A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media

The traditional Richards’ equation implies that the wetting front in unsaturated soil follows Boltzmann scaling, with travel distance growing as the square root of time. This study proposes a fractal Richards’ equation (FRE), replacing the integer-order time derivative of water content by a fractal derivative, using a power law ruler in time. FRE solutions exhibit anomalous non-Boltzmann scaling, attributed to the fractal nature of heterogeneous media. Several applications are presented, fitting the FRE to water content curves from previous literature.

► A novel generalization of Richards’ equation is proposed.
► The power-law time ruler captures both the super-diffusion and sub-diffusion.
► Analytical approximations and numerical scheme of the new model are developed.