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 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 3/8Rule 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.
Highlights
The classical Fickian advection-dispersion transport equation application to groundwater systems to model transport relies on the availability of information to characterise the physical system in its entirety
The simulated concentration at a distance of 400 m from the sources is approximately 600 mg/l at the initial time already, compared to the classical simulation where this concentration was only reached at the end of the simulation period of 1 000 days. This transport could be characterised as superdiffusion, which is characterised by a growth rate faster than linear growth
The use of the classical advection-dispersion equation tends to inaccurately simulate the observed contaminant transport because the information to characterise a fractured system to an appropriate level of detail is often not available
Summary
The classical Fickian advection-dispersion transport equation application to groundwater systems to model transport relies on the availability of information to characterise the physical system in its entirety. Where preferential flow pathways, such as fractures, faults or dykes, are present there is often insufficient information to accurately characterise the heterogeneity of these systems In these environments with an incomplete characterisation of heterogeneity, the classical Fickian advection-dispersion transport equation is unable to accurately simulate the observed plume [1,2]. Traditional solutions to the problem of dispersivity scale-dependency consist in establishing scaling relationships [2], or alternatively models are used to incorporate fractures (or preferential flow pathways), namely the dual porosity model; random array of fractures; multiple continuum; and others [6,7] These models cannot account for the fractal nature of fractured systems. Considering the current limitations of transport modelling, especially in fractured media, the development of a fractal advection-dispersion equation for groundwater transport is investigated
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