Global existence of weak solutions to unsaturated poroelasticity

¹ University of Bergen, Department of Mathematics, Allégaten 41, 5007 Bergen, Norway
² Hasselt University, Faculty of Sciences, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium
³ University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

^* Corresponding author: jakub.both@uib.no

Received: 4 August 2020
Accepted: 2 October 2021

Abstract

We study unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in variably saturated porous media, here modeled by a non-linear extension of Biot’s well-known quasi-static consolidation model. The coupled elliptic-parabolic system of partial differential equations is a simplified version of the general model for multi-phase flow in deformable porous media, obtained under similar assumptions as usually considered for Richards’ equation. In this work, existence of weak solutions is established in several steps involving a numerical approximation of the problem using a physically-motivated regularization and a finite element/finite volume discretization. Eventually, solvability of the original problem is proved by a combination of the Rothe and Galerkin methods, and further compactness arguments. This approach in particular provides the convergence of the numerical discretization to a regularized model for unsaturated poroelasticity. The final existence result holds under non-degeneracy conditions and natural continuity properties for the constitutive relations. The assumptions are demonstrated to be reasonable in view of geotechnical applications.