Sequential Methods for Coupled Geomechanics and Multiphase Flow

124 CHAPTER 5. FIXED-STRAIN AND FIXED-STRESS SPLITS
different from those for the undrained split. Interestingly, they are identical to those for the
fully coupled method.
Remark 5.2. When the mechanical problem has complicated boundary conditions in
multiple dimensions, we might not obtain the exact local value of Kdr in the flow problem
of Equation 5.20 for the fixed-stress split. Let K est
dr be an estimated local Kdr. We define η =
Kdr/K est
dr
as the deviation factor between the true and estimated local drained bulk moduli.
Then, following the same procedure of the Von Neumann method with the backward Euler
time discretization, we obtain the condition for the linear stability as
η ≥ 1
1 −
2
1
, (5.26)
τ
where one of the amplification factors is negative if η < 1, but all the amplification factors
are positive if η > 1. From Equation 5.26, η ≥ 0.5 provides unconditional stability for linear
problems.
Suppose we use K est
dr
= K3D
dr for incompressible fluid and solid grains when the true Kdr
is K1D dr . In this case, τ = ∞ and η = 3(1 − ν)/(1 + ν), where ν is Poisson’s ratio. Since
0 ≤ ν ≤ 0.5, 1.0 ≤ η ≤ 3.0, which gives unconditional stability with monotonic behavior.
On the other hand, we obtain η = (1 + ν)/3(1 − ν) where we use K est
dr
= K1D
dr
while the
true Kdr is K3D dr . In this case, instabilities may occur because 0.33 ≤ η ≤ 1.0 (e.g., η = 0.33
when ν = 0.0), and we may lead to oscillatory behaviors, even though the scheme is stable.
Hence, a less stiff K est
dr than the true Kdr yields unconditional stability with monotonicity
because it guarantees 1.0 ≤ η ≤ 3.0. Thus, it is appropriate to choose K 1D
dr
as Kest dr ’s for one, two, and three dimensional problems, respectively.
, K2D
dr , and K3D
dr