Nonlinear Quasi-static Poroelasticity

TL;DR

This paper develops a mathematical framework for analyzing nonlinear quasi-static poroelastic systems, focusing on weak solutions, a priori estimates, and conditions for uniqueness, advancing understanding of complex pressure-dilation interactions.

Contribution

It introduces a full weak solution construction for nonlinear poroelasticity with nonlinear coupling, using semi-discretization and fixed point methods, and provides criteria for solution uniqueness.

Findings

Successful construction of weak solutions for nonlinear poroelastic models

Establishment of a priori estimates for the nonlinear system

Regularity criteria for the uniqueness of solutions

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 both the elastic displacement and pressure variables and give a full treatment of weak solutions. Our construction of weak solutions for the nonlinear problem is based on a priori estimates, a requisite feature in addressing the nonlinearity. We utilize a spatial semi-discretization and employ a multi-valued fixed point argument for a clear construction of weak solutions. We also provide regularity criteria for uniqueness of solutions.