Sequential Methods for Coupled Geomechanics and Multiphase Flow

172 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
γ k ee i(j)θ ˆ
Sw to Equations 6.26 – 6.28, where γe is the error amplification factor. Following a
similar procedure of the error analysis for single-phase flow, we obtain
where
e k+1
fs
B m dr =
⎡
⎢
⎣
Kdr
h 2(1 − cos θ)γe b2i
From det(Bm dr ) = 0, we have
γe = 0, −
Kdr
B m dr
⎡ ⎤
Û
⎢ ⎥
⎢ ⎥
⎢ ˆPo ⎥
⎣ ⎦
Sw
ˆ
=
⎡ ⎤
0
⎢ ⎥
⎢ ⎥
⎢ 0 ⎥,
⎣ ⎦
(6.29)
0
sin θ
2
A o,k
u 2i sin θ
2 γe A oo,k
f hγe + T k o ∆t
h 2(1 − cos θ)γe A ow,k
f hγe
A w,k
u 2i sin θ
2γe A wo,k
f hγe + T k w ∆t
h 2(1 − cos θ)γe A ww,k
f
b 2
φ(coSo + cwSw) + b−φ
Ks + (BoT k o + BwT k w) ∆t
h2 . (6.30)
2(1 − cos θ)
Convergence of the drained split is obtained when max |γe| < 1 for all θ, yielding
< . This condition is also used in Schrefler et al. (1997). e k+1
<
e k fs
indicates that x k+1
s
0
hγe
fs
⎤
⎥
⎦ .
e k fs
is closer to the converged solution x n+1
f , shown in Figure 6.1. The
convergence condition, max |γe| < 1 for all θ, yields
τ ≡ b2Mmp < 1,
Kdr
(6.31)
1
b − φ
= φ(coSo + cwSw) + , (6.32)
Mmp
where we extend the definition of the coupling strength to two-phase flow, using the total
fluid compressibility cf = coSo + cwSw. For three phase flow with capillarity, we can define
the coupling strength as
Ks