Sequential Methods for Coupled Geomechanics and Multiphase Flow

5.6. NUMERICAL EXAMPLES 143
coupling strength is less than one (τ = 0.95, Fig. 5.4 (top)). The stability of the fixed-strain
split is not improved by reducing the time step size. Thus, physical problems with large
coupling strength (τ > 1) cannot be solved by the fixed-strain split (Fig. 5.4 (bottom)).
The results from Figures 5.2–5.4 indicate that the stability criterion of the fixed-strain
split is quite sharp. The fixed-strain split yields severe oscillatory behaviors even in its
stability range, which is explained by the negative amplification factor.
Midpoint rule for time integration
We consider the midpoint rule for time discretization (α = 0.5 for both mechanics and
flow). Here, Kdr = 1 GPa and the other data are the same as Case 5.2. Figure 5.5 shows
the stability behaviors for α = 0.5 when τ = 0.83 (top) and τ = 1.11 (bottom). The
fixed-strain split is stable when τ < 1 whereas it is unstable when τ > 1, which supports
our a-priori stability estimates from the Von Neumann method. Furthermore, while the
drained split with the midpoint rule is unconditionally unstable, as shown in Chapter 3,
the fixed-strain split with the midpoint rule is conditionally stable. The fixed-stress split,
however, is unconditionally stable when τ ≥ 1, as shown in the bottom plot of Figure 5.5.
This supports the a-priori stability estimate in Equations 5.17 and 5.24.
The deviation factor η
We use Case 5.2 with the backward Euler time discretization in order to validate the stability
estimate of Equation 5.26, where τ = 3.33 and τ = ∞. From Equation 5.26, the stability
condition becomes η ≥ 0.35 for τ = 3.33 and η ≥ 0.5 for τ = ∞. Figure 5.6 shows the
pressure history with respect to time for τ = 3.33 and τ = ∞. On the top of Figure 5.6, η =
0.33 leads to instability, but η = 0.37 yields a stable solution when τ = 3.33. The bottom
of Figure 5.6 also shows that η = 0.48 causes instability whereas η = 0.52 leads to stability
when τ = ∞. Figure 5.6 supports the fact that the stability estimate of Equation 5.26 is
quite sharp.