Sequential Methods for Coupled Geomechanics and Multiphase Flow

5.2. STABILITY ANALYSIS FOR LINEAR POROELASTICITY 121
the variation of the strain rate, that is
Full discretization in one dimension yields
h ∆P
M
n j
∆t
−( Kdr
+ bh
∆t (−
h Un+α
j− 3
2
∆U n−1
j− 1
2
− 2 Kdr
∆ε n = ∆ε n−1 . (5.9)
− ∆U n−1
j+ 1
2 ) −
h
kp
P
µh
n+α n+α
j−1 − 2Pj h Un+α
j− 1
2
+ Kdr
h Un+α
j+ 1
2
) − b(P n+α
j−1
+ P n+α
j+1 = 0, (5.10)
− P n+α
j ) = 0. (5.11)
Introducing solutions of the form U n j = γn e i(j)θ Û and P n j = γn e i(j)θ ˆ P, where γ is the
amplification factor, e (·) = exp(·), i = √ −1, and θ ∈ [−π, π], we have
⎡
⎣ Un j
P n j
⎤
⎦ = γ n
⎡
⎣ ei(j)θ Û
e i(j)θ ˆ P
Substituting Equation 5.12 into Equations 5.10 and 5.11, we obtain
Gsn
⎤
⎦. (5.12)
⎡
⎣ 1
kp∆t
θ
M h(γ − 1)γ + µh 2((1 − α) + αγ)γ(1 − cos θ) b(γ − 1)2i sin 2
b2i sin θ
⎤⎡
⎦⎣
Kdr
2
h 2(1 − cos θ)
ˆ ⎤ ⎡
P
⎦ = ⎣
Û
0
⎤
⎦.
0
(5.13)
Since the matrix needs to be singular, detGsn = 0. Then the characteristic equation is
obtained and can be written as
F α sn(γ) =
Kdr kp∆t
+ Kdr α2(1 − cos θ) γ
M µh2 2
+ − Kdr
kp∆t
+ Kdr (1 − α)2(1 − cos θ) + b2 γ − b
M µh2 2 = 0. (5.14)