Sequential Methods for Coupled Geomechanics and Multiphase Flow

76 CHAPTER 3. STABILITY OF THE DRAINED AND UNDRAINED SPLITS
Case 3.5—Fluid production scenario in 2D with elastoplasticity
The dimensions of the domain are 50 m×50 m with 5×5 grid blocks under the plane strain
mechanical conditions. The domain is homogeneous with an overburden ¯σ = 2.125 MPa,
side burden ¯σh = 2.125 MPa on both sides, and no-vertical and no-horizontal displacement
boundary at the bottom. The bulk density of the porous medium is ρb = 2400 kg m −1 .
Initial fluid pressure is Pi = 2.125 MPa. Fluid density and viscosity are ρf,0 = 1000 kg m −1
and µ = 1.0 cp, respectively. Permeability is kp = 50 md, and porosity is φ0 = 0.3. Young’s
modulus is E = 350 MPa, and Poisson’s ratio is ν = 0.35. The Biot coefficient is b = 1.0.
For the modified Cam-clay model, the virgin compression index is λ = 0.37, the swell index
is κ = 0.054, the critical state slope is Mmcc = 1.4, and the preconsolidation pressure is
pco,0 = −1.0 MPa, where tensile stress is positive. The production and observation wells
are located at the center of the domain (3,3). The production rate is Qprod = 1000 kg day −1 .
A no-flow boundary condition is applied to the domain. There is no gravity in the domain.
The input parameters for Case 3.5 are listed in Table 3.4.
Figure 3.18 shows that the drained split is not stable in the plastic regime, even though
it is stable in the elastic regime. This is because at around td ≈ 0.018, we reach the plastic
regime and the coupling strength increases beyond one. On the other hand, the undrained
split is stable in the plastic regime. Note that there is a small difference between the
undrained-split and the fully coupled results, and this is due to using a single iteration. As
the figure indicates, when two iterations are used, the solution from the undrained split is
quite close to the fully coupled results.
Staggered Newton schemes for Case 3.5
The staggered Newton method has the following solution procedure (Schrefler et al., 1997).
1. Linearize the flow and mechanical problems.
2. Given the linearized coupled system, solve the flow and mechanical problems in a
sequential way.