Sequential Methods for Coupled Geomechanics and Multiphase Flow

92 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS
errors of the form ek Uj = γk eei(j)θ eU, ˆ ek Vj = γk eei(j)θ eV ˆ , and ek Pj = γk eei(j)θ eP, ˆ we obtain
⎡
⎢
⎣
γe −∆tγe 0
∆tKdr
ρbh 2 2(1 − cos θ)γe
γe
∆tb θ
ρbh2i sin 2
0 ∆tMb θ
h 2i sin 2γe γe + ∆tMkp
µh2
B
2(1 − cos θ)γe
∗ dr
⎤⎡
⎥⎢
⎥⎢
⎥⎢
⎦⎣
eU ˆ
eV ˆ
eP ˆ
⎤
⎥
⎦ =
⎡ ⎤
0
⎢ ⎥
⎢ ⎥
⎢ 0 ⎥
⎣ ⎦
0
.
(4.37)
From det(B∗ dr ) = 0, the error amplification factors of the coupled flow and dynamics for
the drained split are given by
γe = 0, −
Equation 4.38 indicates that
∆t 2 Mb 2
ρbh2 2(1 − cos θ)
1 + ∆t2Kdr ρbh2
2(1 − cos θ) 1 + ∆tMkp
µh2 . (4.38)
2(1 − cos θ)
lim
∆t→0 max |γe| = 0. (4.39)
When we follow a similar procedure for the drained split in coupled flow and stat-
ics (Equations 4.5 – 4.7) and add vk to the unknowns, convergence is obtained for the
drained split of the coupled flow and dynamics from Equations 4.1 and 4.39 because
lim e n+1
= 0. As explained by Armero and Simo (1992), when we use the staggered
∆t→0
fs
method (i.e., one iteration), convergence and first-order accuracy can be directly estimated
by Lie’s formula (Chorin et al., 1978; Lapidus, 1981), written as
lim
n→∞ [exp(A1t/n)exp(A2t/n)] n = exp[(A1 + A2)t] , (4.40)
because Equation 4.29 is split by the drained split as
˙ξ = A1ξ + s1, ˙ ξ = A2ξ + s2, (4.41)