Separate-phase model and its lattice Boltzmann algorithm for liquid-vapor two-phase flows in porous media

In this article, we formulate a set of the separate-phase governing equations at the representative-elementary-volume scale and develop its double-distribution function lattice Boltzmann (LB) algorithm to describe liquid-vapor two-phase flows with or without phase change in porous media. Different from those previous studies, the mathematical description in this article involves the Darcy force, viscous force, and pressure gradient, and the resulting LB simulations can well describe two-phase flows and mass transfer throughout porous media under the compounding effects of these forces. The LB algorithm was validated by simulating single-phase flows in porous media. Its results are in good agreement with those available analytical solutions. We also applied it to model water flows through a semi-infinite porous region bounded by a heated solid wall, where liquid-vapor phase change takes place. The numerical simulations recover the previous results in the limit of the zero Darcy number. Significantly, it reveals much richer two-phase flow and mass transfer characteristics in porous media adjacent to solid walls. The separate-phase model and its lattice Boltzmann algorithm in this article are effective means to gain more profound and clearer understandings of complex two-phase transport processes in a porous system.

Figure 1

Schematic of the single-phase flows through a square porous channel. The origin of the coordinates is at the bottom-left corner of the channel.

Figure 2

The $L_1$-norm of errors obtained on the $100\times 100$, $200\times 200$, $400\times 400$, and $800\times 800$ grids for the single-phase flows through a square porous channel.

Figure 3

The dimensionless streamwise velocity $U^{*}$ in the single-phase flows through a square porous channel. (a) A: $\mathrm{Da}=5\times {10}^{-5}$, $\varepsilon =0.5$; B: $\mathrm{Da}=5\times {10}^{-4}$, $\varepsilon =0.5$; C: $\mathrm{Da}=5\times {10}^{-3}$, $\varepsilon =0.5$. (b) D: $\mathrm{Da}=5\times {10}^{-4}$, $\varepsilon =0.7$; E: $\mathrm{Da}=5\times {10}^{-4}$, $\varepsilon =0.5$; F: $\mathrm{Da}=5\times {10}^{-4}$, $\varepsilon =0.3$.

Figure 4

The dimensionless streamwise velocity $U^{*}$ in the single-phase flows through a square porous channel. Solid lines: analytical solutions to Eq. (28); open circles: the LB results. A: $\mathrm{Da}/\phantom{\mathrm{Da}\varepsilon }\varepsilon ={10}^{-2}$; B: $\mathrm{Da}/\phantom{\mathrm{Da}\varepsilon }\varepsilon ={10}^{-3}$; C: $\mathrm{Da}/\phantom{\mathrm{Da}\varepsilon }\varepsilon ={10}^{-4}$.

Figure 5

The dependence of ${\delta }^{*}$ on different $\varepsilon$ and $\mathrm{Da}$. Diamonds: $\varepsilon =0.3$; circles: $\varepsilon =0.5$; squares: $\varepsilon =0.7$.

Figure 6

Schematic of the liquid-vapor two-phase flows through a semi-infinite porous region. The origin of the coordinates is at the bottom-left corner.

Figure 7

The streamwise mixture momenta ${\rho }_0^{*}{U}^{*}$ at $X^{*}=0.5$ with $\mathrm{Da}=1\times {10}^{-3}$ and $\varepsilon =0.5$. Squares: $100\times 100$; open circles: $200\times 200$; crosses: $400\times 400$; solid line: $800\times 800$.

Figure 8

The $L_1$-norm of errors obtained on the $100\times 100$, $200\times 200$, and $400\times 400$ grids for the liquid-vapor two-phase flows through a semi-infinite porous region.

Figure 9

Dependence of ${\left|\lambda \right|}_{\max }$ of the matrix $\tilde{\mathbf{A}}$ on the wave number $k$. (a) ${\tau }_f=0.75$; (b) ${\tau }_f=1.0$; (c) ${\tau }_f=1.25$; (d) ${\tau }_f=1.5$. Circles: $\mathrm{Da}=10^{-6}$; diamonds: $\mathrm{Da}=10^{-5}$; crosses: $\mathrm{Da}=10^{-4}$; squares: $\mathrm{Da}=10^{-3}$.

Figure 10

Dependence of ${\left|\lambda \right|}_{\max }$ of the matrix $\tilde{\mathbf{B}}$ on the wave number $k$. (a) ${\tau }_f=0.75$; (b) ${\tau }_f=1.0$; (c) ${\tau }_f=1.25$; (d) ${\tau }_f=1.5$. Circles: $\mathrm{Da}=10^{-6}$; diamonds: $\mathrm{Da}=10^{-5}$; crosses: $\mathrm{Da}=10^{-4}$; squares: $\mathrm{Da}=10^{-3}$.

Figure 11

The saturation $s$ with different Darcy numbers. A: Results from Ref. [35]; B: $\mathrm{Da}=10^{-6}$; C: $\mathrm{Da}=10^{-5}$; D: $\mathrm{Da}=10^{-4}$; E: $\mathrm{Da}=10^{-3}$.

Figure 12

The profiles of ${\rho }_0^{*}{U}^{*}$ with different Darcy numbers. A: Results from Ref. [35]; B: $\mathrm{Da}=10^{-6}$; C: $\mathrm{Da}=10^{-5}$; D: $\mathrm{Da}=10^{-4}$; E: $\mathrm{Da}=10^{-3}$.

Figure 13

The profiles of ${\rho }_0^{*}{V}^{*}\xi$ with different Darcy numbers. A: Result from Ref. [35]; B: $\mathrm{Da}=10^{-6}$; C: $\mathrm{Da}=10^{-5}$; D: $\mathrm{Da}=10^{-4}$; E: $\mathrm{Da}=10^{-3}$.

Figure 14

The profiles of the streamwise velocities and saturation at $\mathrm{Da}=10^{-5}$. A: Saturation $s$; B: mixture $U^{*}$; C: liquid $U_l^{*}$; D: vapor $U_v^{*}$.

Figure 15

The profiles of the cross-streamwise velocities and saturation at $\mathrm{Da}=10^{-5}$. A: Saturation $s$; B: mixture $V^{*}\xi$; C: liquid $V_l^{*}\xi$; D: vapor $V_v^{*}\xi$. Inset: the details of $V_v^{*}\xi$.