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The dispersion of solute in porous media shows a non-linear increase in thetransition from diffusion to advection dominated dispersion as the flowvelocity is raised. In the past, the behavior in this intermediate regime hasbeen explained with a variety of models. {We present and use a simplifiednumerical model which does not contain any turbulence, Taylor dispersion, orfractality. With it, we show that the non-linearity in the intermediate regimenevertheless occurs. Furthermore,} we show that that the intermediate regimecan be regarded as a phase transition between random, diffusive transport atlow flow velocity and ordered transport controlled by the geometry of the porespace at high flow velocities. This phase transition explains the first-orderbehavior in the intermediate regime. A new quantifier, the ratio of the amountof solute in dominantly advective versus dominantly diffusive pore channels,plays the role of `order parameter' of this phase transition. Taylordispersion, often invoked to explain the supra-linear behavior of longitudinaldispersion in this regime, was found not to be of primary importance. The noveltreatment of the intermediate regime paves the way for a more accuratedescription of dispersion as a function of flow velocity, spanning the wholerange of P\'eclet numbers relevant to practical applications, such as groundwater remediation.

Simplified numerical model for clarifying scaling behavior in the intermediate dispersion regime in homogeneous porous media