Sequential Methods for Coupled Geomechanics and Multiphase Flow

140 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
The first three cases assume linear poroelastic behavior. The numerical results are based
on a one-pass (implicit-implicit) strategy per time step (i.e., staggered method) unless noted
explicitly otherwise.
Case 5.1—The Terzaghi problem
The numerical values of the parameters for Case 5.1 are the same as Case 3.1. The Biot
modulus M is left unspecified to test the performance of the fixed-strain and fixed-stress
splits for different values of the coupling strength τ.
Figure 5.2 shows the results of the numerical experiments. Both sequential methods
are stable for τ = 0.83 < 1 (Figure 5.2 (top)). The fixed-strain split, however, is unstable
for τ = 1.21 > 1 (Figure 5.2 (bottom)). On the other hand, the fixed-stress split is stable
for τ = 1.21 > 1 (Figure 5.2 (bottom) ). Even in the region of stability, the fixed-strain
split produces wildly oscillatory solutions, in agreement with the predictions of the Von
Neumann stability analysis.
Case 5.2—1D fluid injection and production
The parameter values for Case 5.2 are the same as Case 3.2, and the Biot modulus M is left
unspecified as for Case 5.1. Figure 5.3 shows the results from the numerical simulations,
which support the same conclusion as in Case 5.1. When the coupling strength, τ, is less
than one, both sequential methods are stable. The fixed-strain split is unstable when τ is
greater than one, and it produces an oscillatory solution even when it is stable (in this case,
the oscillations are relatively small). On the other hand, the fixed-stress split is stable and
nonoscillatory in all cases.
Independence of stability limit on time step size for the fixed-strain split
Equation 5.17 indicates that the stability limit for the fixed-strain split is independent of
time step size when 0.5 ≤ α ≤ 1.0. Two time step sizes are considered: ∆td = 4.4 × 10 −2
and ∆td = 4.4 × 10 −5 . The fixed-strain split is stable for both time step sizes when the