Onset of convection in a porous medium in the presence of chemical reaction

Using scaling, we show that the stability of a buoyant boundary layer in a porous medium in the presence of a first-order chemical reaction is fully determined by the nondimensional number $\mathrm{Da}/{\mathrm{Ra}}^2=k_r\mathit{aD}\varphi {\mu }^2/(k\Delta {\rho }_{0}g){}^2$, where $\mathrm{Da}=k_{\mathrm{r}}{\mathit{aL}}_{\mathrm{Z}}^2/k_{\mathrm{r}}{\mathit{aL}}_{\mathrm{Z}}^2(D\varphi )$ is the Damköhler number and $\mathrm{Ra}=k\Delta {\rho }_0{\mathit{gL}}_{\mathrm{Z}}/k\Delta {\rho }_0{\mathit{gL}}_{\mathrm{Z}}(\mu D\varphi )$ is the solutal Rayleigh number. The time for onset of convection is shown to increase with rising $\mathrm{Da}/{\mathrm{Ra}}^2$. Above a critical $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2\approx 2\times {10}^{-3}\hspace{0.5em}{\mathrm{Ra}}^2\approx 2\times {10}^{-3}$, no convection occurs as reaction stabilizes the diffusive layer at a finite thickness. This thickness decreases with increasing $\mathrm{Da}/{\mathrm{Ra}}^2$, becoming zero at $\mathrm{Da}/{\mathrm{Ra}}^2\approx \mathrm{O}(1)$. As applied to CO${}_2$ geostorage, our results suggest distinct regimes for CO${}_2$ transport in saline aquifers.

Figure 1

Injected CO${}_2$ (shaded) rises in a brine-saturated aquifer and spreads laterally under a caprock. Inset: Free CO${}_2$ (shaded) gradually dissolves and diffuses in the underlying brine, resulting in a denser CO${}_2$-rich solution (shaded gradient) which may later sink into the pure brine as fingers, enhancing further dissolution.

Figure 2

Effects of $\mathrm{Da}/{\mathrm{Ra}}^2$ on dimensionless CO${}_2$ concentration $C^{\prime }$ at dimensionless times (i) $t^{\prime }=3.3\times {10}^3$, (ii) $t^{\prime }=3.8\times {10}^3$, and (iii) $t^{\prime }=5.7\times {10}^3$. Inert case A ($\mathrm{Da}=0$, $\mathrm{Ra}=750$, $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2=0{\mathrm{Ra}}^2=0$) shows a faster growth rate of CO${}_2$-rich fingers than in reactive cases B ($\mathrm{Da}=86$, $\mathrm{Ra}=750$, $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2=1.5\times {10}^{-4}{\mathrm{Ra}}^2=1.5\times {10}^{-4}$), C ($\mathrm{Da}=345$, $\mathrm{Ra}=1500$, $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2=1.5\times {10}^{-4}{\mathrm{Ra}}^2=1.5\times {10}^{-4}$) and D ($\mathrm{Da}=571$, $\mathrm{Ra}=750$, $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2=1\times {10}^{-3}{\mathrm{Ra}}^2=1\times {10}^{-3}$). Significant finger interaction is seen in A, B, and C, where larger merged fingers grow deeper, while reaction in D is strong enough to inhibit growth of merged fingers, stabilizing their lengths at a constant depth. Dimensionless wavelength at onset of instability is similar in all four cases. Each picture shows half of the domain width used in simulations, and pictures of C show only the active, upper half of the simulated depth.

Figure 3

Evolution of CO${}_2$ mass flux $J^{\prime }={\mathit{JL}}_{\mathrm{C}}/{\mathit{JL}}_{\mathrm{C}}(D\varphi C_{\mathrm{S}})(D\varphi C_{\mathrm{S}})$ dissolving at the top boundary for selected $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2\hspace{0.5em}{\mathrm{Ra}}^2$, where $t^{\prime }=\mathit{tD}/L_{\mathrm{C}}^2$. Dimensionless time for onset of convection $t_{\mathrm{oc}}^{\prime }$ corresponds to $t^{\prime }$ at minimum $J^{\prime }$. Cases A–D refer to those shown in Fig. 2.

Figure 4

Dimensionless time for onset of convection $t_{\mathrm{oc}}^{\prime }$ increases with $\mathrm{Da}/{\mathrm{Ra}}^2$ until $\mathrm{Da}/\mathrm{Da}{\mathrm{Ra}}^2\approx 2\times {10}^{-3}{\mathrm{Ra}}^2\approx 2\times {10}^{-3}$ (blue dashed line), above which convection no longer occurs. Y-error bars reflect the error in locating the minimum value of $J^{\prime }$ in the curves in Fig. 3, i.e., flatter $J^{\prime }$ curves bring about larger error bars. Results of cases A–D (from Fig. 2) are shown.