Sequential Methods for Coupled Geomechanics and Multiphase Flow

36 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
3.3.2 Drained split
In the drained split, the solution is obtained sequentially by first solving the mechanics
problem, and then the flow problem. The pressure field is frozen when the mechanical
problem is solved. The drained-split approximation of the operator A can be written as
⎡
⎤
⎡
⎣ un
pn ⎦ Au dr
−→ ⎣ un+1
pn ⎤
⎦ Ap
dr
−→
⎡
⎣ un+1
pn+1 ⎤
⎦, where
⎧
⎪⎨ Au dr : Div σ + ρbg = 0, δp = 0,
⎪⎩ A p
dr : ˙m + Div w = ρf,0f, ˙ε : prescribed,
(3.32)
where the superscript n indicates the time level. One solves the mechanical problem with
no pressure change, thus allowing the fluid to drain. Then, the fluid flow problem is solved
with a frozen displacement field.
3.3.3 Undrained split
In contrast to the drained split, the undrained split uses a different pressure predictor for
the mechanical problem, which is computed by imposing that the fluid mass in each grid
block remains constant during the mechanical step (δm = 0). The original operator A is
split as follows:
⎡
⎤
⎡
⎣ un
pn ⎦ Au ud
−→ ⎣ un+1
p ∗
⎤
⎦ Ap
ud
−→
⎡
⎣ un+1
pn+1 ⎤
⎦, where
⎧
⎪⎨ Au ud : Div σ + ρbg = 0, δm = 0,
⎪⎩ A p
ud : ˙m + Div w = ρf,0f, ˙ε : prescribed.
(3.33)
The undrained strategy allows the pressure to change locally when the mechanical prob-
lem is solved. From Equation 3.3, the undrained condition (δm = 0) yields
and the pressure is updated locally in each element using
0 = bδεv + 1
δp, (3.34)
M
p ∗ = p n − bM(ε n+1
v − ε n v). (3.35)