Sequential Methods for Coupled Geomechanics and Multiphase Flow

3.7. NUMERICAL EXAMPLES 61
Table 3.2: Input data for Case 3.2
Property Value
Permeability (kp) 50 md
Porosity (φ0) 0.3
Drained (constrained) modulus (Kdr) 100 MPa
Biot coefficient (b) 1.0
Bulk density (ρb) 2400 kg m −3
Fluid density (ρf,0) 1000 kg m −3
Fluid viscosity (µ) 1.0 cp
Injection rate (Qinj) 100 kg day −1
Production rate (Qprod) 100 kg day −1
Boundary pressure (Pbc) 2.125 MPa
Overburden (¯σ) 2.125 MPa
Grid spacing (∆z) 10 m
kp = 50 md, porosity is φ0 = 0.3, the constrained modulus is Kdr = 100 MPa, and the Biot
coefficient is b = 1.0. No production and injection of fluid is applied. An observation well is
located at the fifth grid block from the top. Gravity is neglected. The Biot modulus is left
unspecified to test the performance of the drained and undrained splits for different values
of the coupling strength, τ, in Equation 3.50. The numerical values of the parameters for
Case 3.1 are also listed in Table 3.1. We determine the pressure and displacement fields,
using the fully coupled method with very small time step size in order to minimize the
temporal error.
Figures 3.9 shows the results of the numerical experiments, as well as, the reference an-
alytical solution. The backward Euler time discretization is used. Both sequential methods
are stable for τ = 0.83 < 1 (Figs. 3.9 (top)). The drained split, however, is unstable for
τ = 1.21 > 1 (Figs. 3.9 (bottom)). On the other hand, the undrained split is stable for
τ = 1.21 > 1 (Figs. 3.9 (bottom)). Even in the region of stability, the drained split pro-
duces wildly oscillatory solutions, which is in agreement with the predictions from the Von
Neumann stability analysis. The validity of the simulations is supported by the agreement
between the (stable) numerical solutions and the analytical solution to the problem (see,
e.g., (Wang, 2000)).