Preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow in porous media

An improved preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow (IPMRT-LBE) in porous media is proposed. Motivated by previous LBE models [Guo et al., Phys. Rev. E 70, 066706 (2004); Premnath et al., J. Comput. Phys. 228, 746 (2009); Guo et al., J. Comput. Phys. 165, 288 (2000)], the current model is demonstrated to have the advantages of accurate implementation of the no-slip boundary condition, reducing the compressible effect as well as fast convergence rate compared with standard LBE models. To validate the IPMRT-LBE model, flows in two- and three-dimensional synthetic porous media (square array of cylinders and body-centered cubic array of spheres) are simulated. The results show that the current model can predict the macroscopic property (such as permeability) accurately with significantly accelerated convergence rate. Furthermore, simulations of flow through a three-dimensional sandpack confirm the applicability and advantages of the IPMRT-LBE model.

Figure 1

Flow in 2D fracture: velocity profiles for different preconditioning factors ($\gamma$) at two Reynolds numbers. (a) $\text{Re}=0.1$; (b) $\text{Re}=1$.

Figure 2

Flow in 2D fracture: convergence histories for different preconditioning factors ($\gamma$) at two Reynolds numbers. (a) $\text{Re}=0.1$; (b) $\text{Re}=1$.

Figure 3

Flow in 2D fracture: the effect of relaxation factors ($s_4/s_6$) in the IPMRT-LBE on the velocity profiles. (a) $\gamma =0.1$; (b) $\gamma =1$.

Figure 4

2D computational domain: circular cylinders in square array.

Figure 5

Flow in 2D porous media: computed permeabilities ($K/K_a$) for fluids with different viscosities ($\nu$). (a) Comparison among three models for $\gamma =1$; (b) tests on different preconditioning factors $\gamma$ (red dashed: PSRT-LBE; blue dotted: PMRT-LBE; black solid: IPMRT-LBE; black symbols: $\gamma$).

Figure 6

Flow in 2D porous media simulated by the IPMRT-LBE: convergence histories for fluids with different viscosities ($\nu$). Different values of the preconditioning factor $\gamma$ are also tested. (a) $\nu =1/12$; (b) $\nu =1/6$; (c) $\nu =1/4$; (d) $\nu =1/3$.

Figure 7

3D computational domain: periodic body-centered cubic array of spheres.

Figure 8

Flow in 3D porous media simulated by the IPMRT-LBE: Computed permeabilities ($K/K_a$) for fluids with different viscosities ($\nu$). Different values of the preconditioning factor $\gamma$ are also tested.

Figure 9

Flow in 3D porous media simulated by the IPMRT-LBE: convergence histories for fluids with different viscosities ($\nu$). (a) $\nu =1.0\times {10}^{-4}$; (b) $\nu =2.0\times {10}^{-4}$; (c) $\nu =4.0\times {10}^{-4}$; (d) $\nu =5.0\times {10}^{-4}$.

Figure 10

Flow in touching spheres simulated by the IPMRT-LBE. (a) Computed permeabilities ($K/K_a$) for fluids with different viscosities ($\nu$); (b) convergence histories for $\nu =1.0\times {10}^{-4}$.

Figure 11

Pore structure of the sandpack [5] (red: solid phase; blue: void space). (a) LV60A; (b) F42A.

Figure 12

Computed permeabilities for flows through the Sandpack LV60A (solid: $\nu =2\times {10}^{-5}$; dotted: $\nu =5\times {10}^{-5}$).

Figure 13

Convergence histories for flows through LV60A. (a) $\nu =2\times {10}^{-5}$; (b) $\nu =5\times {10}^{-5}$.

Figure 14

Computed permeabilities and convergence histories for flows through F42A. (a) $\nu =2\times {10}^{-5}$; dotted: $\nu =5\times {10}^{-5}$; (b) $\nu =5\times {10}^{-5}$.

Figure 15

Schematic of force driven flow and lattice arrangement.