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Abstract
Abstract. In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.
Highlights
70 years ago in his hydrologic studies of the Aswan High Dam, Hurst (1951) discovered that the flow time series of the Nile River demonstrated fluctuations whose rescaled range may not be proportional to the square root of the observation duration but may be proportional to the duration raised to a power H that is larger than 0.5 but less than 1
From the standard governing Eq (23) of unconfined groundwater flow in integer time–space the saturated hydraulic conductivity may be interpreted as a diffusion coefficient for the nonlinear diffusion of groundwater in an unconfined aquifer
The basic difference between confined and unconfined groundwater flow is that the former can be interpreted as a linear diffusion of groundwater while the latter is a nonlinear diffusion of groundwater within an aquifer
Summary
70 years ago in his hydrologic studies of the Aswan High Dam, Hurst (1951) discovered that the flow time series of the Nile River demonstrated fluctuations whose rescaled range may not be proportional to the square root of the observation duration but may be proportional to the duration raised to a power H (the so-called Hurst coefficient) that is larger than 0.5 but less than 1. Mehdinejadiani et al (2013) expanded the approach of Wheatcraft and Meerschaert (2008) to the derivation of a governing equation of groundwater flow in an unconfined aquifer in fractional space but in integer time In their derivation, they used the conventional Darcy formulation for the water flux with an integer spatial derivative, while utilizing fractional spatial derivatives in their continuity equation. Around that time, Kavvas et al (2017a) utilized the mean value formulation from Usero (2008), Odibat and Shawagfeh (2007), and Li et al (2009) to derive a complete governing equation of transient groundwater flow in an anisotropic confined aquifer with fractional time and multi-fractional space derivatives which addressed the continuity and the water flux (motion) in fractional time–space and the effect of a sink/source term. Combining the fractional motion Eq (19) of groundwater flow in an unconfined aquifer with the fractional continuity Eq (14) of unconfined groundwater flow results in the equawww.earth-syst-dynam.net/11/1/2020/
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