Sequential Methods for Coupled Geomechanics and Multiphase Flow

1.6. OUTLINE 11
(e.g., Coussy (1995), Lewis and Schrefler (1998), Borja (2006)), honoring the thermody-
namics of the problem. Then, we extend the formulation to multiphase flow and transport
in reservoirs.
In Chapter 3, we perform detailed stability analyses of the drained and undrained splits
for coupled flow and geomechanics using the generalized midpoint time integration rule.
We first employ the Von Neumann method to obtain a-priori stability estimates for the
linear coupled problem. To complete the stability analysis of the undrained split for non-
linear problems (i.e., poro-elasto-plasticity), we use the energy method with the generalized
midpoint rule.
In Chapter 4, we investigate the convergence properties of the drained and undrained
splits. The backward Euler time discretization (i.e., implicit time integration) is used. To
obtain a-priori error estimates for both the drained and undrained splitting schemes, we
use matrix and spectral methods.
In Chapter 5, we perform stability analysis of sequential methods that solve the flow
problem first, namely, the fixed-strain and fixed-stress splits. The fixed-strain split freezes
the rate of total volumetric strain whereas the fixed-stress split freezes the rate of total
volumetric stress. We perform a thorough stability analysis for poro-elasticity and poro-
elasto-plasticity for single-phase flow of a slightly compressible fluid. The Von Neumann
and energy methods are used to obtain the stability limits, where the generalized midpoint
rule is used for time discretization. In Chapter 5, we also perform convergence analysis of
the linear coupled problem. Matrix and spectral methods are used to derive a-priori error
estimates of the fixed-strain and fixed-stress splits.
Chapter 6 extends the sequential coupling strategies from single-phase flow to multiphase
flow. Since the sequential splits can be used in staggered Newton methods (Schrefler et al.,
1997), we perform spectral analysis to analyze the convergence properties of the various
splits using the backward Euler time discretization. For nonlinear operator splitting (i.e.,
no linearization is performed of the full problem), we perform stability analysis of the
sequential implicit solution schemes using the energy method with an appropriately defined
norm. In Chapter 7, we summarize our findings and suggest future work.