Sequential Methods for Coupled Geomechanics and Multiphase Flow

178 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
sequentially, additional restriction to ensure convergence may be required, depending on
the characteristics of the particular sequential method used.
6.3.1 Drained split
In the drained split, the predictor of pressure change in the mechanical problem is eval-
uated from the previous iteration, and the error equation for mechanics is the same as
Equation 6.26. The the error equation for flow is
hA s,n
e
p
k+1
Po,j
∆t
hAs,n u
+
∆t (−
e k+1
U
j− 1
2
− e k+1
U j+ 1 2
h
) − T n s
h
e k+1
− 2ek+1 + ek+1 = 0. (6.52)
Po,j+1 Po,j Po,j−1
Using the spectral method and introducing errors of the form e k Uj = γk ee i(j)θ Û and
e k Po,j = γk ee i(j)θ ˆ Po, we obtain the matrix equation as
⎡
Kdr
⎣ h 2 (1 − cos θ) γe b2i sin θ
A
2
s,n
u 2i sin θ
2γe A s,n
p hγe + T n s ∆t
⎤
⎦
h 2 (1 − cos θ)γe
Bs ⎡
⎣
dr
Û
⎤ ⎡
⎦ = ⎣
ˆPo
0
⎤
⎦.
0
(6.53)
From det(Bs dr ) = 0, we have
γe = 0, −
Kdr
b 2
1
Mmp + (BoT n o + BwT n w) ∆t
h2 . (6.54)
2 (1 − cos θ)
The convergence condition for the drained split with IMPES is obtained as
τ ≡ b2 Mmp
Kdr
which is the same as the drained split of FIM, as shown previously.
6.3.2 Undrained split
< 1, (6.55)
In the IMPES formulation, the undrained split yields the error equations for mechanics
and flow as Equations 6.35 and 6.52, respectively. Using the spectral method, the matrix