Delay mechanisms of non‐Fickian transport in heterogeneous media

Abstract

Fickian models of diffusion often fail to describe transport phenomena in heterogeneous environments due to their inability to capture the sub‐scale fluctuations. We present an effective description of non‐Fickian behavior that reflects the dichotomy between the continuum nature of Fick's law and the finite (effective) observation scale associated with experimental studies of transport phenomena in heterogeneous systems. This dichotomy gives rise to a time delay between the cause and effect, i.e. between the concentration gradient and the mass flux. Evolving scales of heterogeneity induce a spectrum of such delay times that can lead to anomalous behavior. The presented model is a direct generalization of Fick's law and the well‐established delay diffusion model. It complements effective modeling frameworks based on stochastic non‐local theories and continuous time random walks.