Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.4. STABILITY ANALYSIS FOR LINEAR POROELASTICITY 43
geomechanics. This mixed time discretization yields the following amplification factors:
γ = 1,
−b2 (1 + cos θ)
2Kdr
kp∆t
3M (2 + cos θ) + µh2 , (3.55)
2(1 − cos θ)
from which the stability condition is τ ≤ 1, which is the same as the backward Euler dis-
cretization with finite-volume (for flow) and finite-element (for mechanics) methods. More-
over, this stability condition is sharper than that proposed by Armero and Simo (1992),
which is
2 kp∆tM
≥ τ −
µh2 4
. (3.56)
3
Equation 3.50 clearly satisfies Equation 3.56. The mixed time discretization with the finite-
volume and finite-element methods also yields the same stability criterion as the backward
Euler time discretization.
3.4.2 Undrained split
The undrained split freezes the variation of fluid mass during the mechanical problem, which
from Equations 3.35 and 3.36 leads to
P n+α = −αbM∆ε n + P n , (3.57)
where ∆ε n = ε n+1 − ε n . Then the discretization of the undrained split becomes
− Kdr
h (Un+α
j− 3
2
− 2U n+α
j− 1
2
−(U n
j− 3 − 2U
2
n
j− 1 + U
2
n
j+ 1
2
+ U n+α
j+ 1 ) − α
2
b2M h
)
(U n+1
j− 3
2
− 2U n+1
j− 1
2
+ U n+1
)
j+ 1
2
− b(P n j−1 − P n j ) = 0, (3.58)