  We study a random walk infiltration (RWI) model, in homogeneous and in
fractal media, with localized sources at their boundaries. The particles
released at a source, which is maintained at a constant density, execute
unbiased random walks over a lattice; it represents solute infiltration by
diffusion into a medium in contact with a reservoir. 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 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 relation and improves results of lattice random walk simulations.
In a Menger sponge in which D_F>D_W (a fractal with a dimension close to 3), a
linear time increase of the infiltrated volume is predicted and confirmed
numerically. Thus, no evidence of fractality can be observed in infiltrated
volumes or masses in media where random walks are not recurrent, although the
tracer diffusion is anomalous. We compare our results with 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 or numerical
solutions in two and three dimensions and 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.
