Eur. Phys. J. Plus (2017) 132: 165
https://doi.org/10.1140/epjp/i2017-11434-8

Regular Article

Modelling groundwater fractal flow with fractional differentiation via Mittag-Leffler law

Institute for Groundwater Studies, Faculty of Natural and Agricultural Science, University of Free State, 9300, Bloemfontein, South Africa

^* e-mail: AtanganaA@ufs.ac.za

Received: 11 February 2017
Accepted: 6 March 2017
Published online: 13 April 2017

Abstract

Modelling the flow of groundwater within a network of fractures is perhaps one of the most difficult exercises within the field of geohydrology. This physical problem has attracted the attention of several scientists across the globe. Already two different types of differentiations have been used to attempt modelling this problem including the classical and the fractional differentiation. In this paper, we employed the most recent concept of differentiation based on the non-local and non-singular kernel called the generalized Mittag-Leffler function, to reshape the model of groundwater fractal flow. We presented the existence of positive solution of the new model. Using the fixed-point approach, we established the uniqueness of the positive solution. We solve the new model with three different numerical schemes including implicit, explicit and Crank-Nicholson numerical methods. Experimental data collected from four constant discharge tests conducted in a typical fractured crystalline rock aquifer of the Northern Limb (Bushveld Complex) in the Limpopo Province (South Africa) are compared with the numerical solutions. It is worth noting that the four boreholes (BPAC1, BPAC2, BPAC3, and BPAC4) are located on Faults.