We study the temporal evolution of solute dispersion in
three-dimensional porous rocks of different heterogeneity and pore
structure. To this end, we perform direct numerical simulations of
pore-scale flow and transport in a sand-like medium, which exhibits mild
heterogeneity, and a Berea sandstone, which is characterized by strong
heterogeneity as measured by the variance of the logarithm of the flow
velocity. Solute dispersion is quantified by effective and ensemble
dispersion coefficients. The former is a measure for the typical width
of the plume, the latter for the deformation, that is, the spread of the
mixing front. Both dispersion coefficients evolve from the molecular
diffusion coefficients toward a common finite asymptotic value. Their
evolution is governed by the interplay between diffusion, pore-scale
velocity fluctuations and the medium structure, which determine the
characteristic diffusion and advection time scales. Dispersion in the
sand-like medium evolves on the transverse diffusion time across a
characteristic streamtube diameter, which is the mechanism by which
pore-scale flow variability is sampled by the solute. Dispersion in the
Berea sandstone in contrast is governed by both the diffusion time
across a typical streamtube, and the diffusion time along a pore
conduit. These insights shed light on the evolution of mixing fronts in
porous rocks, with implications for the understanding and modeling of
transport phenomena of microbes and reactive solutes in porous media.