Scaling relations in the diffusive infiltration in fractals

Time increasing rates of infiltration and reaction in porous media at the percolation thresholds

The infiltration of a solute in a fractal porous medium is usually anomalous, but chemical reactions of the solute and that material may increase the porosity and affect the evolution of the infiltration. We study this problem in two- and three-dimensional lattices with randomly distributed porous sites at the critical percolation thresholds and with a border in contact with a reservoir of an aggressive solute. The solute infiltrates that medium by diffusion and the reactions with the impermeable sites produce new porous sites with a probability r, which is proportional to the ratio of reaction and diffusion rates at the scale of a lattice site. Numerical simulations for r≪1 show initial subdiffusive scaling and long time Fickean scaling of the infiltrated volumes or areas, but with an intermediate regime with time increasing rates of infiltration and reaction. The anomalous exponent of the initial regime agrees with a relation previously applied to infinitely ramified fractals. We develop a scaling approach that explains the subsequent time increase of the infiltration rate, the dependence of this rate on r, and the crossover to the Fickean regime. The exponents of the scaling relations depend on the fractal dimensions of the critical percolation clusters and on the dimensions of random walks in those clusters. The time increase of the reaction rate is also justified by that reasoning. As r decreases, there is an increase in the number of time decades of the intermediate regime, which suggests that the time increasing rates are more likely to be observed is slowly reacting systems.

Models of infiltration into homogeneous and fractal porous media with localized sources

We study a random-walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t^{D_{F}/D_{W}} (Alexander-Orbach relation), where D_{F} is the fractal dimension of the medium and D_{W} is the random-walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random-walk simulations. In a Menger sponge in which D_{F}>D_{W} (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media.