Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.6. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 57
Integrating over the domain and using the generalized midpoint rule, Equation 3.101 yields
Ω
n+α 1
dp
M
=
= −
Equation 3.102 implies that
(dpn+1 − dpn )
dΩ
∆t
Graddp
Ω
n+α dv n+α dΩ
We introduce the following identity
Ω
dv n+α · µk −1 dv n+α dΩ . . . Grad dp n+α = −µk −1 dv n+α . (3.103)
n+1
dΣ 2 E = dΣn 2
E . (3.104)
n+α 1 n+1 n
dp dp − dp
Ω M
dΩ
= 1
2M ( n+1
dp 2 L2 − dp n 2 1
L2) + (2α − 1)
2M
dp n+1 − dp n 2
L 2 . (3.105)
Then, from Equations 3.103 – 3.105, the evolution of the norm during the flow step is
written as
dχ n+1 2
N − dχn 2
N
= dζ n+1 2
T − dζn 2
T
= 1
2M ( dp n+1 2
L 2 − dp n 2
L 2)
= − (2α − 1) 1
2M
dp n+1 − dp n 2
L 2 − ∆t
dv
Ω
n+α · µk −1 dv n+α dΩ,
(3.106)
where the stability condition is 0.5 ≤ α ≤ 1. From Equation 3.106, the stability condition
during the flow step is identical to the uncoupled problem, where the proper norm to
show stability is a weighted L 2 norm of the pressure (Thomée, 2006; Simo, 1991). From
Equations 3.100 and 3.106, the B-stability condition of the undrained split is 0.5 ≤ α ≤ 1.