Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.7. NUMERICAL EXAMPLES 65
cannot be solved by the drained split. The sharpness of the stability estimate is also valid
in multidimensional problems, and this is shown later when we study Case 3.3.
The results from Figures 3.9–3.11 indicate that the stability criterion of Equation 3.50
is quite sharp. The drained split yields severe oscillatory behaviors even when it is stable,
which is expected due to the presence of a negative amplification factor.
Other time discretization schemes
We investigate the midpoint time discretization (α = 0.5 for both mechanics and flow)
and the mixed time discretization (α = 1.0 for mechanics and α = 0.5 for flow). Here,
Kdr = 1GPa and other data are the same as Table 3.2. Figure 3.12 shows the stability
behaviors of the midpoint (α = 0.5) and backward Euler (α = 1.0) time discretizations
for a very low coupling strength τ = 0.083. The drained split is unstable even though the
coupling is very weak (see the top figure), whereas the undrained split is stable.
In Figure 3.13, the mixed time discretization is applied to the mechanics (α = 1.0)
and flow (α = 0.5). When the coupling strength is less than one, both the drained and
undrained split are stable. On the other hand, the drained split is unstable when the
coupling strength is 1.11, while the undrained split is stable. Therefore, the numerical
results support all stability estimates obtained from the Von Neumann method.
Case 3.3—The Mandel problem
Mandel’s problem is commonly used to show the validity of simulators for coupled flow and
geomechanics. A description and analytical solution to the Mandel problem is presented in
Abousleiman et al. (1996).
For this case, the homogeneous domain has dimensions of 0.02 m × 100 m discretized
using 4 × 25 grid blocks. The domain has an overburden ¯σ = 2 × 2.125 MPa at the
top, no-horizontal displacement boundary on the left side, and no vertical displacement
boundary at the bottom. The right side has a side burden σh ¯ = 3 × 2.125 MPa, where
we impose constant horizontal displacement. The bulk density of the porous medium is
ρb = 2400 kg m −1 . Initial fluid pressure is Pi = 2.125 MPa. Fluid density and viscosity