Sequential Methods for Coupled Geomechanics and Multiphase Flow

5.6. NUMERICAL EXAMPLES 149
and 5.10). This behavior is the same as the drained split. However, the solutions from
the fixed-stress split are stable regardless of the coupling strength, and they match the
analytical solutions.
Case 5.4—Fluid production scenario in 2D with elastoplasticity
We consider a nonlinear poroelastoplastic problem using the staggered method, where the
numerical stability is compared with a-priori stability estimates of the fixed-strain and fixed-
stress splits. The modified Cam-clay model is used for plastic modeling (Borja and Lee,
1990; Borja, 1991). We adopt the associative plasticity formulation (Coussy, 1995; Simo,
1991; Simo and Hughes, 1998) for Case 5.4. We use the backward Euler time discretization.
The input values for Case 5.4 are the same as Case 3.5.
In Case 5.4, compaction occurs because of production. As a result, subsidence occurs and
the fluid pressure decreases. Figure 5.11 shows that the fixed-strain split becomes unstable
when it enters the plastic regime, even though it is stable in the elastic regime. This is
because plasticity increases the coupling strength beyond unity. The fixed-stress split is,
however, stable in the plastic regime. Around td = 0.018, we observe some roughness in the
pressure solution because we treat the relaxation term b 2 /Kdr explicitly. The solution from
fixed-stress split is slightly different from the fully coupled solution, since one iteration is
performed. But, when two iterations are taken, the solutions by the fixed-stress split match
those from the fully coupled method, as shown in Figure 5.11. That figure also shows that
plasticity can slow down the rate of decline in the reservoir pressure.
Staggered Newton schemes for Case 5.4
We apply a staggered Newton scheme to Case 5.4. Figure 5.12 shows that the fixed-strain
split is stable during the early-time elastic regime, which has a weak coupling strength
(τ < 1). However, when plasticity is reached, the solution by the fixed-strain method
is no longer stable because the coupling strength increases beyond unity. The bottom of
Figure 5.11 shows the variation of the coupling strength during the simulation. It is clear
that when the coupling strength increases beyond unity, the solution by the fixed-strain