Diffusion in Deforming Porous Media - Department of Mathematics

6 R.E. SHOWALTERthat is, where β = 0, only the effective or elastic component of stress is specified,since there the fluid pressure does not contribute to the support of the matrix. Onthe entire boundary there is a transverse flow that is given by the input h 1 (·) andthe relative normal displacement of the structure. This input could be specifiedin the form h 1 (t) = −(1 − β)v(t) · n, where v(t) is the given velocity of fluid orboundary flux on Γ t . The first term and right side of this flux balance is nullwhere β = 1, so the same holds for the second terms in (2e), that is, we have theimpermeable conditions k ∂p(t) = 0 on a completely sealed portion of Γ ∂nt. We alsonote that in (2e) the first term on the left side and the right side of the equationare null on Γ 0 , so the same necessarily holds for the second term on the left side.That is, we always have the null flux condition k ∂p∂n = 0 on Γ 0.We proved in [48] that the initial-boundary-value problem (2) is essentially a parabolicsystem which has a strong solution under minimal smoothness requirementson the initial data and source h(·). In particular, the dynamics of the problemcorresponds to an analytic semigroup in L 2 for strong solutions and in H −1 forweak solutions.2.2. Composite Deformable Porous Media. The simplest and most frequentlyused model of flow in a rigid fully–saturated but heterogeneous medium with severaldistinct spatial scales which allows for qualitatively different properties is theBarenblatt double–diffusion model. This consists of the combined effects of two distinctcomponents in parallel. Both components occur locally in any representativevolume element, and they behave as two independent diffusion processes which arecoupled by a distributed exchange term that is proportional to the difference inpressure between fluids in the two components. In the special case which is usedto model naturally fractured media, the first component of the model is the highlydeveloped fracture system and the second is the porous matrix structure. SeeBarenblatt et al [6], Bai et al [5], Bear [7], Warren-Root [59]. In the more complexmodels of double–diffusion combined with deformation, both of the pressure fieldscontribute to the stress field of the structure, so it is necessary to incorporate Biot’sconcepts into the Barenblatt model. The momentum equations contain contributionsto total stress from each of the two pressure fields, and the two equations offluid transport follow from the continuity of fluid mass and consideration of theeffects of dilation of the structure on the flow in both of the components. The fluidtransport within this composite deformable porous medium is described by a pairof pressure equations for diffusion in the respective components of the mediumtogether with an exchange term. This simplistic combination of the Barenblattdouble–diffusion model with the Biot diffusion–deformation model has been developedand used extensively in the engineering literature, and it takes the form(3a)(3b)(3c)−(λ + µ)∇(∇ · u(x, t)) − µ∆u(x, t) + α 1 ∇p 1 (t) + α 2 ∇p 2 (t) = 0 ,c 1 ṗ 1 (t) − ∇ · k 1 ∇p 1 (t) + α 1 ∇ · ˙u(t) + γ(p 1 (t) − p 2 (t)) = h 1 (t) ,c 2 ṗ 2 (t) − ∇ · k 2 ∇p 2 (t) + α 2 ∇ · ˙u(t) + γ(p 2 (t) − p 1 t)) = h 2 (t) .