Sequential Methods for Coupled Geomechanics and Multiphase Flow

154 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
split becomes unstable during iterations (Figures 5.12). No solution by the fixed-strain
split is possible after plasticity. On the other hand, the fully coupled and fixed-stress split
methods provide stable results well into the plastic regime, as shown in Figure 5.12.
Dimensionless pressure (P/P i )
1
0.9
0.8
0.7
0.6
0.5
0.4
Case 5.4 elastoplasticity (Full iterations )
Fixed strain
Fixed stress
Fully coupled
0.3
0 0.01 0.02 0.03 0.04 0.05
Dimensionless time (pore volume produced)
Figure 5.12: Case 5.4 (2D production problem in elastoplastic regime). Shown is the evolution
of the dimensionless pressure at the observation well against dimensionless time (pore
volume produced). Once the model enters the plastic regime, the fixed-strain split becomes
unstable and fails to produce a solution at all.
5.6.2 Convergence
The Terzaghi problem (Case 5.1) - one-dimensional consolidation - is used for convergence
analysis. The coupling strength τ is 0.95, and other input data are same as Case 5.1. We
determine true solutions of pressure and displacement by the fully coupled method with
very small time step sizes.
Figure 5.13 shows the errors between the true and numerical solutions using the fixed-
strain, fixed-stress, and fully coupled methods with respect to time step size when a fixed
number of iterations is performed. The errors in dimensionless pressure and displacement
are measured by the L 2 norm.