Diffusion in Deforming Porous Media - Department of Mathematics

DEFORMING POROUS MEDIA 5require estimates or properties of solutions for comparative analysis of the appropriatenessof each of these systems as a model for the intended application from amathematical and numerical point of view. Selected quantitative results deliveredby numerical simulations will be validated when possible against available fielddata or examples.2. Recent Results2.1. The Linear Quasi-Static Case. Although there is an enormous amount ofliterature concerning the applications of the coupled quasi-static case of the Biotsystem (1) to engineering and geophysics problems in poroelasticity, one finds veryfew references devoted to the fundamental mathematical issues, even for this simplestsystem. The first results on well-posedness of the quasi-static case of thebasic system (1) appeared in the fundamental work of J.-L Auriault and Sanchez-Palencia [3]. There the meaning of the various coefficients was given by means ofhomogenization. The later paper of Zenisek [64] deals with existence of solutions,and there one has not only ρ = 0 but also the additional degeneracy of the incompressiblecase, c 0 = 0. These seem to be the only such references which addressthe fundamental well-posedness for the coupled quasi-static case, but additionalissues of analysis and approximation of this case were already raised in [39], [38],[63]. We started our study with the analysis of existence, uniqueness and regularityproperties of solutions to the linear quasi-static Biot problem [48]. The modelwas extended to include the exposed pore fraction on the boundary and secondaryviscosity effects. If we denote the characteristic function of the traction boundary,Γ t by χ t , the initial boundary value problem takes the form(2a)(2b)(2c)(2d)(2e)(2f)(2g)−(λ + µ)∇(∇ · u(t)) − µ∆u(t) + ∇p(t) = 0 and∂∂t (c 0p(t) + ∇ · u(t)) − ∇ · k∇(p(t)) = h 0 (t) in Ω ,u(t) = 0 on Γ 0 ,σ ij (u(t))n j − p(t) n i β = 0, 1 ≤ i ≤ 3, on Γ t ,− ∂ ∂t (u(t) · n) (1 − β)χ t + k ∂p(t)∂n = h 1(t) χ t on Γ ,limt→0 +(c 0p(t) + ∇ · u(t)) = v 0 in L 2 (Ω) ,limt→0 +(1 − β)(u(t) · n) = v 1 in L 2 (Γ t ) .The partial differential equations (2a), (2b) comprise the quasi-static case of theBiot system (1). The boundary conditions (2c), (2d) are the complementary pairconsisting of null displacement on the clamped boundary, Γ 0 , and a balance offorces on the traction boundary, Γ t , and (2e) requires a balance of fluid mass.The function β(·) is defined on that portion of the boundary Γ t which is neitherdrained nor clamped, and it specifies the surface fraction of the pores which aresealed along Γ t . Here the hydraulic pressure contributes to the total stress withinthe structure. The remaining portion 1 − β(·) of the pores are exposed along Γ t ,and these contribute to the flux. On any portion of Γ t which is completely exposed,