Effect of dispersion on convective mixing in porous media

We investigate the effect of dispersion on convection in porous media by performing direct numerical simulations (DNS) in a 2D Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system is not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, $Ra_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $Ra_d = H/\alpha_t$ and the dispersivity ratio $r = \alpha_l/\alpha_t$, where $H$ is the domain height, $\alpha_t$ and $\alpha_l$ are the transverse and longitudinal dispersivities, respectively. For $Ra_m \ll Ra_d$, the effect of dispersion on convection is negligible; for $Ra_m \gg Ra_d$, however, the flow pattern is determined by $Ra_d$ while the mass transport flux $F\sim Ra_m$ at high-$Ra_m$ regime. Our DNS results also show that the increase of the mechanical dispersion (i.e. decreasing $Ra_d$) will broaden the plume spacing and coarsen the convective pattern. Moreover, for $r \gg 1$ the anisotropy of dispersion destroys the slender columnar structure of the primary plumes at large $Ra_m$ and therefore reduces the mass transport rate.