New twofold saddle-point formulations for Biot poroelasticity with porosity-dependent permeability

TL;DR

This paper introduces new saddle-point formulations for nonlinear Biot poroelasticity with permeability depending on pressure and dilation, providing theoretical analysis and finite element discretizations with confirmed convergence.

Contribution

It develops four- and five-field Hu--Washizu-type mixed formulations for nonlinear poroelasticity considering porosity-dependent permeability, including solvability analysis and finite element discretizations.

Findings

Proved unique solvability of the formulations.

Established a priori error estimates for discretizations.

Numerical examples confirm theoretical convergence rates.

Abstract

We propose four-field and five-field Hu--Washizu-type mixed formulations for nonlinear poroelasticity -- a coupled fluid diffusion and solid deformation process -- considering that the permeability depends on a linear combination between fluid pressure and dilation. As the determination of the physical strains is necessary, the first formulation is written in terms of the primal unknowns of solid displacement and pore fluid pressure as well as the poroelastic stress and the infinitesimal strain, and it considers strongly symmetric Cauchy stresses. The second formulation imposes stress symmetry in a weak sense and it requires the additional unknown of solid rotation tensor. We study the unique solvability of the problem using the Banach fixed-point theory, properties of twofold saddle-point problems, and the Banach--Ne\v{c}as--Babu\v{s}ka theory. We propose monolithic Galerkin discretisations based on conforming Arnold--Winther for poroelastic stress and displacement, and either PEERS or Arnold--Falk--Winther finite element families for the stress-displacement-rotation field variables. The wellposedness of the discrete problem is established as well, and we show a priori error estimates in the natural norms. Some numerical examples are provided to confirm the rates of convergence predicted by the theory, and we also illustrate the use of the formulation in some typical tests in Biot poroelasticity.