Sequential Methods for Coupled Geomechanics and Multiphase Flow

34 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
is the undrained split shown on the right diagram of Figure 3.2.
Figure 3.2: Sequential methods for flow and geomechanics. Left: drained split. Right:
undrained split.
3.3.1 Fully coupled method
Let us denote by A the operator of the original problem (Equations 3.1 and 3.9). The
discrete approximation of this operator corresponding to the fully coupled method can be
represented as:
⎡
⎣ un
p n
⎤
⎦ Afc
−→
⎡
⎣ un+1
pn+1 ⎤
⎧
⎪⎨ Div σ + ρbg = 0,
⎦, where Afc :
⎪⎩ ˙m + Div w = ρf,0f,
(3.25)
where ˙
( ) denotes the material time derivative with respect to the solid skeleton, d ( ) /dt.
Using the backward Euler time discretization in Equations 3.19 and 3.20, the residual form