Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.6. DISCRETE STABILITY OF THE NONLINEAR PROBLEM 55
Adding Equations 3.94 and 3.95,
≪ dΣ n − dΣ n+α , −dΣ n+α ≫
+ ≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫ ≤ 0 , (3.96)
where again d(·) = (·) − ˜
(·), (e.g., ∆dε n = ∆ε n −∆˜ε n ). The first term of Equation 3.96 can
be written as
≪ dΣ n − dΣ n+α , −dΣ n+α ≫
= − ≪ α(dΣ n − dΣ n+1 ), dΣ n+1/2 +
= α
dΣ n+1 2
E − dΣn 2
E
α − 1
dΣn+1 n
− dΣ
2
≫
+ α (2α − 1) dΣ n+1 − dΣ n 2
E
, (3.97)
where Σ n+1/2 = (Σ n + Σ n+1 )/2. The second term of Equation 3.96 can be written as
≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫
= −
Ω
= −α
Ω
= −α
Ω
.. .
= α
Ω
α∆dε n : dσ ′n+α dΩ
∆dε n : (dσ n+α + bdp n+α 1)dΩ
∆dε n : bdp n+α 1dΩ
∆dε
Ω
n : dσ n+α dΩ = 0 from Equation 3.93
1
M (dpn+1 − dp n )dp n+α dΩ
.. . b∆dε n : 1 = − 1
M (dpn+1 − dp n ) from ∆dm = 0
= α 1
2M ( dp n+1 2
L2 − dp n 2
1
L2) + α (2α − 1)
2M
dp n+1 − dp n 2
L 2 , (3.98)