Sequential Methods for Coupled Geomechanics and Multiphase Flow

4.2. SPECTRAL ANALYSIS 89
from the fully coupled method, and U n+1,k and P n+1,k are those from the sequential methods
at the k th iteration step. Let e n,niter
P
and e n,niter
U
be the difference between the solutions from
the fully coupled and sequential methods at tn for pressure and displacement, respectively.
Subtracting Equations 4.19 and 4.20 from Equations 4.17 and 4.18, respectively, gives
− Kdr
h ek+1
U j− 3 2
h
M
We set e n,niter
P
− h
M
e k+1
Pj
∆t
e n,niter
Pj
∆t
+ 2 Kdr
h ek+1
U j− 1 2
h
+ b
∆t (−
e k+1
U
j− 1
2
and e n,niter
U
− b h
∆t (−
− Kdr
h ek+1
U j+ 1 2
− e k+1
U j+ 1 2
h
e n,niter
U j− 1 2
) − kp
µ
− e n,niter
U j+ 1 2
h
− b(e k Pj−1 − ekPj ) = 0, (4.21)
1
h (ek+1 − 2ek+1 + ek+1
Pj+1 Pj Pj−1 )
) = 0. (4.22)
to zero in order to investigate D in Equation 4.4 only.
This implies that we drop the second term in Equation 4.4. Introducing errors of the
form ek Uj = γk eei(j)θ eU ˆ and ek Pj = γk eei(j)θ eP, ˆ where γe is the amplification factor of error,
e (·) = exp(·), i = √ −1, and θ ∈ [−π, π], we obtain from Equations 4.21 and 4.22
⎡
Kdr
h 2γe(1 − cos θ) b2i sin θ
2
⎤⎡
⎣
bγe2i sin θ
2 [ 1
⎦⎣
kp∆t 1
M h + µ h2(1 − cos θ)]γe
Bdr
ˆ
e k U
ˆ
e k P
⎤ ⎡
⎦ = ⎣ 0
⎤
⎦.
0
(4.23)
Since the matrix Bdr is required to be singular (i.e., detBdr = 0), this leads to
γe = 0, −
Kdr
1
M
b 2
kp∆t
χ = . (4.24)
+ χ2(1 − cos θ), µh2 The γe is equivalent to the eigenvalue of the error amplification matrix G defined by
⎡
⎣ ek+1
Uj
e k+1
Pj
⎤
⎦ = G
⎡
⎣ ek Uj
e k Pj
⎤
⎦. (4.25)
The two γe’s in Equation 4.24 are distinct, then G can be decomposed as G = PΛP −1