Sequential Methods for Coupled Geomechanics and Multiphase Flow

98 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS
|γ e,ud |−|γ e,dr |
0.05
0
−0.05
0
−5
log 10 (χ)
−10
|γ e,ud |−|γ e,dr |
−15
0
1
2
θ (0~π)
Figure 4.2: The difference of the magnitude of the error amplification factors between the
drained and undrained splits.
The true (reference) solutions are computed using the fully coupled method with very
small time step sizes to minimize the temporal error. For space discretization, we employ
the finite-volume and finite-element methods for flow and mechanics, respectively. The
analytical solution is available for Case 1. The reference solution matches the analytical
solution within tight error tolerances.
Case 4.1—The Terzaghi problem
We reuse Case 3.1 in Chapter 3. The coupling strength τ is 0.95 where cf = 3.5×10 −8 Pa −1 .
The other data are the same as those for Case 3.1.
For Case 3.1, Figure 4.4 illustrates the errors between the true and numerical solutions
from the drained, undrained, and fully coupled methods with respect to time step size
when a fixed number of iterations is performed. The errors of dimensionless pressure and
displacement are measured by the L 2 norm. The undrained and fully coupled methods are
convergent when a staggered method is used. As the time step size is refined, the errors
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