Sequential Methods for Coupled Geomechanics and Multiphase Flow

118 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
5.1.1 Fixed-strain split
For the fixed-strain approach, the original operator A is split as follows.
⎡
⎣ un
p n
⎤
⎦ Ap sn
−→
⎡
⎣ u∗
pn+1 ⎤
⎦ Au sn
−→
⎡
⎣ un+1
pn+1 ⎤
⎦, where
⎧
⎪⎨ A p sn : ˙m + Div w = ρf,0f, δ ˙εv = 0,
⎪⎩ Au sn : Div σ + ρbg = 0, p : prescribed,
(5.1)
where we solve the flow problem first using implicit time discretization, followed by solution
of the mechanical problem using an appropriate implicit time discretization scheme. In the
fixed-strain split, δ ˙εv = 0 means that the volumetric strain term b ˙εv in the accumulation
term for the flow problem (Equation 3.10) is evaluated explicitly. We first solve the flow
problem while freezing the rate of the strain everywhere (i.e., δ ˙εv = 0). Then we solve the
mechanical problem. Note that the pressure is prescribed when we solve the mechanical
problem because we determine the pressure at tn+1 from the previous flow problem. It is
worth noting that the mechanical problem uses the drained rock properties, and that the
pressure corrections act as “loads” (Settari and Mourits, 1998).
5.1.2 Fixed-stress split
In this scheme, the flow problem is solved first while freezing the rate of the total mean
stress (δ ˙σv = 0). That is, the volumetric stress term (b/Kdr)˙σv in the accumulation term
of Equation 3.12 is evaluated explicitly when solving the flow problem.
⎡
The original operator A is split as follows:
⎤
⎡
⎣ un
pn ⎦ Ap ss
−→ ⎣ u∗
pn+1 ⎦ Auss −→ ⎣ un+1
pn+1 ⎤
⎡
⎤
⎦, where
⎧
⎪⎨ A p ss : ˙m + Div w = ρf,0f, δ ˙σv = 0,
⎪⎩ Au ss : Div σ + ρbg = 0, p : prescribed.
(5.2)
The initial conditions of A p ss are determined from the initial time conditions of the
original coupled problem, which satisfy
Div ˙σt=0 = 0 , Div σt=0 = 0. (5.3)