Nonlocal Diffusion in Porous Media: A Spatial Fractional Approach.

One-dimensional diffusion problems in bounded porous media characterized by the presence of nonlocal interactions are investigated by assuming a Darcy's constitutive equation of convolution integral type. A power law attenuation function is implemented: Analogies and differences of the flow-rate-pressure law with respect to other nonlocal and fractal models are outlined. By means of the continuity relationship, the fractional diffusion equation is then derived. It involves spatial Riemann-Liouville derivatives with a noninteger order consisting of between 1 and 2. The solution is obtained numerically using fractional finite differences, and results are presented in both the transient and the steady-state regimes. Eventually, the physical meaning of fractional operators is discussed and potential applications of the analysis are suggested. [ABSTRACT FROM AUTHOR]