D(∇ψj + ~ej) · ∇ψi〉 − 〈Q~λ · ∇ψj ψi〉 = −〈φψi〉

〈φ〉〈Q〉~λ · ~ej + 〈Q~λψi〉 · ~ej.

From div(Q~λ) = 0 it follows 〈Q~λ ·∇ψi ψj〉 = −〈Q~λ ·∇ψj ψi〉, and using the symmetryof the matrix D, the two previous equations lead to

〈D(∇ψi + ~ei)〉 · ~ej +〈φψj〉

〈φ〉〈Q〉~λ · ~ei − 〈Q~λψj〉 · ~ei

= 〈D(∇ψj + ~ej)〉 · ~ei −〈φψi〉

〈φ〉〈Q〉~λ · ~ej + 〈Q~λψi〉 · ~ej ,

that is exactly Dh(~λ)j,i = Dh(−~λ)i,j .


Proposition 4.4. Assume that for some i ∈ 1, . . . , d

∀z ∈ Rd, CTi K(Ciz)Ci = K(z), CT

i D(Ciz)Ci = D(z), φ(Ciz) = φ(z). (4.2)

Then for all z ∈ Rd it holds

ψi(Ciz;~λ) = −ψi(z;CTi~λ), ψj(Ciz;~λ) = ψj(z;C

Ti~λ), for j 6= i, (4.3)

Dh(~λ) = CiDh(CT

i~λ)CT

i . (4.4)

Moreover if (4.2) holds for all i ∈ 1, . . . , d, then

Dh(−~λ) = Dh(~λ) = Dh(~λ)T . (4.5)

Proof. We introduce a Y -periodic function v(z;~λ) = ψj(Ciz;~λ), then by thechange of variables y = Ciz, using Proposition 4.1 and (4.2) we get:

−divz(

D(z)(∇zv(z) + CTi ~ej)

)

+Q(z)CTi~λ · ∇zv(z)

=

[

φ(z)

〈φ〉〈Q〉CT

i~λ−Q(z)CT

i~λ

]

· CTi ~ej .

By the uniqueness of the solution we obtain (4.3).To prove (4.4) we use the change of variables y = Ciz in the integrals over the

periodic cell Y . By (4.3), (4.2) and Proposition 4.1 we get

〈D(∇ψj(·;~λ) + ~ej)〉 = Ci〈D(∇ψj(·;Cτi~λ) + ~ej)〉, for j 6= i,

〈ψj(·;~λ)Q~λ〉 = Ci〈ψj(·;CTi~λ)Q〉CT

i~λ,

〈φψj(·;~λ)〉

〈φ〉〈Q〉~λ =

〈φψj(·;CTi~λ)〉

〈φ〉Ci〈Q〉CT

i~λ.

Therefore, we conclude that for j 6= i, Dh(~λ)~ej = CiDh(CT

i~λ)CT

i ~ej , since CTi ~ej = ~ej .

In the same way we can see that the conclusion holds for j = i, and therefore (4.4)follows. Finally, (4.5) follows by iterating (4.4) for i = 1, . . . , d.

5. Numerical simulations. In this section, we present some numerical simu-lations of a two-dimensional aquifer using the effective parameters (hydraulic con-ductivity and macrodiffusivity) computed as in Sections 3 and 4. We compare thenumerical simulations based on the macroscopic effective transport equations (3.26),(3.27), using the effective macrodiffusivity tensor given by (3.23), to the numericalsimulations based on the local heterogeneous transport equations (2.3), (2.4). Inthe following we will call in short the numerical simulations of the effective trans-port equations (3.26), (3.27) homogeneous simulations, and the numerical simulationsof the local heterogeneous transport equations (2.3), (2.4) will be called heteroge-neous simulations. For both simplicity and less computational task we will considera 2D heterogeneous aquifer, assuming symmetry of the heterogeneities and diago-nal hydraulic conductivity. This last assumption will lead, by Proposition 4.4, to asymmetric effective macrodiffusivity tensor (3.23).

To perform the homogeneous simulation we have to take the following steps:


1. Compute the effective hydraulic conductivity Kh, by solving problems (3.3),and build the matrix Q using (3.5).

2. Compute the effective Darcy’s velocity ~q 0 by solving the problem (3.27).

3. For each x ∈ Ω compute Dh(~λ), where ~λ = Pe∇H0(x) by solving the problems

(3.21) and using formula (3.22). Then from (3.23) compute the effective

macrodiffusivity tensor Dh(x) = Dh(Pe∇H0(x)).

4. Perform the macroscopic concentration simulation by solving (3.26).

To obtain an accurate approximation of the local problems (3.3), which are el-liptic, we can use either a mixed finite element method or a classical finite volumescheme to solve those problems and compute the effective hydraulic conductivity, asfor example in [4]. But the local problems (3.21) are of convection-diffusion typewith periodic boundary conditions, and we prefer to use a finite volume scheme overuniform rectangular grids, with the convective term treated by upwinding for the ap-proximation of these problems. This approach leads to a robust scheme, conservative,efficient, and stable. Our theory is based on the assumption that the local Pecletnumber Pe is of order one in the convection-diffusion cell problem (3.21), while theglobal convection-diffusion equation (2.3) is governed by the global Peclet numberPe/ε, and hence convection dominated. Careful treatment of the convective termin the cell problem (3.21) is necessary when the model is used at the border of itstheoretical limits, for this we use the full upwinding.

Various discretization methods exist for the numerical simulation of miscible flowin porous media (see, e.g., [12], [17]). Since the transport term is governed by thefluid velocity, accurate simulation requires an accurate approximation of the velocityfield. The concentration equation is convection dominated, and special care shouldbe taken in order to avoid oscillations.

The simulations were performed using an IMPES simulator which applies themixed finite element method for computing hydraulic head and velocity and a finitevolume method for the concentration equation. The code uses a suitable method forhandling the full diffusivity tensors produced by the homogenization method. Only ashort description of the method employed in this work will be given. The interestedreader is referred to [20], [1] and [2] for more details.

The numerical methods feature the mixed finite element method over triangles asa solver to the Darcy flow equation. Our implementation uses a triangular mesh andthe lowest order Raviart-Thomas elements that specify piecewise constant hydraulichead and piecewise continuous flux (see, e.g., [15], [35]). Mixed methods provide a veryaccurate determination of the velocity field, but they also allow for a natural treatmentof practical heterogeneous conditions. This method conserves mass cell by cell andproduces a direct approximation of the two variables hydraulic head and Darcy’svelocity. The popularity of the mixed hybrid method has increased considerably asconsequence of the progress made in recent years in developing efficient algorithmsfor solving this system (see, e.g., [17]).

A conservative finite volume scheme is used for the concentration equation (see[3]). The convective term is approximated with the aid of a Godunov scheme consid-ered over the finite volume mesh dual to a triangular grid, whereas the diffusion termis discretized by piecewise linear conforming triangular finite elements. In the presentcase of vertex-centred finite volume, the secondary mesh is constructed by connect-ing element barycenters with edge midpoints. Time discretization is explicit for theconvection term and implicit for the diffusion part. This approach leads to a robust,conservative and stable scheme applicable for unstructured grids and the approximate


solution has various interesting properties which correspond to the properties of thephysical solution.

The approximation of the Darcy velocity field ~q 0 is constant by triangle sincediv~q 0 = 0, and therefore the local problem (3.21) should be solved for each trianglein the domain Ω. From a practical point of view this is prohibitively expensive andunnecessary. In order to reduce the amount of the computations we use a kind ofclustering. Namely, after the homogeneous Darcy’s flow simulation is performed andthe Darcy velocity vectors ~q 0

T are given in all triangles T , we divide them into groupsor clusters. All vectors in the same group are replaced by their mean value vector,say ~q 0

mean. The tensor (3.22) is then calculated for ~λ = Pe(Kh)−1~q 0mean and applied to

all triangles corresponding to the vectors in the group. We fix a priori the number ofgroups, or clusters, and this number will determine the amount of calculations.

The clustering of the vectors is performed in polar coordinates (r, φ). We calculatethe rectangle (φmin, φmax)×(rmin, rmax) spanned by all Darcy’s velocity vectors in thetriangulation and divide it uniformly into n×m rectangles, each one representing onegroup (possibly empty) of vectors.

In order to compare the solutions of equation (2.3), denoted chet, and the solutionsof equation (3.26), denoted chom, for small times, we use the relative L2-error:

l2err =‖chet(·, t)− chom(·, t)‖L2(Ω)

‖chet(·, t)‖L2(Ω). (5.1)

For a comparison of the long time behavior we use the first three spatial momentsfrequently used in the Lagrangean approach to the effective transport problem:

M0(t) =

∫

Ω

φ(x)c(x, t) dx, M i1(t) =

1

M0(t)

∫

Ω

xiφ(x)c(x, t) dx, (5.2)

M i,j2 (t) =

1

M0(t)

∫

Ω

(xi −M i(t))(xj −M j(t))φ(x)c(x, t) dx, (5.3)

for i, j = 1, 2. They will be calculated in both homogeneous and heterogeneoussimulations and compared to each other.

The homogenization technique used in this paper to derive the effective macro-transport equation belongs to an Eulerian approach since the effective macrodiffusionis given as the average flux due to a given mean Darcy’s velocity (see (3.21), (3.22)and (3.23)). An alternative approach consists of the study of the motion of a singlesolute particle transported due to the incompressible velocity field ~q and diffused dueto diffusivity tensor D, which is then taken to be constant. The equation of the motionof the solute particle, located at x = x0 and t = 0, is a stochastic differential equationof the form:

dX(t) = ~q(X(t))dt+ BdW(t), (5.4)

where the matrix B is given by D = 12BB

T , and W(t) represents a d-dimensionalBrownian motion. Using (5.4) we can define the Lagrangean effective diffusivity as

(DhL)i,j = lim

t→∞

φ

2

d

dt

[

(〈X(t)− 〈X(t)〉W 〉W )i(〈X(t)− 〈X(t)〉W 〉W )j]

(5.5)

where 〈·〉W represents the expectation with respect to the Brownian motion, andthe porosity is taken to be constant. It was argued in [25] that Dh

L is equal to the


~q · ~n = D∇c · ~n = 0

~q · ~n = D∇c · ~n = 0

~q · ~n = −q0

D∇c · ~n = 0

H = 0D∇c · ~n = 0

600

100

Fig. 5.1. Domain Ω and boundary conditions for test problem 1.

symmetric part of the Eulerian effective diffusivity Dh. If c(x, t) is the transitionprobability density associated to (5.4) then (Dh

L)i,j can be formulated as follows:

(DhL)i,j = lim

t→∞

φ

2

d

dtM i,j

2 (t). (5.6)

Therefore we will choose the initial condition as a Gauss pulse and we will use thesecond order moments (5.3) and the estimates

di,j(t) =1

2〈φ〉

d

dtM i,j

2 (t); (5.7)

to measure the quality of the upscaling procedure.Remark 5.1. All computations presented in Sections 5.2 and 5.3 have been

performed with dimensional form of the mesoscale and the macroscale equations. Inthat form the Peclet number does not appear any more in formula (3.23).

5.1. Simulation data. Inspired from an example of a pollutant spreading inan aquifer given in [5], we consider a rectangular domain Ω = (0, Lx)× (0, Ly), withLx = 600 m, Ly = 100 m, that represents a vertical soil cross-section (Figure 5.1).

The domain is covered by a grid of 40×20 cells of the form (0, lx) × (0, ly), withlx = 15, ly = 5 (Figure 5.2). Each cell contains symmetrically placed square inclusionof dimensions presented in Figure 5.2. All physical properties have constant values inthe matrix and in the inclusions.

The porosity is taken to be a constant: φ = 0.3; the hydraulic conductivity tensor(m/day) is given by:

Kmatrix =

[

1 00 0.1

]

, Kinclusion = 10−4

[

1 00 0.1

]

.

The diffusivity tensor (m2/day) is given by:

Dmatrix = 10−2

[

5 00 1

]

, Dinclusion = 10−4

[

5 00 5

]

.

We consider two different boundary conditions. In Section 5.2, we impose anunidirectional flux on the right side boundary, as shown in Figure 5.1, making theproblem almost one-dimensional. This case will be used, essentially to measure theinfluence of the numerical diffusion in the two simulations. In Section 5.3, the inflowboundary will be a part of the upper boundary of the aquifer, as shown on Figure 5.7.

In order to measure the diffusion/dispersion by means of the first and the secondmoments (5.2), (5.3), we choose the initial condition to be a Gauss pulse (A = 5π,σ2 = 2.5, (x0, y0) = (500, 85))

c(x, 0) =A

2πσ2exp(−

1

2σ2[(x − x0)

2 + (y − y0)2]).


15

5

1.5

5

matrix

inclusion

Fig. 5.2. Periodic cell.

5.2. The first test problem: unidirectional flow. In this case we run threesimulations with different Peclet numbers. Outcomes from heterogeneous and homo-geneous simulations are compared through the function d1,1(t), given by (5.7).

It is not difficult to show that for a flow with constant Darcy’s velocity, constantporosity and diffusivity, the estimates di,j(t), given by (5.7), are constant and equal toDi,j , until the solvent starts leaving the domain. We compute the obtained d1,1(t) fromthe heterogeneous and homogeneous simulations and compare them to the effectivemacrodiffusivity Dh

1,1, computed by (3.23).The ratio of convection and diffusion is measured by a local Peclet number which

is calculated as follows:

Pe =|〈qx〉|lx〈D1,1〉

,

where 〈·〉 is the mean value over Ω. The Peclet numbers are generally different inthe heterogeneous and homogeneous simulations (because of the enhancement of theeffective macrodiffusion by convection), and for sufficiently fast convection, the Pecletnumber in the homogeneous simulation will be smaller than in the correspondingheterogeneous one. We will therefore refer to the Peclet number in homogeneoussimulation (respectively in heterogeneous simulation) as homogeneous Peclet number(respectively heterogeneous Peclet number).

We perform three simulations with the local heterogeneous Pe = 1.728, 3.45 and6.192, corresponding respectively to the global heterogeneous Peclet numbers 69.12,138 and 247.6. The results are given in Figures 5.3, 5.4 and 5.5.

The difference between the effective macrodiffusivity Dh11, computed from (3.23),

and the estimate d1,1(t) of effective macrodiffusion in homogeneous simulation com-puted from (5.7), appearing in Figures 5.3, 5.4 and 5.5, could be seen as a consequenceof the effect of numerical diffusion in the simulation. For this we quantify the numer-ical diffusion in our first order scheme by roughly estimating it as in classical finitedifference scheme by the cell Peclet number:

Penum =|〈qx〉|h

2〈D1,1〉= Pe

h

2lx,

where h is the mesh size. In our example, having 89511 triangles in the domain wehave h ≈ 1.414 and the homogeneous Penum ≈ 0.079, 0.136 and 0.176 respectively.In all three figures the numerical diffusion is measured by the difference between theeffective macrodiffusivity (3.23) and the estimated effective macrodiffusivity (5.7) inhomogeneous simulation. We verify that the homogeneous cell Peclet number is a goodestimate for the numerical diffusion: in Figure 5.3, 5.4 and 5.5 we have respectively6.6%, 11.8% and 14.0% of numerical diffusion.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dxx

t/days

macroscopic effective diffusionheterogeneous estimatehomogeneous estimate

Fig. 5.3. Heterogeneous and homogeneous d1,1(t) and Dh1,1 for Pe = 1.728.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dxx

t/days



The results of the heterogeneous computations are different. At this level, an addi-tional numerical diffusion (depending linearly on the cell Peclet number) will diminishartificially the importance of the mechanical dispersion, leading finally to underesti-mate the effective macrodiffusivity computed by (3.23) (as seen in Figure 5.5). Theunderestimate of the effective macrodiffusivity Dh

1,1 in the heterogeneous simulationsby d1,1(t) is a consequence of a quadratic dependency of the mechanical dispersion onthe Peclet number (as seen in Figure 5.6) which implies that for large Peclet numbersthe decrease of mechanical dispersion prevails over additional numerical diffusion inthe heterogeneous simulation. We illustrate that point by the following computation.

We solve the cell problem (3.21) with ~λ = v~e1, for different values of v in order toshow the effective macrodiffusivity behavior for a given flow regime. In Figure 5.6 weshow that the longitudinal effective macrodiffusivity Dh

11(v~e1), calculated by (3.22),grows quadratically with v (that is with the Peclet number). Note that in this examplethe velocity in the range 0 to 1 corresponds to the local Peclet number in the range0 to 330.


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Dxx

t/days



0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1

Dxx

h

v

Fig. 5.6. Longitudinal effective macrodiffusivity Dh11(v~e1).

In other words, in this example we are in the situation where the convectionenhanced effective macrodiffusion is maximal (see [10], [21]). It is already observedin the literature (see for example [26]) that this maximal enhancement is due to thefact that the mean flow is parallel to one lattice axis, but when the flow is inclinedto the lattice axis the longitudinal effective diffusivity may vary linearly with thevelocity, which is the most often used assumption for natural media in the engineeringliterature.

5.3. The second test problem: upper inflow. In this example we need todo a clustering of the computed Darcy’s velocity ~q 0 in order to reduce the amountof computation in the upscaling procedure, leading to the definition of a new velocityfield ~q app, constant in each cluster. In this example, we see that the error due tothe clustering does not have a strong influence on the quality of the overall upscalingprocedure. This is mostly due to the fact that the most rapid change in Darcy’svelocity field ~q 0 is localized around the point (400, 100), where we have a strong


~q · ~n = D∇c · ~n = 0

~q · ~n = D∇c · ~n = 0 ~q · ~n = −q0

D∇c · ~n = 0

H = 0D∇c · ~n = 0

600

100

400

Fig. 5.7. Domain Ω and boundary conditions for test problem 2.

0

0.02

0.04

0.06

0.08

0.1

0 20 40 60 80 100 120 140 160

l2er

r

time/days

Fig. 5.8. Relative L2-error.

change in the velocity direction. More refined clustering will improve the effectivemacrodiffusivity tensor only locally, in the neighbourhood of that point only. Forexample, a clustering based on 33 different clusters gives a relative error in Darcy’svelocity of 12.76%, and the clustering with 80 different clusters will reduce the errordown to only 8.28%. The error of the approximation is measured in the relativeL2-norm.

In Figure 5.8 we show the relative L2-error of the concentration (5.1) for the first150 days, the time needed for the solution amplitude to drop to 5% of its initial value.Since the parameter ε = lx/Lx = 0.025, we see that our L2-error is of order ε, aspredicted by the theory, according to the fact that the homogenized solution does notcontain neither the oscillatory terms nor the boundary layer correctors.

In Figures 5.9, 5.10, 5.11 and 5.12, we consider the moments of order one and twoas defined in (5.2), (5.3), according to [14] and [6], in order to compare the homoge-neous to the heterogeneous simulations. We see in Figures 5.9 and 5.10 that the firstorder moments M1

1 (t) and M21 (t) in homogeneous and heterogeneous simulations are

in good agreement, i.e. the Darcy velocity is well scaled up.

The difference in the computation of the second order moments, that appears inFigures 5.11 and 5.12, is a consequence of the computed mechanical dispersion be-ing lower than the enhancement of the macrodiffusion by convection (quadraticallygrowing with the Peclet number) in formula (3.23). In the one hand there is addi-tional numerical diffusion diminishing the importance of the mechanical dispersion


0

50

100

150

200

250

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Xm

ean

time/days

heterogeneoushomogeneous

Fig. 5.9. The first order moment M1

1(t).

45

50

55

60

65

70

75

80

85

90

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Ym

ean

time/days


Fig. 5.10. The first order moment M2

1(t).

in the heterogeneous simulations, as explained before. In the other hand, for thehomogeneous simulation, the mechanical dispersion is already included in the effec-tive macrodiffusivity Dh(x), the homogeneous Darcy’s velocity is not oscillating, andtherefore numerical diffusion does not modify the already computed global dispersion.Having these effects in mind we may conclude that the second order moments are ingood agreement.

In Figure 5.13 we show the dispersivity coefficients (see definition in [5]) in thelongitudinal direction as functions of time.

d1 = φM1,1

2 (t)

2M1(t). (5.8)

We see for both simulations that this coefficient is not constant but is increasing withtime (with traveled distance) which is consistent with the quadratic dependency on


0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

sigm

a xx

time/days


Fig. 5.11. The second order moment M1,12

(t).

0

50

100

150

200

250

300

350

400

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

sigm

a yy

time/days


Fig. 5.12. The second order moment M2,22

(t).

the Peclet number, and with the fact that the Peclet number grows slightly with thetraveled distance.

Finally, we compare the isovalues of the heterogeneous and homogeneous soluteconcentrations, given at t = 3889 days, in Figures 5.14 and 5.15, respectively.

In conclusion the simulations presented here show the good accuracy of the ho-mogenization procedure for a global Peclet number of order 1/ε. The results of thesimulations should be carefully interpreted with respect to the numerical diffusionappearing in the computations.

Simulations with larger Peclet numbers would require higher order numericalmethod for convection-diffusion equation in order to avoid excessive numerical diffu-sion.

Acknowledgements. This work was supported partially from the GdR MoMaSCNRS-2439 ANDRA BRGM CEA whose support is gratefully acknowledged. This


0

0.5

1

1.5

2

2.5

3

3.5

4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

phi*

sigm

a xx/

2x

t/days


Fig. 5.13. Dispersivity coefficient φM1,12

(t)/2M1(t).

Fig. 5.14. Heterogeneous simulation: c(·, t), t = 3889 days.

Fig. 5.15. Homogeneous simulation: c(·, t), t = 3889 days.


paper was completed when M. Jurak was visiting the Applied Mathematics Labora-tory of the University of Pau. He is grateful for the invitation.

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