  We develop a time domain random walk approach for conservative solute
transport in heterogeneous media where medium properties vary over a
distribution of length scales. The spatial transition lengths are equal to the
heterogeneity length scales, and thus determined by medium geometry. We derive
analytical expressions for the associated transition times and probabilities in
one spatial dimension. This approach determines the coarse-grained solute
concentration at the interfaces between regions; we derive a generalized master
equation for the evolution of the coarse-grained concentration and reconstruct
the fine-scale concentration using the propagator of the subscale transport
mechanism. The performance of this approach is demonstrated for diffusion under
random retardation in power-law media characterized by heavy-tailed lengthscale
and retardation distributions. The coarse representation preserves the correct
late-time scaling of concentration variance, and the reconstructed fine-scale
concentration is essentially identical to that obtained by direct numerical
simulation by random walk particle tracking.
