ECOLE POLYTECHNIQUE. Two Asymptotic Models for Arrays of Underground Waste Containers CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS 7641

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1 ECOLE POLYTECHNIQUE CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS PALAISEAU CEDEX FRANCE. Tél: Fax: Two Asymptotic Models for Arrays of Underground Waste Containers Gregoire Allaire, Marc Briane, Robert Brizzi, Yves Capdeboscq R.I. 657 January 29

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3 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS GREGOIRE ALLAIRE, MARC BRIANE 2, ROBERT BRIZZI, YVES CAPDEBOSCQ 3 Centre de Mathématiques Appliquées, École Polytechnique, 928 Palaiseau, France. 2 Centre de Mathématiques, INSA de Rennes et IRMAR, 2 avenue des Buttes de Coësmes Rennes, France. 3 University of Oxford, OXFORD OX 3LB, UK Abstract. We study the homogenization of two models of an underground 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 rigorously 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. 2 Mathematics Subject Classication: 35B27, 74Q.. 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 ρ x u t div A x u + ω x u = f t, x in for a.e. t >, u = on for a.e. t >, u t = = in, where is a smooth open set of R d, with d = 2 or 3, which intersects the horizontal hyperplane passing through the origin, i.e., {x d = } = for any x R d we write x = x, x d with x R d. The unknown is u t, x L 2 R + ; H. The repository, where the inhomogeneities and the source term f are located, is an horizontal strip of thickness and midplane {x d = }. Furthermore, the small parameter is the periodicity of the repository heterogeneities too. Outside the

4 2 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ Oxfordian L z Storage zone Callovo Oxfordian Callovo Oxfordian Η Dogger L x Figure. Sketch of the medium. repository, the diusion tensor A and the scalar porosities ω and ρ are functions of the macroscopic or slow variable x only. More precisely, we assume that there exist functions µ, ω and ρ in L such that { for any x = x 2, x d with x d > /2, f t, x =, t and A x resp. ω x, ρ x = µ x resp. ω x, ρ x. Inside the repository, we assume that all coecients, as well as the source term, are periodic in the rst d variables. We furthermore consider a highly contrasted diusion tensor compared to µ. Namely, we assume 3 { for any x = x, x d with x d < /2, f t, x = f t, x, t and A x = a µ x, ω x = ω x and ρ x = ρ x, with periodic functions µ L #,d Y d d, ω, ρ L #,d Y and f L R + ; L #,dy. We assume that ω, ρ c > resp. ω, ρ c > where the constant c is independent of and that both diusion tensors µ and µ are uniformly elliptic and bounded, that is, there exists two positive constants β α > such that, for all ξ R d, 4 α ξ 2 µ i ξ ξ β ξ 2 i =,. The exponent a in 3 is either or 2. The periodicity cell is the unit cube Y = /2, +/2 d, and by L #,dy we denote the following space L #,dy = { f L R d, s.t. y = y, y d R d and p Z d, fy + p, y d = fy, y d }. Note that the scaling of the source term is such that its integral over the repository is of order one. We denote by χ the characteristic function of the repository, that is, χ x = χ x d where χ is the indicator function of the interval /2, +/2. We denote by T > the nite time-scale for which the model under consideration is valid. We also use the notations = {x d = }, + = {x d > }, = {x d < }, and Y = /2, +/2 d.

5 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS 3 The motivation for this work is the study on the long time behavior of an underground waste repository in the presence of leakage. A sketch of the medium under consideration is presented in Figure. The spent fuel cells are stored in the centre of the middle geological layer. Inside each module, to 5 spent fuel cells are stored horizontally normally to the section represented. A description of the geometry is presented in Section 6.. 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 containment area. The second case, is that of a large permeabilty, a =, 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 = and a = 2 in the limit as tends 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 case a = 2 in Section 4 and for the case a = in Section 5. A related result appeared in [4] 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, 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 parameter has 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 Main Results The rst result addresses the case when a = 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, /2 and r > such that µ y C,r # Y { xd > x d } d d. Our result describes the limit behavior of u both outside the fuel repository and inside. The 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.

6 4 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ Theorem. Under assumptions 2,3,4,5, when a = 2, the solution u of is such that χ u converges strongly in L 2, T ; L 2 to u, given by u := u + in + and u := u in, and χ u converges weakly in L 2, T ; L 2 d to ξ, with ξ = u ± in ±, where u ± is the unique solution of ρ x u± div µ x u ± + ω x u ± = in ±, t u 6 ± = on ±, µ x u ± e d = s ± on, u ± t = = in. The eective source terms s + and s are given by 7 s ± t = ± µ y, ±/2 u 2 t, y, ±/2 e d dy, Y where u 2 is the limit of u in the sense of two-scale boundary layers convergence see Proposition 5. The limit u 2 t, y is the unique solution in L 2, T ; H Y of the following problem 8 ρ y u 2 t div y µ y y u 2 + ω yu 2 = f t, y in, T Y, u 2 t, y = on, T { y d = /2}, y,..., y d u 2 t, y Y -periodic, u 2 t = =. Asymptotically, the eect of the strip is therefore to transform the source term f into two eective time-dependent boundary conditions s + or s 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 is related to a recent result in [4], for nonlinear reactiondiusion systems. However the scaling, as well as the analysis, are quite dierent. We now address the second case when a =. Theorem 2. Under assumptions 2,3,4, when a =, the solution u of is such that it converges strongly in L 2, T ; L 2 to u and u converges weakly in L, 2 T ; L 2 d to u, where u H is the unique solution of the coupled system u ρ t div µ u + ω u = in + 9 ut = = in +, u = on, u = u on = {x d = }, and u is the solution in H of div x µ x u t, x + [µ u n] = f,

7 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS 5 where [j] stands for the jump of a quantity j between + and. The eective coecients µ and f are given by ft = f t, y dy, and 2 µ = Y Y µ y I d + P y dy. For each i =,..., d, P e i is given by P e i = ϕ i where ϕ i H#,d Y is Y -periodic in its d rst variables and is the unique solution, up to an additive constant, of 3 div µ y i + y ϕ i = in Y, µ y i + y ϕ i e d = on Y {+/2}, µ y i + y ϕ i e d = on Y { /2}. Of course, the two partial dierential equations 9 and are strongly coupled. Asymptotically, the eect of the strip is therefore to transform the source term f into an eective time-dependent tangential diusion along the interface given by. The homogenized problem 9- is the inspiration for an approximate numerical method in Section 6. For a homogeneous thin strip, Theorem 2 was already obtained in [2]. 3. A priori estimates and two-scale boundary layer convergence A prerequisite for the proofs of Theorems and 2 is to check that the solution u of is uniformly bounded in appropriate norms. The necessary a priori estimates are established in the following Proposition. Proposition 3. Assume the coecients A, ω, ρ satisfy assumptions 2, 3, 4. i If, instead of assumption 3, we suppose that f t, x = b χ xd f t, x with f L R + ; L #,d Y /2, /2, and b R, we have 4 u L,T ;L 2 + χ u L2,T ;L 2 d + a 2 χ u L2,T ;L 2 d C 2 +b min, a 2, ii Alternatively, instead of assumption 3 assume that f t, x = s t, x δ { xd =/2} xd where s C R + R d is bounded above and below independently of. Then we have 5 u L,T,L 2 + χ u L2,T ;L 2 d + a 2 χ u L2,T ;L 2 d C. Corollary 4. Assume the coecients A, ω, ρ satisfy assumptions 2, 3 and 4. If a =, then we have 6 u L,T ;L 2 + χ u L,T ;L 2 d C, and 7 χ u L 2,T ;L 2 + χ u L2,T ;L 2 d C.

8 6 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ If a = 2, then w = u satises 8 w L,T ;L 2 + χ w L2,T ;L 2 d + χ w L2,T ;L 2 d C which, in particular, implies that 9 χ w L,T ;L 2 + χ w L2,T ;L 2 d C. Proof of Corollary 4. Estimates 6, 8 and 9 are obvious consequences of Proposition 3. Inequality 7 is easily deduced from 4 and 22. Proof of Proposition 3. Testing against u we obtain, after integration by parts with respect to the space variable x, d x ρ u 2 2 dt dx + χ µ x u u dx + ω u 2 dx x 2 + a χ µ u u dx = χ f u dx. Let us rst address the case i: there is a constant M such that f L R + b < M < for any. Then 2 yields, because of the positivity of ρ, ω and the coercivity of µ and µ, that d dt ρ 2 u 2 L 2 C χ f L 2 χ ρ 2 L u 2 C 2 +b χ ρ 2 L u, 2 which in turns shows that u t, L 2 Ct 2 +b, the constant C, depending a priori on min ρ, and M being chosen independently of. Integrating 2 with respect to time, and using Cauchy-Schwarz inequality again we obtain, 2 χ u 2 L 2,T ;L 2 d + a χ u 2 L 2,T ;L 2 d C+2b. This implies 4. On the other hand, thanks to the Dirichlet boundary condition satised by u in, we have, for all t >, the following Poincaré inequality 22 χ u 2 L 2 d χ u 2 L 2 d, where d depends on only. This can be proved, for example, by integrating u along a vertical or horizontal path connecting x and the boundary of the domain within the repository. Consequently, we have, for all t >, and all λ >, χ f u dx a d 4λα χ f 2 αλ L 2 + a χ u 2 L d 2, 23 a d 4αλ χ f 2 L 2 + a λ For λ =, this last inequality combined with 2 yields d ρ u 2 2 dt dx d 4α a χ f 2 L 2 24 C a++2b, χ µ x u u dx.

9 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS 7 which shows that u t, L 2 C 2 +b 2 a. For λ = /2, using 23 in 2 we obtain that for all t >, d ρ u 2 2 dt dx + χ µ x u u dx + x 2 a χ µ u u dx C a++2b, which in turn implies that χ u 2 L 2,T ;L 2 d + a χ u 2 L 2,T ;L 2 d C a++2b. Estimate 4 is proved, and the proof of the rst part of Proposition 3 is nished. Let us now turn to case ii, and suppose that f = s t, x δ { xd =/2} with s CR + R d, and s L < M where the constant M is independent of. Then f t, xu dx M u dx M Cτ u 2 dx τ 26 M 2 τ {x d =±/2} + χ µ x u u dx for a suciently small constant τ >. Therefore we obtain d ρ xu 2 2 dt dx M 2 τ, and arguing as above, we obtain 5. {x d =±/2} Following the strategy introduced in [3], we shall derive the asymptotic limit of problem using two-scale convergence for boundary layers. This notion of convergence generalizes the usual two-scale convergence [2], [3] to the case when the test functions periodically oscillate in the rst d 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 by G the band Y R. A generic point y G is denoted by y = y, y d with y Y and y d R. We introduce the space L 2 #G of square integrable functions on G which are periodic in the rst d variables, i.e. L 2 #G = { ϕ L 2 G s.t. y ϕy, y d is Y -periodic }. Similarly, we dene H# G the Sobolev space of functions in H G which are Y - 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 by C the space of continuous functions on the closure of, a compact set in R d. Proposition 5. Let v be a sequence in L 2 such that v L 2 C. There exists a subsequence, still denoted by, and a limit v x, y L 2 ; L 2 # G such that v two-scale converges weakly in the sense of boundary layers to v i.e. lim v xϕ x, x dx = Y v x, y ϕ x, y dx dy G

10 8 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ for all test functions ϕ x, y C ; L 2 # G. 2 Let v be a sequence which two-scale converges weakly in the sense of boundary layers to v, and furthermore satises lim v L 2 = Y 2 v L 2 G. Then, v is said to two-scale converge strongly in the sense of boundary layers to v, which means that, for any sequence w in L 2 which two-scale converges weakly in the sense of boundary layers to w x, y L 2 G one has lim v xw xϕ x, x dx = Y 2 for all smooth functions ϕ x, y C ; C #,c G. G v x, yw x, yϕ x, y dx dy, 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. Let v be a bounded sequence in H converging weakly to a limit v. Assume that w = χ v admits a limit w in the sense of boundary layers convergence. Then w = χw and for almost every x, vx, = wx, y dy. Y Proof. Recall that χ = χ x is the characteristic function of the repository zone. The fact that w = χw is obvious. Note that the assumption on w having a twoscale limit in the sense of boundary layers is plausible since, if v is a constant sequence v L, then w satises the a priori estimate w L 2 C which is required for the two-scale convergence in the sense of boundary layers. We have = + χ v x, ϕxdx v x, ϕx, dx χ v x, ϕx, dx v x, ϕx, dx v x, χ ϕx, ϕx dx. The rst term is exactly zero. For the second one, note that χ ϕx, ϕx /2 ϕx, t x d dt ϕ L, /2 therefore, v x, χ ϕx, ϕx dx ϕ L v x, dx C ϕ L v L 2, d

11 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS 9 where this last term is of order O since the sequence v is bounded in H. Similarly, χ v x, v x dx χ /2 v x, s d x d ds d dx /2 C v L 2 d. As a consequence, χ v x ϕxdx vx, ϕx, dx χ v x, ϕxdx vx, ϕx, dx + O v x, vx, ϕx, dx + O Cϕ v, u, L 2 + O. The trace of v on converge strongly in L 2 to the trace of v, thus we have proved that G wx, yϕx, dx dy = lim χ xv xϕx, dx χ xv xϕx dx = lim = vx, ϕx, dx. which implies the desired result, namely that ux, is the average on Y of wx, 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 =. There exists a function u t, x L 2, T ; H and a function wt, x, y L 2, T ; Hloc,# G such that, up to a subsequence, χ u χ u 2SBL χy d u t, x 2SBL χy d x u t, x + y wt, x, y. ii Suppose that a = 2. There exists a function u 2 t, x, y L 2, T ; H loc,# G such that, up to a subsequence, 29 3 Furthermore, u 2 u 2SBL u 2 t, x, y 2SBL χy d y u 2 t, x, y 3 χy d u 2 t, x, y = for a.e. y G.

12 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ Note. We use the notation x =,...,,. x x d Similarly, for a d dimensional vector Ψ we shall write Ψ = Ψ,..., Ψ d,. Proof. Suppose a = 2. The bound 8 shows that w = u satises w L 2,T ;L 2 + w L2,T ;L 2 d C. Consequently, w resp. w admits a limit u 2 resp. ζ 2 in the sense of boundary layer two scale limits. It is a classical result [2], [3] that w two scale converges in the sense of boundary layers to ζ 2 = y u 2. We shall now prove 3. The bounds 8 show that the sequence w satises χ w L 2,T ;L 2 + χ w L 2,T ;L 2 d C. Therefore, up to a subsequence, χ w converges in the sense of boundary layers to ξ 2 t, x, y. Since χ converges strongly in the sense of two-scale boundary layers convergence to χ, we have ξ 2 t, x, y = χy d ξ 2 t, x, y. Let d Ψx, y Cc ; C#,c G be a smooth test function such that Ψx, y χy d = Ψx, y, and ϕt C[, T ]. By an integration by parts we see that This implies that lim T = lim lim ϕt T 2 T ϕt χ w Ψ x, x dt dx w div y Ψ x, x dt dx dy x, x dt dx dy. ϕt w div x Ψ T lim ϕt w div y Ψ x, x dt dx =, that is, χu 2 t, x, y does not depend on y and is therefore, since y u 2,, y L 2 # G. Let us now turn to the case a =. The bound 7 shows that χ u and χ u admit two scale limits in the sense of boundary layers convergence. Denoting by w and ζ these limits, we have for all ψx, y Cc ; Cc# G, Ψx, y d ; Cc# G and ϕt C[, T ], C c lim lim T T ϕtdt ϕtdt χ u ψ x, x dx = x, x dx = χ u Ψ T T ϕtdt ϕtdt G G w t, x, yψ x, y dx dy, ζ t, x, y Ψ x, y dx dy.

13 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS We select a test function Ψ = Ψ,..., Ψ d such that Ψx, yχy d = Ψx, y, integrating by parts and passing to the limit we obtain a condition on w, T ϕtdt w t, x, ydiv y Ψ x, y dx dy =, G which implies that the two-scale limit w of χ u is independent of y within the support of χ: w t, x, y = χy d u t, x. Selecting now Ψ = Ψ,..., Ψ d, = Ψ x, yχy d satisfying additionnally that div y Ψ =, we have T = = T T ϕtdt G ϕtdt ϕtdt G G ξ t, x, y Ψ x, y dx dy χy d u t, x div x Ψ x, y dx dy χy d x u t, x Ψ x, y dx dy. Since the orthogonal set in L 2 Y of all Y -periodic functions Ψ satisfying the constraint div y Ψ = is precisely the set of all gradients y wx, y with w L 2 ; H# Y recall that these last functions are not periodic in the last y d variable, we have proved that T ζ t, x, y = χy d x u t, x + χy d y wt, x, y. 4. Proof of Theorem For any test function ψx, x/ such that ψx, yχy d = ψx, y, and any smooth function ϕt such that ϕ = ϕt =, we obtain from Proposition 7 that T ϕtdt ρ x u T t ψx, x/dx = dt ϕtdt T T 2 A u ψx, x/dx = ϕtdt ϕtdt ω u ψx, x/dx = f xψx, x/dx = T T T ϕdt ϕtdt ϕtdt G G ψ x, y ρ yu 2 t, x, y ϕ t dx dy + o, µ y y u 2 t, x, y y ψ x, y dx dy + o, G G ωyu 2 t, x, y ψ x, y dx dy + o, f t, yψx, ydx dy + o. Here, as in Proposition 7, u 2 is the limit of u. Combining these terms as in we deduce that u 2 satises the limit system ρ y u 2 t div y µ y y u 2 + ω yu 2 = f t, y in Y, for a.e. t 32 u 2 t, x, y = on Y { y d = /2}, for a.e. t u 2 t = =. Note that u 2 is independent of x : thus, system 32 is in fact 8, and our notations are consistent. In the sequel we shall write u 2 t, y, the unique solution in

14 2 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ C[, T ]; H# Y of 8. This rst result, the characterization of the limit behavior of u, does not provide a detailed description of the asymptotic behavior of the solution of, since the support of u 2, shrinks to zero with. Consequently, we turn to the second order term, and dene 33 v t, x = u t, x u 2 t, x for a.e. t, x [, T ]. The following gives the equation satised by v. Proposition 8. The solution u of problem can be written u t, x = u 2t, x/ + v t, x, for a.e. t, x [, T ], where u 2 is the unique solution of system 32. The remainder term v satises 34 ρ x v t div A x v +ω x v = s t, x δ{xd =/2} δ {xd = /2} where 35 st, y = µ y u 2 y d t, y. in, u = on, u t = = in, Furthermore, v satises the following a priori estimate 36 v L,T ;L 2 + χ v L2,T ;L 2 d + χ v L2,T ;L 2 d C, where C is a constant independent of. Proof. System 34 is obtained without diculty from and 32. From assumption 5 we deduce that there exists η > such that u 2 L, T ; C,η Y [ /2, /2 + η] [/2 η, /2] see e.g. [, Thm 8.34]. As a consequence, s t, x/ is continuous, and bounded around x d = ±/2. Since in the right hand side of 34 s is multiplied by Dirac functions located at x d = ±/2, we can assume without loss of generality that st, x/ C[, T ]. Then, estimate 36 is a consequence of 5. The conclusion of the proof of Theorem is given by the following proposition. Proposition 9. The solution v of 34 is such that χ v converges strongly in L 2, T ± to u ± and χ v converges weakly in L 2, T ± d to u ±, where u ± is the solution of the homogenized problem 6. Proof. We focus on the convergence to u +, the convergence to u can be proved by similar arguments. The bound 36 on v L,T,L 2 shows that, up to the possible extraction of a subsequence, there exists v t, x L 2, T ; L 2 + such that for v converges weakly in L 2, T ; L 2 to v. Similarly, the bound 36

15 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS3 on χ v L 2,T ;,L 2 d shows that, up to the possible extraction of a subsequence, there exists ξ t, x L 2, T ; L 2 d such that for χ v converges weakly in L, 2 T ; L 2 d to ξ. For any Φ Cc, T + d, one easily obtains that T T ξ Φ dtdx = v div Φ dtdx, + + or in other words that ξ = v on +. Let θ C 2 R be a non-negative one dimensional cut-o function, such that θs = for s /2 and θs = for s /2, and dene xd θ x = θ x R d. Given a test function ϕ C, T ; H, such that ϕt, =, we test system 34 against ϕθ to obtain T dt dx χ θ xv x + T + dt dx χ µ x v x ϕ + T + dt dx s t, x, /2 ϕ t, x, T = dt dx χ θ v ρ x ϕ t ω x ϕ T x dt dx 2 χ µ v θ ϕ. Thanks to 36, the rst right-hand side term is bounded by ρ x dϕ dt + ω x ϕ ϕ L ρ ϕ t ω v L,T ;L 2 C,,T ;L 2 whereas the second right-hand side term is bounded by χ v L2,T ;L 2 d µ L R d ϕ L,T ;L 2 d + ϕ L,T ;L 2 χ θ L 2 R d C. Passing to the limit in the left-hand side, we obtain T v ρ x ϕ T t + ω x ϕ + µ x v ϕ + + T s + tϕt, x, dx =. 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 6 we know that u is a bounded sequence in L, T ; H. It therefore admits a weak limit u, up to the extraction of a subsequence, in L 2, T ; H. We can now pass to the limit in in any

16 4 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ subdomain ω not intersecting the plane x d =. Furthermore, thanks to Lemma 6, we know that for a.e. x and every t >, ut, x, = χy d u t, x dy = u t, x. G We therefore have proved that, up to a subsequence, u satises ρ x u t div µ x u + ω x u = in, ut, x, = u t, x on, ut = = in, u = on. Let us now turn to the derivation of equation to conclude the proof. The following proposition determines wt, x, y, dened in Proposition 7, in terms of u t, x. Proposition. The two-scale limit of χ u is precisely χy d I d + P y x u t, x where I d is the identity matrix of R d, and P y is the matrix-valued function dened by P e i = ϕ i, for i =,..., d, where ϕ i H#,d Y is the solution of 3. Proof. For any ψx, y C ; L 2 # G and ϕt C c, T, testing against ϕtψx, x/ yields T x ϕtdt χ xµ u x ψ + yψ = o. Since µ y is periodic, it converges strongly with respect to two-scale convergence and we have T x ϕtdt µ χ x u y ψ dx T = ϕtdt µ yχy d x u t, x + y wt, x, y y ψ dx dy + o. G The above limit is just the variational formulation, for a.e. x and t >, of div y µ y x u t, x + y wt, x, y = in y Y, with Neumann boundary conditions on the lower and upper faces as in 3. Let us now test against ϕ Cc, T with ϕt, =. We obtain T dt dx χ u ρ x dϕ dt + ω x ϕ + µ x u ϕ T x dϕ x + dt dxχ u ρ dt + ω ϕ + x µ u ϕ T dt χ ϕf dx =.

17 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS5 Oxfordian L z Callovo Oxfordian Storage zone Callovo Oxfordian Η Dogger L x L H Figure 2. Nuclear waste repository and surrounding geological layers. In order to pass to the limit as tends to zero, remark that χ ϕ or χ ϕ are admissible test functions for two-scale convergence in the sense of boundary layers. As a consequence, we obtain T dt dx u ρ x dϕ dt t, x, x d + ω x ϕ t, x, x d + µ x u ϕ t, x, x d T + dt χy d µ y I d + P y x u t, x x ϕ t, x, x d dydx G This last system is the weak formulation of 9-, T T F u, ϕdtdx+ F ut, x,, ϕt, x, dtdx = T dt dx χy d f t, yϕ t, x, x d dy =. G T where ft is given by, F and F are the bilinear forms dv F u, v = u ρ dt + ω v + µ x u x v, F u, v = µ x u x v, ftϕt, x, dtdx, with µ being the d periodic homogenized diusion coecient given by Numerical simulation of a nuclear waste storage In this section, we revisit the scaling of our model. Introducing appropriate scales, we perform an adimensionalization and introduce the small parameter. Then, using some experimental values of the diusion coecients, we show that the dimensionless derived model is of the same type as. 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 [].

18 6 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ 6.. Nuclear waste repository. The storage of nuclear waste is achieved within glass containers in groups of, inside alveolus situated on both sides of 3 galleries. This set is called module, following the ANDRA French National Radioactive Waste Management Agency terminology. A large number of disposal modules makes up a storage zone. The whole repository system lies at a depth of about four to ve hundred meters in the 3 meters thick geological argillite layer called Callovo-Oxfordian formation. Above this layer, the Oxfordian formation 4 meters height is considered as the rst geological limestone underground layer. Bellow, the Dogger formation 5 meters height is a limestone layer. The computational domain is constituated by these geological layers. This zone is on the whole L z = 684 meters deep see Figure 3. Modeling all the details of a repository is dicult and it is a common practice to use a simplied model based on the homogenization of the storage zone []. In our approach, the cylindrical unit of storage.5 meters height and meter diameter containing the radioactive waste is surrounded by material such as concrete, clay, argilite or bentonite. This is what the simplied elementary cell is composed of. In our 2-d setting the cell has height H = 3 meters and length L = 3 meters. A large number of these elementary cells, 3 meters distant from each other, lie horizontally, overlapping the reference plane z = located in the middle of the Callovo-Oxfordian Scale analysis of the diusion problem. In rst approximation the radionucleide transport is governed by a pure diusion process []. Thus, the radionucleide concentration C satises the diusion equation R ω C t + µ C div{d C} = F in O where R is the latency retardation factor, ω the eective porosity and µ is related to the half life time τ of the element by the relation µ = log2/τ. We denote by D the eective diusion tensor. Here, O is the 2-d Oxz plane section of the repository surrounded with its geological layers and we only consider the section given by L x = γ L z with γ. All the containers are supposed to be cut by this plane see Figure 3. The source term F is related to experimental data ftmoles/year by the relation F t = ft/s where S is the surface occupied by the nuclear waste and is zero outside the containers. In the following, we use the subscript ref for characteristic or reference values of the parameters and variables involved. Setting x a = x γl ref, z a = z, D a = D, R a = L ref D ref R R ref, ω a = ω ω ref, t a = and omitting the subscript a for the dimensionless parameters and variables, the equation for the transport of the radioactive elements in each medium takes the general form 37 R ref ω ref T ref R ω C t + λ C div{ D ref L 2 ref D γ C} = F T ref t. t T ref

19 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS7 Oxfordian Oxfordian + Callovo Oxfordian repository Callovo Oxfordian Callovo Oxfordian Callovo Oxfordian Dogger Dogger Figure 3. The computational domain. where D γ γ 2 = D. Here λ stands for µ T ref. The concentration C takes its values in the rescaled domain = ], [ ], [ see Figure 3 composed by the subdomains denoted sup limestone layer, COX argillite layer and inf limestone layer. After dividing by R ref ω ref /T ref, equation 37 reads R ω C t The adimensionalization relation + λ C div{ D reft ref L 2 ref R D γ C} = F T ref in. refω ref R ref ω ref T ref = L2 ref ρ ref D ref denes the reference time. The surface S occupied by the L x /L containers represents a percentage δ of the thin strip in which they are stored S = δ γ L ref H. Introducing the concentration reference value C ref = max f, δγd ref the dimensionless transport equation for the rescaled concentration U = C/C ref is R ω U t + λ U div{d γ U} = L ref f in, H where f = f/max f inside the containers and zero outside Small parametric dependences. We choose the "observation distance" L ref = L z as the reference distance value and we introduce the small parameter as the ratio of the repository height over the reference length: = H L ref The L x /L containers are distributed periodically over with period. They are surrounded by a coating or buer material.

20 8 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ Typical values for diusion coecients in the case of 29 I Iodine 29 are.89 2 m 2 /year for concrete [5] and [8] for the argillite. The choice of concrete diusion coecient, as reference value, gives a dimensionless coecient for surrounding material equal to 5. 5 which is of the same order of magnitude as and corresponds to the rst case a = 2. Another typical value for diusion coecient of the argillite always in the case of 29 I and with the same value for concrete is.58 4 m 2 /year [5]. With this value as reference value the dimensionless coecient for concrete equal to 2 which is of the same order of magnitude as 228 and corresponds to the second case a =. This simple scaling computation is thus a justication of our choice of -scaling in. In other words, the diusion tensor A scales like or 2, i.e., takes the form γ 2 38 A x, z = a Ax/, z/ with a = or 2, in the thin strip x < /2 where 39 Ax/, z/ = α χ Wx, z + α 2 χ Wx, z. In 39, α denotes the diusion constant for the vitried nuclear waste, α 2 the one of the buer material and χ W x, z = χ W x, z the characteristic function of the waste into the computational domain. For x > /2, the physical values of the diusion constants of the others geological layers dene the diusion tensor A of order. In the same way, we denote by ρ resp. ω the bounded function corresponding to the products R ω resp. ρ λ in the dierent subdomains of. Introducing a right hand side or source term, which is of average of order, f = f χ Wx, z, the adimensionalized diusion equation reads ρ u t div{a u } + λ u = f in where the concentration in the dierent subdomains is denoted u. Of course, we assume perfect transmission conditions at the interfaces between geological layers, namely continuity of the concentration and of the normal ux. As a nal remark, let us mention that we have ignored convection phenomena which can take place, most notably in the upper and lower geological layers. This is just for simplicity and there is no conceptual diculty to include them in our model as well Numerical simulation of a nuclear waste storage. Our asymptotic analysis in the previous sections leads to two dierent homogenized problems according to the value of the exponent a =, 2. The diusion coecients for the storage zone are constant, equal to a with a =, 2. All other physical constants are taken to be equal to one. We make a comparison between direct simulations of the original model and reconstructed solutions of the homogenized models. The simulations concern the direct problems and its asymptotic ones. The source term

21 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS9 is a hat function which takes its maximum value f = in the middle of the time interval, : 2t for t /2 ft = 2 2t for /2 t for t All computations are perform on a unit square domain γ =. The numerical approximation of such parabolic boundary value problems is classical: we use a P nite element method to discretize the variational formulation in space and an Euler implicit scheme for the time integration in the FreeFem++ software []. Time integration is perform until t = in 2 time-steps. For the two direct problems a =, 2, the same triangular mesh is used 34 degrees of freedom. The corresponding Callovo-Oxfordian zone, surrounding the waste, is meshed more nely than the rest of the computational domain The case of small diusion coecient: a = 2. In this case, the asymptotic model 6 has a time-dependent Neumann type boundary condition on the asymptotic limit surface occupied by the storage zone. This emitting surface is the support of eective boundary conditions which supply a source term for the partial dierential equation describing the diusion of the concentration of radionucleides into geological medium surrounding the repository. The numerical simulation of the direct problem shows a decoupling see Figure 4a of the solution at the emitting surface between the upper and lower parts of the computational domain. The solution of the direct problem remains highly concentrated in the storage zone see Figure 4b which weakly emits towards the rest of the domain. The asymptotic problem 6 shows that the numerical integration of the parabolic problem associated to u 2 see 8 allows the computation of the source terms s + and s dened by 7. In the parabolic boundary value problem 8, all coecients are equal to one. Thus, u 2 is computed on a very ne mesh see Figure 5 of the elementary cell a unit square containing the nuclear waste with a classical P 2 nite element method and s + resp. s by a standard quadrature formula. Then, using two dierent triangular meshes for, respectively, the upper and lower parts of the computational domain with respectively 76 and 34 degrees of freedom, the calculus of the solution of the asymptotic problem see Figure 6 is achieved. The decoupling phenomenon, appearing in the asymptotic problem formulation, can be illustrated by Figure 7 which allows to compare the truncated solution of the direct problem in the storage zone and the one of the asymptotic problem. Outside the repository, the asymptotic solution overestimates the concentration found in the numerical computation of the direct problem The case of large diusion coecient: a =. In this case, the asymptotic model features a diusive transmission condition on the asymptotic limit surface occupied by the storage zone. The numerical simulation of the direct problem shows isovalues see g. 8a in the shape of a "smiling conguration". The same shape see Figure 8b is also found in the numerical simulation of the homogenized problem 9. This numerical

22 2 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ a b Figure 4. Small diusion coecient a = 2. In Figure a, isoval-

23 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS2 Figure 5. Mesh 2 degrees of freedom of the unit cell containing the waste. approximation degrees of freedom requires the computing of the eective coecient µ 2 by a quadrature formula, after calculating the test functions φ i a P 2 nite element approximation is done, and the computing of the eective source f on the unit cell. The cross section of the solutions are represented in Figure 9 and show that both solution behave similarly. Remark. It is worth mentionning that the retardation factors as well as the eective porosity of the media linked with the repository doesn't play any part in the homogenized problem. Acknowledgments. This work has been supported by the Groupement de Recherches MoMaS sponsored by ANDRA, BRGM, CEA, CNRS, EDF, and IRSN. G. Allaire is a member of the DEFI project at INRIA Saclay Ile de France. References [] Dossier 25 Argile - Évolution phénoménologique du stockage géologique, Collection Les Rapports, Juin 25. ANDRA Agence nationale pour la gestion des dã chets radioactifs. [2] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., , pp [3] G. Allaire and C. Conca, Boundary layers in the homogenization of a spectral problem in uid-solid structures, SIAM J. Math. Anal., , pp [4] A. Bourgeat, O. Gipouloux, and E. Marusic-Paloka, Mathematical modeling of an array of underground waste containers, C. R. Mécanique, 33 22, pp [5] A. Bourgeat, O. Gipouloux and E. Marusic-Paloka, Modelling of an underground waste disposal site by upscaling, Math. Methods Appl. Sci., , pp [6] A. Bourgeat and E. Marusic-Paloka, A homogenized model of an underground waste repository including a disturbed zone, SIAM J. Multiscale Model. Simul , pp [7] A. Bourgeat, I. Boursier and D. Tromeur-Dervout, Modelling of an underground waste disposal site by upscaling, simulation with domain decomposition method, in Domain Decomposition methods in Science and Enginering, R. Kornhuber, R. Hoppe, J. Periaux,

24 22 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ Figure 6. Small diusion coecient a = 2: Isovalues of the solution of the asymptotic problem 6 in the two parts of the computational domain t = /2.

25 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS Figure 7. Small diusion coecient a = 2: Cross sections of the truncated dashed direct and asymptotic solutions at x =.5 t = /2. O. Pironneau, O. Widlung and J. Xu, Eds., Lecture Notes in Computational Science and Engineering, Vol 4, pp , Springer 24. [8] A. Bourgeat, M. Kern, S. Schumacher, J. Talandier The COUPLEX Test Cases: Nuclear Waste Disposal Simulation, Computational Geosciences, Vol. 8, No. 2, 24. [9] S. Del Pino and O. Pironneau, Asymptotic analysis and layer decomposition for the couplex exercise, Computational Science 8, 24, pp [] FreeFem++, version2.3.,

26 24 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ a b Figure 8. Large diusion coecient a =. In Figure a Iso-

27 TWO ASYMPTOTIC MODELS FOR ARRAYS OF UNDERGROUND WASTE CONTAINERS Figure 9. Large diusion coecient a = : Cross sections of the direct versus homogenized dashed solutions at x =.5 t = /2. [] D. Gilbarg and N. S. Trudinger, Elliptic Partial Dierential Equations of Second Order, Comprehensive Studies in Mathematics, Springer-Verlag, 2nd ed., 983. [2] H. P. Huy and E. Sánchez-Palencia, Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., , pp [3] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 23, 989, pp

28 26 GREGOIRE ALLAIRE, MARC BRIANE, ROBERT BRIZZI, YVES CAPDEBOSCQ [4] M. Neuss-Radu and W. Jäger, Eective transmission conditions for reaction diusion processes in domains separated by an interface, SIAM J. Math. Anal., 3927, pp [5] Test cases definition: Homogenization of the source terms Momas Document 27