Lattice Boltzmann model for three-phase viscoelastic fluid flow

A lattice Boltzmann (LB) framework is developed for simulation of three-phase viscoelastic fluid flows in complex geometries. This model is based on a Rothman-Keller type model for immiscible multiphase flows which ensures mass conservation of each component in porous media even for a high density ratio. To account for the viscoelastic effects, the Maxwell constitutive relation is correctly introduced into the momentum equation, which leads to a modified lattice Boltzmann evolution equation for Maxwell fluids by removing the normal but excess viscous term. Our simulation tests indicate that this excess viscous term may induce significant errors. After three benchmark cases, the displacement processes of oil by dispersed polymer are studied as a typical example of three-phase viscoelastic fluid flow. The results show that increasing either the polymer intrinsic viscosity or the elastic modulus will enhance the oil recovery.

Figure 1

Sketch of a three-phase system on solid walls, with $r$, $g$, $b$ representing red, green, and blue fluids.

Figure 2

Three typical wall boundary configurations for concentrations: (a) flat case, (b) corner case 1, and (c) corner case 2.

Figure 3

Sketch of a multilayer Couette flow.

Figure 4

Momentum validations for the multilayer Couette flow with (a) ${\beta }^0=0.6$ and (b) ${\beta }^0=0.85$.

Figure 5

Validation results of the three-phase wettability with (a) ${\theta }^{gr}={60}^{\circ }$, ${\theta }^{br}={90}^{\circ }$, ${\theta }^{gb}={60}^{\circ }$ and (b) ${\theta }^{gr}={90}^{\circ }$, ${\theta }^{br}={150}^{\circ }$, ${\theta }^{gb}={30}^{\circ }$.

Figure 6

Physical model of a Newtonian bubble (fluid b) rising in a viscoelastic fluid (r).

Figure 7

Comparisons of the bubble rising processes (a) in a Newtonian fluid and (b) in a viscoelastic fluid.

Figure 8

Comparisons of the streamlines during bubble rising (a) in a Newtonian fluid and (b)–(d) in a viscoelastic fluid. In (c) and (d), the time steps used for calculations are changed to $\mathrm{\Delta }t=2.5\times {10}^{-6}\mathrm{s}$ and $\mathrm{\Delta }t=5\times {10}^{-6}\mathrm{s}$, respectively.

Figure 9

Effects of time step and the excess LB viscosity ${\eta }_{\mathrm{LB}}$ on the present model, where the cross-sectional velocities are extracted through the white dash lines shown in Fig. 8, and the nondimensional time step $\mathrm{\Delta }{t}^{*}=\mathrm{\Delta }t/{\tau }^{el}$ is used.

Figure 10

(a) Initial setup for the simulation of oil displacement by the dispersed polymer system. (b) An enlarged view of the first cross for illustrations of the nondimensional displacing front position in the lower channel and mean pressures in the upper channel.

Figure 11

Comparisons of oil displacement processes by (a) water and by (b) the dispersed polymer system. To record the displacing time, cumulative injection volume of fluids normalized by the pore volume (PV) of the main porous region is used.

Figure 12

Evolutions of the lower displacing front position with pore volume injected: (a) effect of ${\eta }^p$ and (b) effect of $E$.

Figure 13

(a) Evolutions of the upper channel pressure difference with pore volume injected. (b) Relation between the average upper channel pressure difference and polymer elastic modulus.