Super-diffusive transport processes in porous media, in: Modelling and Simulation

Abstract The basic assumption of models for the transport of contaminants through soil is that the movements of solute particles are characterized by the Brownian motion. However, the complexity of pore space in natural porous media makes the hypothesis of Brownian motion far too restrictive in some situations. Therefore, al-ternative models have been proposed. One of the models, many times encountered in hydrology, is based in fractional differential equations, which is a one dimen-sional fractional advection diffusion equation where the usual second-order deriva-tive gives place to a fractional derivative of order α, with 1 < α ≤ 2. When a frac-tional derivative replaces the second order derivative in a diffusion or dispersion model, it leads to anomalous diffusion, also called super-diffusion. We derive ana-lytical solutions for the fractional advection diffusion equation with different initial and boundary conditions. Additionally, we analyze how the fractional parameter α affects the behavior of the solutions. 1 Fractional advection diffusion equation An equation commonly used to describe solute transport is the classical advection diffusion (or dispersion) equation ∂u ∂