Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.2. STAGGERED NEWTON SCHEMES FOR MULTIPHASE FLOW 173
τ ≡ btMb . (6.33)
Kdr
The convergence estimate of the drained split for multiphase flow shows conditional
convergence, where the coupling strength needs to be less than one. Thus, even though
the drained split can be stable at the beginning when a compressible fluid is flowing, it can
be unstable after we inject an incompressible fluid in later time due to an increase in the
coupling strength. Water injection in an oil reservoir is an example of this situation, which
will be shown in numerical experiments.
6.2.3 Undrained split
The undrained split predicts P k+1
o
where e t 1
in Equation 6.22 as
P k+1 = P k − (ε k+1
v − ε k v)e t 1A −1
f Au ,
= P k − bMmp(ε k+1
v − ε k v), (6.34)
= [1, 0]. Then the error equation for mechanics is written as
−( Kud
h ek+1
U j− 3 2
+( b2Mmp e
h
k+1
U
j− 3
2
− 2 Kud
h ek+1
U j− 1 2
Kud = Kdr + b 2 Mmp.
− 2 b2Mmp e
h
k+1
U
j− 1
2
+ Kud
h ek+1
U ) (6.35)
j+ 1
2
+ b2Mmp e
h
k+1
U ) − b(e
j+ 1
2
k Po,j−1 − ekPo,j ) = 0,
where Kud is the undrained bulk modulus for multiphase flow. The error equations for flow
in the undrained split remain the same as Equations 6.27 and 6.28. Taking the same