  This paper presents a time-space Hausdorff derivative model for depicting
solute transport in aquifers or water flow in heterogeneous porous media. In
this model, the time and space Hausdorff derivatives are defined on
non-Euclidean fractal metrics with power law scaling transform which,
respectively, connect the temporal and spatial complexity during transport. The
Hausdorff derivative model can be transformed to an advection-dispersion
equation with time- and space-dependent dispersion and convection coefficients.
This model is a fractal partial differential equation (PDE) defined on a
fractal space and differs from the fractional PDE which is derived for
non-local transport of particles on a non-fractal Euclidean space. As an
example of applications of this model, an explicit solution with a constant
diffusion coefficient and flow velocity subject to an instantaneous source is
derived and fitted to the breakthrough curves of tritium as a tracer in porous
media. These results are compared with those of a scale-dependent dispersion
model and a time-scale dependent dispersion model. Overall, it is found that
the fractal PDE based on the Hausdorff derivatives better captures the early
arrival and heavy tail in the scaled breakthrough curves for variable transport
distances. The estimated parameters in the fractal Hausrdorff model represent
clear physical mechanisms such as linear relationships between the orders of
Hausdorff derivatives and the transport distance. The mathematical formulation
is applicable to both solute transport and water flow in porous media.
