Sequential Methods for Coupled Geomechanics and Multiphase Flow

160 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
We investigate the convergence behavior of the fixed-stress split with K est
dr
= K3D
dr and
τ = ∞, where M = ∞. We assume a less stiff bulk modulus compared with the true
bulk modulus. From ν = 0.0, we obtain η = 3.0, which is the maximum deviation factor
according to the dimension-based estimation of Kest dr . Figure 5.17 shows the convergence
behavior of pressure. On the top of Figure 5.17, we confirm first-order accuracy explained
in Remark 5.4. The bottom of Figure 5.17 shows clearly that the pressure profile along the
X-Y line in Figure 5.16 converges to the true solution by the fixed-stress split when η = 3.0.
For Case 5.6, the domain has a side burden ¯σh = 2.125 MPa on both sides instead
of the no-horizontal displacement condition in Case 5.5. The other data are the same
as Case 5.5. We take two stiffer and one less stiff bulk moduli for K est
dr
the true bulk modulus. If we select K 1D
dr for Kest dr
compared with
, η is close to the stability limit of 0.5,
because the true Kdr is close to K2D dr . For this reason, we consider Kest
dr
and K est
dr
K est
dr
= 0.95 × K1D
dr
= 1.05 × K1D
dr , yielding η ≈ 0.48 and η ≈ 0.52, respectively. In Figure 5.18,
= 0.95×K1D
dr
provides stability, but with severe oscillations. Using Kest
dr
causes instability. Figure 5.18 shows that using K est
dr
= 1.05×K1D
dr
= K3D
dr , we match the true solution,
capturing the initial rise of pressure (Mandel-Cryer effect). Thus, it is better to take a less
stiff bulk modulus when the true Kdr is not known a-priori. The fixed-stress split uses the
dimension-based estimation for Kest dr , which yields high accuracy without oscillations.