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

Abstract

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 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. *

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

“…Scaling relations are well known for the structural properties of percolation clusters and the diffusion of tracers in those media [16,33,36]. Relations between infiltration and diffusion exponents in regular fractals are also known [5,8] and can be tested in the infiltration model without reactions at p = p c . Moreover, scaling approaches for the interplay of diffusion and alteration reactions (interfacial dissolution reprecipitation mechanism [37]) were developed in Ref.…”

“…Eq. ( 9) was confirmed in several infinitely ramified fractals, including some cases with superdiffusive infiltration (1/2 < n < 1) [8,14]. However, it was not formerly tested in stochastic fractals.…”

“…where n is termed infiltration exponent. If the medium in d dimensions is homogeneous and the source is a border of dimension d−1, the infiltration is Fickean, with n = 1/2, as shown in models and experiments [6][7][8]. However, anomalous exponents (n = 1/2) were already obtained in studies of moisture infiltration in construction materials [9][10][11][12][13], in several models of infiltration in regular fractals [5,6,14], and in the infiltration of glycerin in Hele-Shaw cells with fractal pore geometry (a problem that is equivalent to diffusive infiltration) [7].…”

“…However, note that the infiltration exponent n and the random walk exponent 1/d W may be different. For instance, in infinitely ramified fractals such as Sierpinski carpets and Menger sponges, the connection of diffusion driven infiltration and the diffusion anomalies led to a relation between n, d W , the fractal dimension d F of the medium, and the dimension d B of the infiltration border [5,8].…”

Effects of the growth kinetics on solute diffusion in porous films

“…When a molecule leaves the source, another molecule is immediatly inserted at its previous position, and when a molecule crosses the top frontier, it is removed from the system; this maintains the concentrations in the source and in the drain. In the hydrodynamic limit, the average concentration obeys the diffusion equation with the same boundary conditions 29 . The currents J in and J out are defined as the average numbers of molecules that cross the frontiers per unit time of infiltration (this time does not include the growth time t g ).…”