Sequential Methods for Coupled Geomechanics and Multiphase Flow

122 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
Note that the constant term of F α sn(γ), −b 2 , is negative, so one root is positive and the
one is negative. The condition for linear stability is max(|γ|) ≤ 1, which is obtained for
F α sn(γ = 1) ≥ 0 and F α sn(γ = −1) ≥ 0. From Equation 5.14, we get
F α sn(γ = 1) = kp∆t
α2(1 − cos θ) ≥ 0 , (5.15)
µh2 F α dr (γ = −1) = 2Kdr + (2α − 1)Kdr
M
kp∆t
µh 2 2(1 − cos θ) − 2b2 ≥ 0. (5.16)
Equation 5.15 is valid for all θ. Equation 5.16 is valid for all θ depending on the weight
α, as follows:
For 0.5 ≤ α ≤ 1 : τ = b2 M
Kdr
For 0 < α < 0.5 : τ = b2 M
Kdr
≤ 1 , (5.17)
µh 2
≤ 1 and ∆t ≤ ( Kdr
M − b2 )
. (5.18)
2(1 − 2α)Kdrkp
Equation 5.17 indicates that the stability of the fixed-strain split depends on the coupling
strength only and is independent of time step size, when 0.5 ≤ α ≤ 1. In the case that
0 < α < 0.5, we obtain an additional condition for stability with restriction on the time step
size. Since one of the γ’s is negative, oscillation is anticipated even when the fixed-strain
split is stable.
Remark 5.1. For the backward Euler time discretization, α = 1, the characteristic equa-
tion of the fixed-strain split (Equation 5.14) is identical to that of the drained split. Notice,
however, that the fixed-strain split with the midpoint rule α = 0.5 is conditionally stable,
even though the drained split with the midpoint rule is unconditionally unstable.
5.2.2 Fixed-stress split
The fixed-stress split freezes the variation of the total stress rate, which yields
∆ε n = b
(∆P
Kdr
n − ∆P n−1 ) + ∆ε n−1 . (5.19)