Journal of Scientific Computing

, Volume 59, Issue 1, pp 80–103 | Cite as

Hydrodynamics in Porous Media: A Finite Volume Lattice Boltzmann Study

  • Ahad Zarghami
  • Chiara Biscarini
  • Sauro Succi
  • Stefano Ubertini


Fluid flow through porous media is of great importance for many natural systems, such as transport of groundwater flow, pollution transport and mineral processing. In this paper, we propose and validate a novel finite volume formulation of the lattice Boltzmann method for porous flows, based on the Brinkman–Forchheimer equation. The porous media effect is incorporated as a force term in the lattice Boltzmann equation, which is numerically solved through a cell-centered finite volume scheme. Correction factors are introduced to improve the numerical stability. The method is tested against fully porous Poiseuille, Couette and lid-driven cavity flows. Upon comparing the results with well-documented data available in literature, a satisfactory agreement is observed. The method is then applied to simulate the flow in partially porous channels, in order to verify its potential application to fractured porous conduits, and assess the influence of the main porous media parameters, such as Darcy number, porosity and porous media thickness.


LBM Porous media Brinkmann–Forchheimer equation  Darcy number 


  1. 1.
    Satuffer, F.: Groundwater I. ETH University Press, Zurich (2011)Google Scholar
  2. 2.
    Arora, K.R.: Soil Mechanics and Foundation Engineering. Standard Publishers Distributors, Delhi (2009)Google Scholar
  3. 3.
    Narvaez, A., Yazdchi, K., Luding, S., Harting, J.: From creeping to inertial flow in porous media: a lattice Boltzmann finite-element study. J. Stat. Mech-Theory E. P02038 (2013)Google Scholar
  4. 4.
    Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particle. Appl. Sci. Res. A1, 27–34 (1974)Google Scholar
  5. 5.
    Joodi, A.S., Sizaret, S., Binet, S., Bruand, A., Alberic, P., Lepiller, M.: Development of a Darcy-Brinkman model to simulate water flow and tracer transport in a heterogeneous karstic aquifer. Hydrogeol. J. 18, 295–309 (2010)CrossRefGoogle Scholar
  6. 6.
    Liu, H., Patil, P.R., Narusawa, U.: On Darcy-Brinkman equation: viscous flow between two parallel plates packed with regular square arrays of cylinders. Entropy 9, 118–131 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Rao, P.R.M., Venkataraman, P.: Validation of Forchheimer’s law for flow through porous media with converging boundaries. J. Hydraul. Eng. 126, 63–71 (2000)CrossRefGoogle Scholar
  8. 8.
    Montillet, A.: Flow through a finite packed bed of spheres: a note on the limit of applicability of the Forchheimer-type equation. J. Fluids Eng. 126, 139–143 (2004)CrossRefGoogle Scholar
  9. 9.
    Pan, H., Rui, H.: Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput. 52, 563–587 (2012)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Whitaker, S.: The Forchheimer equation: a theoretical development. Transp. Porous Med. 25, 27–61 (1996)CrossRefGoogle Scholar
  11. 11.
    Vafai, K., Tien, C.L.: Boundary and inertia effects on flow and heat transfer in porous media. Int. J. Heat Mass Transf. 24, 195–203 (1981)CrossRefMATHGoogle Scholar
  12. 12.
    Guo, Z., Zhao, T.S.: Lattice Boltzmann model for incompressible flows through porous media. Phys. Rev. E 66, 036304 (2002)CrossRefGoogle Scholar
  13. 13.
    Hamdan, M.O., Al-Nimr, M.A., Alkam, M.K.: Enhancing forced convection by inserting porous substrate in the core of a parallel-plate channel. Int. J. Numer. Method H. 10, 502–517 (2000)Google Scholar
  14. 14.
    Alkam, M.K., Al-Nimr, M.A.: Transient non-Darcian forced convection flow in a pipe partially filled with a porous material. Int. J. Heat Mass Transf. 41, 347–356 (1998)CrossRefMATHGoogle Scholar
  15. 15.
    Sukop, M.C., Thorne, D.T.: Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers. Springer, Berlin (2006)Google Scholar
  16. 16.
    Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, Oxford (2001)MATHGoogle Scholar
  17. 17.
    Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145–197 (1992)CrossRefGoogle Scholar
  18. 18.
    Martys, N., Chen, H.: Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. Phys. Rev. E 53, 743–750 (1996)CrossRefGoogle Scholar
  19. 19.
    Shan, X., Chen, H.: Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47, 1815–1819 (1993)CrossRefGoogle Scholar
  20. 20.
    Artoli, A., Hoekstra, A., Sloot, P.: Mesoscopic simulations of systolic flow in the Human abdominal aorta. J. Biomech. 39, 873–884 (2006)CrossRefGoogle Scholar
  21. 21.
    Shan, X., Yuan, X.F., Chen, H.: Kinetic theory representation of hydrodynamics: a way beyond the Navier-Stokes equation. J. Fluid Mech. 550, 413–441 (2006)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Biscarini, C., Di Francesco, S., Mencattini, M.: Application of the lattice Boltzmann method for large-scale hydraulic problems. Int. J. Numer. Method H. 21, 584–601 (2011)Google Scholar
  23. 23.
    Falcucci, G., Ubertini, S., Biscarini, C., Di Francesco, S., Chiappini, D., Palpacelli, S., De Maio, A., Succi, S.: Lattice Boltzmann methods for multiphase flow simulations across scales. Commun. Comput. Phys. 9, 269–296 (2011)Google Scholar
  24. 24.
    Falcucci, G., Ubertini, S., Succi, S.: Lattice Boltzmann simulations of phase-separating flows at large density ratios: the case of doubly-attractive pseudo-potentials. Soft Matter 6, 4357–4365 (2010)CrossRefGoogle Scholar
  25. 25.
    Succi, S., Foti, E., Higuera, F.: Three-dimensional flows in complex geometries with the lattice Boltzmann method. Europhys. Lett. 10, 433–438 (1989)CrossRefGoogle Scholar
  26. 26.
    Cancelliere, A., Chang, C., Foti, E., Rothman, D.H., Succi, S.: The permeability of a random medium: comparison of simulation with theory. Phys. Fluids A 2, 2085–2089 (1990)CrossRefGoogle Scholar
  27. 27.
    Sukop, M.C., Huang, H., Lin, C.L., Deo, M.D., Oh, K., Miller, J.D.: Distribution of multiphase fluids in porous media: comparison between lattice Boltzmann modeling and micro-x-ray tomography. Phys. Rev. E. Stat. Nonlin. Soft Matter Phys. 77, 026710 (2008)CrossRefGoogle Scholar
  28. 28.
    Parmigiani, A., Huber, C., Bachmann, O., Chopard, B.: Pore-scale mass and reactant transport in multiphase porous media flows. J. Fluid Mech. 686, 40–76 (2011)CrossRefMATHGoogle Scholar
  29. 29.
    Prasianakis, N.I., Rosén, T., Kang, J., Eller, J., Mantzaras, J., Büchi, F.N.: Simulation of 3D porous media flows with application to polymer electrolyte fuel cells. Commun. Comput. Phys. 13, 851–866 (2013)Google Scholar
  30. 30.
    Kang, Q., Zhang, D., Chen, S.: Unified lattice Boltzmann method for flow in multi-scale porous media. Phys. Rev. E 66, 056307 (2002)CrossRefGoogle Scholar
  31. 31.
    Anwar, S., Sukop, M.C.: Regional scale transient groundwater flow modeling using lattice Boltzmann methods. Comput. Math. Appl. 58, 1015–1023 (2009)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Chau, J.F., Or, D., Sukop, M.C.: Simulation of gaseous diffusion in partially saturated porous media under variable gravity with lattice Boltzmann methods. Water Resour. Res. 41, W08410 (2005)Google Scholar
  33. 33.
    Seta, T., Takegoshi, E., Okui, K.: Lattice Boltzmann simulation of natural convection in porous media. Math. Comput. Simul. 72, 195–200 (2006)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Geller, S., Krafczyk, M., Tölke, J., Turek, S., Hron, J.: Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows. Comput. Fluids 35, 888–897 (2006)CrossRefMATHGoogle Scholar
  35. 35.
    Bhatnagar, P.L., Gross, E.P., Krook, M.: A Model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94, 511–525 (1954)CrossRefMATHGoogle Scholar
  36. 36.
    Higuera, F.J., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9, 345–349 (1989)CrossRefGoogle Scholar
  37. 37.
    Chen, H., Chen, S., Matthaeus, W.H.: Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Phys. Rev. A 45, 5339–5342 (1992)CrossRefGoogle Scholar
  38. 38.
    Qian, Y.H., D’Humieres, D., Lallemand, P.: Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17, 479–484 (1992)CrossRefMATHGoogle Scholar
  39. 39.
    Succi, S., Karlin, I.V., Chen, H.: Role of the H-theorem in lattice Boltzmann hydrodynamic simulations. Rev. Mod. Phys. 74, 1203–1220 (2002)CrossRefGoogle Scholar
  40. 40.
    D’Humières, D.: Generalized lattice Boltzmann equations. Prog. Aeronaut. Astronaut. 159, 450–458 (1992)Google Scholar
  41. 41.
    Lallemand, P., Luo, L.S.: Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance and stability. Phys. Rev. E 61, 6546–6562 (2000)CrossRefMathSciNetGoogle Scholar
  42. 42.
    Kaehler, G., Wagner, A.J.: Derivation of hydrodynamics for multi-relaxation time lattice Boltzmann using the moment approach. Commun. Comput. Phys. 13, 614–628 (2013)MathSciNetGoogle Scholar
  43. 43.
    D’Humières, D.: Multiple-relaxation-time lattice Boltzmann models in three dimensions. Phil. Trans. R. Soc. Lond. A 360, 437–451 (2002)CrossRefMATHGoogle Scholar
  44. 44.
    Geier, M.C.: Ab Initio Derivation of the Cascade Lattice Boltzmann. Ph.D. Thesis, University of Freiburg, Germany (2006)Google Scholar
  45. 45.
    Ricot, D., Marié, S., Sagaut, P., Bailly, C.: Lattice Boltzmann method with selective viscosity filter. J. Comput. Phys. 228, 4478–4490 (2009)CrossRefMATHGoogle Scholar
  46. 46.
    Ansumali, S., Arcidiacono, S., Chikatamarla, S.S., Prasianakis, N.I., Gorban, A.N., Karlin, I.V.: Quasi-equilibrium lattice Boltzmann method. Eur. Phys. J. B 56, 135–139 (2007)CrossRefGoogle Scholar
  47. 47.
    Asinari, P., Karlin, I.V.: Quasi-equilibrium lattice Boltzmann models with tunable bulk viscosity for enhancing stability. Phys. Rev. E 81, 016702 (2010)CrossRefGoogle Scholar
  48. 48.
    Ansumali, S., Karlin, I.V.: Stabilization of the lattice Boltzmann method by the H-theorem: a numerical test. Phys. Rev. E 62, 7999–8003 (2002)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Ansumali, S., Karlin, I.V.: Single relaxation time model for entropic lattice Boltzmann methods. Phys. Rev. E 65, 056312 (2002)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Singh, S., Krithivasan, S., Karlin, I.V., Succi, S., Ansumali, S.: Energy conserving lattice Boltzmann models for incompressible flow simulations. Commun. Comput. Phys. 13, 603–613 (2013)Google Scholar
  51. 51.
    Tosi, F., Ubertini, S., Succi, S., Karlin, I.V.: Optimization strategies for the entropic lattice Boltzmann method. J. Sci. Comput. 30, 369–387 (2007)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Lee, T., Lin, C.-L.: A characteristic Galerkin method for discrete Boltzmann equation. J. Comput. Phys. 171, 336–356 (2001)CrossRefMATHGoogle Scholar
  53. 53.
    Imamura, T., Suzuki, K., Nakamura, T., Yoshida, M.: Acceleration of teady-state lattice Boltzmann simulations on non-uniform mesh using local time step method. J. Comput. Phys. 202, 645–663 (2005)CrossRefMATHGoogle Scholar
  54. 54.
    Cao, N., Chen, S., Jin, S., Martinez, D.: Physical symmetry and lattice symmetry in the lattice Boltzmann method. Phys. Rev. E 55, R21–R24 (1997)CrossRefGoogle Scholar
  55. 55.
    Mei, R., Shyy, W.: On the finite difference-based lattice Boltzmann method in curvilinear coordinates. J. Comput. Phys. 143, 426–448 (1998)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Jiang, B.N.: In the Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, New York (1998).Google Scholar
  57. 57.
    Li, Y., LeBoeuf, E.J., Basu, P.K.: Least-squares finite-element scheme for the lattice Boltzmann method on an unstructured mesh. Phys. Rev. E 72, 046711 (2005)CrossRefGoogle Scholar
  58. 58.
    Nannelli, F., Succi, S.: The lattice Boltzmann equation on irregular lattices. J. Stat. Phys. 68, 401–407 (1992)CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Ubertini, S., Succi, S., Bella, G.: Lattice Boltzmann schemes without coordinates. Phil. Trans. R. Soc. A 362, 1763–1771 (2004)CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Ubertini, S., Rossi, N., Succi, S., Bella, G.: Unstructured lattice Boltzmann method in three dimensions. Int. J. Numer. Methods Fluids 49, 619–633 (2005)CrossRefMATHGoogle Scholar
  61. 61.
    Ubertini, S., Bella, G., Succi, S.: Unstructured lattice Boltzmann equation with memory. Math. Comput. Simulat. 72, 237–241 (2006)CrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    Peng, G., Xi, H., Duncan, C., Chou, S.H.: Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys. Rev. E 59, 4675–4682 (1999)CrossRefGoogle Scholar
  63. 63.
    Stiebler, M., Tolkeand, J., Krafczyk, M.: An upwind discretization scheme for the finite volume lattice Boltzmann method. Comput. Fluids 35, 814–819 (2006)CrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    Bernaschi, M., Succi, S., Chen, H.: Accelerated lattice Boltzmann schemes for steady-state flow simulations. J. Sci. Comput. 16, 135–144 (2001)CrossRefMATHMathSciNetGoogle Scholar
  65. 65.
    Ricot, D., Marié, S., Sagaut, P., Bailly, C.: Lattice Boltzmann method with selective viscosity filter. J. Comput. Phys. 228, 4478–4490 (2009)CrossRefMATHGoogle Scholar
  66. 66.
    Du, R., Liu, W.: A new multiple-relaxation-time lattice Boltzmann method for natural convection. J. Sci. Comput. (2012). doi: 10.1007/s10915-012-9665-9
  67. 67.
    Patil, D.V., Lakshmisha, K.N.: Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh. J. Comput. Phys. 228, 5262–5279 (2009)CrossRefMATHMathSciNetGoogle Scholar
  68. 68.
    Patil, D.V., Lakshmisha, K.N.: Two-dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation. Int. J. Numer. Methods Fluids 69, 1149–1164 (2012)CrossRefMATHMathSciNetGoogle Scholar
  69. 69.
    Zarghami, A., Ubertini, S., Succi, S.: Finite-volume lattice Boltzmann modeling of thermal transport in nanofluids. Comput. Fluids 77, 56–65 (2013)CrossRefMathSciNetGoogle Scholar
  70. 70.
    Ubertini, S., Bella, G., Succi, S.: Lattice Boltzmann method on unstructured grids: further developments. Phys. Rev. E 68, 016701 (2003)CrossRefMathSciNetGoogle Scholar
  71. 71.
    Ubertini, S., Asinari, P., Succi, S.: Three ways to lattice Boltzmann: a unified time-marching picture. Phys. Rev. E 81, 016311 (2009)CrossRefGoogle Scholar
  72. 72.
    Zarghami, A., Maghrebi, M.J., Ubertini, S., Succi, S.: Modeling of bifurcation phenomena in suddenly expanded flows with a new finite volume lattice Boltzmann method. Int. J. Mod. Phys. C 22, 977–1003 (2011)CrossRefMATHGoogle Scholar
  73. 73.
    Guo, Z., Zheng, C., Shi, B.: Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308 (2002)CrossRefGoogle Scholar
  74. 74.
    Zou, Q., He, X.: On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9, 1591–1598 (1997)CrossRefMATHMathSciNetGoogle Scholar
  75. 75.
    Zarghami, A., Maghrebi, M.J., Ghasemi, J., Ubertini, S.: Lattice Boltzmann finite volume formulation with improved stability. Commun. Comput. Phys. 12, 42–64 (2012)Google Scholar
  76. 76.
    Ghia, U., Ghia, K.N., Shin, C.T.: High-Re solutions for incompressible flow using Navier-Stokes equations and a multigrid method. J. Comp. Phys. 48, 387–411 (1982)CrossRefMATHGoogle Scholar
  77. 77.
    Cook, P.G.: A Guide to Regional Groundwater Flow in Fractured Rock Aquifers. Seaview Press, South Australia (2003)Google Scholar
  78. 78.
    Hoffmann, K.A., Chiang, S.T.: Computational Fluid Dynamics for Engineers. Engineering Education System, Kansas (1993)Google Scholar
  79. 79.
    Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1988)Google Scholar
  80. 80.
    Gibb, J.P., Barcelona, M.J., Ritchey, J.D., Lefaivre, M.H.: Effective Porosity of Geological Materials. ISWS Report No. 351, Illinois (1984)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Ahad Zarghami
    • 1
    • 2
  • Chiara Biscarini
    • 3
  • Sauro Succi
    • 4
  • Stefano Ubertini
    • 5
  1. 1.Honors Center of Italian Universities, H2CUUniversity of Rome La SapienzaRomeItaly
  2. 2.Science and Research BranchIslamic Azad UniversityFarsIran
  3. 3.WARREDOCUniversity for ForeignersPerugiaItaly
  4. 4.Istituto per le Applicazioni del Calcolo, CNRRomeItaly
  5. 5.DEIM, Industrial Engineering SchoolUniversity of TusciaViterboItaly

Personalised recommendations