Transport in Porous Media

, Volume 113, Issue 3, pp 457–469 | Cite as

Adoption of Extended Dupuit–Forchheimer Assumptions to Non-Darcy Flow Problems

Article

Abstract

Seepage flow is one of the most practical problems in the field of civil, geology, hydrology and agricultural engineering. Nonlinear flow through permeable deposits takes place when the mean size of the aggregates is coarse. In this paper, an analytical solution is presented for unconfined fully developed turbulent flow under “extended Dupuit–Forchheimer (D–F)” assumptions. A dimensionless form of the solution is therefore presented, and then to show the accuracy of the solution, its results are compared with new experimental data and a Darcy-based solution. According to the results of the present study, it may be concluded that the proposed analytical solution could be used to analyze satisfactorily water surface profiles and normal depth in such sloping permeable porous media under fully developed turbulent flow condition.

Keywords

Extended Dupuit–Forchheimer Fully developed turbulent flow Izbash equation Analytical solution 

List of symbols

A

Cross-sectional area \((\hbox {L}^{2})\)

a

Coefficient of Izbash equation

b

Power constant of Izbash equation

\(C_{1}\)

Constants of integral

C

Constants of integral

h

Piezometric head (L)

\(h_{0}\)

Normal depth (L)

i

Bed slope

L

Length of rock drain (L)

Q

Discharge through porous medium \(({\hbox {L}}^{3}/{\hbox {T}})\)

q

Discharge per unit width \(({\hbox {L}}^{2}/{\hbox {T}})\)

V

Flow velocity (L/T)

x

Longitudinal coordinate (L)

z

Depth of low normal to bed (L)

t

Arbitrary variable

T

The depth of water perpendicular to bottom of channel (L)

\({\theta }\)

Angle of bed related to horizontal

\({\eta }\)

Dimensionless depth for non-Darcy flow

\({\phi }\,({\eta })\)

Dimensionless depth function for non-Darcy flow (positive slope)

\({\varsigma }\)

Dimensionless depth Darcy flow

\({\psi }\,({\varsigma })\)

Dimensionless depth function for Darcian flow

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Soil Science, College of AgricultureYasouj UniversityYasoujIran
  2. 2.Department of Irrigation, College of Agricultural EngineeringSari University of Agricultural Sciences and Natural ResourcesSariIran
  3. 3.Department of Civil Engineering, Yasouj BranchIslamic Azad UniversityYasoujIran

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