Archives of Computational Methods in Engineering

, Volume 22, Issue 4, pp 637–653 | Cite as

Finite Element Modeling of Free Surface Flow in Variable Porosity Media

  • Antonia LareseEmail author
  • Riccardo Rossi
  • Eugenio Oñate
Original Paper


The aim of the present work is to present an overview of some numerical procedures for the simulation of free surface flows within a porous structure. A particular algorithm developed by the authors for solving this type of problems is presented. A modified form of the classical Navier–Stokes equations is proposed, with the principal aim of simulating in a unified way the seepage flow inside rockfill-like porous material and the free surface flow in the clear fluid region. The problem is solved using a semi-explicit stabilized fractional step algorithm where velocity is calculated using a 4th order Runge–Kutta scheme. The numerical formulation is developed in an Eulerian framework using a level set technique to track the evolution of the free surface. An edge-based data structure is employed to allow an easy OpenMP parallelization of the resulting finite element code. The numerical model is validated against laboratory experiments on small scale rockfill dams and is compared with other existing methods for solving similar problems.


Porous Medium Fractional Step Free Surface Flow Mixed Finite Element Method Coastal Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The research was supported by the FP7-Capacities ULITES project GA-314891 and ERC Advance Grant SAFECON project AdG-267521. The authors wants to acknowledge Prof. Miguel Angel Toledo, Dr. Rafa Moran and Mr. Hibber Campos of the Technical University of Madrid (UPM), Mr. Angel Lara and Mrs. Pilar Viña of CEDEX for the experimental results provided for this work.


  1. 1.
    Andrade JA Jr, Costa UMS, Almeida MP, Makse HA, Stanley HE (1998) Inertial effects on fluid flow through disordered porous media. Phys Rev Lett 82(26):5249–5252CrossRefGoogle Scholar
  2. 2.
    Bear J (1972) Dynamics of fluids in porous media, chapter. American Elsevier, New YorkGoogle Scholar
  3. 3.
    Bear J (1988) Dynamics of fluids in pouros media. Elsevier, USAGoogle Scholar
  4. 4.
    Chen Y, Hu R, Lu W, Li D, Zhou C (2011) Modeling coupled processes of non-steady seepage flow and non-linear deformation for a concrete faced rockfill dam. Comput Struct 89:1333–1351CrossRefGoogle Scholar
  5. 5.
    Chilton TH, Colburn AP (1931) Pressure drop in packed tubes. Ind Eng Chem 23:913–919CrossRefGoogle Scholar
  6. 6.
    Codina R (2000) A nodal-based implementation of a stabilized finite element method for incompressible flow problems. Int J Numer Methods Fluids 33:737–766zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Codina R (2000) On stabilized finite element methods for linear system of convection–diffusion-reaction equations. Comput Methods Appl Mach Eng 188:61–82zbMATHCrossRefGoogle Scholar
  8. 8.
    Codina R (2000) Pressure stability in fractional step finite element methods for incompressible flows. J Comput Phys 170:112–140MathSciNetCrossRefGoogle Scholar
  9. 9.
    Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element method. Comput Methods Appl Mach Eng 190:1579–1599zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191:4295–4321zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Codina R, Soto O (2004) Approximation of the incompressible navier–stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes. Comput Methods Appl Mach Eng 193:1403–1419zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17:253–297zbMATHCrossRefGoogle Scholar
  13. 13.
    de Lemos MJS (2006) Turbulence in porous media. Elsevier, UKGoogle Scholar
  14. 14.
    del Jesus M, Lara JL, Losada IJ (2012) Three-dimensional interaction of waves and porous coastal structures. Part i: nomerical model formulation. Coast Eng 64:57–72CrossRefGoogle Scholar
  15. 15.
    Da Deppo L, Datei C, Salandin P (2002) Sistemazione dei corsi d’acqua. Libreria Internazionale Cortina, PadovaGoogle Scholar
  16. 16.
    Donea J, Huerta A (2003) Finite elements methods for flow problems. Wiley, New YorkCrossRefGoogle Scholar
  17. 17.
    Du Plessis JP, Masliyah JH (1988) Mathematical modeling of flow through consolidated isotropic porous media. Transport porous media 3:145–161CrossRefGoogle Scholar
  18. 18.
    Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
  19. 19.
    Fancher GH, Lewis JA (1933) Flow of simple fluids through porous materials. Ind Eng Chem 25:1139–1147CrossRefGoogle Scholar
  20. 20.
    Ghetti A (1984) Idraulica. Ed. Cortina, MilanoGoogle Scholar
  21. 21.
    Hansen D (1992) The behaviour of flowthrough Rockfill dams. PhD Thesis: Univerity of Ottawa, CanadaGoogle Scholar
  22. 22.
    Hsu T-J, Sakakiyama T, Liu PL-F (2002) A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coast Eng 46(1):25–50Google Scholar
  23. 23.
    Khoei AR, Mohammadnejad T (2011) Numerical modeling of multiphase fluid flow in deforming porous media: a comparison between two- and three-phase models for seismic analysis of earth and rockfill dams. Comput Geotech 38(2):142–166Google Scholar
  24. 24.
    Kratos, multiphysics opensource fem code.
  25. 25.
    Larese A (2012) A coupled Eulerian-PFEM model for the simulation of overtopping in rockfill dams. Phd thesis: Universitat Politècnica de Catalunya (UPC BarcelonaTech), Barcelona, SpainGoogle Scholar
  26. 26.
    Larese A, Rossi R, Oñate E (2011) Coupling eulerian and lagrangian models to simulate seepage and evolution of failure in prototype rockfill dams. In: Proceeding of the XI ICOLD benchmark workshop on numerical analysis of dams, SpanCOLD, Madrid, Spain, ISBN: 978-84-695-1816-8Google Scholar
  27. 27.
    Larese A, Rossi R, Oñate E (2011) Theme b: simulation of the behavior of prototypes of rockfill dams during overtopping scenarios: seepage evolution and beginning of failure. In: Proceeding of the XI ICOLD benchmark workshop on numerical analysis of Dams, SpanCOLD Madrid, Spain, ISBN: 978-84-695-1816-8Google Scholar
  28. 28.
    Larese A, Rossi R, Oñate E, Idelsohn SR (2012) A coupled pfem–eulerian approach for the solution of porous fsi problems. Comput Mech 50(6):805–819. doi: 10.1007/s00466-012-0768-9 zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Larese A, Rossi R, Oñate E, Toledo MA, Moran R, Campos H (2013) Numerical and experimental study of overtopping and failure of rockfill dams. Int J Geomech (ASCE). doi: 10.1061/(ASCE)GM.1943-5622.0000345
  30. 30.
    Li B (1995) Flowthrough and overtopped rockfill dams. Phd thesis: Universty of Ottawa ISBN: 0-612-15734-2Google Scholar
  31. 31.
    Li B, Garga VK (1998) Theoretical solution for seepage flow in overtopped rockfill. J Hydraul Eng Comput 124:213–217CrossRefGoogle Scholar
  32. 32.
    Li B, Garga VK, Davies MH (1998) Relationships for non-darcy flow in rockfill. J Hydraul Eng 124(2):206–212CrossRefGoogle Scholar
  33. 33.
    Li T, Troch P, De Rouck J (2004) Wave overtopping over a sea dike. J Comput Phys 198(2):686–726Google Scholar
  34. 34.
    Lin P (1998) Numerical modeling of breaking waves. Ph.D. thesis, Cornell UniversityGoogle Scholar
  35. 35.
    Löhner R (2001) Applied computational fluid dynamics techniques. Wiley, UKGoogle Scholar
  36. 36.
    Lubin P, Vincent S, Caltagirone J-P, Abadie S (2003) Fully three-dimensional direct numerical simulation of a plunging breaker. Comptes Rendus Mecanique 331(7):495–501Google Scholar
  37. 37.
    Martins JP, Milton-Taylor D, Leung HK (1990) The effects of non-darcy flow in propped hydraulic fractures. In: Proceedings of the SPE Annual Technical ConferenceGoogle Scholar
  38. 38.
    McCorquodale JA, Hannoura AA, Nasser MS (1978) Hydraulic conductivity of rockfill. J Hydraul Res 16(2):123–137CrossRefGoogle Scholar
  39. 39.
    Morán R (2013) Mejora de la seguridad de las presas de escollera frente a percolación accidental mediante protecciones tipo rapié. PhD Thesis: Universidad Politécnica de Madrid, Madrid, SpainGoogle Scholar
  40. 40.
    Morán R, Toledo MA (2011) Reserarch into protection of rockfill dams from overtopping using rockfill downstream toes. Can J Civil Eng 38(12):1314–1326Google Scholar
  41. 41.
    Nield DA, Bejan A (1992) Convection in porous media. Springer, New YorkGoogle Scholar
  42. 42.
    Niessner J, Hassanizadeh SM (2008) A model for two-phase flow in porous media including fluid-fluid interfacial area. Water Resourc Res 44(8). doi: 10.1029/2007WR006721
  43. 43.
    Nithiarasu P, Seetharamu KN, Sundararajan T (1997) Natural convective heat transfer in a fluid saturated variable porosity medium. Int J Heat Mass Transf 40:3955–3967Google Scholar
  44. 44.
    Nithiarasu P, Sujatha KS, Ravindran K, Sundararajan T, Seetharamu KN (2000) Non-darcy natural convection in a hydrodynamically and thermally anisotropic porous medium. Comput Methods Appl Mech Eng 188:413–430zbMATHCrossRefGoogle Scholar
  45. 45.
    Nithiarasu P, Sundararajan T, Seetharamu KN (1997) Double-diffusive natural convection in a fluid saturated porous cavity with a freely convecting wall. Int Commun Heat Mass Transf 24:1121–1130Google Scholar
  46. 46.
    Osher S, Fedkiw RP (2003) Level set methods and dynamic implicit surfaces. Springer, New YorkGoogle Scholar
  47. 47.
    Rossi R, Larese A, Dadvand P, Oñate E (2013) An efficient edge-based level set finite element method for free surface flow problems. Int J Numer Meth Fluids 71(6):687–716. doi: 10.1002/fld.3680 CrossRefGoogle Scholar
  48. 48.
    Ryzhakov P, Rossi R, Oñate E (2012) An algorithm for the simulation of thermally coupled low speed flow problems. Int J Numer Methods Fluids 70(1):1–19Google Scholar
  49. 49.
    Scheidegger AE (1974) The physics of flow through porous media. University of Toronto Press, TorontoGoogle Scholar
  50. 50.
    Schrefler BA, Scotta R (2001) A fully coupled dynamic model for two-phase fluid flow in deformable porous media. Comput Methods Appl Mech Eng 190(24–25):3223–3246zbMATHCrossRefGoogle Scholar
  51. 51.
    Sheng D, Sloan SW, Gens A, Smith DW (2003) Finite element formulation and algorithms for unsaturated soils. Part i: Theory. Int J Numer Anal Methods Geomech 27:745–765zbMATHCrossRefGoogle Scholar
  52. 52.
    Soto O, Lohner R, Cebral J, Camelli F (2004) A stabilized edge-based implicit incompressible flow formulation. Comput Methods Appl Mech Eng 193:2139–2154zbMATHCrossRefGoogle Scholar
  53. 53.
    Stephenson D (1969) Rockfill Hydraul Eng. Elsevier Scientific, AmsterdamGoogle Scholar
  54. 54.
    Szymkiewicz A (2013) Modelling water flow in unsaturated porous media, chapter mathematical models of flow in porous media. Springer, BerlinCrossRefGoogle Scholar
  55. 55.
    Taylor DW (1948) Fundamentals of soil mechanics. Wiley, New YorkGoogle Scholar
  56. 56.
    Toledo MA (1997) Presas De Escollera Sometidas a Sobrevertido. Estudio del Movimientos dal Agua a Través de la Escollera e de la Estabilidad Frente al Deslizamiento en Masa. PhD thesis: Universidad Politécnica de MadridGoogle Scholar
  57. 57.
    Toledo MA (1998) Safety of rockfill dams subject to overtopping. In: Proceding of the international symposium on new trend an guidelines on dam safety, Taylor & Francis Inc. isbn 9054109742, 9789054109747Google Scholar
  58. 58.
    Uzuoka R, Unno T, Sento N, Kazama M (2014) Effect of pore air pressure on cyclic behavior of unsaturated sandy soil. Proceedings of the 6th International Conference on Unsaturated Soils, UNSAT 2014, Sydney, vol 1, pp 783–789Google Scholar
  59. 59.
    Wang Z, Zou Q, Reeve D (2009) Simulation of spilling breaking waves using a two phase flow cfd model. Comput Fluids 38(10):1995–2005Google Scholar
  60. 60.
    Wilkins JK (1956) Flow of water through rockfill and its aplication to the desing of dams. In: Proceedings of 2nd Australia—New Zealand conference on soil mechanics and foundation engineeringGoogle Scholar
  61. 61.
    Zeng Z, Grigg R (2006) A criterion for non-darcy flow in porous media. Transp Porous Media 63:57–69CrossRefGoogle Scholar
  62. 62.
    Zienkiewicz OC, Chan AHC, Pastor M, Schrefler BA, Shiomi T (1999) Computational geomechanics with special reference to earthquake engineering. Wiley, New YorkzbMATHGoogle Scholar
  63. 63.
    Zienkiewicz OC, Taylor RL (2004) The finite element method, vol 3, fluid dynamics. Butterworth-Heinemann, OxfordGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  • Antonia Larese
    • 1
    • 2
    Email author
  • Riccardo Rossi
    • 1
    • 2
  • Eugenio Oñate
    • 1
    • 2
  1. 1.Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE)BarcelonaSpain
  2. 2.Universitat Politècnica de Catalunya, UPC Barcelona TechBarcelonaSpain

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