ABSTRACT
This chapter is dedicated to mass-time and space-time fractional partial differential equations (fPDEs) for solute transport in soils and solute exchange between large and small pores of soils. Mass-time fPDEs are formulated in a material coordinate for analyzing solute transport in swelling-shrinking soils, whereas space-time fPDEs are used for non-swelling soils. These fPDEs complement a set of mass-time and space-time fPDEs for water flow in soils (Su 2014), and enable the analysis of soils with mobile and immobile pores. Symmetrical mass- (or space-) fractional derivatives explain the forward-backward motion of solute particles, and the multi-term time-fractional derivatives account for differentiated particle movements in various pores of the fractal media.
As an application of the fPDEs, an analytical solution of the fPDE has been derived for an arbitrary input, and the solutions are in the form of the hyper-gamma distribution. The asymptotic solutions have also been derived from the solutions of this example and presented for solute exchange between large and small pores, subject to two types of initial conditions and the main solution for solute movement.
