Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities

Abstract

Abstract. 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 advectiondispersion 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 advectiondispersion 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.

Study of one‐dimensional contaminant transport in soils using fractional calculus

“…A new transport model for groundwater had been proposed using fractured acquifier by Allwright and Atangana. 29 Authors have developed and simulated a fractal advection-dispersion equation (ADE) for the groundwater transportation exploiting fractal derivatives and fractal integrals. Mirza et al in 2016 have solved fractional one-dimensional ADE by taking time-varying pulses as the boundary condition and found the fundamental solution using Duhamel's principle.…”

Fractional and fractal advection-dispersion model

“…An important difference defined in terms of practical simulation, was that the fractal model is a local operator and can be solved with local approximation techniques more efficiently than the fractional model. Recently, a fractal advection-dispersion equation, and a fractional space-time advection-dispersion equation have been developed to improve the simulation of groundwater transport specifically in fractured aquifers [2,3].…”

Analytical and numerical approaches to nerve impulse model of fractional‐order

“…These fractional operators are being used on a daily basis in analyzing several nonlinear real-world problems. For example, Allwright et al [6] and Atangana et al [7] obtained important results in groundwater flow dynamics by using a derivative which has a nonlocal-nonsingular type kernel. Singh et al [8] analyzed a moderate epidemiological model to describe computer viruses with an arbitrary order derivative that has a nonsingular kernel.…”