We devise an efficient methodology to provide a universal statistical description of advection-dominated dispersion (Péclet→∞) in natural porous media including carbonates. First, we investigate the dispersion of tracer particles by direct numerical simulation (DNS). The transverse dispersion is found to be essentially determined by the tortuosity and it approaches a Fickian limit within a dozen characteristic scales. Longitudinal dispersion was found to be Fickian in the limit for bead packs and superdiffusive for all other natural media inspected. We demonstrate that the Lagrangian velocity correlation length is a quantity that characterizes the spatial variability for transport. Finally, a statistical transport model is presented that sheds light on the connection between pore-scale characteristics and the resulting macroscopic transport behavior. Our computationally efficient model accurately reproduces the transport behavior in longitudinal direction and approaches the Fickian limit in transverse direction.