%0 Journal Article
%T Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation
%A Abdon Atangana
%A P. D. Vermeulen
%J Abstract and Applied Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/381753
%X The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate and monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given. 1. Introduction The real problem encounter in groundwater studies up to now is the real shape of the geological formation in which water flows in the aquifer under investigation. However, there are many fractured rock aquifers where the flow of groundwater does not fit conventional geometries [1], for example, in South Africa, the Karoo aquifers, characterized by the presence of a very few bedding parallel fractures that serve as the main conduits of water in the aquifers [2]. With a challenge to fit the solution of the groundwater flow equation with experimental data from field observation in particular, the observed drawdown see [3], for all time yields a fit that undervalues the observed drawdown at early times and overvalues it at later times. The variation of observations from theoretically predictable values is usually an indication of uncertainties in the predictable. To investigate the first possibility Botha et al. [2] developed a three-dimensional model for the Karoo aquifer on the campus of the University of the Free State. This model is based on the conventional, saturated groundwater flow equation for density-independent flow: where is the specific storativity, the hydraulic conductivity tensor of the aquifer, the piezometric head, ) the strength of any sources or sinks, with and the usual spatial and time coordinates; the gradient operator, and the time derivative. This model showed that the dominant flow field in these aquifers is vertical and linear and not horizontal and
%U http://www.hindawi.com/journals/aaa/2014/381753/