SOME PROBLEMS IN SCALING UP THE SOURCE TERMS IN AN UNDERGROUND WASTE REPOSITORY. A.Bourgeat

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1 SOME PROBLEMS IN SCALING UP THE SOURCE TERMS IN AN UNDERGROUND WASTE REPOSITORY A.Bourgeat Oberwolfach Conference - October 31, November 4; Reactive Flow and Transport in Complex Systems Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 1

2 Figure 1: Situation of the underground laboratory site Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 2

3 Figure 2: A part of a Waste Repository Site, with two different zones; a zone I, with 11 long storage tunnels; a part of the zone II, with 3 storage units Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 3

4 Figure 3: A Storage Unit (or Repository Module) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 4

5 Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / Figure 4: The Far Field, with geological layers containing the units 5 Oberwolfach Conference - October 31/November Reactive Flow and Transport in Complex Systems

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7 Figure 5: A Repository Zone : from Waste Packages(or containers sets) to Storage Units Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 7

8 There are several levels of upscaling from waste packages to a storage unit global model from storage units to a zone model from similar zones to the repository site global model The use of Re-iterated Homogenization,could not be not straightforward! (the phenomena to be taken in account at each level could be different leading to different equations parameters or boundary conditions) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 8

9 1 General Equations Rω ρ t R the latency retardation factor, (A ρ)+(v )ρ + λrωρ =0 (1) ω the porosity, v the Darcy s velocity λ = log2 T ; T the element radioactivity half life time according to the units width and their length we consider a storage 2D vertical section Iodine 129 I has half life time T = years and is releasing during a time t m =810 3 years, with intensity Φ =10 1. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 9

10 (I) From STORAGE UNITS to a ZONE model (OR) From Similar ZONES to the REPOSITORY SITE model Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 10

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12 Figure 6: A Repository Zone : from Waste Packages to Storage Units Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 12

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14 1000 m S + storage unit Σ Upper Layer 150 m 100 m low permeability layer containing the units lower layer S Figure 7: According to the units width and their length we consider a Repository Zone 2D vertical section Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 14

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16 ε β ο(ε) ε Figure 8: Storage Units (of Waste Packages), after renormalization x = x L Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 16

17 2 The Equations ω ε ϕ ε t div (Aε ϕε)+(v ε )ϕε + λω ε ϕε = 0 in Ω T ε (2) ϕε(0,x)=ϕ0(x) x Ωε (3) n σ = n (A ε ϕε v ε ϕε) =Φ(t) on Γ T ε (4) ϕε =0 on S1, (5) n (A ε ϕε v ε ϕε) =0 on S2 (6) with A ε (x2) =A( x 2 ε ); vε (x, t) =v(x, x 2 ε,t); ωε (x2) =ω(x2/ε). (7) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 17

18 3 A priori Energy estimates give ϕε ϕ weak* in L (0,T; L 2 (Ω)) (8) ϕε ϕ weakly in L 2 (0,T; L β (Ω)) (9) β = 2β 3β 2. with ϕ L 2 (0,T; H 1 (Ω)) L (0,T; L 2 (Ω)); Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 18

19 ω 2 ϕ t div (A2 ϕ)+(v 2 )ϕ + λω 2 ϕ =0in Ω T (10) ϕ(x, 0) = ϕ0(x) x Ω =Ω\Σ (11) ϕ =0 on S1 (12) n (A 2 ϕ v 2 ϕ)=0 on S2 (13) [ [ϕ] =0, e 2 (A 2 ϕ v 2 ϕ) ] = M Φ on Σ, (14) where [ ] denotes the jump over Σ, and M stands for the limit of a storage unit area; (M ε ) area = M + O(ε β 1 ) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 19

20 Remark 1 We do not need exact periodicity in space, of the units. The same proof holds whenever each unit is randomly placed in a mesh of an ε net. The units do not even need to have the same shape as long as their thickness is small enough ( ε). We may extend to a general case where the flux Φ depends also on the space Φ(x, t) and the units have different shapes Mε(x), then the right hand side of (14) has to be replaced by lim M ε(x) Φ(x,t). ε 0 Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 20

21 The long time behavior: Figure 9: Global homogenized solution ϕ vs. real solution ϕε at 200 Kyears. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 21

22 4 Asymptotic expansion and Matching for the Short time L=O(1) 2 ε log 1 ε Σ Σ + ε ε G ε O(1) Figure 10: Gε The inner layer; and Ω\Gε the outer domain Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 22

23 In Gε, the inner domain, we look for an asymptotic expansion of ϕε: ( ϕε ϕ 0 ε + ε χ k ε( x ε ) ϕ0 ε + wε( x xk ε )Φ ϕ0 ερ k ε( x ) ε )v1 k ϕ 1 ε, (15) where ϕ 0 ε mimics the behaviour of ϕ but has two jumps respectively on Σ + ε = {ε log (1/ε)} ] δ/2,δ/2 [ and on Σ ε = { ε log (1/ε)} ] δ/2,δ/2 [, instead of only one on Σ. The functions χ k ε,ρ k ε and wε are 1-periodic solutions in y1 of three auxiliary stationary diffusion type problems posed in an infinite strip Gε =(] 1/2, 1/2[ R )\Mε. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 23

24 4.1 Error estimates for the Matched expansion With the approximation: ϕ 0 ε in Ω\Gε ; ( outer expansion) Fε = ϕ 0 ε + ε ( χ k ε( x ε ) ϕ0 ε xk + wε( x ε )Φ ϕ0 ερ k ε( x ε )v1 k ) in Gε. (16) Theorem 1 For any 0 <τ<1 there exists a constant Cτ > 0 non dependent on ε, such that ϕε Fε L 2 (0,T ;H 1 (B ε)) Cτ ε τ, (17) where Bε =Ω\ Gε. The same estimate holds in L (0,T; L 2 (Ωε)) norm. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 24

25 5 Conclusion of Part One The expansion (16) clearly points out two important terms: the zero order term ϕ 0 ε and the first order term εwε( x ε )Φ. On one hand the diffusion in the low permeable layer around the units is small and on the other hand the leaking is intensive during a short time; then: during that short time the first order term εwε( x ε )Φ will dominate in ϕ ε; and after this short time the diffusion will become dominant, i.e. ϕ 0 ε is now the most important term in the expansion. Remark: The effects of the boundary layer caused by the non periodicity of the geometry on Gε could be neglected for a ε order approximation. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 25

26 6 Numerical Simulations Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 26

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33 (II) From WASTE PACKAGES to a STORAGE UNIT Global model, with a possibly damaged zone Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 33

34 Figure 11: A Storage Unit (or Repository Module) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 34

35 Seeking a mathematical model describing the global behavior of one Storage Unit of an underground waste Repository Zone, Assuming it is made of a high number of Waste Packages (or containers sets), located inside a low permeable rock, lying on a hypersurface Σ and linked by parallel filled shafts; all the parallel shafts being connected at the top to a main shaft, also filled. All the repository is embedded in a thin (100 m.) layer, called host layer, which is included between two higher permeability layers, The convection field (Hydrology regime) is given. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 35

36 Figure 12: A part of a Storage Unit, with 5 rows of waste packages (or containers sets) along shafts Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 36

37 Denoting ε the ratio between the width of a unit (500 m.) and distance (50 m.)between two shafts The containers set have a diameter, of order ε γ, γ close to three. In the renormalized model there are three scales: 1 for a disposal unit scale, ε for both the scale of a containers row and the shafts period, and ε γ for the containers diameter. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 37

38 Figure 13: Cell of periodicity Y containing a cylindrical shaft S = ] 1/2, 1/2[ Cand a waste package Pε; ε γ 1 =diameter of Pε. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 38

39 7 The model and equations The Darcy s velocity: v ε v h (x) in the host rock Ωε\Sε (x) =. ε β v d (x,x2/ε; x3/ε) in the shafts Sε The Diffusion/Dispersion A ε A h (x) in the host rock Ωε\Sε (x) =. d(x) I + ε β A d (x2,x2/ε, x3/ε) in the shafts Sε Because the convection in a storage unit goes mainly in the direction of the shafts: A d (x2,y2,y3) =a(x2,y2,y3) (e1 e1 ) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 39

40 Microspic model of a storage unit ω ε ϕ ε t div (Aε ϕε)+(v ε )ϕε + λω ε ϕε = 0 in Ω T ε(18) ϕε(0,x)=ϕ0(x) x Ωε (19) n (A ε ϕε v ε ϕε) = Φε(t) on Γ T ε (20) n (A ε ϕε v ε ϕε) =κ (ϕε gε) on K T ε H T ε (21) ϕε =0 on Z T ε. (22) with H ε T the shafts tops surface, Z ε T the Shafts Bottoms (sealed), K ε T the rest of the exterior boundary of Ω, and Γε the Waste Packages boundary (0,T). gε will measure the concentration entering at the shafts tops; and ε β the Darcy s velocity range inside the shafts. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 40

41 8 Results Depending on β(the Darcy s velocity range) we have three different cases : 0 β<1 With a sufficiently strong source : lim ε 0 εγ 1 Φε(t) =Φ(t) uniformly in t. (23) and gε = g = g h on the shafts cylindrical surfaces Kε. g on the shafts tops Hε Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 41

42 The shafts do not make any contribution, i.e. the repository behaves as if they were not there. ϕε ϕ the unique solution of a problem, of same type as the microscopic problem: ω h ϕ t div (Ah ϕ)+(v h )ϕ + λω h ϕ =0 in Ω T ϕ(x, 0) = f0(x) x Ω =Ω\Σ, (24) n (A h ϕ v h ϕ)=κ (ϕ g) on S T (25) [ [ϕ] =0, e 3 A h ϕ (v h e3) ϕ ] = ΦM on Σ. (26) ΩT =(Ω\Σ) ]0,T[; S T = Ω ]0,T[ [w](x )=w(x, 0+) w(x, 0 ), denotes the jump over Σ and M denotes the limit of the rescaled containers surface area, i.e. M = lim ε 0 ε1 γ Pε. (27) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 42

43 β =1 With a source term, lim ε(t) =Φ(t) ε 0 uniformly in t, (28) and some concentration entering the shafts tops gε g h on the shafts cylindrical surfaces Kε gε =. (29) ε 1 g d on the shafts tops Hε ϕε ϕ weakly in L 2 (0,T; W 1,γ (Ω)) and ϕε ϕ 0 = ϕ(x1,x2, 0), dµ ε (x)2 scale, where ϕ is the unique solution of a coupled problem. The transport processes, inside and outside the damaged shafts are comparable and there are interactions between them. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 43

44 β = 1; The model could be seen as representing connected shafts, galleries and drifts with damaged sealings. ω h ϕ t div (Ah ϕ)+(v h )ϕ + λω h ϕ =0 in Ω T ; (30) ϕ(0,x)=ϕ0(x)in Ω; (31) n (A h ϕ v h ϕ)=κ(ϕ g h ) on S T (32) [e 3 (A h ϕ v h ϕ)] = MΦ x1 ( a ϕ0 x1 )+ v d 1 ϕ0 x1 on Σ T (33) a ϕ0 x1 (t, L, x2, 0) + v d 1 ϕ 0 (t, L, x2, 0) = κg d. (34) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 44

45 2 >β>1 With a sufficiently strong source: lim ε β 2 +γ 3 2 Φε(t) =Φ(t) uniformly in t, (35) ε 0 and some concentration entering the shafts tops gε: g h everywhere on the boundary except on the shafts tops gε = ε β+1 2 g d on the shafts tops Hε (36). Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 45

46 The Transport process in the shafts is dominant and we do not see anything else in the corresponding global model. Then ε (1 β)/2 ϕε φ, dµ ε (x)2 scale, ε 1 β 2 ϕε ϕ 0 (37) ε 1 β 2 ϕε x1 ϕ0 x1, (38) The global concentration ϕ 0 is the unique solution of a 1-dimensional problem defined for any x ]0,L[ (assuming g d = Cte). x1 ( ϕ 0 (0) = 0, A d 11 A d 11 ϕ 0 x1 ϕ 0 x1 ) + v d 1 ϕ 0 x1 = 0 in ]0,L[ (39) (L)+(v d 1 + κ) ϕ 0 (L) =κg d. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 46

47 9 Proofs The starting point is a priori estimates obtained from (18)-(22): ϕε L 2 (0,T ;L 2 (Ω ε)) C (40) ϕε L (0,T ;L 2 (Ω ε)) C (41) ϕε L 2 (0,T ;L 2 (C ε)) Cε β/2 (42) ϕ ε x1 L 2 (0,T ;L 2 (C ε)) Cε β/2. (43) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 47

48 The global models obtained, at the limit, are defined on the hypersurface Σ and the general two-scale convergence has to be adapted to this situation We use the two-scale convergence with respect to the rescaled measure dµ ε (x) =ε 1 1Cε measure. dx, where dx is the Lebesgue Definition 1 A sequence {ϕε}ε>0, ϕε L p (Ωε) is said to converge two-scale, with respect to the singular measure dµ(x), to ϕ 0 L p (Σ C) if for any ψ C(Ω; L p (C) ) limε 0 1 ε Σ dx 1 dx2 Sε ϕ ε(x)ψ(x, x ε ) dx = C ϕ0 (x1,x2,y2,y3) ψ(x1,x2, 0,y2,y3) dy2 dy3 Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 48

49 Additional technical difficulty comes from the Dirichlet condition on Zε, the sealed Shafts Bottoms, for the test functions, zε, used in the proofs of the above convergence theorems. For this,we start from a z C 1 (Ω) and we construct the test functions zε satisfying the Dirichlet condition on Zε, as needed in the proofs. We use the existence for any z H 1 (Ω) of a sequence of functions {zm}m N, zm C 1 (Ω) such that zm(0,x2, 0) = 0 and zm z in H 1 (Ω) since on a 1-dimensional line c = {x R 3 ; x1 =0,x3 =0} the trace of a function from H 1 (Ω) cannot be specified. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 49

50 We need also sharp estimate of the flux through all the containers sets boundaries Γε for the a priori estimates. Mainly, for L ε [H 1 (Ω)] defined for any ψ H 1 (Ω), by:, we should prove L ε ψ = ε 1 γ Γε ψ L ε MδΣ strongly in [H 1 (Ω)]. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 50

51 (III) From a LONG STORAGE UNITS model to a Global REPOSITORY ZONE model with a possibly damaged zone Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 51

52 Figure 14: A part of a Waste Repository Site, with two different zones; a zone I, with 11 long storage tunnels; a part of the zone II, with 3 storage units Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 52

53 we study a simplified but typical repository zone, assuming it is made of a high number of similar long waste filled storage units, lying on a hypersurface Σ and linked by backfilled working and haulage drifts. The units are periodically distributed on both sides of a backfilled drift ( haulage and working drift) with period ε The working and haulage drifts are represented by a single circular cylinder Cε =]0,L[ ε S, where the cross section of this cylinder is S = {(y2,y3) R 2 ; y y 3 2 <s 2 } the set of all units is denoted Uε: Uε = N(ε) j=1 U j ε, N(ε) =O(ε 1 ). Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 53

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56 Like previously, the parameter β will characterize de degree of damaging ( Darcy s velocity and consequently dispersion will be scaled by means of ε β ). The main difference and difficulties compared to the two situations we studied previously, in [?] or in[?] are coming from the singular behavior of the drift. In [?] there vas no damaged zone at all, while in our second paper [?] the damaged drifts were periodically repeating, allowing to use the technique of singular measures. But, the global models will only slightly differ; depending on β. The first approximation (the weak limit) is independent of the choice of β and only further order correctors will differ, depending on β. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 56

57 10 Definition of the problem: We assume the convection, i.e. the Darcy s velocity, to be given by the hydrology and to have the form: v = v h + χ C ε ε β v d e1, (44) with χ standing for the characteristic function of the drift, C C ε ε, and where v d, the absolute value of the velocity inside the drift, depends only on r = x = x x2 3. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 57

58 The evolution of the pollutant s concentration ϕε in the Repository Zone Ω is governed by the equation: ϕε t div (Aε ϕε)+(v ε )ϕε + λϕε = f ε in Ω T =Ω ]0,T[ (45), ϕε(0,x)=φ0(x) x Ω ;ϕε = 0 on Γ T ;, (46) where f ε the sources density is: f ε = ε 1 f ( t, x, x 1 ε, x 3 ε ) (47) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 58

59 11 Zero order approximation: ϕε ϕ0 weakly in L 2 (0,T; H 1 (Ω)). (48) The limit ϕ0 is the unique solution of the problem ϕ0 t d ϕ 0 +(v h )ϕ0 + λϕ0 = f δσ in Ω T (49) ϕ0(0,x)=φ0(x) x Ω (50) ϕε = 0 on Γ T (51) f (t, x) = εs f(t, x, y1,y3) dy1 dy3. (52) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 59

60 12 asymptotic expansion,with matching, of the solution Outside the drift we cut-off the limit and we pose: ( ) ϕε ϕ0(t, x)+ 1 log r log (sε) 1 log (sε) ϕ 1(t, x1). (53) with: r = x x2 3 sε Inside the drift Cε we seek an expansion: ϕε 1 log (sε) Ψε 1(t, x1, x 2 ε, x 3 ε ). (54) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 60

61 Matching the concentration s values and the flux on the drift surface, we conclude: ϕ1(t, x1) =Ψ ε 1(t, x1,s) s Ψε 1 ρ (t, x 1,s)= ϕ0(r = sε) ϕ0(t, x1, 0, 0) ; with,the interior approximation in the drift Ψ ε 1 : v d 2 Ψ ε ( 1 x 2 ε β 2 2 d 1 ρ + 1 ) Ψ ε 1 + v d Ψε 1 ρ ρ x1 =0 in S s Ψε 1 ρ = ϕ 0(t, x1, 0, 0) for ρ = s, Ψ ε 1 = 0 for x1 =0, 1 (55). Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 61

62 Finally, with the above matching (by computing exactly the solution Ψ ε 1 )we obtain the strong convergence : denoting : R ε = ϕε ϕ(ε); then limε 0 R ε L 2 (0,T ;H 1 (Ω)) =0. and, for any 1 <q<2 limε 0 ϕε ϕ0 L 2 (0,T ;W 1,q )(Ω)) =0. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 62

63 13 More about convergence From the results above we conclude that the contribution of our corrector inside the damaged drift is not very significant. Indeed, what we gain is the L 2 (0,T; H 1 (Ω)) estimate, while the convergence without corrector is in L 2 (0,T; W 1,q (Ω)) for q<2. That is because the norm of the corrector is negligible in W 1,q for q<2 but not in H 1. All this is due to the fact that integral norms in L p and W 1,p spaces are not suitable for a precise asymptotic analysis of tiny objects like the present drift. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 63

64 Although uniform estimates are not expected for the part of the error that comes from the sources, since the limit of the source function f ε is only a measure, we want to prove the uniform convergence to zero for the other part of the error coming from the matching. And finally we obtain the result: Theorem 2 Rε = ϕε ϕ(ε), the error of the approximation can be decomposed as Rε = rε + Wε, where rε L 2 (0,T ;H 1 (Ω) C ε and Wε tends uniformly to zero on Ω T. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 64

65 14 Conclusion of Part III It appears at the end that whatever was the magnitude of the convection (i.e. how big is the power β), it does not make an important difference. As we can see, on the macroscopic scale, there is barely a mild logarithmic singularity around the drift; this is mainly due to the fact that the units are long comparing to the drift diameter and, in the limit, there is a uniform density of the source everywhere on Σ, which is relatively important compared to the effect of the strong convection, which was localized only in the vicinity of the drift itself, i.e. localized on a very thin cylinder. Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 65

66 15 bibliography A.Bourgeat, O. Gipouloux, E. Marusic-Paloka. Mathematical Modeling of an underground waste disposal site by upscaling. Math. Meth. Appli. Sci., Volume 27, Issue 4; March 2004, p A. Bourgeat, E. Marusic-Paloka. A homogenized model of an underground waste repository including a disturbed zone. To appear in SIAM J. on Multiscale Modeling and Simulation, Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 66

67 (IV) Positions and contents of the Waste Packages are Random work in progress, with A. Piatnitski Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 67

68 The local sources f ε are periodically repeated, lying on a plan Σ; the emission starting time and the emission time evolution, of each local source, are both random : f ε 1 (x, t) =1IBε ε f(t γ x /ε ω, t). Theorem 1 lim ε 0 uε u 0 L 2 (0, ;H 1 (G)) =0 a.s.; with : tu 0 div(a(x) u 0 ) + div(b(x)u 0 )=F (t)δσ(x); (56) F (t) =s1s2e{f(,t)}. (57) Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 68

69 THE END Thank you Alain Bourgeat / GdR MoMaS / UCB-Lyon1 / 69