Sequential Methods for Coupled Geomechanics and Multiphase Flow

170 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
where (·) k = (·) n+1,k . The subscripts J and K denote the fluid phases. Equation 6.21
implies that the solutions at the k th iteration for the fully coupled or sequential methods
are located near the solution of Equations 6.18 – 6.20, as illustrated in Figure 6.1.
Figure 6.1: The region for the linear theory.
Using Equation 6.21 and applying the fully coupled method, we can linearize Equa-
tions 6.18 – 6.20, which can be written as
U n+1
j− 3
2
−( Kdr
− U
h
n
j− 3
2
−b (P n+1
o,j−1 − P n
o,j−1 −
hA oo,k P
f
n+1
o,j − P n o,j
− T k o
h
− 2 Kdr
h
U n+1
j− 1
2
P n+1
o,j − P n o,j)
+ hA ow,k S
f
n+1
w,j − Sn w,j
− U n
j− 1
+
2
Kdr
h
U n+1
j+ 1
2
− U n
j+ 1
)
2
= 0, (6.22)
+ hAo,k u
∆t (−
U n+1
j− 1
2
− U n
j− 1
2
h
U n+1
j+ 1
2
− Un j+ 1
2
)
h
+
∆t
∆t
P n+1
n+1 n+1
o,j−1 − 2Po,j + Po,j+1 = 0, (6.23)
hA wo,k P
f
n+1
o,j − P n o,j
− T k w
h
+ hA ww,k S
f
n+1
w,j − Sn w,j
+ hAw,k u
(−
∆t
U n+1
j− 1
2
− U n
j− 1
2
h
U n+1
j+ 1
2
− Un j+ 1
2
)
h
+
∆t
∆t
∆P n+1
n+1 n+1
o,j−1 − ∆Po,j + ∆Po,j+1 = 0, (6.24)
where one iteration with the fully coupled method is required to obtain the converged