Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.3. STAGGERED NEWTON SCHEME WITH IMPES 177
where B m ss =
⎡
⎢
⎣
A oo,k
f hγe + T k o ∆t
h 2(1 − cos θ)γe + b2So KdrBo (γe − 1) A ow,k
f hγe A o,k
u 2i sin θ
2
A wo,k
f hγe + T k w ∆t
h 2(1 − cos θ)γe + b2Sw KdrBw (γe − 1) A ww,k
f hγe A w,k
u 2i sin θ
2
b2i sin θ
2 γe
From det (B m ss) = 0, the γe is
which yields unconditional convergence.
0
Kdr
h 2(1 − cos θ)γe
⎤
⎥
⎦ .
γe = 0, (6.49)
6.3 Staggered Newton scheme with IMPES
From the one-dimensional flow equations, the IMPES pressure equation is expressed as
− Aww
f
A oo
f
+ Awo
f
Aow f
= Aww
f
Aow f
As p
wo
∂
∂x
ρo,0
dpo
dt +
− Aww
f
− ∂
ww
∂x ρw,0
A o u + A w u
Aow f
A s u
− Aww
f
Aow f
dεv
dt
f f
+ , (6.50)
B o B w
where all the coefficients and saturation values are explicit except for the pressure. Neglect-
ing the source terms of Equation 6.50, we discretize Equation 6.50 based on the IMPES
method, which produces
hA s,n∆P
p
n o,j
− T n s
h
hAs,n u
+
∆t (−
∆Un j− 1
2
− ∆U n
j+ 1
2
)
∆t h
P n+1
n+1 n+1
o,j+1 − 2Po,j + Po,j−1 = 0, (6.51)
where Ts = − Aww
f
Aow To + Tw. The discretized equation for mechanics is the same as Equa-
f
tion 6.18. The flow problem is itself conditionally stable because we treat saturation and
all the coefficients explicitly (Aziz and Settari, 1979). When the coupled problem is solved