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**Two Asymptotic Models for Arrays of Underground** **Waste Containers**

### Grégoire Allaire, M. Briane, R. Brizzi, Y. Capdeboscq

**To cite this version:**

### Grégoire Allaire, M. Briane, R. Brizzi, Y. Capdeboscq. Two Asymptotic Models for Arrays of Un-

### derground Waste Containers. Applicable Analysis, Taylor & Francis, 2009, 88, pp.1445-1467. �hal-

### 00784062�

## ECOLE POLYTECHNIQUE

### CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS 7641

91128 PALAISEAU CEDEX (FRANCE). Tél: 01 69 33 46 00. Fax: 01 69 33 46 46 http://www.cmap.polytechnique.fr/

## Two Asymptotic Models for Arrays of Underground Waste

## Containers

### Gregoire Allaire, Marc Briane, Robert Brizzi, Yves Capdeboscq

### R.I. 657 January 2009

TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS

GREGOIRE ALLAIRE^{1}, MARC BRIANE^{2}, ROBERT BRIZZI^{1}, YVES CAPDEBOSCQ^{3}

1 Centre de Mathématiques Appliquées, École Polytechnique, 91128 Palaiseau, France.

2 Centre de Mathématiques, INSA de Rennes et IRMAR, 20 avenue des Buttes de CoÃ«smes - 35043 Rennes, France.

3 University of Oxford, OXFORD OX1 3LB, UK

Abstract. We study the homogenization of two models of an under- ground nuclear waste repository. The nuclear waste cells are periodically stored in the middle of a geological layer and are the only source terms in a parabolic evolution problem. The diusion constants have a very large contrast between the fuel repository and the soil. It is thus a combined problem of homogenization and singular perturbation. For two dierent asymptotic contrats we give the homogenized limit problem which is rig- orously justied by using two-scale convergence. Eventually we perform 2-d numerical computations to show the eectiveness of using the limit model instead of the original one.

Key words: Homogenization, diusion, porous media.

2000 Mathematics Subject Classication: 35B27, 74Q10.

1. Introduction

In this paper, we consider a simplied model used to describe the diusion process of radioactive elements from a repository into surrounding geological layers. It is a parabolic evolution problem with very contrasted coecients along a thin strip.

Namely, we study the following model (1)

ρ_{ε}(x)∂u_{ε}

∂t −div (A_{ε}(x)∇uε) +ω_{ε}(x)u_{ε} = f_{ε}(t, x) inΩfor a.e. t >0,
uε = 0on∂Ωfor a.e. t >0,
u_{ε}(t= 0) = 0in Ω,

whereΩis a smooth open set ofR^{d}, withd= 2or3, which intersects the horizontal
hyperplane passing through the origin, i.e.,Ω∪ {xd = 0} 6=∅ (for any x∈R^{d} we
write x = (x^{0}, xd) with x^{0} ∈ R^{d−1}). The unknown is uε(t, x) ∈ L^{2}(R^{+};H_{0}^{1}(Ω)).
The repository, where the inhomogeneities and the source term fε are located, is
an horizontal strip of thickness εand midplane {xd= 0}. Furthermore, the small
parameter ε is the periodicity of the repository heterogeneities too. Outside the

1

**Oxfordian**

**Lx****Lz**

**Storage zone**

* Dogger*
Callovo−Oxfordian
Callovo−Oxfordian

Η

Figure 1. Sketch of the medium.

repository, the diusion tensorA_{ε}and the scalar porositiesω_{ε}andρ_{ε}are functions
of the macroscopic or slow variablexonly. More precisely, we assume that there
exist functionsµ_{0}, ω_{0} andρ_{0} inL^{∞}(Ω)such that

(2)

for anyx= (x^{0}, x_{d})∈Ωwith |xd|> ε/2, f_{ε}(t, x) = 0,∀t≥0
andA_{ε}(x) (resp. ω_{ε}(x), ρ_{ε}(x)) =µ_{0}(x) (resp. ω_{0}(x), ρ_{0}(x)).

Inside the repository, we assume that all coecients, as well as the source term, are periodic in the rst(d−1)variables. We furthermore consider a highly contrasted diusion tensor compared toµ0. Namely, we assume

(3)

for anyx= (x^{0}, xd)∈Ωwith |xd|< ε/2, fε(t, x) =ε^{−1}f1 t,^{x}_{ε}

, ∀t≥0
andAε(x) =ε^{a}µ1 x

ε

, ωε(x) =ω1 x ε

andρε(x) =ρ1 x ε

,

with periodic functionsµ_{1}∈L^{∞}_{#,d}(Y)^{d×d},ω_{1},ρ_{1}∈L^{∞}_{#,d}(Y)andf_{1}∈L^{∞}(R^{+};L^{∞}_{#,d}(Y)).
We assume thatω_{0}, ρ_{0}≥c >0(resp. ω_{1}, ρ_{1}≥c >0) where the constantc is inde-
pendent ofε and that both diusion tensorsµ_{0} andµ_{1} are uniformly elliptic and
bounded, that is, there exists two positive constantsβ ≥α > 0 such that, for all
ξ∈R^{d},

(4) α|ξ|^{2}≤µ_{i}ξ·ξ≤β|ξ|^{2} i= 0,1.

The exponent a in (3) is either −1 or 2. The periodicity cell is the unit cube
Y = (−1/2,+1/2)^{d}, and byL^{∞}_{#,d}(Y)we denote the following space

L^{∞}_{#,d}(Y) =

f ∈L^{∞}(R^{d}), s.t. ∀y= (y^{0}, yd)∈R^{d}
and∀p∈Z^{d−1}, f(y^{0}+p, yd) =f(y^{0}, yd) .

Note that the scaling of the source term is such that its integral over the repository
is of order one. We denote byχεthe characteristic function of the repository, that
is, χε(x) =χ ^{x}_{ε}^{d}

where χis the indicator function of the interval(−1/2,+1/2).
We denote byT >0 the nite time-scale for which the model under consideration
is valid. We also use the notations Ω^{0} = Ω∩ {xd = 0}, Ω^{+} = Ω∩ {xd > 0},
Ω^{−}= Ω∩ {xd<0}, andY^{0} = (−1/2,+1/2)^{d−1}.

The motivation for this work is the study on the long time behavior of an under- ground waste repository in the presence of leakage. A sketch of the medium under consideration is presented in Figure 1. The spent fuel cells are stored in the centre of the middle geological layer. Inside each module, 10 to 15 spent fuel cells are stored horizontally (normally to the section represented). A description of the geometry is presented in Section 6.1. This model problem, introduced as a benchmark case for the COUPLEX exercise [8], has been previously studied by Bourgeat et al. in [4], [9], [5], [6], [7]. The novelty here is that we allow a strong contrast between the fuel repository and the soil. The rst case is that of a very small permeability of the engineered structure, a= 2, where the repository is designed as a sealed con- tainment area. The second case, is that of a large permeabilty,a=−1, where the repository structure is the weak point of the apparatus, the surrounding soil being a better natural barrier. This last case is what happens with a concrete structure is a clay layer. The source term represents a leakage (which would happen in these worst case scenari in every module).

The rst part of the paper, sections 3 to 5, is devoted to the asymptotic study of two models (a=−1 anda= 2) in the limit asεtends to zero. After establishing a priori estimates and recalling the notion of two scale convergence for boundary layers [3] in Section 3, we derive the homogenized limits for the casea= 2in Section 4 and for the case a=−1 in Section 5. A related result appeared in [14] with a dierent application to transmission through thin membrane.

The second part of the paper is dedicated to the application of this study in the
nuclear repository context. The model used by the so-called MoMaS group^{1}, in
charge of the mathematical aspects of the long term underground waste repository
behavior predictions, is slightly more complex than the one studied in the rst part,
and is exposed rst. Because the coecients are measured physical quantities, the
relevant small parameterεhas to be identied, and evaluated. With that in mind,
we conduct a dimensional analysis of the available data, and express the MoMaS
group model in this setting. The two cases we have considered correspond to limit
situations, with very strong or very weak diusion. The corresponding numerical
results are presented in Section 6.

2. Main Results

The rst result addresses the case whena = 2, i.e. of very weak diusion in the
repository zone. In such a case, we need an additional technical assumption on the
smoothness of the coecients in the vicinity of the repository boundaries, namely
(5) ∃x^{d}_{0}∈(0,1/2)andr >0 such thatµ1(y)∈C_{#}^{0,r} Y ∩ {|xd|> x^{d}_{0}}^{d×d}

. Our result describes the limit behavior ofuε both outside the fuel repository and inside.

1The research group MoMaS (Mathematical Modeling and Numerical Simulation for Nuclear Waste Management Problems) is part of the PACE Research Federation. MoMaS's sponsors are: ANDRA National Radioactive Waste Management Agency, BRGM Bureau des recherches géologiques et minières, CEA Atomic Energy Commission, CNRS National Center for Scientic Research, EDF Électricité de France, IRSN Institute for Radiological Protection and Nuclear Safety.

Theorem 1. Under assumptions (2,3,4,5), when a= 2, the solution uε of (1) is
such that(1−χε)uεconverges strongly inL^{2} (0, T);L^{2}(Ω)

tou, given byu:=u^{+}
inΩ^{+}andu:=u^{−}inΩ^{−}, and(1−χ_{ε})∇uεconverges weakly inL^{2}

(0, T);L^{2}(Ω)^{d}
toξ, with ξ=∇u^{±} inΩ^{±}, whereu^{±} is the unique solution of

(6)

ρ_{0}(x)∂u^{±}

∂t −div (µ_{0}(x)∇u^{±}) +ω_{0}(x)u^{±}= 0 inΩ^{±},
u^{±}= 0 on∂Ω∩∂Ω^{±},

∓µ_{0}(x)∇u^{±}·e_{d}=s^{±} onΩ^{0},
u^{±}(t= 0) = 0 inΩ.

The eective source terms s^{+} ands^{−} are given by
(7) s^{±}(t) =±

Z

Y^{0}

µ_{1}(y^{0},±1/2)∇u2(t, y^{0},±1/2)·e_{d}dy^{0},

where u2 is the limit ofεuε in the sense of two-scale boundary layers convergence
(see Proposition 5). The limit u2(t, y)is the unique solution inL^{2} (0, T);H^{1}(Y)
of the following problem

(8)

ρ_{1}(y)∂u2

∂t −div_{y}(µ_{1}(y)∇_{y}u_{2}) +ω_{1}(y)u_{2} = f_{1}(t, y)in(0, T)×Y,
u_{2}(t, y) = 0 on(0, T)× {|y_{d}|= 1/2},
(y_{1}, . . . , y_{d−1}) → u_{2}(t, y)Y^{0}-periodic,

u_{2}(t= 0) = 0.

Asymptotically, the eect of the strip is therefore to transform the source term
f_{1} into two eective time-dependent boundary conditionss^{+} ors^{−} for the normal
derivatives given by (7). Note that system (8) is independent of system (6). Such
a limit problem is the inspiration for an approximate numerical method presented
in Section 6. Theorem 1 is related to a recent result in [14], for nonlinear reaction-
diusion systems. However the scaling, as well as the analysis, are quite dierent.

We now address the second case whena=−1.

Theorem 2. Under assumptions (2,3,4), when a=−1, the solution u_{ε} of (1) is
such that it converges strongly inL^{2} (0, T);L^{2}(Ω)touand∇uεconverges weakly
inL^{2}

(0, T);L^{2}(Ω)^{d}

to∇u, whereu∈H^{1}(Ω)is the unique solution of the coupled
system

(9)

ρ_{0}∂u

∂t −div(µ_{0}∇u) +ω_{0}u = 0in Ω^{+}∪Ω^{−}
u(t= 0) = 0in Ω^{+}∪Ω^{−},

u = 0on ∂Ω,

u =u_{−1} on Ω^{0}= Ω∩ {xd= 0},
andu_{−1} is the solution in H^{1}(Ω^{0})of

(10) −divx^{0}(µ^{∗}∇x^{0}u_{−1}(t, x)) + [µ_{0}∇u·n] =f ,

where [j] stands for the jump of a quantity j between Ω^{+} and Ω^{−}. The eective
coecientsµ^{∗} andf are given by

(11) f(t) =

Z

Y

f1(t, y)dy, and

(12) µ^{∗}=

Z

Y

µ1(y) (Id+P(y))dy.

For each i = 1, . . . , d, P(·)ei is given by P(·)ei = ∇ϕi where ϕi ∈ H_{#,d}^{1} (Y) is
Y^{0}-periodic in its(d−1)rst variables and is the unique solution, up to an additive
constant, of

(13)

div (µ1(∇yi+∇yϕi)) = 0in Y,

µ1(∇yi+∇yϕi)·ed = 0on Y^{0}× {+1/2},
µ_{1}(∇yi+∇yϕ_{i})·e_{d} = 0on Y^{0}× {−1/2}.

Of course, the two partial dierential equations (9) and (10) are strongly coupled.

Asymptotically, the eect of the strip is therefore to transform the source term f1 into an eective time-dependent tangential diusion along the interface given by (10). The homogenized problem (9)-(10) is the inspiration for an approximate numerical method in Section 6. For a homogeneous thin strip, Theorem 2 was already obtained in [12].

3. A priori estimates and two-scale boundary layer convergence
A prerequisite for the proofs of Theorems 1 and 2 is to check that the solutionu_{ε}
of (1) is uniformly bounded in appropriate norms. The necessary a priori estimates
are established in the following Proposition.

Proposition 3. Assume the coecients A_{ε}, ω_{ε}, ρ_{ε} satisfy assumptions (2, 3, 4).

(i) If, instead of assumption (3), we suppose thatfε(t, x) =ε^{b}χε xd

ε

f1 t,^{x}_{ε}with
f1∈L^{∞}(R^{+};L^{∞}_{#,d}(Y^{0}×(−1/2,1/2))), andb∈R, we have

kuεk_{L}∞((0,T);L^{2}(Ω))+k(1−χ_{ε})∇uεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})

+ε^{a}^{2} kχ_{ε}∇u_{ε}k_{L}2(^{(0,T}^{);L}^{2}^{(Ω)}^{d}) ≤ Cε^{1}^{2}^{+b}min 1, ε^{−}^{a}^{2}
(14) ,

(ii) Alternatively, instead of assumption (3) assume thatfε(t, x) =sε(t, x^{0})δ_{{|x}_{d}_{|=ε/2}} ^{x}_{ε}^{d}
where s_{ε} ∈ C R^{+}×R^{d−1}

is bounded above and below independently of ε. Then we have

(15)

kuεk_{L}∞(0,T ,L^{2}(Ω))+k(1−χε)∇uεk_{L}2(^{(0,T}^{);L}^{2}^{(Ω)}^{d}) +ε^{a}^{2}kχε∇uεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C.

Corollary 4. Assume the coecientsA_{ε}, ω_{ε}, ρ_{ε} satisfy assumptions (2, 3 and 4).

If a=−1, then we have

(16) kuεk_{L}∞((0,T);L^{2}(Ω))+k(1−χε)∇uεk_{L}∞((0,T);L^{2}(Ω)^{d})≤C,
and

(17) kχεu_{ε}k_{L}2((0,T);L^{2}(Ω))+kχε∇uεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C√
ε.

If a= 2, thenwε=εuεsatises (18)

kwεk_{L}∞((0,T);L^{2}(Ω))+k(1−χε)∇wεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})+εkχε∇wεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C√
ε
which, in particular, implies that

(19) kχ_{ε}w_{ε}k_{L}∞((0,T);L^{2}(Ω))+εkχ_{ε}∇w_{ε}k_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C√
ε.

Proof of Corollary 4. Estimates (16), (18) and (19) are obvious consequences of Proposition 3. Inequality (17) is easily deduced from (14) and (22).

Proof of Proposition 3. Testing (1) againstu_{ε}we obtain, after integration by parts
with respect to the space variablex,

1 2

d dt

Z

Ω

ρεu^{2}_{ε}dx+
Z

Ω

(1−χε)µ0(x)∇uε· ∇uεdx+ Z

Ω

ωx ε

u^{2}_{ε}dx
+ε^{a}

Z

Ω

χεµ1

x ε

∇uε· ∇uεdx = Z

Ω

χεfεuεdx.

(20)

Let us rst address the case (i): there is a constantM such thatkfεk_{L}∞(R^{+}×Ω)ε^{−b}<

M < ∞ for any ε. Then (20) yields, because of the positivity of ρε, ωε and the coercivity ofµ0 andµ1, that

d dt

ρε^{1}^{2}u_{ε}

2
L^{2}(Ω)

≤Ckχ_{ε}f_{ε}k_{L}2(Ω)

χ_{ε}ρε^{1}^{2}u_{ε}

_{L}_{2}_{(Ω)}≤Cε^{1}^{2}^{+b}

χ_{ε}ρε^{1}^{2}u_{ε}
_{L}_{2}_{(Ω)},
which in turns shows that kuε(t,·)k_{L}2(Ω) ≤Ctε^{1}^{2}^{+b}, the constantC, depending a
priori onminΩ|ρε|, andM being chosen independently ofε. Integrating (20) with
respect to time, and using Cauchy-Schwarz inequality again we obtain,

(21) k(1−χε)∇uεk^{2}_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d}) +ε^{a}kχε∇uεk^{2}_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤Cε^{1+2b}.
This implies (14). On the other hand, thanks to the Dirichlet boundary condition
satised byuεin (1), we have, for allt >0, the following Poincaré inequality
(22) kχεuεk^{2}_{L}2(Ω)≤dΩkχε∇uεk^{2}_{L}2(Ω)^{d},

where dΩdepends on Ωonly. This can be proved, for example, by integrating uε

along a vertical or horizontal path connecting xand the boundary of the domain within the repository. Consequently, we have, for allt >0, and allλ >0,

Z

Ω

χεfεuεdx ≤ ε^{−a} dΩ

4λαkχεfεk^{2}_{L}2(Ω)+ε^{a}αλ

d_{Ω}kχεuεk^{2}_{L}2(Ω),

≤ ε^{−a} dΩ

4αλkχεfεk^{2}_{L}2(Ω)+ε^{a}λ
Z

Ω

χεµ1(x

ε)∇uε· ∇uεdx.

(23)

Forλ= 1, this last inequality combined with (20) yields 1

2 d dt

Z

Ω

ρ_{ε}u^{2}_{ε}dx ≤ d_{Ω}

4αε^{−a}kχεf_{ε}k^{2}_{L}2(Ω)

≤ Cε^{−a+1+2b},
(24)

which shows thatkuε(t,·)k_{L}2(Ω) ≤Cε^{1}^{2}^{+b−}^{1}^{2}^{a}. Forλ= 1/2, using (23) in (20) we
obtain that for allt >0,

1 2

d dt

Z

Ω

ρεu^{2}_{ε}dx+
Z

Ω

(1−χε)µ0(x)∇uε· ∇uεdx +1

2ε^{a}
Z

Ω

χ_{ε}µ_{1}x
ε

∇uε· ∇uεdx ≤ Cε^{−a+1+2b},
which in turn implies that

k(1−χε)∇uεk^{2}_{L}2((0,T);L^{2}(Ω)^{d})+ε^{a}kχε∇uεk^{2}_{L}2((0,T);L^{2}(Ω)^{d})≤Cε^{−a+1+2b}.
Estimate (14) is proved, and the proof of the rst part of Proposition 3 is nished.

Let us now turn to case (ii), and suppose that fε = sε(t, x^{0})δ_{{|x}_{d}_{|=ε/2}} with
s_{ε} ∈ C(R^{+} ×R^{d−1}), and ks_{ε}k_{L}^{∞} < M where the constant M is independent
ofε. Then

Z

Ω

fε(t, x)uεdx ≤ M Z

{xd=±ε/2}

|uε|dx^{0}≤ M^{2}
τ +Cτ

Z

{xd=±ε/2}

|uε|^{2}dx^{0}
(25)

≤ M^{2}
τ +

Z

Ω

(1−χε)µ0(x)∇uε· ∇uεdx (26)

for a suciently small constantτ >0. Therefore we obtain 1

2 d dt

Z

Ω

ρε(x)u^{2}_{ε}dx≤ M^{2}
τ ,

and arguing as above, we obtain (15).

Following the strategy introduced in [3], we shall derive the asymptotic limit of problem (1) using two-scale convergence for boundary layers. This notion of con- vergence generalizes the usual two-scale convergence [2], [13] to the case when the test functions periodically oscillate in the rst(d−1) space directions and simply decay at innity in the last direction. The following proposition summarizes the denition and properties of two-scale convergence in the sense of boundary layers.

We denote byGthe bandY^{0}×R. A generic pointy∈Gis denoted byy= (y^{0}, yd)
with y^{0} ∈ Y^{0} and yd ∈ R. We introduce the space L^{2}_{#}(G) of square integrable
functions onGwhich are periodic in the rstd−1variables, i.e.

L^{2}_{#}(G) =

ϕ∈L^{2}(G)s.t. y^{0}7→ϕ(y^{0}, yd)isY^{0}-periodic .

Similarly, we deneH_{#}^{1}(G)the Sobolev space of functions inH^{1}(G)which areY^{0}-
periodic, and C_{#,c}^{∞} (G) and C_{#}^{∞}(G) the space of innitely dierentiable functions
with compact support in the rst case on G. We denote byC(Ω^{0})the space of
continuous functions on the closure ofΩ^{0}, a compact set inR^{d−1}.

Proposition 5. (1) Letvε be a sequence inL^{2}(Ω) such that
kv_{ε}k_{L}2(Ω)≤C√

ε.

There exists a subsequence, still denoted byε, and a limitv0(x^{0}, y)∈L^{2}

Ω^{0};L^{2}_{#}(G)
such that vε two-scale converges weakly in the sense of boundary layers tov0 i.e.

ε→0lim 1 ε

Z

Ω

v_{ε}(x)ϕ
x^{0},x

ε

dx = 1

|Y^{0}|
Z

Ω^{0}

Z

G

v_{0}(x^{0}, y)ϕ(x^{0}, y)dx^{0}dy

for all test functionsϕ(x^{0}, y)∈C

Ω^{0};L^{2}_{#}(G)
.

(2) Letvε be a sequence which two-scale converges weakly in the sense of boundary layers tov0, and furthermore satises

ε→0lim

√1

εkvεk_{L}2(Ω) = 1

|Y^{0}|^{1}^{2}

kv0k_{L}2(Ω^{0}×G).

Then,vεis said to two-scale converge strongly in the sense of boundary layers tov0,
which means that, for any sequencew_{ε} in L^{2}(Ω) which two-scale converges weakly
in the sense of boundary layers tow_{0}(x^{0}, y)∈L^{2}(Ω^{0}×G)one has

ε→0lim 1 ε

Z

Ω

vε(x)wε(x)ϕ
x^{0},x

ε

dx = 1

|Y^{0}|^{1}^{2}
Z

Ω^{0}

Z

G

v0(x^{0}, y)w0(x^{0}, y)ϕ(x^{0}, y)dx^{0}dy,
for all smooth functionsϕ(x^{0}, y)∈C Ω^{0};C_{#,c}(G)

.

Note that two-scale boundary layer convergence (called 2SBL in the sequel) can be also used to characterize the precise behavior of a non-vanishing function, restricted to a small domain, as it is shown by the following simple lemma.

Lemma 6. Letvε be a bounded sequence inH_{0}^{1}(Ω) converging weakly to a limitv.
Assume that wε =χεvε admits a limit w in the sense of boundary layers conver-
gence. Then w=χwand for almost everyx^{0}∈Ω^{0},

v(x^{0},0) =
Z

Y

w(x^{0}, y)dy.

Proof. Recall thatχε=χ(^{x}_{ε})is the characteristic function of the repository zone.

The fact thatw=χw is obvious. Note that the assumption onwε having a two-
scale limit in the sense of boundary layers is plausible since, if vε is a constant
sequence v ∈ L^{∞}(Ω), then wε satises the a priori estimate kwεk_{L}2(Ω) ≤ C√

ε which is required for the two-scale convergence in the sense of boundary layers. We have

1 ε

Z

Ω

χ_{ε}v_{ε}(x^{0},0)ϕ(x)dx−
Z

Ω^{0}

v_{ε}(x^{0},0)ϕ(x^{0},0)dx^{0}

= 1 ε

Z

Ω

χ_{ε}v_{ε}(x^{0},0)ϕ(x^{0},0)dx−
Z

Ω^{0}

v_{ε}(x^{0},0)ϕ(x^{0},0)dx^{0}
+

Z

Ω

vε(x^{0},0)1

εχε(ϕ(x^{0},0)−ϕ(x))dx.

The rst term is exactly zero. For the second one, note that

1

εχε(ϕ(x^{0},0)−ϕ(x))

≤ 1 ε

Z ε/2

−ε/2

∂

∂xd

ϕ(x^{0}, t)

dt≤ k∇ϕk_{L}∞(Ω),
therefore,

Z

Ω

vε(x^{0},0)1

εχε(ϕ(x^{0},0)−ϕ(x))dx

≤ k∇ϕk_{L}∞(Ω)ε
Z

Ω^{0}

vε(x^{0},0)dx^{0}

≤ Cεk∇ϕk_{L}∞(Ω)k∇v^{ε}k_{L}2(Ω)^{d},

where this last term is of order O(ε)since the sequence vε is bounded in H_{0}^{1}(Ω).
Similarly,

Z

Ω

1

εχ_{ε}(v_{ε}(x^{0},0)−v_{ε}(x))

dx ≤ Z

Ω

χ_{ε}
ε

Z ε/2

−ε/2

∂

∂xd

v_{ε}(x^{0}, s_{d})

ds_{d}dx

≤ C√

εk∇v^{ε}k_{L}2(Ω)^{d}.
As a consequence,

Z

Ω

1

εχεv_{ε(x)}ϕ(x)dx−
Z

Ω^{0}

v(x^{0},0)ϕ(x^{0},0)dx^{0}

≤ Z

Ω

1

εχεvε(x^{0},0)ϕ(x)dx−
Z

Ω^{0}

v(x^{0},0)ϕ(x^{0},0)dx^{0}

+O(√ ε)

≤ Z

Ω^{0}

|vε(x^{0},0)−v(x^{0},0)| |ϕ(x^{0},0)|dx^{0}+O(√
ε)

≤ C(ϕ)kvε(·,0)−u(·,0)k_{L}2(Ω^{0})+O(√
ε).

The trace of vε onΩ^{0} converge strongly in L^{2}(Ω^{0}) to the trace ofv, thus we have
proved that

Z

Ω^{0}

Z

G

w(x^{0}, y)ϕ(x^{0},0)dx^{0}dy = lim

ε→0

Z

Ω

1

εχ_{ε}(x)v_{ε}(x)ϕ(x^{0},0)dx

= lim

ε→0

Z

Ω

1

εχ_{ε}(x)v_{ε}(x)ϕ(x)dx

= Z

Ω^{0}

v(x^{0},0)ϕ(x^{0},0)dx^{0}.

which implies the desired result, namely thatu(x^{0},0)is the average onY ofw(x^{0}, y).
The two scale convergence for boundary layers was introduced for time independent
problems. It naturally extends to time dependent problems.

Proposition 7. (i) Suppose that a = −1. There exists a function u_{−1}(t, x^{0}) ∈
L^{2} (0, T);H^{1}(Ω^{0})

and a function w(t, x^{0}, y) ∈ L^{2}

(0, T)×Ω^{0};H_{loc,#}^{1} (G)
such
that, up to a subsequence,

χ_{ε}u_{ε} −−−^{2SBL}* χ(y_{d})u_{−1}(t, x^{0})
(27)

χε∇uε

−−−2SBL* χ(yd) (∇x^{0}u−1(t, x^{0}) +∇yw(t, x^{0}, y)).
(28)

(ii) Suppose thata= 2. There exists a functionu2(t, x, y)∈L^{2}

(0, T)×Ω^{0};H_{loc,#}^{1} (G)
such that, up to a subsequence,

εu_{ε} −−−^{2SBL}* u_{2}(t, x^{0}, y)
(29)

ε^{2}∇u_{ε} −−−^{2SBL}* χ(y_{d})∇_{y}u_{2}(t, x^{0}, y)
(30)

Furthermore,

(31) (1−χ(yd))u2(t, x^{0}, y) = 0 for a.e. y∈G.

Note. We use the notation

∇_{x}^{0} =
∂

∂x1

, . . . , ∂

∂xd−1

,0

. Similarly, for addimensional vectorΨwe shall write

Ψ^{0}= (Ψ_{1}, . . . ,Ψ_{d−1},0).

Proof. Supposea= 2.The bound (18) shows thatwε=εuεsatises
kw_{ε}k_{L}2((0,T);L^{2}(Ω))+εk∇w_{ε}k_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C√

ε.

Consequently,w_{ε}(resp. ε∇w_{ε}) admits a limitu_{2}(resp.ζ_{2}) in the sense of boundary
layer two scale limits. It is a classical result [2], [3] thatε∇w_{ε}two scale converges in
the sense of boundary layers to ζ2=∇yu2. We shall now prove (31). The bounds
(18) show that the sequencewε satises

k(1−χ_{ε})w_{ε}k_{L}2(0,T;L^{2}(Ω))+k(1−χ_{ε})∇wεk_{L}2(0,T;L^{2}(Ω)^{d})≤C√
ε.

Therefore, up to a subsequence, (1−χε)∇wε converges in the sense of bound-
ary layers to ξ2(t, x^{0}, y). Since χε converges strongly in the sense of two-scale
boundary layers convergence toχ, we have ξ2(t, x^{0}, y) = (1−χ(yd))ξ2(t, x^{0}, y). Let
Ψ(x^{0}, y) ∈ C_{c}^{∞}

Ω^{0};C_{#,c}^{∞} (G)^{d}

be a smooth test function such that Ψ(x^{0}, y)(1−
χ(yd)) = Ψ(x^{0}, y), and ϕ(t)∈C[0, T]. By an integration by parts we see that

ε→0lim 1 ε

Z T 0

ϕ(t) Z

Ω

(1−χε)∇wε·Ψ
x^{0},x

ε

dt dx

= −lim

ε→0

1
ε^{2}

Z T 0

ϕ(t) Z

Ω

wεdivyΨ
x^{0},x

ε

dt dx^{0}dy

−lim

ε→0

1 ε

Z T 0

ϕ(t) Z

Ω

w_{ε}div_{x}^{0}Ψ
x^{0},x

ε

dt dx^{0}dy.

This implies that

ε→0lim 1 ε

Z T 0

ϕ(t) Z

Ω

wεdivyΨ
x^{0},x

ε

dt dx= 0,

that is, (1−χ)u2(t, x^{0}, y) does not depend on y and is therefore 0, since y 7→

u2(·,·, y)∈L^{2}_{#}(G).

Let us now turn to the casea=−1. The bound (17) shows that χεuε andχε∇uε

admit two scale limits in the sense of boundary layers convergence. Denoting by
w_{−1} and ζ_{−1} these limits, we have for allψ(x^{0}, y)∈C_{c}^{∞}

Ω^{0};C_{c#}^{∞}(G)

, Ψ(x^{0}, y)∈
C_{c}^{∞}

Ω^{0};C_{c#}^{∞}(G)d

andϕ(t)∈C[0, T],

ε→0lim 1 ε

Z T 0

ϕ(t)dt Z

Ω

χεuεψ
x^{0},x

ε

dx = Z T

0

ϕ(t)dt Z

Ω^{0}×G

w_{−1}(t, x^{0}, y)ψ(x^{0}, y)dx^{0}dy,

ε→0lim 1 ε

Z T 0

ϕ(t)dt Z

Ω

χ_{ε}∇uε·Ψ
x^{0},x

ε

dx = Z T

0

ϕ(t)dt Z

Ω^{0}×G

ζ_{−1}(t, x^{0}, y)·Ψ (x^{0}, y)dx^{0}dy.

We select a test function Ψ = (Ψ1, . . . ,Ψd) such that Ψ(x^{0}, y)χ(yd) = Ψ(x^{0}, y),
integrating by parts and passing to the limit we obtain a condition onw−1,

Z T 0

ϕ(t)dt Z

Ω^{0}×G

w_{−1}(t, x^{0}, y)divyΨ (x^{0}, y)dx^{0}dy= 0,

which implies that the two-scale limit w_{−1} ofχ_{ε}u_{ε} is independent ofy within the
support ofχ:

w−1(t, x^{0}, y) =χ(yd)u−1(t, x^{0}).

Selecting nowΨ^{0}= (Ψ1, . . . ,Ψ_{d−1},0) = Ψ^{0}(x^{0}, y)χ(yd)satisfying additionnally that
divyΨ^{0} = 0, we have

Z T 0

ϕ(t)dt Z

Ω^{0}×G

ξ_{−1}(t, x^{0}, y)·Ψ^{0}(x^{0}, y)dx^{0}dy

= −

Z T 0

ϕ(t)dt Z

Ω^{0}×G

χ(yd)u−1(t, x^{0})divx^{0}Ψ^{0}(x^{0}, y)dx^{0}dy

= Z T

0

ϕ(t)dt Z

Ω^{0}×G

χ(y_{d})∇_{x}^{0}u_{−1}(t, x^{0})·Ψ^{0}(x^{0}, y)dx^{0}dy.

Since the orthogonal set in L^{2}(Ω^{0} ×Y) of all Y^{0}-periodic functions Ψ^{0} satisfying
the constraint div_{y}Ψ^{0} = 0 is precisely the set of all gradients ∇yw(x^{0}, y) with
w∈L^{2}(Ω^{0};H_{#}^{1}(Y))(recall that these last functions are not periodic in the last y_{d}
variable), we have proved that

ζ_{−1}(t, x^{0}, y) =χ(yd)∇x^{0}u_{−1}(t, x^{0}) +χ(yd)∇yw(t, x^{0}, y).

4. Proof of Theorem 1

For any test functionψ(x^{0}, x/ε)such thatψ(x^{0}, y)χ(yd) =ψ(x^{0}, y), and any smooth
functionϕ(t)such thatϕ(0) =ϕ(T) = 0, we obtain from Proposition 7 that

ε Z T

0

ϕ(t)dt Z

Ω

ρε(x)∂uε

∂t ψ(x^{0}, x/ε)dx = −
Z T

0

dt Z

Ω^{0}

Z

G

ψ(x^{0}, y)ρ1(y)u2(t, x^{0}, y)∂ϕ

∂tdx^{0}dy+o(1),
Z T

0

ϕ(t)dt Z

Ω

ε^{2}Aε∇uε· ∇ψ(x^{0}, x/ε)dx =
Z T

0

ϕdt Z

Ω^{0}

Z

G

µ1(y)∇yu2(t, x^{0}, y)· ∇yψ(x^{0}, y)dx^{0}dy+o(1),
ε

Z T 0

ϕ(t)dt Z

Ω

ωεuεψ(x^{0}, x/ε)dx =
Z T

0

ϕ(t)dt Z

Ω^{0}

Z

G

ω(y)u2(t, x^{0}, y)ψ(x^{0}, y)dx^{0}dy+o(1),
Z T

0

ϕ(t)dt Z

Ω

fε(x)ψ(x^{0}, x/ε)dx =
Z T

0

ϕ(t)dt Z

Ω^{0}

Z

G

f1(t, y)ψ(x^{0}, y)dx^{0}dy+o(1).

Here, as in Proposition 7, u2 is the limit ofεuε. Combining these terms as in (1) we deduce thatu2satises the limit system

ρ_{1}(y)∂u_{2}

∂t −div_{y}(µ_{1}(y)∇_{y}u_{2}) +ω_{1}(y)u_{2} = f_{1}(t, y)inΩ^{0}×Y, for a.e. t

u2(t, x^{0}, y) = 0 onΩ^{0}×Y^{0}× {|yd|= 1/2}, for a.e. t
(32)

u2(t= 0) = 0.

Note that u_{2} is independent ofx^{0}: thus, system (32) is in fact (8), and our nota-
tions are consistent. In the sequel we shall write u_{2}(t, y), the unique solution in

C([0, T];H_{#}^{1}(Y))of (8). This rst result, the characterization of the limit behavior
of εuε, does not provide a detailed description of the asymptotic behavior of the
solution of (1), since the support ofu2(·,_{ε}^{·}) shrinks to zero withε. Consequently,
we turn to the second order term, and dene

(33) vε(t, x) =uε(t, x)−1 εu2

t,x

ε

for a.e. (t, x)∈[0, T]×Ω.

The following gives the equation satised byv_{ε}.

Proposition 8. The solution uε of problem (1) can be written uε(t, x) =1

εu2(t, x/ε) +vε(t, x), for a.e. (t, x)∈[0, T]×Ω,

whereu_{2} is the unique solution of system (32). The remainder termv_{ε}satises
(34)

ρε(x)∂vε

∂t −div (Aε(x)∇vε) +ωε(x)vε =s t,^{x}_{ε}

δ_{{x}_{d}_{=ε/2}}−δ_{{x}_{d}_{=−ε/2}} inΩ,

uε = 0 on∂Ω,

uε(t= 0) = 0 inΩ,

where

(35) s(t, y) =µ_{1}(y)∂u_{2}

∂yd

(t, y). Furthermore,vε satises the following a priori estimate (36)

kvεk_{L}∞((0,T);L^{2}(Ω))+k(1−χε)∇vεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d}) +εkχε∇vεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})≤C,
whereC is a constant independent ofε.

Proof. System (34) is obtained without diculty from (1) and (32). From assump- tion (5) we deduce that there existsη >0 such that

u2∈L^{∞} (0, T);C^{1,η}(Y^{0}×[−1/2,−1/2 +η]∪[1/2−η,1/2])

(see e.g. [11, Thm 8.34]). As a consequence,s(t, x/ε)is continuous, and bounded
around x_{d} =±ε/2. Since in the right hand side of (34) s is multiplied by Dirac
functions located at x_{d} = ±ε/2, we can assume without loss of generality that
s(t, x/ε)∈C([0, T]×Ω). Then, estimate (36) is a consequence of (15).

The conclusion of the proof of Theorem 1 is given by the following proposition.

Proposition 9. The solutionvεof (34) is such that(1−χε)vε converges strongly
in L^{2}((0, T)×Ω^{±})to u^{±} and (1−χε)∇vε converges weakly in L^{2}((0, T)×Ω^{±})^{d}
to∇u^{±}, whereu^{±} is the solution of the homogenized problem (6).

Proof. We focus on the convergence tou^{+}, the convergence tou^{−}can be proved by
similar arguments. The bound (36) onkvεk_{L}∞(0,T ,L^{2}(Ω))shows that, up to the pos-
sible extraction of a subsequence, there existsv0(t, x)∈ L^{2} (0, T);L^{2}(Ω^{+})such
that forv_{ε} converges weakly inL^{2} (0, T);L^{2}(Ω)

to v_{0}. Similarly, the bound (36)

onk(1−χε)∇vεk_{L}2((0,T);,L^{2}(Ω)^{d})shows that, up to the possible extraction of a sub-
sequence, there existsξ_{0}(t, x)∈L^{2}

(0, T);L_{2}(Ω)^{d}

such that for(1−χε)∇vεcon-
verges weakly inL^{2}

(0, T);L^{2}(Ω)^{d}

toξ0. For anyΦ∈C_{c}^{1}((0, T)×(Ω^{+}∪Ω^{−}))^{d},
one easily obtains that

Z T 0

Z

Ω^{+}∪Ω^{−}

ξ0·Φdtdx=− Z T

0

Z

Ω^{+}∪Ω^{−}

v0div Φdtdx,

or in other words thatξ_{0} = ∇v_{0} on Ω^{+}∪Ω^{−}. Let θ ∈ C^{2}(R)be a non-negative
one dimensional cut-o function, such that θ(s) = 1 fors≥1/2 andθ(s) = 0for
s≤ −1/2, and dene

θε(x) =θxd

ε

x∈R^{d}.

Given a test functionϕ∈C^{1} (0, T);H_{0}^{1}(Ω), such thatϕ(T,·) = 0, we test system
(34) againstϕθεto obtain

Z T 0

dt Z

Ω^{+}

dx(1−χ_{ε})θ_{ε}(x)v_{ε}(x)

−ρ0(x)dϕ

dt +ω_{0}(x)ϕ

+ Z T

0

dt Z

Ω^{+}

dx(1−χ_{ε})µ_{0}(x)∇vε(x)· ∇ϕ
+

Z T 0

dt Z

Ω^{0}

dx^{0}s

t,x^{0}
ε,1/2

ϕ(t, x^{0}, ε)

= Z T

0

dt Z

Ω

dx χεθεvε

ρε(x)∂ϕ

∂t −ωε(x)ϕ

− Z T

0

dt Z

Ω

dx ε^{2}χεµ1

x ε

∇vε· ∇(θεϕ). Thanks to (36), the rst right-hand side term is bounded by

√ε

ρ_{1}∂ϕ

∂t −ω_{1}ϕ

L^{∞}((0,T);L^{2}(Ω))

kvεk_{L}∞((0,T);L^{2}(Ω))≤C√
ε,
whereas the second right-hand side term is bounded by

εkχε∇vεk_{L}2(^{(0,T);L}^{2}^{(Ω)}^{d})kµ1k_{L}^{∞}_{(}

R^{d})

εk∇ϕk_{L}^{∞}(^{(0,T);L}^{2}^{(Ω)}^{d}) +

√εkϕk_{L}∞((0,T);L^{2}(Ω))kχ∇θk_{L}2(R^{d})

≤C√ ε. Passing to the limit in the left-hand side, we obtain

Z T 0

Z

Ω^{+}

v_{0}

−ρ_{0}(x)∂ϕ

∂t +ω_{0}(x)ϕ

+ Z T

0

Z

Ω^{+}

µ_{0}(x)∇v_{0}·∇ϕ−

Z T 0

Z

Ω^{0}

s^{+}(t)ϕ(t, x^{0},0)dx^{0}= 0.

which is the system (6) satised by u^{+}. The limit system for u^{−} is obtained

similarly.

5. Proof of Theorem 2

Thanks to the a priori estimate (16) we know that uε is a bounded sequence in
L^{∞}((0, T);H_{0}^{1}(Ω)). It therefore admits a weak limit u, up to the extraction of a
subsequence, in L^{2}((0, T);H_{0}^{1}(Ω)). We can now pass to the limit in (1) in any

subdomain ω ⊂ Ω not intersecting the plane xd = 0. Furthermore, thanks to
Lemma 6, we know that for a.e. x^{0}∈Ω^{0} and everyt >0,

u(t, x^{0},0) =
Z

G

χ(yd)u_{−1}(t, x^{0})dy=u_{−1}(t, x^{0}).

We therefore have proved that, up to a subsequence,usatises

ρ_{0}(x)∂u

∂t −div (µ_{0}(x)∇u) +ω_{0}(x)u = 0 inΩ,
u(t, x^{0},0) =u_{−1}(t, x^{0}) onΩ^{0},

u(t= 0) = 0 inΩ,

u = 0 on∂Ω.

Let us now turn to the derivation of equation (10) to conclude the proof.

The following proposition determinesw(t, x, y), dened in Proposition 7, in terms
ofu_{−1}(t, x^{0}).

Proposition 10. The two-scale limit ofχε∇uεis preciselyχ(yd) (Id+P(y))∇x^{0}u−1(t, x^{0})
whereI_{d}is the identity matrix ofR^{d}, andP(y)is the matrix-valued function dened
byP(·)ei=∇ϕi, for i= 1, . . . , d, whereϕ_{i}∈H_{#,d}^{1} (Y)is the solution of (13).

Proof. For any ψ(x^{0}, y) ∈ C(Ω^{0};L^{2}_{#}(G)) and ϕ(t) ∈ C_{c}^{1}(0, T), testing (1) against
εϕ(t)ψ(x^{0}, x/ε)yields

Z T 0

ϕ(t)dt Z

Ω

χε(x)µ1

x ε

∇uε·

∇x^{0}ψ+1
ε∇yψ

= o(1).

Sinceµ1(y)is periodic, it converges strongly with respect to two-scale convergence and we have

1 ε

Z T 0

ϕ(t)dt Z

Ω

µ1

x ε

χε(x)∇uε· ∇yψ dx

= Z T

0

ϕ(t)dt Z

Ω^{0}×G

µ_{1}(y)χ(y_{d}) (∇_{x}^{0}u_{−1}(t, x^{0}) +∇_{y}w(t, x^{0}, y))· ∇_{y}ψ dx^{0}dy+o(1).

The above limit is just the variational formulation, for a.e. x∈Ω^{0} andt >0, of
divy(µ1(y) (∇x^{0}u_{−1}(t, x^{0}) +∇yw(t, x^{0}, y))) = 0iny∈Y,

with Neumann boundary conditions on the lower and upper faces as in (13).

Let us now test (1) againstϕ∈C_{c}^{1}((0, T)×Ω)withϕ(T,·) = 0. We obtain
Z T

0

dt Z

Ω

dx(1−χ_{ε})

u_{ε}

−ρ0(x)dϕ

dt +ω_{0}(x)ϕ

+µ_{0}(x)∇uε· ∇ϕ

+ Z T

0

dt Z

Ω

dxχ_{ε}

u_{ε}

−ρ1

x ε

dϕ

dt +ω_{1}x
ε

ϕ

+1
εµ_{1}x

ε

∇uε· ∇ϕ

− Z T

0

dt Z

Ω

χ_{ε}1

εϕf_{1}dx = 0.

**Storage zone**

**Dogger****Oxfordian**

**L x****L z**

**Callovo−Oxfordian**
**Callovo−Oxfordian**

Η

**L****H**

Figure 2. Nuclear waste repository and surrounding geological layers.

In order to pass to the limit as ε tends to zero, remark that χεϕ or χε∇ϕ are admissible test functions for two-scale convergence in the sense of boundary layers.

As a consequence, we obtain Z T

0

dt Z

Ω

dx

u

−ρ0(x)dϕ

dt (t, x^{0}, x_{d}) +ω_{0}(x)ϕ(t, x^{0}, x_{d})

+µ_{0}(x)∇u· ∇ϕ(t, x^{0}, x_{d})

+ Z T

0

dt Z

Ω^{0}

Z

G

χ(y_{d}) (µ_{1}(y) (I_{d}+P(y)))∇x^{0}u_{−1}(t, x^{0})· ∇x^{0}ϕ(t, x^{0}, x_{d})dydx^{0}

− Z T

0

dt Z

Ω^{0}

dx^{0}
Z

G

χ(yd)f1(t, y)ϕ(t, x^{0}, xd)dy = 0.

This last system is the weak formulation of (9)-(10), Z T

0

Z

Ω

F(u, ϕ)dtdx+

Z T 0

Z

Ω^{0}

F^{0}(u(t, x^{0},0), ϕ(t, x^{0},0))dtdx^{0} =
Z T

0

Z

Ω^{0}

f¯(t)ϕ(t, x^{0},0))dtdx^{0},
wheref(t) is given by(11),F andF^{0} are the bilinear forms

F(u, v) =u

−ρ0

dv dt +ω0v

+µ0∇xu· ∇xv,
F^{0}(u, v) =µ^{∗}∇x^{0}u· ∇x^{0}v,

withµ^{∗} being thed−1periodic homogenized diusion coecient given by (12).

6. Numerical simulation of a nuclear waste storage

In this section, we revisit the scaling of our model (1). Introducing appropriate scales, we perform an adimensionalization and introduce the small parameter ε. Then, using some experimental values of the diusion coecients, we show that the dimensionless derived model is of the same type as (1). Relying on the previously obtained homogenized problems we perform numerical simulations of the long time behavior of the nuclear waste storage. These computations are done with the FreeFem++ software which may be downloaded from [10].