Crossover from anomalous to Fickean behavior in infiltration and reaction in fractal porous media

Abstract

When the solution in a porous medium is in chemical equilibrium with mineral grains and an external surface is put in contact with water, the infiltration of unsaturated solution may be accompanied by mineral reactions that change the morphology of that medium. Here we study the evolution of this infiltration process considering that the initial porous media are fractal and that the reactions form additional porous material plus soluble products. A Sierpinski carpet and a Menger sponge are the fractal models, the mineral inclusions are represented by the non-fractal blocks (squares and cubes) formed in their iterative constructions, and reaction rates are described by a thermally activated model, in conditions of slow reactions in comparison with the diffusion. The interplay between the infiltration by diffusion and the structural changes by the reactions is explained by a combination of kinetic Monte Carlo simulations of lattice models and a scaling approach. At short times, the infiltration is subdiffusive, but the dissolved mass increases anomalously fast due to delocalization of the reaction, which is distributed through a time increasing infiltrated region. At long times, normal (Fickean) infiltration and reaction are observed. The crossover between the two regimes occurs when the thickness of the altered layer formed around the mineral inclusions is of the same order of magnitude of the width a of the smallest gaps between those inclusions. The order of magnitude of the crossover time tc is estimated as a/(vk), where k is the reaction rate constant and v is the molar volume of the mineral; this time does not depend on the diffusion coefficient of reaction products or on the fractal geometry of the initial medium, which suggests the application to a variety of systems. Estimates of tc are obtained in two recently studied rocks with reported fractal properties and justify the Fickean diffusion assumption of previous models for their weathering in geological time scales. The morphological evolution of the mineral blocks that partially reacted qualitatively agrees with experiments.