Abstract
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 rateQand 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.
Highlights
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
The analytical solution obtained via Frobenius method fit the experimental data or have described successfully the events taking place in the vicinity of the borehole, on one hand, and the analytical solutions obtained via Adomian decomposition method was successfully compared to the solution proposed by Barker, on the other hand, the problem of choosing an appropriate geometry for the geological system in which the flow occurs still remains a challenge in groundwater studies
It is believed that the field test gives the characteristic of an aquifer, but we believe that the field test gives both uncertainties and characteristics of the aquifer; quantify uncertainties in this measurement lead us to the real picture of aquifer characteristics, we propose that the studies in groundwater should focus on both uncertainties and fields observations, because what is known is bounded by what is not known; knowing what is not known give a real picture to what was known, and it follows that the knowledge of uncertainties in groundwater study will give a clear picture of what we already know in groundwater
Summary
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. 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]. 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: S0 (x, t) ∂tΦ (x, t) = ∇ ⋅ [K (x, t) ∇Φ (x, t)] + f (x, t) , (1)
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