Sequential Methods for Coupled Geomechanics and Multiphase Flow

4.3. CONVERGENCE RATE WITH FULL ITERATIONS 95
ek Uj = γk eei(j)θ eU ˆ and ek Pj = γk eei(j)θ eP, ˆ Equations 4.47 and 4.22 yield
⎡
⎣ (Kdr+b 2M h
γe − b2M θ
h )2(1 − cos θ) b2i sin 2
⎤⎡
bγe2i sin θ
2 [ 1
⎦⎣
kp∆t 1
M h + µ h2(1 − cos θ)]γe
Bud
ˆ
e k U
ˆ
e k P
⎤ ⎡
⎦ = ⎣ 0
⎤
⎦.
0
From detBud = 0, the error amplification factors of the undrained split are obtained as
γe = 0,
b 2 χM2(1 − cos θ)
(Kdr + b2M) 1
M + 2χ(1 − cos θ).
(4.49)
(4.48)
From Equation 4.49, when ∆t is refined, max |γe| approaches zero. Since Dud =
max |γe|, e n fs
disappears when the time step size is refined even though a fixed iteration
number is used, including one iteration. Hence, the undrained split is convergent with a
fixed number of iterations for a compressible system, yielding first-order accuracy.
Remark 4.7. We obtain first-order accuracy for the undrained split only for a compress-
ible system. From Equation 4.49, if the fluid and solid grains become incompressible, we
have M → ∞ and max |γe| = 1 regardless of time step size. Thus, we expect severe re-
ductions of accuracy if the system is incompressible (M = ∞), or nearly incompressible
(M ≈ ∞).
Remark 4.8. The undrained split is always stable during iterations, since max |γe| ≤ 1
and γe’s are distinct. Global unconditional stability is rigorously shown in Chapter 3.
4.3 Convergence Rate with Full Iterations
Solutions of sequential methods become the same as the fully coupled method when full iter-
ations are performed if the solutions are stable and convergent during iterations. Then, the
next question is which sequential method is more efficient in terms of the rate of convergence
when full iterations are performed. The error amplification factors, shown in Equations 4.24