Sequential Methods for Coupled Geomechanics and Multiphase Flow

38 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
3.4.1 Drained split
From the generalized midpoint rule, we have for Equations 3.19 and 3.20
σ n+α
h
V n+α
h
= (1 − α)σ n h
= (1 − α)V n
h
+ ασn+1
h , (3.39)
+ αV n+1
h . (3.40)
Based on the finite-volume and finite-element methods, the one-dimensional space dis-
cretization is shown in Figure 3.3.
Figure 3.3: One-dimensional element with the finite-volume and finite-element methods. h
is the grid spacing, and j is the element index
The drained split freezes the variation of the pressure, so that
Full discretization in one dimension yields
−( Kdr
h Un+α
j− 3
2
− 2 Kdr
h P
M
n+1
j − P n j
∆t
− kp
µh
h Un+α
j− 1
2
P n+α = P n . (3.41)
+ Kdr
+ bh
∆t [(−
P n+α n+α
j−1 − 2Pj h Un+α
j+ 1 ) − b(P
2
n j−1 − P n j ) = 0, (3.42)
U n+1
j− 1
2
+ P n+α
j+1
− U n+1
j+ 1
2
U
) + (
n
j− 1 − U
2
n
j+ 1
2 )]
h
h
= 0, (3.43)