Sequential Methods for Coupled Geomechanics and Multiphase Flow

138 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
as
where
γe = 0. (5.76)
Then G can be expressed using a similarity transform as G = PJP −1 , (Strang, 1988),
⎡ ⎤
0 1
J = ⎣ ⎦. (5.77)
0 0
Since γ e ss = 0, the fixed-stress split is a convergent scheme with a fixed number of
iterations. Furthermore, for the linear coupled flow-geomechanics problem, exactly two
iterations of the fixed-stress scheme are needed to converge to the fully coupled solution
because G 2 = PJ 2 P −1 = 0. This assumes that the local Kdr is estimated exactly in the
flow problem.
Remark 5.4. The local Kdr may not be estimated exactly in the flow problem when
complex boundary conditions are present, as pointed out in Remark 5.2. Introducing the
deviation factor η and following the same procedure of the spectral method described in
Chapter 4, we obtain
γe = 0,
b2 (η − 1)
Kdr
M + ηb2 , (5.78)
+ Kdrχ2(1 − cosθ)
lim
∆t→0 max|γe|
|η − 1|
=
1
τ
+ η . (5.79)
Even though lim
∆t→0 max|γe| = 0, max|γe| is much smaller than one because 1 ≤ η ≤ 3,
where we follow the dimension-based estimation for K est
dr
explained after Remark 5.2. Hence,
efs decreases exponentially as we increase the number of time steps and reduce the time
step size. Hence, first-order accuracy in time is obtained eventually.