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 CO2 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.

中文翻译：

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多孔力学显式耦合模拟中的压力稳定性及其在 CO[公式省略] 封存中的应用


我们详细研究了非迭代固定应力分裂在几乎不排水且不可压缩的多孔力学问题中的压力稳定效应。当与不满足离散 inf-sup 条件的空间离散化（即混合分段线性-分段常数空间离散化）结合使用时，显式固定应力方案可以在瞬态问题中具有压力稳定作用。但是，这种效果会在时间步长细化或达到稳态时消失。将该方案解释为类似于不可压缩流的 Uzawa 迭代的增广拉格朗日方法有助于解释这些结果。此外，由于在不排水的密封区域内缓慢演变的解，我们表明显式固定应力方案需要非常大的时间步长才能在地质 CO2 封存的例子中揭示其压力稳定作用。我们注意到，较大的时间步长会导致排水区域（例如这些示例中的含水层或储层区域）出现较大的误差，并且会阻止多相流情况下非线性求解器的收敛，这会使显式方案成为不可靠的压力稳定源。最后，我们证明了压力跃变稳定在显式固定应力设置中与在未排水问题的完全隐式设置中一样有效，同时保持了排水问题的固定应力分裂的稳定性和收敛性。