# The fractional Boussinesq equation of groundwater flow and its applications

Su, Ninghu (2017) The fractional Boussinesq equation of groundwater flow and its applications. Journal of Hydrology, 547. pp. 403-412.

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## Abstract

This paper presents a set of fractional Boussinesq equations (fBEs) for groundwater flow in confined and unconfined aquifers and demonstrates the application of one of the fBEs for groundwater discharges known as recession curves. The fBEs are formulated with two-term distributed fractional orders in time and symmetrical fractional derivatives (SFD) in space applicable to both confined and unconfined aquifers. The SFD in theory consists of the forward fractional derivative (FFD) and the backward fractional derivative (BFD). The FFD represents the forward movement of water along the direction of mainstream flow while the BFD accounts for the backward motion of water in the direction opposite to the mainstream flow. The backward flow at the pore level can be referred to as the micro-scale backwater effect. The analogue of the backwater effect on a micro-scale using the BFD coincides with the wandering processes based on the continuous-time random walk (CTRW) theory which results in the fractional governing equation. With the analytical solutions of the fBE for given initial and boundary conditions of the first type for a finite depth, a set of formulae for groundwater recession has been derived using approximate solutions of the fBE. The examples of the applications of the recession curves are graphically illustrated and the effects of the orders of fractional derivatives on the geometry of the flow curves examined.

Item ID: 47752 Article (Research - C1) 0022-1694 fractional Boussinesq equation; CTRW; symmetrical fractional derivatives; micro-scale backwater effect; anomalous groundwater flow; new formulae for recession curves Distinguished Expert of Ningxia 14 Mar 2017 04:35 40 ENGINEERING > 4012 Fluid mechanics and thermal engineering > 401208 Geophysical and environmental fluid flows @ 40%37 EARTH SCIENCES > 3707 Hydrology > 370703 Groundwater hydrology @ 40%37 EARTH SCIENCES > 3706 Geophysics > 370601 Applied geophysics @ 20% 96 ENVIRONMENT > 9609 Land and Water Management > 960999 Land and Water Management of Environments not elsewhere classified @ 40%96 ENVIRONMENT > 9609 Land and Water Management > 960905 Farmland, Arable Cropland and Permanent Cropland Water Management @ 40%96 ENVIRONMENT > 9609 Land and Water Management > 960903 Coastal and Estuarine Water Management @ 20% Total: 1180 Last 12 Months: 21 More Statistics

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