Méthode des Éléments Virtuels pour le calcul de la déformation mécanique couplée aux écoulements en milieux poreux

Julien Coulet 1, 2
2 ALPINES - Algorithms and parallel tools for integrated numerical simulations
INSMI - Institut National des Sciences Mathématiques et de leurs Interactions, Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : This manuscript focuses on the design and the study of numerical schemes 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. Depending on the selected finite volume method, we create two different coupled schemes: the vem-tpfa scheme, using a two-point flux approximation for the fluid flow and for which we provide a mathematical analysis, and the vem-mpfa scheme, which uses a multipoint flux approximation and can thus handle more general meshes. In addition to the discretization, we also investigate the way to solve the coupled problem in order to improve efficiency and robustness in the context of parallel computing. For this purpose, we consider the fixed-stress split strategy in order to fall back to the resolution of two subproblems for which we can use efficient solvers.
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Julien Coulet. Méthode des Éléments Virtuels pour le calcul de la déformation mécanique couplée aux écoulements en milieux poreux. Analyse numérique [math.NA]. Sorbonne-Université, 2019. Français. ⟨tel-02416511v2⟩

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