Sequential Methods for Coupled Geomechanics and Multiphase Flow

6.5. NUMERICAL EXAMPLES WITH THE STAGGERED NEWTON METHODS 203
Table 6.2: Input data for Case 6.2
Property Value
Permeability (kp) 500 md
Porosity (φ0) 0.3
Young modulus (E) 300 MPa
Poisson ratio (ν) 0.35
Biot coefficient (b) 1.0
Bulk density (ρb) 2400 kg m −3
Oil compressibility (co) 3.5×10 −8 Pa −1
Oil density (ρo,0) 1000 kg m −3
Oil viscosity (µo) 1.0 cp
Water compressibility (cw) 3.5×10 −10 Pa −1
Water density (ρw,0) 1000 kg m −3
Water viscosity (µw) 1.0 cp
Initial oil pressure (Po,i) 2.125 MPa
Side burden 2.125 MPa
Overburden (¯σ) 3×2.125 MPa
Grid spacing (∆x) 10 m
Grid spacing (∆z) 5 m
Grid 10 × 4
Water injection rate 5000 kg day −1
Total production rate 5000 kg day −1
splits, the undrained and fixed-stress splits yield stable and accurate solutions. The bottom
of Figure 6.6 shows the water saturation profile for the bottom layer at td = 0.3. The
high water saturation reduces the total fluid compressibility, and leads to violating the
convergence conditions of the drained and fixed-strain splits.
To analyze the convergence of each scheme, we investigate the behaviors as a function of
iterations, as shown in Figure 6.7. We choose the first time step for convergence behaviors
because the effect of instantaneous loading is considerable for the first time step. The
drained and fixed-strain splits show severe oscillations and slow convergence. The undrained
split is monotonic and yields better convergence. The fixed-stress split is monotonic and
yields the best convergence behavior. There is a large difference between the fixed-stress
and fully coupled solutions for the first iteration of the first time step because the fixed-
stress split does not capture the instantaneous loading. But, in the second iteration, the