Sequential Methods for Coupled Geomechanics and Multiphase Flow

130 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
Using the following identity,
Equation 5.46 yields
n+α 1 n+1 n
dp dp − dp
Ω M
dΩ
= 1
2M (dp n+1 2 L2 − dp n 2 1
L2) + (2α − 1)
1
dp 2M
n+1 2
L2 − dp n 2
L2
=
−(2α − 1) 1
2M
−∆t
dp n+1 − dp n 2
L 2 −
where dp n+α = −µk −1 dv n+α from Darcy’s law.
2M
dp n+1 − dp n 2
L 2 , (5.47)
dp n+α b∆dε n vdΩ
dv n+α · kp −1 µdv n+α dΩ, (5.48)
Then, when we solve the flow problem by the fixed-stress split, the evolution of the norm
· N at the discrete time level is shown from Equations 5.45 and 5.48 as
dχ n+1 − dχ N n N = dΣ n+1 2 1
+ dp E 2M
n+1 2
L2 − dΣ n 2 1
E −
2M dpn 2
L2
dΣ
≤ −(2α − 1)
n+1 − dΣ n 2 1
+ dp E 2M
n+1 − dp n 2 L2
−∆t
dv n+α · kp −1 µdv n+α dΩ , (5.49)
from which the condition for unconditional stability is 0.5 ≤ α ≤ 1.
Remark 5.3 The return mapping with maximum plastic dissipation is required in order
to treat b 2 /Kdr implicitly in the flow problem. In this case, the flow problem would be
iterated according to the updated b 2 /Kdr by the return mapping. However, the implicit
treatment of b 2 /Kdr leads to considerable computational cost due to iterations in the flow
problem. Hence, as described in Section 4.2, we treat b 2 /Kdr explicitly (i.e., the previous
time step). Accordingly, the return mapping is not required when solving the flow problem,