Scaling relations in the diffusive infiltration in fractals

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 $F\sim t^n$, 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 $n=\nu \left(D_F-D_B\right)$, where $D_F$ and $D_B$ are the fractal dimensions of the bulk and 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.