Nonlinear Quasi-static Poroelasticity

Abstract

We analyze a quasi-static Biot system of poroelasticity for both compressible and incompressible constituents. The main feature of this model is a nonlinear coupling of pressure and dilation through the system's permeability tensor. Such a model has been analyzed previously from the point of view of constructing weak solutions through a fully discretized approach. In this treatment, we consider simplified Dirichlet type boundary conditions in the elastic displacement and pressure variables and give a full treatment of weak solutions. Our construction of weak solutions for the nonlinear problem is natural and based on a priori estimates, a requisite feature in addressing the nonlinearity. This is in contrast to previous work which exploits linearity or monotonicity in the permeability, both of which are not available here. We utilize a spatial semi-discretization and employ a multi-valued fixed point argument in for a clear construction of weak solutions. We also provide regularity criteria for uniqueness of solutions.