Abstract
For non-Darcy flow in coarse porous media, an analytical solution based on the Homotopy Perturbation Method (HPM) is proposed. Subsurface water profile data of a laboratory model are used for six different inlet discharges in both rounded and crushed coarse porous media types. The model slope is S = 0.00001 close to the horizon. Different upstream and downstream water level boundary conditions are considered. The results of the analytical solution of non-Darcy flow by the HPM method are compared with experimental data. The normal objective function (NOF) is used for better comparison between the results of analytical solutions and experimental data. Results depict that the HPM method provides acceptable solutions in both rounded and crushed media types. The analytical results of flow rates q = 26.25 lit/s with NOF of 0.000294586 percent in rounded porous media and q = 30 lit/s with NOF of 0.00028660 percent in crushed one are the most consistent with the experimental data. The proposed HPM solution performs very well, particularly at higher flow rates, in both rounded and crushed porous media.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- a :
-
non-Darcy flow coefficient
- c :
-
constant of integration
- q :
-
Discharge per unit width
- NOF :
-
Normal Objective Function
- s :
-
Flume slope
- x :
-
longitudinal distance
- y :
-
Flow depth
- X :
-
Overal mean
- θ :
-
Bed slope
- y obs(i) :
-
Values of subsurface water profiles depth
- y HPM(i) :
-
Values computed by the final HPM solution
References
Bear J (1972) Dynamics of fluids in porous media. Elsevier Science, New York
Eck BJ, Barrett ME, Charbeneau RJ (2012) Forchheimer flow in gently sloping layers: Application to drainage of porous asphalt. Water Resources Research 48(1):1530–1554, DOI: https://doi.org/10.1029/2011WR010837
Faraz N (2011) Study of the effects of the Reynolds number on circular porous slider via variational iteration algorithm-II. Computers & Mathematics with Application 61(8):1991–1994, DOI: https://doi.org/10.1016/j.camwa.2010.08.048
Faraz N, Khan Y, Anjum A, Kahshan M (2019b) Three-dimensional hydro-magnetic flow arising in a long porous slider and a circular porous slider with velocity slip. Mathematics 7(8):748, DOI: https://doi.org/10.3390/math7080748
Faraz N, Khan Y, Lu DC, Goodarzi M (2019a) Integral transform method to solve the problem of porous slider without velocity slip. Symmetry 11(6):791, DOI: https://doi.org/10.3390/sym11060791
Ghotbi AR, Omidvar M, Barari A (2011) Infiltration in unsaturated soils — An analytical approach. Computers and Geotechnics 38(6):777–782, DOI: https://doi.org/10.1016/j.compgeo.2011.05.007
Gikas GD, Yiannakopoulou T, Tsihrintzis VA (2006) Modeling of non-point source pollution in a Mediterranean drainage basin. Environmental Modeling and Assessment 11(3):219–233, DOI: https://doi.org/10.1007/s10666-005-9017-3
Gill MA (1977) Perturbation solution of spatially varied flow in open channels. Journal of Hydraulic Research 15(4):337–350, DOI: https://doi.org/10.1080/00221687709499639
He JH (1999) Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 178(3):257–262, DOI: https://doi.org/10.1016/S0045-7825(99)00018-3
He JH (2003) Homotopy perturbation method: A new nonlinear analytical technique. Applied Mathematics & Computation 135(1):73–79, DOI: https://doi.org/10.1016/S0096-3003(01)00312-5
He JH (2004) The homotopy perturbation method for nonlinear oscillators with discontinuities. Applied Mathematics& Computation 151(1): 287–292, DOI: https://doi.org/10.1016/S0096-3003(03)00341-2
He JH (2005) Homotopy perturbation method for bifurcation of nonlinear problems. International Journal of Nonlinear Science and Numerical Simulation 6(2):207–208, DOI: https://doi.org/10.1515/IJNSNS.2005.6.2.207
He JH (2012) Homotopy perturbation method with an auxiliary term. Abstract and Applied Analysis Article ID 857612:1-7, DOI: https://doi.org/10.1155/2012/857612
He JH (2014) Homotopy perturbation method with two expanding parameters. Indian Journal of Physics 88(2):193–196, DOI: https://doi.org/10.1007/s12648-013-0378-1
Hosseini SM, Joy DM (2006) Calibration of hydraulic parameters for flows through. Rock-fill Structures. Dam Engineering 17(2):85–111
Khan Y (2013) A method for solving nonlinear time-dependent drainage model. Neural Computing & Applications 23(2):411–415, DOI: https://doi.org/10.1007/s00521-012-0933-2
Khan Y (2014) Two-dimensional boundary layer flow of chemical reaction MHD fluid over a shrinking sheet with suction and injection. Journal of Aerospace Engineering 27(5), DOI: https://doi.org/10.1061/(ASCE)AS.1943-5525.0000274
Khan Y (2017) Magneto hydrodynamic flow of linear visco-elastic fluid model above a shrinking/stretching sheet: A series solution. Scientia Iranica 24(5):2466–2477, DOI: https://doi.org/10.24200/SCI.2017.4305
Khan Y, Faraz N, Yildirim A, Wu Q (2011) A series solution of the long porous slider. Tribology Transactions 54(2):187–191, DOI: https://doi.org/10.1080/10402004.2010.533818
Khan Y, Usman M (2014) Modified homotopy perturbation transform method: A paradigm for nonlinear boundary layer problems. International Journal of Nonlinear Science and Numerical Simulation 15(1):19–25, DOI: https://doi.org/10.1515/ijnsns-2011-0065
Khan Y, Vazquez-Leal H, Faraz N (2013) An auxiliary parameter method using Adomian polynomials and Laplace transformation for nonlinear differential equations. Applied Mathematical Modelling 37(5):2702–2708, DOI: https://doi.org/10.1016/j.apm.2012.06.026
Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Computers and Mathematics with Applications 61(8):1963–1967, DOI: https://doi.org/10.1016/j.camwa.2010.08.022
Li WH (1955) Open channels with non-uniform discharge. Transactions of the American Society of Civil Engineers 120(1):255–274, DOI: https://doi.org/10.1080/00221687709499639
Madani M, Khan Y, Mahmodi G, Faraz N, Yildirim A, Nasernejad B (2012) Application of homotopy perturbation and numerical methods to the circular porous slider. International Journal of Numerical Methods for Heat & Fluid Flow 22(6):705–717, DOI: https://doi.org/10.1108/09615531211244844
Moutsopoulos KN (2009) Exact and approximate analytical solutions for unsteady fully developed turbulent flow in porous media and fractures for time dependent boundary conditions. Journal of Hydrology 369(1–2):78–89, DOI: https://doi.org/10.1016/j.jhydrol.2009.02.025
Moutsopoulos KN, Tsihrintzis VA (2005) Approximate analytical solutions of the Forchheimer equation. Journal of Hydrology 309(1–4):93–103, DOI: https://doi.org/10.1016/j.jhydrol.2004.11.014
Sedghi-Asl M (2010) Investigation of dupuit approximate limits for gradually varied flow in coarse grained porous media. PhD Thesis, Department of Irrigation and Reclamation, College University of Agriculture and Natural Resources, University of Tehran, Iran Sedghi-Asl M, Ansari E (2016) Adoption of extended dupuit—forchheimer assumptions to non-Darcy flow problems. Transport in Porous Media 113(3):457–468, DOI: https://doi.org/10.1007/s11242-016-0703-1
Sedghi-Asl M, Farhoudi J, Rahimi H, Hartmann S (2014c) An analytical solution for 1-D Non-Darcy flow through slanting coarse deposits. Transport in Porous Media 104(3):565–579, DOI: https://doi.org/10.1007/s11242-014-0350-3
Sedghi-Asl M, Rahimi H (2011) Adoption of Manning’s equation to 1D non-Darcy flow problems. Journal of Hydraulic Research 49(6): 814–817, DOI: https://doi.org/10.1080/00221686.2011.629911
Sedghi-Asl M, Rahimi H, Farhoudi J, Hoorfar A, Hartmann S (2014a) 1-D fully-developed turbulent flow through coarse porous medium. Journal of Hydrologic Engineering 19(7):1491–1496, DOI: https://doi.org/10.1061/(ASCE)HE.1943-5584.0000937
Sedghi-Asl M, Rahimi H, Salehi R (2014b) Non-Darcy flow of water through a packed column test. Transport in Porous Media 101(2): 215–227, DOI: https://doi.org/10.1007/s11242-013-0240-0
Stephenson D (1979) Rockfill in hydraulic engineering. Elsevier Scientific, Amsterdam
Xu L (2007) Homotopy perturbation method for a boundary layer equation in unbounded domain. Computers and Mathematics with Applications 54(7):1067–1070, DOI: https://doi.org/10.1016/j.camwa.2006.12.052
Acknowledgments
This article was excerpted from a Ph.D. thesis in Water Resources Engineering held in department of civil engineering, Marvdasht branch, Islamic Azad University, Marvdasht, Iran.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arvin, A., Fattahi, M.H., Sedghi-Asl, M. et al. Adoption of Homotopy Perturbation Method (HPM) for non-Darcy Flow in Porous Media. KSCE J Civ Eng 27, 1551–1557 (2023). https://doi.org/10.1007/s12205-023-1866-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12205-023-1866-2