Diffusion in disordered media with long-range correlations: Anomalous, Fickian, and superdiffusive transport and log-periodic oscillations

We present the results of extensive Monte Carlo simulation of diffusion in disordered media with long-range correlations, a problem which is relevant to transport of contaminants in field-scale porous media, such as aquifers, gas transport in soils, and transport in composite materials. The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. For H>1/2 the correlations appear to have no effect, and the transport process is diffusive. However, for H<1/2 and depending on the morphology of the medium, three distinct types of transport processes, namely, anomalous, Fickian, and superdiffusive transport may emerge. Moreover, if the medium is anisotropic and stratified, biased diffusion in it is characterized by power-law growth of the mean square displacements with the time in which the effective exponents characterizing the power-law oscillates log periodically with the time. This result cannot be predicted by any of the currently available continuum theories of transport in disordered media.