Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.4. STABILITY ANALYSIS FOR LINEAR POROELASTICITY 37
For the mechanical problem, Equation 3.2 with the generalized midpoint rule at tn+α is
discretized as follows:
σ n+α − σ0 = Cdr : ε n+α − b(p n+α − p0)1, (3.36)
p n+α = αp ∗ + (1 − α)p n ,
where α ∈ (0, 1]. After substituting Equation 3.35 in Equation 3.36, the mechanical problem
can be expressed in terms of displacements using the undrained bulk modulus, Cud =
Cdr + b 2 M1 ⊗1. The additional computational cost is negligible because the calculation of
p ∗ is explicit. However, we have a stiff mechanical problem due to the undrained constraint,
which requires a more robust linear solver compared with the mechanical problem associated
with the drained split.
3.4 Stability Analysis for Linear Poroelasticity
We employ the Von Neumann method to analyze the stability of sequential schemes, which
is frequently used for stability analysis of linear (or linearized versions of) problems (e.g.,
Strikwerda (2004), Wan et al. (2005), Miga et al. (1998)). Miga et al. (1998) show that the
fully coupled method for the coupled flow and mechanics problem is unconditionally stable
for α ≥ 0.5. The governing equations of coupled flow and geomechanics in one dimension
without source terms are
∂
∂x
Kdr
∂u
∂x
1 ∂p ∂
+ b
M ∂t ∂t
where ∂u
∂x is εxx, which is equal to εv.
− bp = 0 (for mechanics), (3.37)
∂u k ∂
−
∂x µ
2p = 0 (for flow), (3.38)
∂x2