Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities
Institute for Groundwater Studies, Faculty of Agricultural and Natural Sciences, University of the Free State, 9301, Bloemfontein, South Africa
* e-mail: AllwrightAJ@ufs.ac.za
Accepted: 9 January 2018
Published online: 13 February 2018
Groundwater transport within a fractured aquifer with a fractal nature exhibiting self-similarity cannot be accurately simulated by the classical Fickian advection-dispersion transport equation without a detailed characterisation of the fracture network and heterogeneity of the system. Because the information to characterise such a system to an appropriate level of detail is often not available, most applications fail to accurately simulate the observed contaminant transport. In response to the current limitations of transport modelling using the advection-dispersion equation, especially in fractured media, a fractal advection-dispersion groundwater transport equation is developed. Fractal differentiation is discussed in terms of the fractal derivative and the fractal integral. The fractal derivative is commonly applied and known, yet the fractal integral is developed in this paper along with the appropriate theorem and proof. The numerical approximation of the fractal derivative and integral are given, where Simpson’s 3/8 Rule and Boole’s Rule for numerical integration are applied for the fractal integral, and the forward and Crank-Nicolson finite difference schemes are applied for the fractal derivative. Upon the given foundation, the fractal advection-dispersion transport equation is formulated to develop a new groundwater transport model. The qualitative properties of the fractal advection-dispersion equation are investigated to determine boundedness, existence and uniqueness of the solution. To validate the developed fractal advection-dispersion equation, a numerical simulation is performed with different fractal dimensions to demonstrate the applicability of the model to fractal groundwater systems. The numerical simulations found that anomalous diffusion could be better modelled by incorporating a fractal dimension, especially where limited information is available on preferential pathways.
© The Author(s), 2018