The classical integer-order Fickian diffusion PDE was formulated based on
assumptions of particle movements characterized by mean free path and mean
waiting time. These assumptions are valid for transport in homogeneous media,
where solute plumes were observed to display exponentially decaying tails.
However, field tests have unveiled that transport in heterogeneous media
exhibits markedly nonlocal behavior, featuring highly skewed power-law
decaying tails. Modeling such behavior using Fickian diffusion PDE across a
broad parameter range poses significant challenges.
Fractional diffusion PDE was developed with the assumption that its solutions
manifest power-law decaying tails. This framework presents a competitive
approach for modeling transport in heterogeneous media. Nevertheless,
fractional PDEs introduce novel modeling, computational, and mathematical
challenges distinct from those encountered with integer-order PDEs.
In this session, we will delve into the modeling and analysis, addressing the
complexities and challenges unique to this domain.