Sequential Methods for Coupled Geomechanics and Multiphase Flow

200 CHAPTER 6. COUPLED MULTIPHASE FLOW AND GEOMECHANICS
the error decreases rapidly with iterations. In contrast to pressure, all the schemes shows
the same convergence behavior for saturation at td = 0.08.
6.5.2 Case 6.2—Water injection and oil production in 2D with poroelas-
ticity
Figure 6.2 illustrates a 2D oil production problem. The injection and production wells
are located at the right bottom and left top, respectively. The monitoring well is at (3,2)
grid block. The input data for the 2D problem are shown in Table 6.2. The dimension
of the domain is 100 m × 20 m with 10 × 4 grid blocks under the plane strain mechanical
condition. The water injection rate Qw,inj = 5000 kg day −1 is the same as the production
rate Qo,prod = 5000 kg day −1 . The domain is homogeneous. The domain has an overburden
¯σ = 3 × 2.125 MPa on the top, no horizontal displacement on the left side, a side burden
¯σh = 2.125 MPa on the right side, and no vertical displacement at the bottom. The bulk
density of the porous medium is ρb = 2400 kg m −1 . Initial oil pressure is Po,i = 2.125 MPa,
where oil fully saturates the formation. The density, viscosity and compressibility of water
are ρw,0 = 1000 kg m −1 , µw = 1.0 cp, and cw = 3.5 × 10 −10 Pa −1 , respectively. The
density, viscosity, and compressibility of oil are ρo,0 = 1000 kg m −1 , µo = 1.0 cp, and
co = 3.5 × 10 −8 Pa −1 , respectively. The permeability, kp, is 500 md, and the porosity,
φ0, is 0.3. Young’s modulus is E = 300 MPa, and Poisson’s ratio is ν = 0.35. The Biot
coefficient is b = 1.0. There is no capillarity. No-flow boundary conditions are applied at
all sides. Gravity is neglected. The residual oil and water saturations are zero.
The oil and water compressibilities differ from each other by two orders of magnitude.
It is anticipated that the drained and fixed-strain splits will face instability because of water
injection. The relative permeability is the same as Equation 6.120. Figure 6.5 shows the
distributions of pressure and water saturation at td = 0.3.
As shown in the top of Figure 6.6, the drained and fixed-strain splits deviate from
the true solution around td = 0.2 and become unstable. From the pressure behaviors
around td = 0.22, the solutions by the drained and fixed-strain splits are not reliable, even
though they provide finite pressure values. In contrast to the drained and fixed-strain