A fully coupled scheme using Virtual Element Method and Finite Volume for poroelasticity

Julien Coulet 1, 2, * Isabelle Faille 1 Vivette Girault 2 Nicolas Guy 1 Frédéric Nataf 2, 3
* Corresponding author
3 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL - Laboratoire Jacques-Louis Lions
Abstract : In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modelled 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 underground media such as heterogeneities, discontinuities and faults, meshing procedures commonly lead to badly shaped cells for finite element based modelling. 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 the both mechanical and fluid flow modelling. 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.
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Julien Coulet, Isabelle Faille, Vivette Girault, Nicolas Guy, Frédéric Nataf. A fully coupled scheme using Virtual Element Method and Finite Volume for poroelasticity. 2018. ⟨hal-01947455⟩

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