Stable cell-centered finite volume discretization for Biot equations

TL;DR

This paper introduces a novel stable finite volume discretization for Biot equations that directly couples deformation and flow, ensuring stability, convergence, and local computability without artificial stabilization.

Contribution

The paper presents a new coupled discretization for Biot equations with co-located variables, stability in limiting cases, and explicit local expressions, which is a novel approach.

Findings

The discretization is stable for incompressible fluids and small time steps.

The method converges as proven by stability and consistency analysis.

Numerical examples verify the theoretical results.

Abstract

In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate sub-problems. The coupled discretization has the following key properties, the combination of which is novel: 1) The variables for the pressure and displacement are co-located, and are as sparse as possible (e.g. one displacement vector and one scalar pressure per cell center). 2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. 3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces as well as mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and convergence of the method.