Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.3. OPERATOR SPLITTING 33
The stress–strain relation for linear poroelasticity takes the form:
σh = σ ′ h − bph1, δσ ′ h = Dpsδεh, (3.23)
where σ ′ is the effective stress tensor, and Dps is the elasticity matrix which, for 2D plane
strain conditions, can be written as
Dps =
E(1 − ν)
(1 + ν)(1 − 2ν)
⎡
1
⎢ ν
⎣
ν
1−ν
ν
1−ν
ν
1−ν 1 1−ν
ν
1−ν
ν
1−ν
where E is the Young modulus, and ν is the Poisson ratio.
1
⎤
⎥ , (3.24)
⎦
The coupled equations that describe quasi-static poromechanics form an elliptic–parabolic
system. A fully discrete system of equations can be obtained by further discretizing in time
the mass accumulation term in Equations 3.19–3.20. We use the generalized mid-point rule
for time discretization.
3.3 Operator Splitting
The fully coupled method solves the equations of flow and mechanics simultaneously and
obtains a converged solution through iteration, typically using the Newton-Raphson method
(Lewis and Sukirman, 1993; Wan et al., 2003; Jha and Juanes, 2007; Jean et al., 2007). The
fully coupled method is unconditionally stable, but requires high computational cost and
the development of a unified simulator for flow and mechanics. It is thus desirable to develop
solution methods that have stability properties comparable to the fully coupled approach,
but that are computationally more efficient and easier to implement. Sequential solution
methods offer such a possibility. With sequential methods, we can employ separate flow
and mechanics simulators.
There are two representative sequential strategies, in which one solves the mechanical
problem first. One is the drained split shown in the left diagram of Figure 3.2. The other