Sequential Methods for Coupled Geomechanics and Multiphase Flow

74 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
for the modified Cam-clay model is shown in the top picture of Figure 3.19 (refer to Borja
and Lee (1990); Borja (1991) for more details).
Case 3.4—1D fluid injection and production in elastoplasticity
The constrained modulus is Kdr = 1 GPa and the Biot modulus is M = 83.3 MPa. The
hardening moduli are unspecified for the numerical tests. The other data and geometry of
the domain are the same as in Case 3.2. The coupling strength τ is 0.083 when the me-
chanical problem is elastic. An iterative staggered method is used, whereby. Each iteration
involves solving two problems in sequence, such that each problem is solved implicitly. So,
for a given time step, a single-pass strategy would entail solving the first sub-problem im-
plicitly (subject to a given tolerance), updating the appropriate terms to set up the second
problem, and finally solving the second problem implicitly. We then move to the next time
step. In the iterative approach, the implicit-implicit solution sequence is repeated until
convergence.
Figure 3.17 shows the numerical results for Case 3.4. On the top figure, the yield stress
and hardening modulus are σ ′ Y
= 1.5 MPa and H = 250 MPa, respectively. This condition
yields a coupling strength less than one for plasticity, τ = 0.417. The drained split is stable
in the elastic regime, since the coupling strength is less than one. After the simulation
enters the plastic regime, at about td = 0.05, the drained split is still stable, shown on the
top of Figure 3.17, because the coupling strength is still less than one.
The case that the yield stress and hardening modulus are σ ′ Y
= 1.5 MPa and H =
25 MPa, respectively, is shown in Figure 3.17 (bottom). These conditions yield a coupling
strength greater than one for plasticity, τ = 3.41. The bottom figure shows that the drained
split is unstable in the plastic regime because the coupling strength is higher than one.
This supports the a-priori stability estimates from the Von Neumann method. Moreover,
the instability of the drained split cannot be fixed by tuning the time step size, since the
stability limit is independent of the time step size. However, the undrained split is stable
regardless of the coupling strength, as shown in Figure 3.17.