  The dispersion of solute in porous media shows a non-linear increase in the
transition from diffusion to advection dominated dispersion as the flow
velocity is raised. In the past, the behavior in this intermediate regime has
been explained with a variety of models. {We present and use a simplified
numerical model which does not contain any turbulence, Taylor dispersion, or
fractality. With it, we show that the non-linearity in the intermediate regime
nevertheless occurs. Furthermore,} we show that that the intermediate regime
can be regarded as a phase transition between random, diffusive transport at
low flow velocity and ordered transport controlled by the geometry of the pore
space at high flow velocities. This phase transition explains the first-order
behavior in the intermediate regime. A new quantifier, the ratio of the amount
of solute in dominantly advective versus dominantly diffusive pore channels,
plays the role of `order parameter' of this phase transition. Taylor
dispersion, often invoked to explain the supra-linear behavior of longitudinal
dispersion in this regime, was found not to be of primary importance. The novel
treatment of the intermediate regime paves the way for a more accurate
description of dispersion as a function of flow velocity, spanning the whole
range of P\'eclet numbers relevant to practical applications, such as ground
water remediation.
