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 tDF/DW (Alexander-Orbach relation), where DF is the fractal dimension of the medium and DW 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 DF>DW (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.
Bibliographical noteFunding Information:
F.D.A.A.R. acknowledges support by the Brazilian agencies CNPq (304766/2014-3) and FAPERJ (E-26/ 202941/2015) and thanks the hospitality of the Department of Civil, Environmental, and Geo-Engineering of the University of Minnesota, where part of this work was done. V.R.V. acknowledges support from the James L. Record Professorship.
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