Development of a fractal advection-despersion equation and new numerical schemes for the classical, fractal and fractional advection-despersion transport equations

Abstract

Groundwater is a vital source of fresh water to many people across the globe, but it is prone to contamination by human activities. In the last few decades, great strides have been made in legislation protecting and controlling the quality of groundwater, creating awareness of potential groundwater contamination and the importance of prevention and mitigation. The accurate representation of contaminant movement within a groundwater system is important, because misrepresentation could increase the environmental impact due to inadequate mitigation or remediation measures. However, groundwater transport is particularly complex due to the inherent heterogeneity of aquifers. Where, predicting the movement of contaminants within groundwater systems, especially in fractured systems, is prone to discrepancies between modelled and observed. There are two general approaches to improve the simulation of groundwater transport: develop the physical characterisation of the heterogeneous system, or redefine the formulation of the governing equations. The focus of this research is to advance the simulation of groundwater transport by examining the formulation of the governing advection-dispersion equation. To achieve this aim, improved numerical approximation schemes for the classical advection-dispersion equation are developed, fractal and fractional derivatives are incorporated into the formulation, and fractional and fractal derivatives are combined. Augmented upwind finite difference numerical approximation schemes, which are better suited for advection-dominated systems, are applied to the solution of the classical advection-dispersion equation for fractured groundwater systems. The simulation of anomalous transport in fractured aquifer systems is improved by providing a fractal advection-dispersion equation with numerical integration and approximation methods for solution. The fractal advection-dispersion equation is proven to simulate superdiffusion and subdiffusion by varying the fractal dimension, without explicit characterisation of fractures or preferential pathways. To improve the governing equation for groundwater transport modelling, the Caputo and Atangana-Baleanu in Caputo sense (ABC) fractional derivatives are applied to the advection-dispersion equation with a focus on the advection term to account for anomalous advection. Appropriate numerical approximation methods for the fractional advection-dispersion equations are provided and analysed for stability requirements. A fractional-fractal advection-dispersion equation is developed to provide an efficient non-local, in both space and time, modelling tool. The fractional-fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide the tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model. A modest step is taken forward to advance the use of fractional calculus, achieve the collective mission of resolving the difference between modelled and observed, and to increase the comprehension and management of natural systems.