Fractional-Order Differential Models for Anomalous Transport Phenomena in Environmental Engineering

Abstract

Abnormal transport processes are common in environmental engineering models, including the flow of contaminants in groundwater, dispersion of pollutants in porous materials, and air quality processes. The more conventional integer-order models, which are subject to advection diffusion equations, are inadequate in modeling long-tailed waiting times, non-local memory and spatial heterogeneity that are characteristic of real world systems. The non-Markovian scale-dependent processes have a mathematically rigorous framework described by the use of fractional-order differential equations (FDEs). This paper constructs new fractional-order equations of anomalous transport in soil systems (ground water) and air systems (atmosphere). Spatio-temporal heterogeneity is modeled by the use of a Caputo Riesz formulation and a hybrid numerical solver between Grunwald-Letnikov discretization and finite element methods is suggested. Comparison to benchmark data sets of solute breakthrough curves show high precision compared to the classical diffusion models, and error margins decrease by 3852 percent in a variety of heterogeneous aquifer conditions. The ability of the fractional models to reproduce the long-range correlations and non-Fickian transport is further demonstrated by case studies of the dispersion of atmospheric pollutants. The results emphasize the promise of the use of fractional calculus as a unifying framework to the models of environmental transport to facilitate better accuracy of predictions, as well as, decision-making to manage sustainably the environmental resources.