Spatial stability for the monolithic and sequential methods with various space discretizations in poroelasticity

Abstract

Summary We investigate spatial stability with various numerical discretizations in displacement and pressure fields for poroelasticity. We study 2 sources of the early time instability: discontinuity of pressure and violation of the inf‐sup condition. We consider both compressible and incompressible fluids by employing the monolithic, stabilized monolithic, and fixed‐stress sequential methods. Four different spatial discretization schemes are used: Q1Q1, Q2Q1, Q1P0, and Q2P0. From mathematic analysis and numerical tests, the piecewise constant finite volume method for flow provides stability at the early time for the case of the pressure discontinuity. On the other hand, a piecewise continuous (or higher‐order) interpolation of pressure shows spatial oscillation, having lower limits of time step size, although lower approximations of pressure than displacement can alleviate the oscillation. For an incompressible fluid, Q2Q1 can be better than Q1P0, because Q1P0 might not satisfy the inf‐sup condition. However, regardless of fluid compressibility and the pressure discontinuity, the fixed‐stress method can effectively stabilize the oscillation without an artificial stabilizer. Even when Q1P0 and Q1Q1 with the monolithic method cannot satisfy the inf‐sup condition, the fixed‐stress method can yield the full‐rank linear system, providing stability. Thus, the fixed‐stress method with Q1P0 can effectively circumvent the aforementioned 2 types of instability.

Dynamic soil consolidation model using a nonlocal continuum poroelastic damage approach

“…The results obtained from their works show good convergence behavior and do not exhibit spurious pressure modes. Further detailed comparative studies of the stability of these elements have been reported in Yoon and Kim 120 …”

Spectral deferred correction methods for high‐order accuracy in poroelastic problems

“…This mixed spatial discretization provides a relatively stable pressure field for the early‐time behavior of poromechanics, allowing a discontinuous pressure field caused by instantaneous pressure build‐up. Whereas the mixed spatial discretization may cause checkerboard‐type pressure oscillations when the monolithic method is employed, resulting in a violation of the inf‐sup condition, the fixed‐stress sequential method, described in the following section, can prevent this violation from occurring 14 . A block‐partitioned preconditioning skill based on the fixed‐stress sequential approach can solve the linear system of the monolith method with more computational efficiency, avoiding violation of the inf‐sup condition, although the direct method of matrix solution is used for the monolithic method 44 .…”

“…Several stabilization techniques have been proposed for spatial stability for the equal‐order approximations 46–48 . In this case, the fixed‐stress method can stabilize spatial solution effectively 14 …”

“…For example, gas hydrate deposits can undergo severe deformation and geological failure induced by hydrate dissociation and gas production from depressurization 1–3 . Coupled processes between flow and geomechanics frequently cause numerical stability and convergence issues that may affect reliable numerical simulation, and therefore require both accuracy and efficiency 1–15 . Thus, an in‐depth analysis of the numerical accuracies of these methods in multiphysics problems is essential.…”

Large deformation poromechanics with local mass conservation: An enriched Galerkin finite element framework

“…This has motivated a number of poromechanical formulations that have combined a locally conservative method for the fluid flow equation with CG discretization of the solid deformation equation. The types of locally conservative methods used for this purpose include the finite volume method, the Raviart‐Thomas mixed finite element method, the discontinuous Galerkin (DG) method, and the enriched Galerkin (EG) method . These studies have shown that the use of a locally conservative method enables more robust solution of the fluid flow equation in a poromechanical problem.…”