Sequential Methods for Coupled Geomechanics and Multiphase Flow

180 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
From det(B s sn) = 0, we have
γe = 0, −
Kdr
which yields the additional condition for convergence as
6.3.4 Fixed-stress split
b 2
1
Mmp + (BoT n o + BwT n w) ∆t
h2 , (6.60)
2 (1 − cos θ)
τ ≡ b2 Mmp
Kdr
< 1. (6.61)
The fixed-stress split with IMPES predicts the strain change of the flow problem as Equa-
tion 6.45. Then, the error equation for flow is
hA s,n
p + b2
− T n s
h
KdrBw
e k+1
Po,j
∆t
e k+1
− 2ek+1 + ek+1
Po,j+1 Po,j Po,j−1
e k Po,j
∆t
hAs,n u
+
∆t (−
ek U
j− 1
2
− e k U j+ 1 2
b2
−
KdrBw
h
)
= 0. (6.62)
The error equation for mechanics is the same as Equation 6.41. Using the spectral
method, the matrix equation is written as
where
⎡
B s ss
⎡
⎣ ˆ ⎤ ⎡
Po
⎦ = ⎣
Û
0
⎤
⎦, (6.63)
0
Bs ss = ⎣ As,n p hγe + T n s ∆t
b2
h 2 (1 − cos θ) + KdrBw (γe − 1) A s,n
u 2i sin θ
2γ From det(B s ss) = 0, we have
b2i sin θ
2 γe
Kdr
h 2 (1 − cos θ)γe
γe = 0 , (6.64)
⎤
⎦.