A fully coupled scheme using virtual element method and finite volume for poroelasticity

In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modeled through Biot's equations. The classical way to numerically solve this system is to use a finite element method for the mechanical equilibrium equation and a finite volume method for the fluid mass conservation equation. However, to capture specific properties of the underground medium such as heterogeneities, discontinuities, and faults, meshing procedures commonly lead to badly shaped cells for finite element-based modeling. Consequently, we investigate the use of the recent virtual element method which appears as a potential discretization method for the mechanical part and could therefore allow the use of a unique mesh for both the mechanical and fluid flow modeling. Starting from a first insight into virtual element method applied to the elastic problem in the context of geomechanical simulations, we apply in addition a finite volume method to take care of the fluid conservation equation. We focus on the first-order virtual element method and the two-point flux approximation for the finite volume part. A mathematical analysis of this original coupled scheme is provided, including existence and uniqueness results and a priori estimates. The method is then illustrated by some computations on two- or three-dimensional grids inspired by realistic application cases.