Sequential Methods for Coupled Geomechanics and Multiphase Flow

88 CHAPTER 4. CONVERGENCE OF THE DRAINED AND UNDRAINED SPLITS
and finite-element methods adopted for flow and mechanics, respectively. The error es-
timates can be extended to multiple dimensions because the coupling between flow and
mechanics is based on the volumetric response, which is a scalar quantity.
In the fully coupled method with the backward Euler time discretization, we have
−( Kdr
h Un+1
j− 3
2
− 2 Kdr
h P
M
n+1
j − P n j
∆t
− kp
µh
where kp denotes permeability.
h Un+1
j− 1
2
+ Kdr
+ bh
∆t [(−
h Un+1
j+ 1
2
U n+1
j− 1
2
) − b(P n+1
− U n+1
j+ 1
2
j−1
− P n+1
j ) = 0, (4.17)
U
) + (
n
j− 1 − U
2
n
j+ 1
2 )]
h
h
P n+1
n+1 n+1
j+1 − 2Pj + P = 0, (4.18)
j−1
4.2.1 Error amplification of the drained split
The drained split treats the pressure term P n+1 in Equation 4.17 explicitly as P n+1,k , which
is obtained from the previous iteration (k th ) step. The other variables in Equations 4.17
and 4.18 are treated implicitly as U n+1,k+1 and P n+1,k+1 , which are unknown at the present
(k + 1) th step. Then the discretized equations for flow and mechanics by the drained split
are written as
h
M
− kp
µh
Kdr
−
P n+1,k+1
j
−b(P n+1,k
j−1
∆t
h Un+1,k+1
j− 3
2
− P n j
− 2 Kdr
h Un+1,k+1
j− 1
2
+ Kdr
h Un+1,k+1
j+ 1
2
− P n+1,k
j ) = 0, (4.19)
+ bh
∆t [(−
P n+1,k+1
j+1 − 2P n+1,k+1
j
U n+1,k+1
j− 1
2
− U n+1,k+1
j+ 1
2
h
+ P n+1,k+1
j−1
U
) + (
n
j− 1 − U
2
n
j+ 1
2 )]
h
= 0. (4.20)
Let e k u = U n+1 −U n+1,k and e k P = P n+1 −P n+1,k , where U n+1 and P n+1 are the solutions