Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

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Abstract

We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.
Original languageEnglish
JournalMathematical Geosciences
Issue numberSpecial Issue
Pages (from-to)1-39
ISSN1874-8961
DOIs
Publication statusPublished - 2020

Keywords

  • Poroelasticity
  • Biot’s equations
  • Local mass conservative
  • Finite element
  • Two-field formulation
  • Three-field formulation

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