TRANSPORT IN POROUS MEDIA

Transcription

1 1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case of bounded domains Ecole CEA-EDF-INRIA, Décembre 20

2 2 -I- INTRODUCTION Based on a joint work with A. Mikelic and A. Piatnitski. Infinite porous medium: (connected) fluid part Ω f, solid part Ω s = R n \ Ω f. Saturated incompressible single phase flow in Ω f and a single solute. The unknown is the concentration u in the fluid. convection diffusion in the fluid: u τ + b yu div y (D y u) = 0 in Ω f (0, T ), Given incompressible and steady-state velocity: div b = 0 in Ω f no-flux boundary condition on the pore boundaries: b n = 0 and D y u n = 0 on Ω f (0, T ),

3 3 Scaling We want to upscale this model, so we define a large macroscopic scale ǫ 1 and we choose a long time scale of order ǫ 2 (parabolic or diffusion scaling) x = ǫy and t = ǫ 2 τ. We define u ǫ (t, x) = u(τ, y) which is a solution of u ǫ t + 1 ǫ b ǫ x u ǫ div x (D ǫ x u ǫ ) = 0 in Ω ǫ (0, T) with T = ǫ 2 T. u ǫ (x, 0) = u 0 (x), x Ω ǫ, D ǫ x u ǫ n = 0 on Ω ǫ (0, T)

4 4 Periodicity assumption ε Ω Periodic unit cell Y = (0, 1) n = Y O with fluid part Y Periodic (infinite) porous media x Ω ǫ y Y ( x ) Stationary incompressible periodic flow b ǫ (x) = b ǫ Y and b n = 0 on O ( x ) Periodic symmetric coercive diffusion D ǫ (x) = D ǫ with div y b = 0 in

5 5 Another approach to scaling: dimensional analysis We write the same equations with dimensional constants denoted by * : c t + b x c div x (D x c ) = 0 in Ω f (0, T ), D x c n = 0 on Ω f (0, T ),

6 6 Dimensional analysis (ctd.) We adimensionalize the equations as follows: Characteristic lengthscale L R and timescale T R. Period l << L R : we introduce a small parameter ǫ = l L R. Characteristic velocity b R. Characteristic concentration c R. Characteristic diffusivity D R. New adimensionalized variables and functions: x = x, t = t, b ǫ (x, t) = b (x, t ), D = D, u ǫ = c L R T R b R D R c R

7 7 Dimensional analysis (ctd.) Dimensionless equation u ǫ t + b RT R b ǫ x u ǫ D RT R L R L 2 div x (D x u ǫ ) = 0 in Ω ǫ (0, T) R and D x u ǫ n = 0 on Ω ǫ (0, T). We choose a diffusion timescale, i.e., we assume T R = L2 R D R. Péclet number: Pe = L Rb R D R. We assume Pe = ǫ 1. u ǫ t + Pe b ǫ x u ǫ div x (D x u ǫ ) = 0 in Ω ǫ (0, T)

8 8 Microscopic model u ǫ t + 1 ǫ b ǫ x u ǫ div x (D ǫ x u ǫ ) = 0 in Ω ǫ (0, T) u ǫ (x, 0) = u 0 (x), x Ω ǫ, D ǫ x u ǫ n = 0 on Ω ǫ (0, T) Assumptions: Stationary incompressible periodic flow div y b = 0 in Y, b n = 0 on O Periodic symmetric coercive diffusion D Goal of homogenization: find the effective diffusion tensor.

9 9 -II- MAIN RESULT Theorem. The solution u ǫ satisfies u ǫ (t, x) u (t, x b ǫ ) t with the effective drift b = 1 Y b(y) dy Y and u the solution of the homogenized problem u t div (A u) = 0 in R n (0, T) u(t = 0, x) = u 0 (x) in R n

10 10 Precise convergence u ǫ (t, x) = u (t, x b ǫ ) t + r ǫ (t, x) with T lim ǫ 0 0 R n r ǫ (t, x) 2 dt dx = 0,

11 11 Homogenized diffusion tensor A ij = D(e i + y χ i ) (e j + y χ j )dy Y where χ i (y), 1 i n, are solutions of the cell problems b(y) y χ i div y (D(y) ( y χ i + e i )) = (b b(y)) e i D(y) ( y χ i + e i ) n = 0 on O y χ i (y) Y -periodic in Y Remark that the value of b is exactly the compatibility condition for the existence of χ i.

12 12 Define ũ ǫ (t, x) = u Equivalent homogenized equation (t, x b ǫ t ). Then, it is solution of ũ ǫ t + 1 ǫ b ũ ǫ div (A ũ ǫ ) = 0 in R n (0, T) ũ ǫ (t = 0, x) = u 0 (x) in R n

13 13 -III- TWO-SCALE ANSATZ WITH DRIFT To motivate our result, let us start with a formal process. Standard two-scale asymptotic expansions should be modified to introduce an unknown large drift b R n u ǫ (t, x) = + i=0 ( ǫ i u i t, x b t ǫ, x ), ǫ with u i (t, x, y) a function of the macroscopic variable x and of the periodic microscopic variable y Y = (0, 1) n.

14 14 We plug these ansatz in the system of equations and use the usual chain rule derivation ( (u i t, x b t ǫ, x )) = ( ) ( ǫ 1 y u i + x u i t, x b t ǫ ǫ, x ), ǫ plus a new contribution t ( (u i t, x b t ǫ, x )) ǫ = u i t ǫ 1 b x u }{{} i ( t, x, x ǫ new term )

15 15 Fredholm alternative in the unit cell Lemma. The boundary value problem b(y) y χ div y (D(y) y χ) = f D(y) y χ n + g = 0 on O y χ(y) Y -periodic in Y admits a unique solution χ H 1 (Y )/R (up to an additive constant), if and only if f(y) dy + g(y) ds = 0. Y O

16 16 Variational formulation of the cell problem b(y) y χ(y)φ(y) dy + D y χ(y) y φ(y) dy = Y Y f(y)φ(y) dy + g(y)φ(y) ds Y O for any test function φ H 1 (Y ). Coercive bilinear form on the orthogonal subspace to its kernel R.

17 17 Cascade of equations Equation of order ǫ 2 : b(y) y u 0 div y (D(y) y u 0 ) = 0 in Y D(y) y u 0 n = 0 on O y u 0 (t, x, y) Y -periodic We deduce u 0 (t, x, y) u(t, x)

18 18 Equation of order ǫ 1 : b x u 0 + b(y) ( x u 0 + y u 1 ) div y (D(y) ( x u 0 + y u 1 )) = 0 in Y D ( x u 0 + y u 1 ) n = 0 on O y u 1 (t, x, y) Y -periodic We deduce u 1 (t, x, y) = n i=1 u 0 x i (t, x)χ i (y)

19 19 Cell problem b(y) y χ i div y (D(y) ( y χ i + e i )) = (b b(y)) e i in Y D(y) ( y χ i + e i ) n = 0 on O y χ i (y) Y -periodic The compatibility condition (Fredholm alternative) for the existence of χ i is b = 1 Y b(y) dy Y

20 20 Equation of order ǫ 0 : u t b x u 1 + b ( x u 1 + y u 2 ) div y (D ( x u 1 + y u 2 )) div x (D( y u 1 + x u)) = 0 in Y D ( x u 1 + y u 2 ) n = 0 on O y u 2 (t, x, y) Y -periodic Compatibility condition for the existence of u 2 homogenized problem. (Recall that u 1 is linear in x u.) u t div (A u) = 0 in R n (0, T) u(t = 0, x) = u 0 (x) in R n,

21 21 -IV- RIGOROUS PROOF The proof is made of 3 steps 1. A priori estimates. 2. Passing to the limit by two-scale convergence with drift. 3. Strong convergence.

22 22 A priori estimates Assume u 0 L 2 (R n ). For any final time T > 0, there exists a constant C > 0 that does not depend on ǫ such that u ǫ L (0,T;L 2 (Ω ǫ )) + u ǫ L 2 ((0,T) Ω ǫ ) C Proof. Multiply the fluid equation by u ǫ and integrate by parts to get the usual parabolic estimates.

23 23 Usual two-scale convergence Proposition. Let w ǫ be a bounded sequence in L 2 (R n ). Up to a subsequence, there exist a limit w(x, y) L 2 (R n T n ) such that w ǫ two-scale converges to w in the sense that ( lim w ǫ (x)φ x, x ) dx = ǫ 0 R ǫ n for all functions φ(x, y) L 2 (R n ; C(T n )). R n T n w(x, y)φ(x, y) dxdy

24 24 Two-scale convergence with drift Proposition (Marusic-Paloka, Piatnitski). Let V R N be a given drift velocity. Let w ǫ be a bounded sequence in L 2 ((0, T) R n ). Up to a subsequence, there exist a limit w 0 (t, x, y) L 2 ((0, T) R n T n ) such that w ǫ two-scale converges with drift weakly to w 0 in the sense that T ( lim w ǫ (t, x)φ t, x + V ǫ 0 0 R ǫ t, x ) dt dx = ǫ n T w 0 (t, x, y)φ(t, x, y) dt dxdy 0 R n T n for all functions φ(t, x, y) L 2 ((0, T) R n ; C(T n )).

25 25 Lemma. Let φ(t, x, y) L 2 ( (0, T) T N ; C c (R N ) ). Then T lim ǫ 0 0 R N (t, φ x + V ( x ) ) 2 ǫ t, dt dx = ǫ T 0 R N T N φ(t, x, y) 2 dt dxdy. Proof. Change of variables x = x + V ǫ t T 0 R N (t, φ x + V ( x ǫ t, ǫ ) ) 2 dt dx = T 0 R N ( φ t, x, x ǫ V ) 2 ǫ 2 t dt dx We mesh R N with cubes of size ǫ, R N = i Z Yi ǫ with Yi ǫ = x ǫ i R N ( φ t, x, x = i Z ǫ V ) 2 ǫ 2 t dx = i Y ǫ i ǫ N T N φ (x ǫ i, y) 2 dy + o(1) = + (0, ǫ)n ( φ t, x ǫ i, x ǫ V ) 2 ǫ 2 t dx + o(1) Ω Y φ (x, y) 2 dxdy + o(1)

26 26 Passing to the limit We multiply the equation by an oscillating test function with drift V = b Ψ ǫ = φ (t, x + Vǫ ) ( t + ǫ φ 1 t, x + V ǫ t, x ) ǫ and we use the two-scale convergence with drift to get the homogenized equation.

27 27 STRONG CONVERGENCE We use the notion of strong two-scale convergence with drift. Proposition. If w ǫ (t, x) two-scale converges with drift weakly to w 0 (t, x, y) (assumed to be smooth enough) and lim w ǫ L2 ((0,T) R ǫ 0 n ) = w 0 L2 ((0,T) R n T n ), then it converges strongly in the sense that w ǫ(t, x) w 0 (t, x b ǫ t, x ) ǫ lim ǫ 0 L 2 ((0,T) R n ) = 0

28 28 -V- NUMERICAL RESULTS Numerical computations done with FreeFem++ in 2-d for circular obstacles. The velocity b(y) is generated by solving the following filtration problem in the fluid part Y of the unit cell Y y p y b = e i in Y ; div y b = 0 in Y ; b = 0 p, b are Y periodic. on O;

29 29

30 Log-log plot of the longitudinal dispersion A 11 as a function of the local Péclet s number (asymptotic slope 1.7).

31 31 -VI- THE CASE OF BOUNDED DOMAINS Consider now a bounded domain Ω with a Dirichlet boundary condition on Ω. To study the long time behavior we consider the eigenvalue problem 1 ( x ) ( x ) ǫ b u ǫ div(d u ǫ ) = λ ǫ u ǫ in Ω ǫ ǫ ǫ u ǫ = 0 on Ω Ω ǫ, ( x ) D u ǫ n = 0 on Ω ǫ \ Ω ǫ where λ ǫ and u ǫ are the first eigenvalue and eigenfunction (which exists by the Krein-Rutman theorem). Assumptions: Stationary incompressible periodic flow div y b = 0 in Y, b n = 0 on O Periodic symmetric coercive diffusion D

32 Typical expected behavior of the first eigenfunction: because of the large drift it behaves like a boundary layer.

33 33 Following Bensoussan-Lions-Papanicolaou and Capdeboscq, for θ R N, we introduce the spectral cell problem div y (D(y) y ψ θ ) + b(y) y ψ θ = λ(θ)ψ θ in Y D(y) y ψ θ n = 0 on O y ψ θ (y)e θ y Y -periodic where λ(θ) and ψ θ are the first eigenvalue and eigenfunction. We also introduce the adjoint spectral cell problem div y (D(y) y ψθ) div y (b(y)ψθ) = λ(θ)ψθ D(y) y ψθ n = 0 on O y ψθ (y)e+θ y Y -periodic in Y The first eigenfunctions ψ θ and ψθ can be chosen positive and normalized by ψ θ (y)e θ y 2 dy = 1 and ψ θ (y)ψθ(y) dy = 1 T N T N

34 34 Lemma. The function θ λ(θ) is strictly concave from R N into R and admits a maximum λ which is obtained for a unique θ = θ. Denoting ψ = ψ θ and ψ = ψ θ, the vector field b(y) = ψ ψ b(y) + ψ D y ψ (y) ψ D y ψ (y) satisfies div y b = 0 in T N, T N b(y) dy = 0.

35 35 Change of unknown Define a new unknown function ũ ǫ (x) = u ǫ(x) ( ψ x ) ǫ ( and multiply the equation by ψ x ) ǫ. We obtain 1 x ) ũ ǫ div( ǫ b( ǫ D ( x ) ( x ) ũ ǫ ) = µ ǫ (ψ ψ ǫ ) ǫ ũ ǫ = 0 on Ω Ω ǫ, ( x ) D ũ ǫ n = 0 on Ω ǫ \ Ω ǫ ũ ǫ in Ω ǫ with D(y) = ψ (y)ψ (y)d(y) and µ ǫ = λ ǫ λ(θ ) ǫ 2

36 36 Homogenization Theorem 1. Since the new velocity b is divergence-free and has zero average, classical periodic homogenization can be applied lim µ ǫ = µ, ǫ 0 ũ ǫ ũ weakly in H 1 0(Ω) where (µ, ũ) is the first eigencouple of div( D ũ) = µũ ũ = 0 on Ω in Ω with D the usual homogenized matrix for D.

37 37 Conclusion Theorem 2. The asymptotic behavior of the first eigencouple of the original problem is λ ǫ = λ(θ ( ) ǫ 2 + µ + o(1), u ǫ (x) e + θ x x ) ǫ φ ũ(x) ǫ where φ (y) = ψ θ (y)e θ y is a Y -periodic function. Remark. The direction of concentration θ is different from the drift b = Y b(y) dy!

38 38 (a) t=0 (b) t=0. (c) t=0.02 (d) t=0.03 (e) t=0.04 (f) t=0.05 Isovalues of u ǫ for various t in the parabolic case (computations by I. Pankratova)

39 39 (g) t=0 (h) t=0.1 (i) t=0.2 (j) t=0.3 (k) t=0.4 (l) t=0.5 Rescaled isovalues of u ǫ for larger times t in the parabolic case (computations by I. Pankratova)