  Many theoretical and experimental results show that solute transport in
heterogeneous porous media exhibits multi-scaling behaviors. To describe such
non-Fickian diffusions, this work provides a distributed order Hausdorff
diffusion model to describe the tracer transport in porous media. This model is
proved to be equivalent with the diffusion equation model with a nonlinear time
dependent diffusion coefficient. In conjunction with the structural derivative,
its mean squared displacement (MSD) of the tracer particles is explicitly
derived as a dilogarithm function when the weight function of the order
distribution is a linear function of the time derivative order. This model can
capture both accelerating and decelerating anomalous and ultraslow diffusions
by varying the weight parameter c. In this study, the tracer transport in
water-filled pore spaces of two-dimensional Euclidean is demonstrated as a
decelerating sub-diffusion, and can well be described by the distributed order
Hausdorff diffusion model with c = 1.73. While the Hausdorff diffusion model
can accurately fit the sub-diffusion experimental data of the tracer transport
in the pore-solid prefractal porous media.
