Sequential Methods for Coupled Geomechanics and Multiphase Flow

6 CHAPTER 1. INTRODUCTION
undrained split, on the other hand, freezes the fluid mass content when solving the mechanics
problem, and then it uses the updated stress and strain fields when solving the flow problem.
The undrained split has been shown to respect the dissipative character of the continuum
problem, leading to unconditional stability (Armero and Simo, 1992; Armero, 1999). The
undrained method can be applied to linear (Zienkiewicz et al., 1988; Huang and Zienkiewicz,
1998) and nonlinear (Armero, 1999) coupled problems of poromechanics and flow. Jha and
Juanes (2007) employed the undrained scheme using a mixed finite-element method for
linear poroelasticity, where the primary unknowns are pressure, velocity, and displacement.
Their scheme is locally mass conservative, and enjoys good stability properties in space and
time. They showed that the undrained method is well suited for reservoir simulation.
One can also solve the flow problem first and then deal with the mechanics. The obvious
split, the so called fixed-strain scheme, corresponds to freezing the displacements during the
solution of the flow problem. This method, however, is conditionally stable. Some sequential
methods add a relaxation term to the compressibility coefficient to improve the stability
and enhance the convergence rate (Bevillon and Masson, 2000; Mainguy and Longuemare,
2002; Jean et al., 2007; Settari and Mourits, 1998; Wheeler and Gai, 2007). It has been
shown that this strategy yields stable numerical behavior in the case of linear poroelasticity.
Gai (2004) concludes using several numerical experiences that iterative solution schemes are
comparable to the fully coupled method, in terms of efficiency and accuracy. Still, there
are no robust stability and convergence analyses of sequential methods of coupled flow and
geomechanics, and this is especially the case for nonlinear deformation.
Even though the undrained split has been shown to be unconditionally stable, the
drained split, which is conditionally stable, has not been studies as extensively. For ex-
ample, Armero and Simo (1992) shows the stability limit of the drained split with a single
wave number, which does not provide a sharp stability limit for practical settings. Fur-
thermore, the algorithmic a-priori stability estimate of the undrained split for nonlinear
problems has not been shown, even though Armero (1999) shows that the undrained split
honors energy decay during the mechanical solution step. In order to show unconditional
stability of a sequential-implicit method by the energy method, the following three steps