Sequential Methods for Coupled Geomechanics and Multiphase Flow

228 APPENDIX B. MISCELLANEOUS DERIVATIONS
From Equations B.4 and B.8, we obtain
R n+1
1
= B
2 ∆t∂2 x n+1
tr
∂t2 + O(∆t2
) . (B.9)
Therefore, the fully coupled method with the backward Euler time discretization is conver-
gent with first-order accuracy in time, O(∆t).
B.3 Drained Split in Coupled Flow and Dynamics
Imposing max |γe| ≤ 1 for all θ, Equation 4.38 provides a necessary condition for stability
of the drained split. Namely,
∆t2Mb2
2(1 − cos θ) ≤ 1 +
ρbh2 ∆t2Kdr 2(1 − cos θ)
ρbh2 which yields
1 + ∆tMkp
2(1 − cos θ)
µh2 for all θ,
(B.10)
4 ∆t2Mb2
≤ 1 + 4
ρbh2 ∆t2Kdr ρbh2
1 + 4 ∆tMkp
µh2
. (B.11)
Since the undrained limit is kp = 0, Equation B.11 reduces to Equation 4.45.
B.4 Non-Contractivity with Different Constitutive Relations
We investigate non-contractivity of the constitutive relations proposed by Lewis and Sukir-
man (1993) and Lewis and Schrefler (1998), which are slightly different from those by Coussy
(1995). The relations for fluid pressure, total and effective stresses by Lewis and Sukirman