Sequential Methods for Coupled Geomechanics and Multiphase Flow

188 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
algorithm for elastoplasticity is also based on the generalized midpoint rule, which satisfies
(Simo, 1991; Simo and Govindjee, 1991)
≪ Σ tr,n+α − Σ n+α , Π − Σ n+α ≫≤ 0 ∀ Π ∈ E , (6.91)
which follows the notation used in single-phase flow.
Same as single-phase flow, Equation 6.91 yields
≪ dΣ n − dΣ n+α , −dΣ n+α ≫
which will be used for elastoplasticity.
+ ≪ (αCdr∆dε n , 0), (−dσ ′n+α , −dκ n+α ) ≫ ≤ 0, (6.92)
6.4.6 B-stability of the undrained split
We show B-stability for two steps, mechanics and flow. When we solve the mechanical prob-
lem by the undrained split A u,m
ud , the discrete form of the mechanical problem is expressed
as
Div dσ n+α = 0, ∆dmJ = 0. (6.93)
When we solve the mechanical problem, Equation 6.92 is satisfied. The first term of
Equation 6.92 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
. (6.94)
E