Pressure-stabilized fixed-stress iterative solutions of compositional poromechanics

TL;DR

The paper analyzes the fixed-stress splitting method for coupled poromechanics in nearly undrained regimes and clarifies that pressure stability arises from the splitting error rather than the mere avoidance of a discrete saddle-point matrix. By connecting concepts from incompressible flow splitting, the authors show that stability hinges on how the splitting introduces a pressure-regularizing effect, which may fail for iterative schemes unless the underlying spatial discretization is inf-sup stable. Through compositional multiphase simulations of CO$_2$ sequestration, they demonstrate that non-stabilized schemes exhibit pressure oscillations akin to fully implicit solvers, while introducing pressure jump stabilization removes these oscillations and markedly improves convergence. The work provides practical stabilization strategies, including a macro-element–informed choice of the stabilization parameter $\tau$ and selective stabilization (burden-only vs full-domain), with results showing reduced iteration counts and robust pressure fields in undrained regions, making fixed-stress splitting more viable for large-scale subsurface simulations with complex geology.

Abstract

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

Figures (16)

• Figure 1: Undrained Barry-Mercer pressure solution after 10 time steps
• Figure 2: Undrained Barry-Mercer pressure solution after 10 time steps with iterative fixed-stress scheme
• Figure 3: Undrained Barry-Mercer pressure solution at various times with non-iterative fixed-stress scheme
• Figure 4: Staircase mesh. Gray regions represent the channel while blue represents the barrier region. Outlined in red are cells where CO$_2$ is injected.
• Figure 5: Staircase: Number of fixed-stress iterations with varying fixed-stress coefficients $\alpha$

Theorems & Definitions (2)

• Remark
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