Diffusion in Deforming Porous Media - Department of Mathematics

12 R.E. SHOWALTERthe characteristic function of Y m for m = 1, 2, extended Y -periodically to all of IR N .Thus, χ 1 (y) + χ 2 (y) = 1. We shall assume that the set {y ∈ IR N : χ 1 (y) = 1} issmooth and connected. The domain Ω is thus partitioned into the two subdomains{ ( x) }Ω ε m = x ∈ Ω : χ m = 1 , m = 1, 2 .εLet Γ ε 12 ≡ ∂Ωε 1 ∩ ∂Ωε 2 ∩ Ω be that part of the interface of Ωε 1 with Ωε 2 that is interiorto Ω, and let Γ 12 ≡ ∂Y 1 ∩ ∂Y 2 ∩ Y be the corresponding interface in the cell Y .Likewise, let Γ 22 ≡ Y 2 ∩ ∂Y and denote by Γ ε 22 its periodic extension which formsthe interface between those parts of the second component Ω ε 2 which lie withinneighboring εY -cells. These are the local inclusions and we denote them by Y2 ε.The second component Ω ε 2 may be connected, but this is not required.The flow in the porous structure Ω ε 1 is described by a Darcy system that is coupledacross the interface Γ ε 12 to a Biot system for the slow flow and deformation inthe inclusions Ω ε 2 . In the region Ωε 2 we scale the permeability by ε2 . Thus, we shalldenote by c m and ε 2(m−1) κ m the compressibility and the permeability, respectively,in Ω ε m , m = 1, 2. The fluid pressure in the macroporous region Ωε 1 is denoted byp ε 1 (x, t) and the corresponding flux there is given by −κ 1∇p ε 1 . The pressure in the regionΩ ε 2 is pε 2 (x, t) with scaled flux −ε2 κ 2 ∇p ε 2 . It is the resulting very high frequencyspatial variations in the pressure gradients in the second component which lead tolocal storage and corresponding local deformations. These are described by the(small) displacement u(x, t) from the position x ∈ Ω ε 2 , and ε kl(u) ≡ 1(∂ 2 ku l + ∂ l u k )is the (linearized) strain tensor. The stress σ(u) is given by the generalized Hooke’slaw σ ij (u) = a m ijkl ε kl(u) with the positive definite symmetric elasticity tensor a ijklfor a general anisotropic material. The boundary conditions will involve the surfacedensity of forces or traction σ ij n j . The normal will be directed out of Ω ε 2 . Theelastic structure is described by the bilinear form∫∫e(u, v) ≡ a ijkl ε kl (u)∂ j v i dx = a ijkl ε kl (u)ε ij (v) dxΩ ε 2on the space V ε ≡ {v ∈ H 1 (Ω) : v = 0 on Ω ε 1} of admissable displacements. Thelocal operator is obtained by means of Stokes’ theorem, that is,∫e(u, v) = E(u(x)) · v(x) dxΩ ε 2where the formal operator is given by E(u) i = −∂ j a ijkl ε kl (u) , 1 ≤ i ≤ 3 , wheneveru, v ∈ V and E(u) ∈ [L 2 (Ω ε 2 )]3 .The Darcy flow in the first component and the Biot system for the second componentare given by the highly heterogeneous systemc 1 ṗ ε 1 − ∇ · (κ 1 ∇p ε 1) = F 1 in Ω ε 1 ,Ω ε 2p ε 1 = pε 2 , κ ∂p ε 11∂n = ∂p εε2 2κ 2∂n , uε = 0 on Γ ε 12 ,c 2 ṗ ε 2 − ∇ · (ε2 κ 2 ∇p ε 2 ) + b ε∇ · ˙u ε = F 2 andρ 2 ü ε + ε 2 E(u ε ) + b ε ∇p ε 2 = f 2 , in Ω ε 2 ,