A fully coupled scheme using virtual element method and finite volume for poroelasticity
A fully coupled scheme using virtual element method and finite volume for poroelasticity
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DOI：
10.1007/s10596-019-09831-w
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发表时间：
2019-07
影响因子：
2.5
通讯作者：
J. Coulet;I. Faille;V. Girault;N. Guy;F. Nataf
中科院分区：
文献类型：
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作者：
J. Coulet;I. Faille;V. Girault;N. Guy;F. Nataf
在本文中，我们设计并研究了通过 Biot 方程建模的孔隙弹性问题的全耦合数值方案。对该系统进行数值求解的经典方法是对机械平衡方程使用有限元法，对流体质量守恒方程使用有限体积法。然而，为了捕获地下介质的特定属性，例如异质性、不连续性和断层，网格划分过程通常会导致有限元建模的单元形状不佳。因此，我们研究了最近的虚拟元素方法的使用，该方法似乎是机械零件的潜在离散方法，因此可以使用独特的网格进行机械和流体流动建模。从对地质力学模拟中应用于弹性问题的虚拟元素方法的初步了解开始，我们另外应用有限体积方法来处理流体守恒方程。我们重点研究有限体积部分的一阶虚拟单元法和两点通量近似。提供了对该原始耦合方案的数学分析，包括存在性和唯一性结果以及先验估计。然后通过受实际应用案例启发的二维或三维网格上的一些计算来说明该方法。
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.