Transport in Porous Media

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

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



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.


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

List of symbols


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


Coefficient of Izbash equation


Power constant of Izbash equation


Constants of integral


Constants of integral


Piezometric head (L)


Normal depth (L)


Bed slope


Length of rock drain (L)


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


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


Flow velocity (L/T)


Longitudinal coordinate (L)


Depth of low normal to bed (L)


Arbitrary variable


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



The authors would like to thank the SimTech cluster of Excellence and Hydraulic Laboratory of Stuttgart University for providing experimental facilities and supporting the research. Also the support of Yasouj University is appreciated.

Funding The authors wish to express their deepest gratitude to Yasouj University and SimTech cluster of Excellence and Hydraulic Laboratory of Stuttgart University for their financial and technical support of the project.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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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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