Groundwater transport modelling: A fractional/fractal perspective

The complexity of real world systems inspire scientists to continually advance methods used to represent these systems as knowledge and technology advances. This fundamental principle has been applied to groundwater transport, a real world problem where the current understanding often cannot describe what is observed in nature. There are two main approaches to improve the simulation of groundwater transport in heterogeneous systems, namely 1) improve the physical characterisation of the heterogeneous system, or 2) improve the formulation of the governing equations used to simulate the system. The latter approach has been pursued by incorporating fractal and fractional derivatives into the governing equation formulation, as well as combining fractional and fractal derivatives. A fractal advection-dispersion equation, with numerical integration and approximation methods for solution, is explored to simulate anomalous transport in fractured aquifer systems. The fractal advection-dispersion equation has been proven to simulate superdiffusion and subdiffusion by varying the fractal dimension, without explicit characterisation of fractures or preferential pathways. A fractional-fractal advection-dispersion equation has also been developed to provide an efficient non-local 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-equation model. In conclusion, the use of fractional calculus and fractal geometry to achieve the collective mission of resolving the difference between modelled and observed is explored for the better understanding and management of fractured systems.