Investigation of the anomalous diffusion in the porous media: a spatiotemporal scaling

The methanol and the methane transport in the mesoporous zeolite-alumina pellet was studied. Varying the pellet’s size and the diffusant amount, the values of the diffusion coefficients were estimated. Previously, it had been demonstrated that the methane transport through the zeolite-containing pellet is described by the Fickian diffusion equation, whereas the methanol transport is described by the non-Fickian diffusion equation with either space-time-fractional or time-fractional derivative. In this respect, for calculating the diffusion coefficients, the standard diffusion and the time-fractional diffusion models were used for the methane and the methanol respectively. The relations between the obtained values of the methanol non-Fickian diffusion coefficients (0.0068–0.0276 cm2/sα) measured for unequal pellet sizes and various diffusant amounts were found to follow the temporal scaling with a fractional exponent equal to 1.17 ± 0.03, which corresponds to the time-fractional diffusion equation, in a concise manner. It supported a conclusion that the anomalous diffusion of the methanol is time-fractional. An essential applicability of the approach based on the analysis of the temporal diffusion coefficient scaling was additionally verified using the standard Fickian diffusion coefficients of the methane (0.00094–0.00376 cm2/s). In addition, we demonstrate that the investigation of the anomalous diffusion regime using the spatial scaling of the diffusion coefficient is restricted by the measurement of the diffusion length of a certain diffusant in a porous material.