Twofold saddle-point formulation of Biot poroelasticity with stress-dependent diffusion

TL;DR

This paper presents a novel stress-dependent diffusion model for poroelasticity, combining a twofold saddle-point formulation with finite element methods, and demonstrates its stability, convergence, and applicability to biological tissue analysis.

Contribution

It introduces a new stress/total-pressure formulation for poroelasticity with nonlinear diffusion, analyzed through fixed-point theory and finite element methods.

Findings

Stable and convergent finite element scheme developed.

Numerical tests validate the model's effectiveness.

Application to brain tissue demonstrates practical relevance.

Abstract

We introduce a stress/total-pressure formulation for poroelasticity that includes the coupling with steady nonlinear diffusion modified by stress. The nonlinear problem is written in mixed-primal form, coupling a perturbed twofold saddle-point system with an elliptic problem. The continuous formulation is analysed in the framework of abstract fixed-point theory and Fredholm alternative for compact operators. A mixed finite element method is proposed and its stability and convergence analysis are carried out. We also include a few illustrative numerical tests. The resulting model can be used to study waste removal in the brain parenchyma, where diffusion of a tracer alone or a combination of advection and diffusion are not sufficient to explain the alterations in rates of filtration observed in porous media samples.