Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.7. NUMERICAL EXAMPLES 69
are ρf,0 = 1000 kg m −1 and µ = 1.0 cp, respectively. Permeability is kp = 50 md, and
porosity is φ0 = 0.3. Young’s modulus is E = 2.9 GPa, and Poisson’s ratio is ν = 0.0. The
Biot coefficient is b = 1.0. An observation well is located at the lower left corner (1,25).
We have a drainage boundary for flow at the top, where the boundary fluid pressure is
Pbc = 2.125 MPa. A no-flow boundary condition is applied to both sides and the bottom.
Gravity is neglected. The Biot modulus can be determined from the different values of the
coupling strength τ, which are used for the various numerical tests. The input parameters
are listed in Table 3.3.
Figure 3.14 shows that the drained split (the top figure) is stable when τ is less than one,
while it is unstable when τ is greater than one (the bottom figure). Furthermore, severe
oscillations are observed even though the drained split is stable. Due to the oscillations,
the early time solution is not computed properly in the drained split even though the late
time solution converges to the analytical results. On the other hand, the undrained split
shows unconditional stability and yields a monotonic solution that matches the analytical
solution for all time.
The Mandel–Cryer effect, where a rise in the pressure during early time is observed,
can be captured accurately by the undrained split. The solutions from the fully coupled
method and the undrained split are in good agreement with the analytical solution.
Figures 3.15 and 3.16 show the vertical and horizontal displacements, respectively. The
drained split suffers from severe oscillations in the displacements, even though it is stable
(top figures). But, when the drained split violates the stability condition, it shows instability
(bottom figures). In contrast, the fully coupled and undrained methods are stable regardless
of the coupling strength, and they agree with the analytical solutions.
3.7.2 Numerical results for elastoplasticity
We investigate the behavior of sequential solution methods for nonlinear poro-elasto-plastic
problems. We compare the numerical solutions using a one-pass strategy (i.e., staggered
method) with our a-priori stability estimates for the drained and undrained splits. After
we investigate the sequential schemes with the staggered method, we perform full iterations