Sequential Methods for Coupled Geomechanics and Multiphase Flow

5.2. STABILITY ANALYSIS FOR LINEAR POROELASTICITY 123
Then the discrete form of the fixed-stress split becomes
1
M
−( Kdr
− kp
µh
+ b2
h Un+α
j− 3
2
Kdr
h ∆P n j
∆t
P n+α n+α
j−1 − 2Pj − 2 Kdr
h Un+α
j− 1
2
b2
− h
Kdr
+ P n+α
j+1
+ Kdr
∆P n−1
j
∆U n−1
j− 1
2
− ∆U n−1
j+ 1
2 )
h
∆t
bh
+
∆t (−
= 0, (5.20)
h Un+α
j+ 1
2
) − b(P n+α
Substituting Equation 5.12 into Equations 5.20 and 5.21, we obtain
where Gss =
⎡
⎣
1 b2
M + γ − Kdr
b2
Kdr
Applying detGss = 0
γ = 0,
Gss
j−1
− P n+α
j ) = 0. (5.21)
⎡
⎣ ˆ ⎤ ⎡
P
⎦ = ⎣
Û
0
⎤
⎦, (5.22)
0
h(γ − 1) + kp∆t
θ
µh 2((1 − α) + αγ)γ(1 − cos θ) b(γ − 1)2i sin 2
1
M
b2i sin θ
2
The condition of linear stability yields
Kdr
h 2(1 − cos θ)
b2 + − Kdr
kp∆t
µh2 (1 − α)2(1 − cos θ)
1 b2
M + + Kdr
kp∆t
µh2 . (5.23)
α2(1 − cos θ)
For 0.5 ≤ α ≤ 1 : Unconditionally stable , (5.24)
µh 2
For 0 < α < 0.5 : ∆t ≤
2(1 − 2α)kp
1
M
+ b2
Kdr
⎤
⎦ .
. (5.25)
Hence, the fixed-stress split is unconditionally stable when 0.5 ≤ α ≤ 1. For 0 < α <
0.5, the time step size is limited for stability. When α = 1, the γ’s are non-negative,
which indicates non-oscillatory behavior. The amplification factors γ’s in Equation 5.23 are