  We investigate the effect of dispersion on convection in porous media by
performing direct numerical simulations (DNS) in a two-dimensional
Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that
the dynamics of this system are 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 $\Delta =
Ra_d/Ra_m > O(1)$, the influence from the mechanical dispersion is minor; for
$\Delta \ll 1$, however, the flow pattern is controlled by $Ra_d$ while the
convective flux is $F\sim Ra_m$ for large $Ra_m$, but with a prefactor that has
a non-monotonic dependence on $Ra_d$. Our DNS results also show that the
increase of mechanical dispersion, i.e. decreasing $Ra_d$, will coarsen the
convective pattern by increasing the plume spacing. Moreover, the inherent
anisotropy of mechanical dispersion breaks the columnar structure of the
mega-plumes at large $Ra_m$, if $Ra_d < 5000$. This results in a fan-flow
geometry that reduces the convective flux.
