Sequential Methods for Coupled Geomechanics and Multiphase Flow

Appendix A
Stability in Space
A.1 Background
Spurious numerical instability and inaccuracy in space at early time may result when nodal
based finite element methods are employed to model consolidation problems. The instability
comes from two parts: the incompressibility of both the fluid and the solid grains, and the
discontinuity of pressure at the drainage boundary.
Since pressure diffusion is negligible at early time, the coupled flow-mechanics problem
converges to a simple mechanical problem with undrained conditions. Hence, the LBB
condition (Fortin and Brezzi, 1991) must be satisfied when both the fluid and the solid
grains are incompressible. This is because, under these stiff conditions, the undrained bulk
modulus is infinite. This numerical instability is obtained with equal-order approximations
of pressure and displacement (e.g., piecewise continuous interpolation), which violates the
LBB condition. Stable results are obtained using the elements satisfying the LBB condition
(Murad and Loula, 1992; Wan, 2002).
However, the instability is still observed at the drainage boundary when the fluid is
compressible even for the elements that satisfy the LBB condition (Vermeer and Verruijt,
1981; Murad and Loula, 1992; Wan, 2002; White and Borja, 2008). For a slightly com-
pressible fluid, the coupled problem at early time reduces to a mechanical problem with a
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