  Convective dissolution is the process by which CO$_2$ injected in deep
geological formations dissolves into the aqueous phase, which allows storing it
perennially by gravity. The process results from buoyancy-coupled Darcy flow
and solute transport. Proper theoretical modeling of the process should
consider in the transport equation a diffusive term accounting for
hydrodynamics (or, mechanical) dispersion, with an effective diffusion
coefficient that is proportional to the local interstitial velocity. A few
two-dimensional (2D) numerical studies, and three-dimensional (3D) experimental
investigations, have investigated the impact of hydrodynamic dispersion on
convection dynamics, with contradictory conclusions. Here, we investigate
systematically the impact of the dispersion strength $S$ (relative to molecular
diffusion), and of the anisotropy $\alpha$ of its tensor, on convective
dissolution in 2D and 3D geometries. We use a new numerical model and analyze
the solute fingers' number density (FND), penetration depth and maximum
velocity; the onset time of convection; the dissolution flux in the
quasi-constant flux regime; the mean concentration of the dissolved CO2; and
the scalar dissipation rate. The efficiency of convective dissolution over long
times is observed to be mostly controlled by the onset time of convection. For
most natural porous media ($\alpha = 0.1$), the onset time is found to increase
as a function of $S$, in agreement with previous experimental findings and in
stark contrast to previous numerical findings. However, if $\alpha$ is
sufficiently large this behavior is reversed. Furthermore, results in 3D are
fully consistent with the 2D results on all accounts, except that in 3D the
onset time is slightly smaller, the dissolution flux in the quasi-constant flux
regime is slightly larger, and the dependence of the FND on the dispersion
parameters is impacted by $Ra$.
