Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.3. STAGGERED NEWTON SCHEME WITH IMPES 179
equation is written as
where
B s ud =
⎡
⎣
B s ud
⎡
⎣ Û
⎤ ⎡
⎦ = ⎣
ˆPo
0
⎤
⎦, (6.56)
0
Kud
h γe − b2
M2p
h 2(1 − cos θ) b2i sin θ
2
A s,n
u 2i sin θ
2 γe
A s,n
p hγe + T n s ∆t
h
From det(Bs ud ) = 0, the error amplification factors are obtained as
γe = 0,
Kud
2 (1 − cos θ)γe
b2Mmp (BoT n o + BwT n w) ∆t
h2 2(1 − cos θ)
1
Mmp + (BoT n o + BwT n w) ∆t
h2 . (6.57)
2(1 − cos θ)
Since 0 ≤ γe < 1, the undrained split with IMPES has no additional restriction on
convergence unless the fluids are incompressible.
6.3.3 Fixed-strain split
The fixed-strain split predicts the strain change of the flow problem from the previous
iteration step as in Equation 6.38. Then the error equation for flow becomes
hA s,n
e
p
k+1
Po,j
∆t
hAs,n u
+
∆t (−
ek U
j− 1
2
− e k U j+ 1 2
h
) − T n s
h
⎤
⎦ .
e k+1
− 2ek+1 + ek+1 = 0. (6.58)
Po,j+1 Po,j Po,j−1
The error equation for mechanics is the same as Equation 6.41. Using the spectral
method, we obtain the matrix equation as
⎡
⎣ As,n p hγe + T n s ∆t
h 2 (1 − cos θ)γe A s,n
u 2i sin θ
b2i sin
2
θ
2γe
B
⎤
⎦
Kdr
h 2 (1 − cos θ) γe
s sn
⎡
⎣ ˆ ⎤ ⎡
Po
⎦ = ⎣
Û
0
⎤
⎦.
0
(6.59)