Differential diffusion effects on density-driven instability of reactive flows in porous media

Based on the lattice Boltzmann (LB) method, pore-scale simulations are performed to investigate the differential diffusion effects on the density-driven instability (DI) with chemical reaction $A+B\to C$ in porous media. A partially miscible stratification is considered, and thus only solutes from the top fluid can diffuse down into the host fluid in pore spaces. Tests with different values of the Rayleigh number ${\text{Ra}}_r$ and the diffusion coefficient $D_r$ of species $r$ ($r=A,B,C$) are considered. The results demonstrate eight distinct scenarios of DI, and four of them are not observed in equal diffusivity simulations. Two differential diffusion effects, namely, the double-diffusive (DD) and the diffusive-layer convection (DLC) mechanisms, can act upon the gravity field and give rise to new fingering phenomena. The DD mechanism comes into play and results in a local minimum density layer if ${\text{Ra}}_B/{\text{Ra}}_C$ is small and $D_B>D_C$; and DLC becomes significant and brings in a local maximum density layer if ${\text{Ra}}_B/{\text{Ra}}_C$ is large and $D_B<D_C$. On one hand, when fluid density increases with dissolved $A$, the DD-induced minimum can act as an inhibiting barrier to suppress fingering propagation, although it can be eventually penetrated by fingering tips; and the DLC-induced maximum can introduce the second DI below the first one. On the other hand, when the dissolution of $A$ contributes to decreasing fluid density, both the DD-induced minimum and the DLC-induced maximum can help trigger the development of DI. Finally, quantitative results are provided to indicate that fingering propagates into the host fluid more deeply with larger $D_B/D_C$, and the dissolution of $A$ decreases with the increasing difference between $D_B$ and $D_C$.

Figure 1

The computational configuration.

Figure 2

Density (a)–(e) and concentration (f)–(h) distributions in homogeneous media for case P2 with ${\text{Ra}}_A={10}^9$, $D_B/D_C=0.1$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at different time instants $t^{*}$. The density scale ${\rho }^{*}$ varies within the range 1–2; the concentrations $C_r^{*}$ scale between 0 and 1; and the reaction rate $R^{*}$ scales between 0 and 0.05.

Figure 3

Horizontally averaged density and species profiles in homogeneous media for case P2 with ${\text{Ra}}_A={10}^9$, $D_B/D_C=0.1$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at five different time instants (a) and time instant $t^{*}=24$ (b).

Figure 4

Density (a)–(e) and concentration (f)–(h) distribution in homogeneous media for case P3 with ${\text{Ra}}_A={10}^9$, $D_B/D_C=5.0$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at different time instants $t^{*}$. The density scale ${\rho }^{*}$ varies within the range 1–2; the concentrations $C_r^{*}$ scale between 0 and 1; and the reaction rate $R^{*}$ scales between 0 and 0.15.

Figure 5

Horizontally averaged density and species profiles in homogeneous media for case P3 with ${\text{Ra}}_A={10}^9$, $D_B/D_C=5.0$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at five different time instants (a) and time instant $t^{*}=6$ (b).

Figure 6

Density (a)–(e) and concentration (f)–(h) distributions in homogeneous media for case P6 with ${\text{Ra}}_A=-{10}^9$, $D_B/D_C=0.1$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at different time instants $t^{*}$. The density scale ${\rho }^{*}$ varies within the range $-0.4$–1.5; the concentrations $C_r^{*}$ scale between 0 and 1; and the reaction rate $R^{*}$ scales between 0 and 0.12.

Figure 7

Horizontally averaged density and species profiles in homogeneous media for case P6 with ${\text{Ra}}_A=-{10}^9$, $D_B/D_C=0.1$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at five different time instants (a) and time instant $t^{*}=6$ (b).

Figure 8

Density (a)–(e) and concentration (f)–(h) distributions in homogeneous media for case P7 with ${\text{Ra}}_A=-{10}^9$, $D_B/D_C=5.0$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at different time instants $t^{*}$. The density scale ${\rho }^{*}$ varies within the range $-0.3$ to 1.3; the concentrations $C_r^{*}$ scale between 0 and 1; and the reaction rate $R^{*}$ scales between 0 and 0.08.

Figure 9

Horizontally averaged density and species profiles in homogeneous media for case P7 with ${\text{Ra}}_A=-{10}^9$, $D_B/D_C=5.0$, and ${\text{Ra}}_B/{\text{Ra}}_C=1.0$ at five different time instants (a) and time instant $t^{*}=6$ (b).

Figure 10

Time evolutions of the mixing length in homogeneous media for tests with ${\text{Ra}}_A={\text{Ra}}_B={\text{Ra}}_C={10}^9$ and $D_B/D_C=0.1,0.5,1,2,5,10$.

Figure 11

Time evolutions of the storage of species $A$ in homogeneous media for tests with ${\text{Ra}}_A={\text{Ra}}_B={\text{Ra}}_C={10}^9$ and $D_B/D_C=0.1,0.5,1,2,5,10$.