Sequential Methods for Coupled Geomechanics and Multiphase Flow

56 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
where · L 2 is the L 2 norm. From Equation 3.96,
dΣ
α
n+1 2
E − dΣn 2 1
E +
2M (dp n+1 2
L2 − dp n 2
L2)
dΣ
+α (2α − 1)
n+1 − dΣ n 2 1
+ dp E 2M
n+1 − dp n 2
L2
≤ 0 . (3.99)
From Equation 3.99, when we solve the mechanical problem, the evolution of the norm at
the discrete time level satisfies
dχ n+1 2
N − dχn 2
N
= dζ n+1 2
T − dζn 2
T
= dΣ n+1 2
E
≤ −(2α − 1)
dp n+1 2
1
+
2M L2 − dΣ n 2 1
E −
2M dpn 2
L2
dΣ n+1 − dΣ n 2 1
+ dp E 2M
n+1 − dp n 2
L2
, (3.100)
which indicates that the stability condition of Equation 3.91 is satisfied during the mechan-
ical step if 0.5 ≤ α ≤ 1.
Remark 3.4. Equation 3.98 contains terms associated with pressure, which is character-
istic of the coupled problem. Equation 3.98 is identically zero in the uncoupled mechanical
problem (Simo, 1991).
yields
After the mechanical problem is solved, we deal with the flow problem. Equation 3.89
1 dp
M
n+1 − dpn + Div(dv
∆t
n+α ) = 0, (3.101)
∆dε = 0, ∆dεp = 0, ∆dξ = 0. (3.102)