Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.4. STABILITY ANALYSIS VIA THE ENERGY METHOD 193
Using the identity of Equation 6.103, Equation 6.112 with Darcy’s law is written as
n+1
dp 2 M − dpn 2
M
= −(2α − 1) dp n+1 − dp n 2
M −
dp n+α
J bJ∆dε n vdΩ
−∆t
dΩ. (6.113)
dv n+α
J · k −1
p,JKdvn+α K
Thus, when we solve the flow problem by the fixed-stress split, we can show the evolution
of the norm · Nm by adding Equations 6.111 and 6.113. So, we have
dχm n+1 2
Nm − dχm n 2
= dζm n+1 2
Nm
Tm − dζm n 2
Tm
= dΣ n+1 2
E + dp n+1 2
M − dΣn 2
E − dpn 2
M
dΣ ≤ −(2α − 1)
n+1 − dΣ n 2
E + dp n+1 − dp n 2
M
−∆t
dΩ. (6.114)
dv n+α
J · k −1
p,JKdvn+α K
From Equation 6.114, we obtain the condition for B-stability if 0.5 ≤ α ≤ 1 when we
solve the flow problem.
After the flow step, we solve the mechanical problem. From Equation 6.85,
Div dσ ′n+α = 0, dp n+α = 0, (6.115)
to which maximum plastic dissipation, Equation 6.92, is applied. The second term of
Equation 6.92 is calculated as