We consider a porous solid, partially filled with a non-wetting fluid, just above the injection threshold: the injected regions have the topology of the infinite cluster in a percolation problem, and they split into a ‘backbone’ part plus ‘dead ends’. Under a steady Darcy flow, a dye molecule moves by convection on the backbone, and by molecular diffusion on the dead ends. The process has some similarity with solute transport in chromatographic columns. However, because the ‘dead ends’ have a broad distribution of sizes, special singularities may occur, which are reminiscent of non-Gaussian transport for charge carriers in amorphous semiconductors.

We ultimately predict the existence of a well-defined diffusion coefficient D|| for motion parallel to the average flow, in the limit of slow molecular diffusion. We find $D_{\|}\approx (\overline{U}\xi)^2/D_{\rm a}$, where ξ is the percolation correlation length, $\overline{U}$ the macroscopic flow velocity and Da the diffusion coefficient of an ‘ant’ on the infinite cluster (i.e. the macroscopic diffusion constant measured in the absence of flow).