A specific form of the Fokker–Planck equation with a time- and scale-dependent dispersivity is presented for modelling solute transport in saturated heterogeneous porous media. By taking a dispersivity in the form of separable power-law dependence on both time and scale, we are able to show the existence of similarity solutions. Explicit closed-form solutions are then derived for an instantaneous point-source (Dirac delta function) input, and for constant concentration and constant flux boundary conditions on a semi-infinite domain. The solutions have realistic behaviour when compared to tracer breakthrough curves observedunder both field and laboratory conditions. Direct comparison with the experimental laboratory data of Pang and Hunt [J. Contam. Hydrol. 53 (2001) 21] shows good agreement between the source solutions and the measured breakthrough curves.