Universal superdiffusive infiltration in layered media with fractal distributions of low conductivity inclusions

The infiltration of fluids into porous media frequently presents anomalous features, in which the fluid front displacement varies in time with an exponent ν different from the expected Fickean value of 1/2. A variety of transport models in fractal media predict the anomalies in this process and in related diffusion problems, but the associated exponents are non universal, i.e they depend not only on the fractal dimensions D_f but also on other geometric properties. Here we study the horizontal infiltration in layered porous media where the matrices of higher conductivity have fractal distributions (D_f<1) of lower conductivity inclusions. When the conductivity contrast is high, this process exhibits universal superdiffusive infiltration with ν=1/1+D_f. This result is first demonstrated for inclusion patterns modeled by Cantor sets, but we argue that it extends to any fractal distributions of the inclusions under the condition of no spatial anisotropy in the rescaling of the observation size. Randomized versions of the Cantor sets are presented as possible realizations of disordered fractal patterns and confirm the universal relation. By considering properties of typical granular media and various soils, numerical calculations indicate that this universal superdiffusive infiltration could be observed in physically realizable laboratory and field settings.