Title: Scaling relations in the diffusive infiltration in fractals,
Abstract: In a recent work on fluid infiltration in a Hele-Shaw cell with the
pore-block geometry of Sierpinski carpets (SCs), the area filled by the
invading fluid was shown to scale as F~t^n, with n<1/2, thus providing a
macroscopic realization of anomalous diffusion [Filipovitch et al, Water
Resour. Res. 52 5167 (2016)]. The results agree with simulations of a diffusion
equation with constant pressure at one of the borders of those fractals, but
the exponent n is very different from the anomalous exponent nu=1/D_W of single
particle diffusion in the same fractals (D_W is the random walk dimension).
Here we use a scaling approach to show that those exponents are related as
n=nu(D_F-D_B), where D_F and D_B are the fractal dimensions of the bulk and of
the border from which diffusing particles come, respectively. This relation is
supported by accurate numerical estimates in two SCs and in two generalized
Menger sponges (MSs), in which we performed simulations of single particle
random walks (RWs) with a rigid impermeable border and of a diffusive
infiltration model in which that border is permanently filled with diffusing
particles. This study includes one MS whose external border is also fractal.
The exponent relation is also consistent with the recent simulational and
experimental results on fluid infiltration in SCs, and explains the approximate
quadratic dependence of n on D_F in these fractals. We also show that the
mean-square displacement of single particle RWs has log-periodic oscillations,
whose periods are similar for fractals with the same scaling factor in the
generator (even with different embedding dimensions), which is consistent with
the discrete scale invariance scenario. The roughness of a diffusion front
defined in the infiltration problem also shows this type of oscillation, which
is enhanced in fractals with narrow channels between large lacunas.