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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�

(2)

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

(3)
(4)

TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS

GREGOIRE ALLAIRE1, MARC BRIANE2, ROBERT BRIZZI1, YVES CAPDEBOSCQ3

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

2 Centre de Mathématiques, INSA de Rennes et IRMAR, 20 avenue des Buttes de Coësmes - 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)









ρε(x)∂uε

∂t −div (Aε(x)∇uε) +ωε(x)uε = fε(t, x) inΩfor a.e. t >0, uε = 0on∂Ωfor a.e. t >0, uε(t= 0) = 0in Ω,

whereΩis a smooth open set ofRd, withd= 2or3, which intersects the horizontal hyperplane passing through the origin, i.e.,Ω∪ {xd = 0} 6=∅ (for any x∈Rd we write x = (x0, xd) with x0 ∈ Rd−1). The unknown is uε(t, x) ∈ L2(R+;H01(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

(5)

Oxfordian

Lx Lz

Storage zone

Dogger Callovo−Oxfordian Callovo−Oxfordian

Η

Figure 1. Sketch of the medium.

repository, the diusion tensorAεand the scalar porositiesωεandρεare functions of the macroscopic or slow variablexonly. More precisely, we assume that there exist functionsµ0, ω0 andρ0 inL(Ω)such that

(2)

for anyx= (x0, xd)∈Ωwith |xd|> ε/2, fε(t, x) = 0,∀t≥0 andAε(x) (resp. ωε(x), ρε(x)) =µ0(x) (resp. ω0(x), ρ0(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= (x0, xd)∈Ωwith |xd|< ε/2, fε(t, x) =ε−1f1 t,xε

, ∀t≥0 andAε(x) =εaµ1 x

ε

, ωε(x) =ω1 x ε

andρε(x) =ρ1 x ε

,

with periodic functionsµ1∈L#,d(Y)d×d11∈L#,d(Y)andf1∈L(R+;L#,d(Y)). We assume thatω0, ρ0≥c >0(resp. ω1, ρ1≥c >0) where the constantc is inde- pendent ofε and that both diusion tensorsµ0 andµ1 are uniformly elliptic and bounded, that is, there exists two positive constantsβ ≥α > 0 such that, for all ξ∈Rd,

(4) α|ξ|2≤µiξ·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(Rd), s.t. ∀y= (y0, yd)∈Rd and∀p∈Zd−1, f(y0+p, yd) =f(y0, 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 xεd

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 Ω0 = Ω∩ {xd = 0}, Ω+ = Ω∩ {xd > 0}, Ω= Ω∩ {xd<0}, andY0 = (−1/2,+1/2)d−1.

(6)

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 group1, 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) ∃xd0∈(0,1/2)andr >0 such thatµ1(y)∈C#0,r Y ∩ {|xd|> xd0}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.

(7)

Theorem 1. Under assumptions (2,3,4,5), when a= 2, the solution uε of (1) is such that(1−χε)uεconverges strongly inL2 (0, T);L2(Ω)

tou, given byu:=u+ inΩ+andu:=uinΩ, and(1−χε)∇uεconverges weakly inL2

(0, T);L2(Ω)d toξ, with ξ=∇u± inΩ±, whereu± is the unique solution of

(6)













ρ0(x)∂u±

∂t −div (µ0(x)∇u±) +ω0(x)u±= 0 inΩ±, u±= 0 on∂Ω∩∂Ω±,

∓µ0(x)∇u±·ed=s± onΩ0, u±(t= 0) = 0 inΩ.

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

Z

Y0

µ1(y0,±1/2)∇u2(t, y0,±1/2)·eddy0,

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 inL2 (0, T);H1(Y) of the following problem

(8)









ρ1(y)∂u2

∂t −divy1(y)∇yu2) +ω1(y)u2 = f1(t, y)in(0, T)×Y, u2(t, y) = 0 on(0, T)× {|yd|= 1/2}, (y1, . . . , yd−1) → u2(t, y)Y0-periodic,

u2(t= 0) = 0.

Asymptotically, the eect of the strip is therefore to transform the source term f1 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 uε of (1) is such that it converges strongly inL2 (0, T);L2(Ω)touand∇uεconverges weakly inL2

(0, T);L2(Ω)d

to∇u, whereu∈H1(Ω)is the unique solution of the coupled system

(9)









 ρ0∂u

∂t −div(µ0∇u) +ω0u = 0in Ω+∪Ω u(t= 0) = 0in Ω+∪Ω,

u = 0on ∂Ω,

u =u−1 on Ω0= Ω∩ {xd= 0}, andu−1 is the solution in H1(Ω0)of

(10) −divx0x0u−1(t, x)) + [µ0∇u·n] =f ,

(8)

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#,d1 (Y) is Y0-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 Y0× {+1/2}, µ1(∇yi+∇yϕi)·ed = 0on Y0× {−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 solutionuε of (1) is uniformly bounded in appropriate norms. The necessary a priori estimates are established in the following Proposition.

Proposition 3. Assume the coecients Aε, ωε, ρε satisfy assumptions (2, 3, 4).

(i) If, instead of assumption (3), we suppose thatfε(t, x) =εbχε xd

ε

f1 t,xεwith f1∈L(R+;L#,d(Y0×(−1/2,1/2))), andb∈R, we have

kuεkL((0,T);L2(Ω))+k(1−χε)∇uεkL2((0,T);L2(Ω)d)

a2ε∇uεkL2((0,T);L2(Ω)d) ≤ Cε12+bmin 1, εa2 (14) ,

(ii) Alternatively, instead of assumption (3) assume thatfε(t, x) =sε(t, x0{|xd|=ε/2} xεd where sε ∈ C R+×Rd−1

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

(15)

kuεkL(0,T ,L2(Ω))+k(1−χε)∇uεkL2((0,T);L2(Ω)d) +εa2ε∇uεkL2((0,T);L2(Ω)d)≤C.

Corollary 4. Assume the coecientsAε, ωε, ρε satisfy assumptions (2, 3 and 4).

If a=−1, then we have

(16) kuεkL((0,T);L2(Ω))+k(1−χε)∇uεkL((0,T);L2(Ω)d)≤C, and

(17) kχεuεkL2((0,T);L2(Ω))+kχε∇uεkL2((0,T);L2(Ω)d)≤C√ ε.

(9)

If a= 2, thenwε=εuεsatises (18)

kwεkL((0,T);L2(Ω))+k(1−χε)∇wεkL2((0,T);L2(Ω)d)+εkχε∇wεkL2((0,T);L2(Ω)d)≤C√ ε which, in particular, implies that

(19) kχεwεkL((0,T);L2(Ω))+εkχε∇wεkL2((0,T);L2(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) againstuεwe obtain, after integration by parts with respect to the space variablex,

1 2

d dt

Z

ρεu2εdx+ Z

(1−χε0(x)∇uε· ∇uεdx+ Z

ωx ε

u2εdx +εa

Z

χεµ1

x ε

∇uε· ∇uεdx = Z

χεfεuεdx.

(20)

Let us rst address the case (i): there is a constantM such thatkfεkL(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

ρε12uε

2 L2(Ω)

≤CkχεfεkL2(Ω)

χερε12uε

L2(Ω)≤Cε12+b

χερε12uε L2(Ω), which in turns shows that kuε(t,·)kL2(Ω) ≤Ctε12+b, the constantC, depending a priori onminε|, andM being chosen independently ofε. Integrating (20) with respect to time, and using Cauchy-Schwarz inequality again we obtain,

(21) k(1−χε)∇uεk2L2((0,T);L2(Ω)d) +εaε∇uεk2L2((0,T);L2(Ω)d)≤Cε1+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) kχεuεk2L2(Ω)≤dε∇uεk2L2(Ω)d,

where ddepends 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

χεfεuεdx ≤ ε−a d

4λαkχεfεk2L2(Ω)aαλ

dεuεk2L2(Ω),

≤ ε−a d

4αλkχεfεk2L2(Ω)aλ Z

χεµ1(x

ε)∇uε· ∇uεdx.

(23)

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

2 d dt

Z

ρεu2εdx ≤ d

4αε−aεfεk2L2(Ω)

≤ Cε−a+1+2b, (24)

(10)

which shows thatkuε(t,·)kL2(Ω) ≤Cε12+b−12a. Forλ= 1/2, using (23) in (20) we obtain that for allt >0,

1 2

d dt

Z

ρεu2εdx+ Z

(1−χε0(x)∇uε· ∇uεdx +1

a Z

χεµ1x ε

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

k(1−χε)∇uεk2L2((0,T);L2(Ω)d)aε∇uεk2L2((0,T);L2(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, x0{|xd|=ε/2} with sε ∈ C(R+ ×Rd−1), and ksεkL < M where the constant M is independent ofε. Then

Z

fε(t, x)uεdx ≤ M Z

{xd=±ε/2}

|uε|dx0≤ M2 τ +Cτ

Z

{xd=±ε/2}

|uε|2dx0 (25)

≤ M2 τ +

Z

(1−χε0(x)∇uε· ∇uεdx (26)

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

2 d dt

Z

ρε(x)u2εdx≤ M2 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 bandY0×R. A generic pointy∈Gis denoted byy= (y0, yd) with y0 ∈ Y0 and yd ∈ R. We introduce the space L2#(G) of square integrable functions onGwhich are periodic in the rstd−1variables, i.e.

L2#(G) =

ϕ∈L2(G)s.t. y07→ϕ(y0, yd)isY0-periodic .

Similarly, we deneH#1(G)the Sobolev space of functions inH1(G)which areY0- 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(Ω0)the space of continuous functions on the closure ofΩ0, a compact set inRd−1.

Proposition 5. (1) Letvε be a sequence inL2(Ω) such that kvεkL2(Ω)≤C√

ε.

There exists a subsequence, still denoted byε, and a limitv0(x0, y)∈L2

0;L2#(G) such that vε two-scale converges weakly in the sense of boundary layers tov0 i.e.

ε→0lim 1 ε

Z

vε(x)ϕ x0,x

ε

dx = 1

|Y0| Z

0

Z

G

v0(x0, y)ϕ(x0, y)dx0dy

(11)

for all test functionsϕ(x0, y)∈C

0;L2#(G) .

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

ε→0lim

√1

εkvεkL2(Ω) = 1

|Y0|12

kv0kL2(Ω0×G).

Then,vεis said to two-scale converge strongly in the sense of boundary layers tov0, which means that, for any sequencewε in L2(Ω) which two-scale converges weakly in the sense of boundary layers tow0(x0, y)∈L2(Ω0×G)one has

ε→0lim 1 ε

Z

vε(x)wε(x)ϕ x0,x

ε

dx = 1

|Y0|12 Z

0

Z

G

v0(x0, y)w0(x0, y)ϕ(x0, y)dx0dy, for all smooth functionsϕ(x0, y)∈C Ω0;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 inH01(Ω) converging weakly to a limitv. Assume that wεεvε admits a limit w in the sense of boundary layers conver- gence. Then w=χwand for almost everyx0∈Ω0,

v(x0,0) = Z

Y

w(x0, y)dy.

Proof. Recall thatχε=χ(xε)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 kwεkL2(Ω) ≤ C√

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

1 ε

Z

χεvε(x0,0)ϕ(x)dx− Z

0

vε(x0,0)ϕ(x0,0)dx0

= 1 ε

Z

χεvε(x0,0)ϕ(x0,0)dx− Z

0

vε(x0,0)ϕ(x0,0)dx0 +

Z

vε(x0,0)1

εχε(ϕ(x0,0)−ϕ(x))dx.

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

1

εχε(ϕ(x0,0)−ϕ(x))

≤ 1 ε

Z ε/2

−ε/2

∂xd

ϕ(x0, t)

dt≤ k∇ϕkL(Ω), therefore,

Z

vε(x0,0)1

εχε(ϕ(x0,0)−ϕ(x))dx

≤ k∇ϕkL(Ω)ε Z

0

vε(x0,0)dx0

≤ Cεk∇ϕkL(Ω)k∇vεkL2(Ω)d,

(12)

where this last term is of order O(ε)since the sequence vε is bounded in H01(Ω). Similarly,

Z

1

εχε(vε(x0,0)−vε(x))

dx ≤ Z

χε ε

Z ε/2

−ε/2

∂xd

vε(x0, sd)

dsddx

≤ C√

εk∇vεkL2(Ω)d. As a consequence,

Z

1

εχεvε(x)ϕ(x)dx− Z

0

v(x0,0)ϕ(x0,0)dx0

≤ Z

1

εχεvε(x0,0)ϕ(x)dx− Z

0

v(x0,0)ϕ(x0,0)dx0

+O(√ ε)

≤ Z

0

|vε(x0,0)−v(x0,0)| |ϕ(x0,0)|dx0+O(√ ε)

≤ C(ϕ)kvε(·,0)−u(·,0)kL2(Ω0)+O(√ ε).

The trace of vε onΩ0 converge strongly in L2(Ω0) to the trace ofv, thus we have proved that

Z

0

Z

G

w(x0, y)ϕ(x0,0)dx0dy = lim

ε→0

Z

1

εχε(x)vε(x)ϕ(x0,0)dx

= lim

ε→0

Z

1

εχε(x)vε(x)ϕ(x)dx

= Z

0

v(x0,0)ϕ(x0,0)dx0.

which implies the desired result, namely thatu(x0,0)is the average onY ofw(x0, 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, x0) ∈ L2 (0, T);H1(Ω0)

and a function w(t, x0, y) ∈ L2

(0, T)×Ω0;Hloc,#1 (G) such that, up to a subsequence,

χεuε −−−2SBL* χ(yd)u−1(t, x0) (27)

χε∇uε

−−−2SBL* χ(yd) (∇x0u−1(t, x0) +∇yw(t, x0, y)). (28)

(ii) Suppose thata= 2. There exists a functionu2(t, x, y)∈L2

(0, T)×Ω0;Hloc,#1 (G) such that, up to a subsequence,

εuε −−−2SBL* u2(t, x0, y) (29)

ε2∇uε −−−2SBL* χ(yd)∇yu2(t, x0, y) (30)

Furthermore,

(31) (1−χ(yd))u2(t, x0, y) = 0 for a.e. y∈G.

(13)

Note. We use the notation

x0 = ∂

∂x1

, . . . , ∂

∂xd−1

,0

. Similarly, for addimensional vectorΨwe shall write

Ψ0= (Ψ1, . . . ,Ψd−1,0).

Proof. Supposea= 2.The bound (18) shows thatwε=εuεsatises kwεkL2((0,T);L2(Ω))+εk∇wεkL2((0,T);L2(Ω)d)≤C√

ε.

Consequently,wε(resp. ε∇wε) admits a limitu2(resp.ζ2) in the sense of boundary layer two scale limits. It is a classical result [2], [3] thatε∇wεtwo 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−χε)wεkL2(0,T;L2(Ω))+k(1−χε)∇wεkL2(0,T;L2(Ω)d)≤C√ ε.

Therefore, up to a subsequence, (1−χε)∇wε converges in the sense of bound- ary layers to ξ2(t, x0, y). Since χε converges strongly in the sense of two-scale boundary layers convergence toχ, we have ξ2(t, x0, y) = (1−χ(yd))ξ2(t, x0, y). Let Ψ(x0, y) ∈ Cc

0;C#,c (G)d

be a smooth test function such that Ψ(x0, y)(1− χ(yd)) = Ψ(x0, 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

(1−χε)∇wε·Ψ x0,x

ε

dt dx

= −lim

ε→0

1 ε2

Z T 0

ϕ(t) Z

wεdivyΨ x0,x

ε

dt dx0dy

−lim

ε→0

1 ε

Z T 0

ϕ(t) Z

wεdivx0Ψ x0,x

ε

dt dx0dy.

This implies that

ε→0lim 1 ε

Z T 0

ϕ(t) Z

wεdivyΨ x0,x

ε

dt dx= 0,

that is, (1−χ)u2(t, x0, y) does not depend on y and is therefore 0, since y 7→

u2(·,·, y)∈L2#(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(x0, y)∈Cc

0;Cc#(G)

, Ψ(x0, y)∈ Cc

0;Cc#(G)d

andϕ(t)∈C[0, T],

ε→0lim 1 ε

Z T 0

ϕ(t)dt Z

χεuεψ x0,x

ε

dx = Z T

0

ϕ(t)dt Z

0×G

w−1(t, x0, y)ψ(x0, y)dx0dy,

ε→0lim 1 ε

Z T 0

ϕ(t)dt Z

χε∇uε·Ψ x0,x

ε

dx = Z T

0

ϕ(t)dt Z

0×G

ζ−1(t, x0, y)·Ψ (x0, y)dx0dy.

(14)

We select a test function Ψ = (Ψ1, . . . ,Ψd) such that Ψ(x0, y)χ(yd) = Ψ(x0, y), integrating by parts and passing to the limit we obtain a condition onw−1,

Z T 0

ϕ(t)dt Z

0×G

w−1(t, x0, y)divyΨ (x0, y)dx0dy= 0,

which implies that the two-scale limit w−1 ofχεuε is independent ofy within the support ofχ:

w−1(t, x0, y) =χ(yd)u−1(t, x0).

Selecting nowΨ0= (Ψ1, . . . ,Ψd−1,0) = Ψ0(x0, y)χ(yd)satisfying additionnally that divyΨ0 = 0, we have

Z T 0

ϕ(t)dt Z

0×G

ξ−1(t, x0, y)·Ψ0(x0, y)dx0dy

= −

Z T 0

ϕ(t)dt Z

0×G

χ(yd)u−1(t, x0)divx0Ψ0(x0, y)dx0dy

= Z T

0

ϕ(t)dt Z

0×G

χ(yd)∇x0u−1(t, x0)·Ψ0(x0, y)dx0dy.

Since the orthogonal set in L2(Ω0 ×Y) of all Y0-periodic functions Ψ0 satisfying the constraint divyΨ0 = 0 is precisely the set of all gradients ∇yw(x0, y) with w∈L2(Ω0;H#1(Y))(recall that these last functions are not periodic in the last yd variable), we have proved that

ζ−1(t, x0, y) =χ(yd)∇x0u−1(t, x0) +χ(yd)∇yw(t, x0, y).

4. Proof of Theorem 1

For any test functionψ(x0, x/ε)such thatψ(x0, y)χ(yd) =ψ(x0, 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

ρε(x)∂uε

∂t ψ(x0, x/ε)dx = − Z T

0

dt Z

0

Z

G

ψ(x0, y)ρ1(y)u2(t, x0, y)∂ϕ

∂tdx0dy+o(1), Z T

0

ϕ(t)dt Z

ε2Aε∇uε· ∇ψ(x0, x/ε)dx = Z T

0

ϕdt Z

0

Z

G

µ1(y)∇yu2(t, x0, y)· ∇yψ(x0, y)dx0dy+o(1), ε

Z T 0

ϕ(t)dt Z

ωεuεψ(x0, x/ε)dx = Z T

0

ϕ(t)dt Z

0

Z

G

ω(y)u2(t, x0, y)ψ(x0, y)dx0dy+o(1), Z T

0

ϕ(t)dt Z

fε(x)ψ(x0, x/ε)dx = Z T

0

ϕ(t)dt Z

0

Z

G

f1(t, y)ψ(x0, y)dx0dy+o(1).

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

ρ1(y)∂u2

∂t −divy1(y)∇yu2) +ω1(y)u2 = f1(t, y)inΩ0×Y, for a.e. t

u2(t, x0, y) = 0 onΩ0×Y0× {|yd|= 1/2}, for a.e. t (32)

u2(t= 0) = 0.

Note that u2 is independent ofx0: thus, system (32) is in fact (8), and our nota- tions are consistent. In the sequel we shall write u2(t, y), the unique solution in

(15)

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 byvε.

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,

whereu2 is the unique solution of system (32). The remainder termvεsatises (34)













ρε(x)∂vε

∂t −div (Aε(x)∇vε) +ωε(x)vε =s t,xε

δ{xd=ε/2}−δ{xd=−ε/2} inΩ,

uε = 0 on∂Ω,

uε(t= 0) = 0 inΩ,

where

(35) s(t, y) =µ1(y)∂u2

∂yd

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

kvεkL((0,T);L2(Ω))+k(1−χε)∇vεkL2((0,T);L2(Ω)d) +εkχε∇vεkL2((0,T);L2(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);C1,η(Y0×[−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 xd =±ε/2. Since in the right hand side of (34) s is multiplied by Dirac functions located at xd = ±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 L2((0, T)×Ω±)to u± and (1−χε)∇vε converges weakly in L2((0, T)×Ω±)d to∇u±, whereu± is the solution of the homogenized problem (6).

Proof. We focus on the convergence tou+, the convergence toucan be proved by similar arguments. The bound (36) onkvεkL(0,T ,L2(Ω))shows that, up to the pos- sible extraction of a subsequence, there existsv0(t, x)∈ L2 (0, T);L2(Ω+)such that forvε converges weakly inL2 (0, T);L2(Ω)

to v0. Similarly, the bound (36)

(16)

onk(1−χε)∇vεkL2((0,T);,L2(Ω)d)shows that, up to the possible extraction of a sub- sequence, there existsξ0(t, x)∈L2

(0, T);L2(Ω)d

such that for(1−χε)∇vεcon- verges weakly inL2

(0, T);L2(Ω)d

toξ0. For anyΦ∈Cc1((0, T)×(Ω+∪Ω))d, one easily obtains that

Z T 0

Z

+∪Ω

ξ0·Φdtdx=− Z T

0

Z

+∪Ω

v0div Φdtdx,

or in other words thatξ0 = ∇v0 on Ω+∪Ω. Let θ ∈ C2(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∈Rd.

Given a test functionϕ∈C1 (0, T);H01(Ω), such thatϕ(T,·) = 0, we test system (34) againstϕθεto obtain

Z T 0

dt Z

+

dx(1−χεε(x)vε(x)

−ρ0(x)dϕ

dt +ω0(x)ϕ

+ Z T

0

dt Z

+

dx(1−χε0(x)∇vε(x)· ∇ϕ +

Z T 0

dt Z

0

dx0s

t,x0 ε,1/2

ϕ(t, x0, ε)

= Z T

0

dt Z

dx χεθεvε

ρε(x)∂ϕ

∂t −ωε(x)ϕ

− Z T

0

dt Z

dx ε2χεµ1

x ε

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

√ε

ρ1∂ϕ

∂t −ω1ϕ

L((0,T);L2(Ω))

kvεkL((0,T);L2(Ω))≤C√ ε, whereas the second right-hand side term is bounded by

εkχε∇vεkL2((0,T);L2(Ω)d)kµ1kL(

Rd)

εk∇ϕkL((0,T);L2(Ω)d) +

√εkϕkL((0,T);L2(Ω))kχ∇θkL2(Rd)

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

Z T 0

Z

+

v0

−ρ0(x)∂ϕ

∂t +ω0(x)ϕ

+ Z T

0

Z

+

µ0(x)∇v0·∇ϕ−

Z T 0

Z

0

s+(t)ϕ(t, x0,0)dx0= 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);H01(Ω)). It therefore admits a weak limit u, up to the extraction of a subsequence, in L2((0, T);H01(Ω)). We can now pass to the limit in (1) in any

(17)

subdomain ω ⊂ Ω not intersecting the plane xd = 0. Furthermore, thanks to Lemma 6, we know that for a.e. x0∈Ω0 and everyt >0,

u(t, x0,0) = Z

G

χ(yd)u−1(t, x0)dy=u−1(t, x0).

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













ρ0(x)∂u

∂t −div (µ0(x)∇u) +ω0(x)u = 0 inΩ, u(t, x0,0) =u−1(t, x0) onΩ0,

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, x0).

Proposition 10. The two-scale limit ofχε∇uεis preciselyχ(yd) (Id+P(y))∇x0u−1(t, x0) whereIdis the identity matrix ofRd, andP(y)is the matrix-valued function dened byP(·)ei=∇ϕi, for i= 1, . . . , d, whereϕi∈H#,d1 (Y)is the solution of (13).

Proof. For any ψ(x0, y) ∈ C(Ω0;L2#(G)) and ϕ(t) ∈ Cc1(0, T), testing (1) against εϕ(t)ψ(x0, x/ε)yields

Z T 0

ϕ(t)dt Z

χε(x)µ1

x ε

∇uε·

x0ψ+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

µ1

x ε

χε(x)∇uε· ∇yψ dx

= Z T

0

ϕ(t)dt Z

0×G

µ1(y)χ(yd) (∇x0u−1(t, x0) +∇yw(t, x0, y))· ∇yψ dx0dy+o(1).

The above limit is just the variational formulation, for a.e. x∈Ω0 andt >0, of divy1(y) (∇x0u−1(t, x0) +∇yw(t, x0, y))) = 0iny∈Y,

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

Let us now test (1) againstϕ∈Cc1((0, T)×Ω)withϕ(T,·) = 0. We obtain Z T

0

dt Z

dx(1−χε)

uε

−ρ0(x)dϕ

dt +ω0(x)ϕ

0(x)∇uε· ∇ϕ

+ Z T

0

dt Z

dxχε

uε

−ρ1

x ε

dt +ω1x ε

ϕ

+1 εµ1x

ε

∇uε· ∇ϕ

− Z T

0

dt Z

χε1

εϕf1dx = 0.

(18)

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

dx

u

−ρ0(x)dϕ

dt (t, x0, xd) +ω0(x)ϕ(t, x0, xd)

0(x)∇u· ∇ϕ(t, x0, xd)

+ Z T

0

dt Z

0

Z

G

χ(yd) (µ1(y) (Id+P(y)))∇x0u−1(t, x0)· ∇x0ϕ(t, x0, xd)dydx0

− Z T

0

dt Z

0

dx0 Z

G

χ(yd)f1(t, y)ϕ(t, x0, xd)dy = 0.

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

0

Z

F(u, ϕ)dtdx+

Z T 0

Z

0

F0(u(t, x0,0), ϕ(t, x0,0))dtdx0 = Z T

0

Z

0

f¯(t)ϕ(t, x0,0))dtdx0, wheref(t) is given by(11),F andF0 are the bilinear forms

F(u, v) =u

−ρ0

dv dt +ω0v

0xu· ∇xv, F0(u, v) =µx0u· ∇x0v,

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].

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