Pressure stability in explicitly coupled simulations of poromechanics with application to CO$_2$ sequestration

TL;DR

This paper investigates the pressure stabilization effects of the explicit fixed-stress splitting scheme in poromechanics, especially for nearly undrained, incompressible problems like CO₂ sequestration, highlighting its limitations and stabilization techniques.

Contribution

It provides a detailed analysis of the pressure stabilization mechanism of the explicit fixed-stress scheme and compares its effectiveness with other stabilization methods in geomechanical simulations.

Findings

The explicit fixed-stress scheme can stabilize pressure in certain conditions.

Large time steps are needed to observe stabilization, which can cause errors.

Pressure jump stabilization is effective in both explicit and implicit schemes.

Abstract

We study in detail the pressure stabilizing effects of the non-iterated fixed-stress splitting in poromechanical problems which are nearly undrained and incompressible. When applied in conjunction with a spatial discretization which does not satisfy the discrete inf-sup condition, namely a mixed piecewise linear - piecewise constant spatial discretization, the explicit fixed-stress scheme can have a pressure stabilizing effect in transient problems. This effect disappears, however, upon time step refinement or the attainment of steady state. The interpretation of the scheme as an Augmented Lagrangian method similar to Uzawa iteration for incompressible flow helps explain these results. Moreover, due to the slowly evolving solution within undrained seal regions, we show that the explicit fixed-stress scheme requires very large time steps to reveal its pressure stabilizing effect in examples of geologic CO$_2$ sequestration. We note that large time steps can result in large errors in drained regions, such as the aquifer or reservoir regions of these examples, and can prevent convergence of nonlinear solvers in the case of multiphase flows, which can make the explicit scheme an unreliable source of pressure stabilization. We conclude by demonstrating that pressure jump stabilization is as effective in the explicit fixed-stress setting as in the fully implicit setting for undrained problems, while maintaining the stability and convergence of the fixed-stress split for drained problems.