Sequential Methods for Coupled Geomechanics and Multiphase Flow

136 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
Since the matrix is required to be singular, detBsn = 0, which leads to
γe = 0, −
Kdr
1
M
b 2
kp∆t
χ = . (5.67)
+ χ2(1 − cos θ), µh2 The γe’s are equivalent to the eigenvalues of the error amplification matrix G defined
by ⎡
⎣ ek+1
Pj
e k+1
Uj
⎤
⎦ = G
⎡
⎣ ek Pj
e k Uj
⎤
⎦ . (5.68)
The two γe’s in Equation 5.67 are distinct, and G can be decomposed as G = PΛP −1
(Hughes, 1987), where Λ = diag {γe,1, γe,2}, as shown in Chapter 4. By recursion and
Equation 5.68, the fixed-strain split yields
where e n+1,0 = x n+1
f
e n+1,niter ≤ (max|γe|) niter e n+1,0 , (5.69)
− xn,niter s . From Equation 5.67, we obtain
lim
∆t→0 max|γe| = b2M (= 0) , (5.70)
Kdr
which yields D = O(1). Therefore, as ∆t approaches zero, e n fs
does not disappear, and
the fixed-strain split with a fixed number of iterations is not a convergent scheme. Non-
convergence is severe when D approaches unity, which is the stability limit. D ≤ 1
is also a necessary condition for stability (Turska et al., 1994), which yields |γe| ≤ 1. The
stability requirement is that the coupling strength should be less than one, which is the
same as Equation 5.17.
5.5.2 Error amplification of the fixed-stress split
When we consider a fixed number of iterations for the fixed-stress split, the constraint of
the fixed-stress rate becomes
σ n+1,k+1
v
= σ n+1,k
v . (5.71)