Geotechnical and Geological Engineering

, Volume 37, Issue 1, pp 413–429 | Cite as

Investigation of Non-Darcy Flow for Fine Grained Materials

  • William Ovalle-VillamilEmail author
  • Inthuorn Sasanakul
Original Paper


Research efforts have been done to determine the limit of validity of Darcy’s Law in various fields of engineering where viscous conditions of flow are likely exceeded. In geotechnical and geological engineering, the application of centrifuge modeling for studying flow through porous materials is complex because the seepage velocity scales proportionally to centrifuge gravity, resulting in greater potential for exceeding the limit of viscous flow. This limit is usually estimated based on Forchheimer’s Law and the concept of critical Reynolds number, Rcritic, but its interpretation remains ambiguous. This study provides new insights and establishes a connection between different theoretical approaches available in the literature. Centrifuge permeability tests were conducted at different gravitational levels for different materials. Effects of the characteristics of the porous media and centrifuge acceleration on the flow behavior were evaluated. Results show that parameters of Forchheimer’s Law remain constant regardless of the centrifuge acceleration. Values of Rcritic were obtained in a range from 0.2 to 11 depending on the characteristics of material. The interpretation of the limit of validity of Darcy’s Law was analyzed based on different definitions of Reynolds number and the Forchheimer number, and critical velocities of flow and hydraulic gradients were estimated.


Porous media Darcy’s Law Forchheimer’s Law Forchheimer number Centrifuge modelling Scaling law 


  1. Abbood DW (2009) An experimental model for flow through porous media using water filter. In: Thirteenth international water and technology conference, Hurghada, Egypt, pp 883–893Google Scholar
  2. Andreasen RR, Canga E, Kjaergaard C, Iversen BV, Poulsen TG (2013) Relating water and air flow characteristics in coarse granular materials. Water Air Soil Pollut 224(4):1469CrossRefGoogle Scholar
  3. Arulanandan K, Thompson PY, Kutter BL, Meegoda NJ, Muraleetharan KK, Yogachandran C (1988) Centrifuge modeling of transport processes for pollutants in soils. J Geotech Eng 114(2):185–205CrossRefGoogle Scholar
  4. Bear J (2013) Dynamics of fluids in porous media. Dover Publications Inc, New YorkGoogle Scholar
  5. Bo-Ming Y, Jian-Hua L (2004) A geometry model for tortuosity of flow path in porous media. Chin Phys Lett 21(8):1569CrossRefGoogle Scholar
  6. Burcharth HF, Christensen C (1991) On stationary and non-stationary porous flow in coarse granular materials: European Community, MAST G6-S: Project 1, Wave Action on and in coastal structures. Aalborg Universitetsforlag, AalborgGoogle Scholar
  7. Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 15:150–166Google Scholar
  8. Carman PC (1956) Flow of gases through porous media. Butterworths Scientific Publications, LondonGoogle Scholar
  9. Carrier WD (2003) Goodbye, Hazen; Hello, Kozeny-Carman. J Geotech Geoenviron Eng 129(11):1054–1056CrossRefGoogle Scholar
  10. Chapuis RP (2004) Predicting the saturated hydraulic conductivity of sand and gravel using effective diameter and void ratio. Can Geotech J 41(5):787–795CrossRefGoogle Scholar
  11. Comiti J, Renaud M (1989) A new model for determining mean structure parameters of fixed beds from pressure drop measurements: application to beds packed with parallelepipedal particles. Chem Eng Sci 44(7):1539–1545CrossRefGoogle Scholar
  12. Comiti J, Sabiri NE, Montillet A (2000) Experimental characterization of flow regimes in various porous media—III: limit of Darcy’s or creeping flow regime for Newtonian and purely viscous non-Newtonian fluids. Chem Eng Sci 55(15):3057–3061CrossRefGoogle Scholar
  13. Dukhan N, Bağcı Ö, Özdemir M (2014) Metal foam hydrodynamics: flow regimes from pre-Darcy to turbulent. Int J Heat Mass Transf 77:114–123CrossRefGoogle Scholar
  14. Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
  15. Ergun S, Orning AA (1949) Fluid flow through randomly packed columns and fluidized beds. Ind Eng Chem 41(6):1179–1184CrossRefGoogle Scholar
  16. Fand RM, Kim BYK, Lam ACC, Phan RT (1987) Resistance to the flow of fluids through simple and complex porous media whose matrices are composed of randomly packed spheres. J Fluids Eng 109(3):268–274CrossRefGoogle Scholar
  17. Fourar M, Radilla G, Lenormand R, Moyne C (2004) On the non-linear behavior of a laminar single-phase flow through two and three-dimensional porous media. Adv Water Resour 27(6):669–677CrossRefGoogle Scholar
  18. Garnier J, Gaudin C, Springman SM, Culligan PJ, Goodings D, Konig D et al (2007) Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling. Int J Phys Modell Geotech 7(3):1CrossRefGoogle Scholar
  19. Gelhar LW, Welty C, Rehfeldt KR (1992) A critical review of data on field-scale dispersion in aquifers. Water Resour Res 28(7):1955–1974CrossRefGoogle Scholar
  20. Goodings DJ (1994) Implications of changes in seepage flow regimes for centrifuge models. In: Proceedings of the international conference centrifuge, vol 94, pp 393–398. ISBN 90-5410-352-3Google Scholar
  21. Kadlec RH, Knight RL (1996) Treatment wetlands. Lewis, Boca Raton, p 893Google Scholar
  22. Khalifa A, Garnier J, Thomas P, Rault G (2000a) Scaling laws of water flow in centrifuge models. In: International symposium on physical modelling and testing in environmental geotechnics, vol 56, pp 207–216Google Scholar
  23. Khalifa MA, Wahyudi I, Thomas P (2000b) A new device for measuring permeability under high gradients and sinusoidal gradients. Geotech Test J 23(4):404–412CrossRefGoogle Scholar
  24. Khalifa MA, Wahyudi I, Thomas P (2002) New extension of Darcy’s law to unsteady flows. Soils Found 42(6):53–63CrossRefGoogle Scholar
  25. Kovács G (1981) Developments in water science—seepage hydraulics, Chap 3.2. Elsevier, New YorkGoogle Scholar
  26. Kozeny J (1927) Uber kapillare leitung der wasser in boden. R Acad Sci Vienna Proc Class I 136:271–306Google Scholar
  27. Kreibich H, Piroth K, Seifert I, Maiwald H, Kunert U, Schwarz J, Merz B, Thieken AH (2009) Is flow velocity a significant parameter in flood damage modelling? Nat Hazards Earth Syst Sci 9(5):1679CrossRefGoogle Scholar
  28. Laut P (1975) Application of centrifuge model tests in connexion with studies of flow patterns of contaminated water in soil structures. Geotechnique 25(2):401–406CrossRefGoogle Scholar
  29. Macdonald IF, El-Sayed MS, Mow K, Dullien FAL (1979) Flow through porous media—the Ergun equation revisited. Ind Eng Chem Fundam 18(3):199–208CrossRefGoogle Scholar
  30. Mathias SA, Butler AP, Zhan H (2008) Approximate solutions for Forchheimer flow to a well. J Hydraul Eng 134(9):1318–1325CrossRefGoogle Scholar
  31. Mesquita M, Testezlaf R, Ramirez JCS (2012) The effect of media bed characteristics and internal auxiliary elements on sand filter head loss. Agric Water Manag 115:178–185CrossRefGoogle Scholar
  32. Moutsopoulos KN, Papaspyros IN, Tsihrintzis VA (2009) Experimental investigation of inertial flow processes in porous media. J Hydrol 374(3):242–254CrossRefGoogle Scholar
  33. Muskat M (1938) The flow of homogeneous fluids through porous media. Soil Sci 46(2):169CrossRefGoogle Scholar
  34. Nielsen P (1992) Coastal bottom boundary layers and sediment transport, vol 4. World Scientific Publishing Co Inc, SingaporeGoogle Scholar
  35. Ovalle-Villamil W, Sasanakul I (2018) A new insight into the behavior of seepage flow in centrifuge modeling. In: 9th International conference of physical modelling in geotechnics, LondonGoogle Scholar
  36. Richardson JF, Harker JH, Backhurst JR (2002) Chemical engineering—particle technology and separation processes, vol 2. Butterworth Heinemann, Oxford. ISBN 0-7506-4445-1Google Scholar
  37. Salahi MB, Sedghi-Asl M, Parvizi M (2015) Nonlinear flow through a packed-column experiment. J Hydrol Eng 20(9):04015003CrossRefGoogle Scholar
  38. Salem HS, Chilingarian GV (2000) Influence of porosity and direction of flow on tortuosity in unconsolidated porous media. Energy Sources 22(3):207–213CrossRefGoogle Scholar
  39. Sedghi-Asl M, Rahimi H, Salehi R (2014) Non-Darcy flow of water through a packed column test. Transp Porous Media 101(2):215–227CrossRefGoogle Scholar
  40. Sidiropoulou MG, Moutsopoulos KN, Tsihrintzis VA (2007) Determination of Forchheimer equation coefficients a and b. Hydrol Process 21(4):534–554CrossRefGoogle Scholar
  41. Singh DN, Gupta AK (2000) Modelling hydraulic conductivity in a small centrifuge. Can Geotech J 37(5):1150–1155CrossRefGoogle Scholar
  42. Snoeijers R (2016) Non-linear flow in unconsolidated sandy porous media—an experimental investigation. Master’s thesis, Utrecht UniversityGoogle Scholar
  43. Stephenson DJ (1979) Rockfill in hydraulic engineering, vol 27. Elsevier, New YorkGoogle Scholar
  44. Trussell RR, Chang M (1999) Review of flow through porous media as applied to head loss in water filters. J Environ Eng 125(11):998–1006CrossRefGoogle Scholar
  45. Wahyudi I, Montillet A, Khalifa AA (2002) Darcy and post-Darcy flows within different sands. J Hydraul Res 40(4):519–525CrossRefGoogle Scholar
  46. Xu P, Yu B (2008) Developing a new form of permeability and Kozeny–Carman constant for homogeneous porous media by means of fractal geometry. Adv Water Resour 31(1):74–81CrossRefGoogle Scholar
  47. Zeng Z, Grigg R (2006) A criterion for non-Darcy flow in porous media. Transp Porous Media 63(1):57–69CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department Civil and Environmental EngineeringUniversity of South CarolinaColumbiaUSA
  2. 2.Department Civil and Environmental EngineeringUniversity of South CarolinaColumbiaUSA

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