Sequential Methods for Coupled Geomechanics and Multiphase Flow

1.3. STABILITY OF SEQUENTIAL SCHEMES 5
coupled flow–mechanics simulation, it is desirable to develop sequential solution methods
that can be competitive with the fully coupled approach. Sequential, or staggered, solution
schemes offer wide flexibility and are highly desirable from a software engineering perspec-
tive. Moreover, sequential schemes allow for using specialized numerical methods for each of
the mechanics and flow problems (Felippa and Park, 1980). In a sequential solution frame-
work, one uses separate software modules for mechanics and flow. Communication between
the modules takes place using a clear, well-defined interface (Felippa and Park, 1980; Gai,
2004; Samier and Gennaro, 2007). In such a setting, the robustness and efficiency of each
simulator (module) — flow and geomechanics — are available for the coupled problem. For
a sequential simulation framework to be competitive in engineering practice, it must be ro-
bust across a wide range of problems and have stability and convergence properties that are
similar to those enjoyed by the fully coupled method. This thesis deals with the challenge of
meeting such requirements with focus on the numerical stability and convergence behaviors
of sequential-implicit solution strategies.
1.3 Stability of Sequential Schemes
Significant efforts to find stable and efficient sequential methods for coupled poromechanics
(or the analogous thermo-mechanics problem) have been pursued in the geotechnical and
computational mechanics communities (Park, 1983; Farhat et al., 1991; Armero and Simo,
1992, 1993; Huang and Zienkiewicz, 1998; Armero, 1999; Soares, 2008). Most of the methods
developed assume that the mechanical subproblem is solved first. Two sequential schemes
are relevant here. One is called the drained split (the isothermal split in the thermo-elastic
and thermo-plastic problems (Armero and Simo, 1992, 1993)), and the other one is the
undrained split (Zienkiewicz et al., 1988; Armero, 1999; Jha and Juanes, 2007) (the adiabatic
split in the thermo-elastic and thermo-plastic problems (Armero and Simo, 1992, 1993)).
The drained split simply freezes the pressure during the mechanical step, thus allowing for
flow to take place. Then, the computed strain and stress fields are used when solving the
flow problem. Despite its simplicity, the drained split is, at best, conditionally stable. The