The diffusion kinetics may experience either standard or anomalous behavior. In a porous medium, a transport process may distinguish between the standard and anomalous diffusion. Herein, we perform an analysis of the diffusion concentration profiles in a hierarchically porous material using the standard and the time-fractional diffusion equations. It is shown that the mass transfer regime changes from the anomalous to the standard one regarding the temporal scale of the concentration profile. Particularly, at relatively short times, the mass transfer regime corresponds to the anomalous diffusion with the superdiffusive anomalous diffusion exponent. In contrast, at relatively long times, the standard Fickian regime is obeyed. The investigated phenomenon is assigned to the diffusion in different components of a porous material. The relevant physical model of the corresponding transition is discussed based on the Cattaneo-type diffusion equation. The relaxation time for the transport of the gaseous molecules in the porous medium was found to be in the range of 10(-8)-10(-6) s.