Diffusion in one-dimensional multifractal porous media

We examine the scaling properties of one-dimensional random walks on media with multifractal diffusivities, which is a simple model for transport in scaling porous media. We find both theoretically and numerically that the anomalous scaling exponent of the walk is d(w) 2 + K(-1) where K(-1) is the scaling exponent of the reciprocal spatially averaged ("dressed") resistance to diffusion. Since K(-1) > 0, the walk is subdiffusive; the walkers are effectively trapped in a hierarchy of barriers. The trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion. We discuss the implications for transport in porous media.