Computational Geosciences

, Volume 19, Issue 2, pp 423–437 | Cite as

Direct numerical simulation of fully saturated flow in natural porous media at the pore scale: a comparison of three computational systems

  • M. Siena
  • J. D. Hyman
  • M. Riva
  • A. Guadagnini
  • C. L. Winter
  • P. K. Smolarkiewicz
  • P. Gouze
  • S. Sadhukhan
  • F. Inzoli
  • G. Guédon
  • E. Colombo


Direct numerical simulations of flow through two millimeter-scale rock samples of limestone and sandstone are performed using three diverse fluid dynamic simulators. The resulting steady-state velocity fields are compared in terms of the associated empirical probability density functions (PDFs) and key statistics of the velocity fields. The pore space geometry of each sample is imaged at 5.06−μm voxel size resolution using X-ray microtomography. The samples offer contrasting characteristics in terms of total connected porosity (about 0.31 for the limestone and 0.07 for the sandstone) and are typical of several applications in hydrogeology and petroleum engineering. The three-dimensional fluid velocity fields within the explicit pore spaces are simulated using ANSYS® FLUENT® ANSYS Inc. (2009), EULAG Prusa et al. (Comput. Fluids 37, 1193–1207 2008), and SSTOKES Sarkar et al. (2002). These computational approaches are highly disperse in terms of algorithmic complexity, differ in terms of their governing equations, the adopted numerical methodologies, the enforcement of internal no-slip boundary conditions at the fluid-solid interface, and the computational mesh structure. As metrics of comparison to probe in a statistical sense the internal similarities/differences across sample populations of velocities obtained through the computational systems, we consider (i) integral quantities, such as the Darcy flux and (ii) main statistical moments of local velocity distributions including local correlations between velocity fields. Comparison of simulation results indicates that mutually consistent estimates of the state of flow are obtained in the analyzed samples of natural pore spaces despite the considerable differences associated with the three computational approaches. We note that in the higher porosity limestone sample, the structures of the velocity fields obtained using ANSYS FLUENT and EULAG are more alike than either compared against the results obtained using SSTOKES. In the low-porosity sample, the structures of the velocity fields obtained by EULAG and SSTOKES are more similar than either is to the fields obtained using ANSYS FLUENT. With respect to macroscopic quantities, ANSYS FLUENT and SSTOKES provide similar results in terms of the average vertical velocity for both of the complex microscale geometries considered, while EULAG tends to render the largest velocity values. The influence of the pore space structure on fluid velocity field characteristics is also discussed.


Pore-scale flow simulation Porous media Eulerian grid-based methods Computational model comparison Immersed boundary method 


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  1. 1.
    Angot, P., Bruneau, C.H., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math. 81, 497–520 (1999)CrossRefGoogle Scholar
  2. 2.
    ANSYS Inc.: \(\text {ANSYS}^{{\circledR }}~\text {FLUENT}^{{\circledR }}\) User’s guide, Rel. 12.1 (2009)Google Scholar
  3. 3.
    Barth, T., Jespersen, D.: The design and application of upwind schemes on unstructured meshes. In: 27th Aerospace Sciences Meeting. AIAA Paper 89-0366, Reno (1989)Google Scholar
  4. 4.
    Blunt, M.J., Bijeljic, B., Dong, H., Gharbi, O., Iglauer, S., Mostaghimi, P., Paluszny, A., Pentland, C.: Pore-scale imaging and modeling. Adv. Water Resour. 51, 197–216 (2013)CrossRefGoogle Scholar
  5. 5.
    Blunt, M.J., Jackson, M.D., Piri, M., Valvatne, P.H.: Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow. Adv. Water Resour. 25, 1069–1089 (2002)CrossRefGoogle Scholar
  6. 6.
    Boek, E.S., Venturoli, M.: Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries. Comput. Math. Appl. 59, 2305–2314 (2010)CrossRefGoogle Scholar
  7. 7.
    Bourbié, T., Zinszner, B.: Hydraulic and acoustic properties as a function of porosity in Fontainebleau sandstone. J. Geophys. Res. 90, 11524–11532 (1985)CrossRefGoogle Scholar
  8. 8.
    Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998)CrossRefGoogle Scholar
  9. 9.
    Coon, E.T., Porter, M.L., Kang, Q.: Taxila LBM: a parallel, modular lattice Boltzmann framework for simulating pore-scale flow in porous media. Comput. Geosci. 18, 17–27 (2014)CrossRefGoogle Scholar
  10. 10.
    Fourie, W., Said, R., Young, P., Barnes, D.L.: The simulation of pore scale fluid flow with real world geometries obtained from X-ray computed tomography. In: Proceedings of the COMSOL Conference (2007)Google Scholar
  11. 11.
    Gerbaux, O., Buyens, F., Mourzenko, V.V., Memponteil, A., Vabre, A., Thovert, J.F., Adler, P.M.: Transport properties of real metallic foams. J. Colloid Interf. Sci. 342, 155–165 (2010)CrossRefGoogle Scholar
  12. 12.
    Goldstein, D., Handler, R., Sirovich, L.: Modeling a no-slip flow boundary with an external force field. J. Comput. Phys. 105, 354–366 (1993)CrossRefGoogle Scholar
  13. 13.
    Gouze, P., Luquot, L.: X-ray microtomography characterization of porosity, permeability and reactive surface changes during dissolution. J. Contam. Hydrol. 120-121, 45–55 (2010)CrossRefGoogle Scholar
  14. 14.
    Gouze, P., Melean, Y., Le Borgne, T., Dentz, M., Carrera, J.: Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res. 44, W11416 (2008)Google Scholar
  15. 15.
    Griebel, M., Klitz, M.: Homogenization and numerical simulation of flow in geometries with textile microstructures. Multiscale Model Sim. 8(4), 1439–1460 (2010)CrossRefGoogle Scholar
  16. 16.
    Guo, Z., Zhao, T.S., Shi, Y.: Preconditioned lattice-Boltzmann method for steady flows. Phys. Rev. E 70, 066706 (2004)CrossRefGoogle Scholar
  17. 17.
    Hyman, J.D., Smolarkiewicz, P.K., Winter, C.L.: Heterogeneities of flow in stochastically generated porous media. Phys. Rev. E 86, 056701 (2012)CrossRefGoogle Scholar
  18. 18.
    Hyman, J.D., Smolarkiewicz, P.K., Winter, C.L.: Pedotransfer functions for permeability: a computational study at pore scales. Water Resour. Res. 49, 2080–2092 (2013)CrossRefGoogle Scholar
  19. 19.
    Hyman, J.D., Winter, C.L.: Hyperbolic regions in flows through three-dimensional pore structures. Phys. Rev. E 88, 063014 (2013)CrossRefGoogle Scholar
  20. 20.
    Hyman, J.D., Winter, C.L.: Stochastic generation of explicit pore structures by thresholding Gaussian random fields. J. Comput. Phys. 277, 16–31 (2014)CrossRefGoogle Scholar
  21. 21.
    Icardi, M., Boccardo, G., Marchisio, D.L., Tosco, T., Sethi, R.: Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media. Phys. Rev. E 90, 013032 (2014)CrossRefGoogle Scholar
  22. 22.
    Issa, R.I.: Solution of the implicitly discretized fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1986)CrossRefGoogle Scholar
  23. 23.
    Kang, Q., Lichtner, P.C., Zhang, D.: Lattice Boltzmann pore-scale model for multicomponent reactive transport in porous media. J. Geophys. Res. 111, B05203 (2006)Google Scholar
  24. 24.
    Lopez Penha, D.J., Geurts, B.J., Stolz, S., Nordlund, M.: Computing the apparent permeability of an array of staggered square rods using volume-penalization. Comput. Fluids 51, 157–173 (2011)CrossRefGoogle Scholar
  25. 25.
    Malico, I., Ferreira de Sousa, P.J.S.A.: Modeling the pore level fluid flow in porous media using the immersed boundary method. In: Delgado, J.M.P.Q., et al. (eds.) Numerical Analysis of Heat and Mass Transfer in Porous Media, Advanced Structured Materials, vol. 27, pp. 229–251. Springer, Heidelberg (2012)Google Scholar
  26. 26.
    Manwart, C., Aaltosalmi, U., Koponen, A., Hilfer, R., Timonen, J.: Lattice-Boltzmann and finite-difference simulations for the permeability for three-dimensional porous media. Phys. Rev. E 66, 016702 (2002)CrossRefGoogle Scholar
  27. 27.
    Mittal, R., Iaccarino, G.: Immersed-boundary methods. Ann. Rev. Fluid Mech. 37, 239–261 (2005)CrossRefGoogle Scholar
  28. 28.
    Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-CT images. Math. Geosci. 45, 103–125 (2013)CrossRefGoogle Scholar
  29. 29.
    Ovaysi, S., Piri, M.: Direct pore-level modeling of incompressible fluid flow in porous media. J. Comput. Phys. 229(19), 7456–7476 (2010)CrossRefGoogle Scholar
  30. 30.
    Pan, C., Hilpert, M., Miller, C.T.: Lattice-Boltzmann simulation of two-phase flow in porous media. Water Resour. Res. 40, W01501 (2004)Google Scholar
  31. 31.
    Pan, D., Shen, T.T.: Computation of incompressible flows with immersed bodies by a simple ghost cell method. Int. J. Numer. Meth. Fluids 60, 1378–1401 (2009)CrossRefGoogle Scholar
  32. 32.
    Peszynska, M., Trykozko, A.: Pore-to-core simulations of flow with large velocities using continuum models and imaging data. Comput. Geosci. 17, 623–645 (2013)CrossRefGoogle Scholar
  33. 33.
    Prusa, J.M., Smolarkiewicz, P.K., Wyszogrodzki, A.A.: EULAG, a computational model for multiscale flows. Comput. Fluids 37, 1193–1207 (2008)CrossRefGoogle Scholar
  34. 34.
    Raoof, A., Hassanizadeh, S.M.: A new formulation for pore-network modeling of two-phase flow. Water Resour. Res. 48, W01514 (2012)Google Scholar
  35. 35.
    Sarkar, S., Toksöz, M.N., Burns, D.R.: Fluid flow simulation in fractured reservoirs. Report, Annual Consortium Meeting. MIT Earth Resources Laboratory (2002)Google Scholar
  36. 36.
    Siena, M., Guadagnini, A., Riva, M., Gouze, P., Smolarkiewicz, P.K., Winter, C.L., Hyman, J.D., Inzoli, F., Guédon, G.R., Colombo, E.: A comparison of body-fitted and immersed boundary methods for pore-scale modeling of fully saturated flow in synthetic porous media. In: Hadi, K., Copty, N.K (eds.) Proceedings of the IAHR International Groundwater Symposium - Modeling and Management Under Uncertainty. Taylor and Francis Group, London (2013)Google Scholar
  37. 37.
    Siena, M., Riva, M., Hyman, J.D., Winter, C.L., Guadagnini, A.: Relationship between pore size and velocity probability distributions in stochastically generated porous media. Phys. Rev. E 89, 003000 (2014)CrossRefGoogle Scholar
  38. 38.
    Smolarkiewicz, P.K.: Multidimensional positive definite advection transport algorithm: an overview. Int. J. Numer. Meth. Fluids 50, 1123–1144 (2006)CrossRefGoogle Scholar
  39. 39.
    Smolarkiewicz, P.K., Margolin, L.G.: MPDATA: a finite-difference solver for geophysical flows. J. Comput. Phys. 140(2), 459–480 (1998)CrossRefGoogle Scholar
  40. 40.
    Smolarkiewicz, P.K., Prusa, J.M.: Forward-in-time differencing for fluids: simulation of geo- physical turbulence. In: Drikakis, D., Geurts, B.J. (eds.) Turbulent Flow Computation, pp. 279–312. Springer, Netherlands (2002)Google Scholar
  41. 41.
    Smolarkiewicz, P.K., Sharman, R., Weil, J., Perry, S.G., Heist, D., Bowker, G.: Building resolving large-eddy simulations and comparison with wind tunnel experiments. J. Comput. Phys. 227, 633–653 (2007)CrossRefGoogle Scholar
  42. 42.
    Smolarkiewicz, P.K., Szmelter, J.: Iterated upwind schemes for gas dynamics. J. Comput. Phys. 228(1), 33–54 (2009)CrossRefGoogle Scholar
  43. 43.
    Smolarkiewicz, P.K., Winter, C.L.: Pore resolving simulation of Darcy flows. J. Comput. Phys. 229, 3121–3133 (2010)CrossRefGoogle Scholar
  44. 44.
    Tartakovsky, A.M., Meakin, P.: A smoothed particle hydrodynamics model for miscible flow in three-dimensional fractures and the two-dimensional Rayleigh–Taylor instability. J. Comput. Phys. 207(2), 610–624 (2005)CrossRefGoogle Scholar
  45. 45.
    Tartakovsky, A.M., Meakin, P., Scheibe, T.D., Eichler West, R.M.: Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. J. Comput. Phys. 222(2), 654–672 (2007)CrossRefGoogle Scholar
  46. 46.
    Tseng, Y.H., Ferziger, J.H.: A ghost-cell immersed boundary method for flow in complex geometry. J. Comput. Phys. 192, 593–623 (2003)CrossRefGoogle Scholar
  47. 47.
    Valvatne, P.H., Blunt, M.J.: Predictive pore-scale modeling of two-phase flow in mixed wet media. Water Resour. Res. 40, W07406 (2004)Google Scholar
  48. 48.
    Wildenschild, D., Vaz, C.M.P., Rivers, M.L., Rikard, D., Christensen, B.S.B.: Using X-ray computed tomography in hydrology: systems, resolutions and limitations. J. Hydrol. 267, 285–297 (2002)CrossRefGoogle Scholar
  49. 49.
    Zaretskiy, Y., Geiger, S., Sorbie, K., Förster, M.: Efficient flow and transport simulations in reconstructed 3D pore geometries. Adv. Water Resour. 33, 1508–1516 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • M. Siena
    • 1
  • J. D. Hyman
    • 2
    • 4
  • M. Riva
    • 1
    • 3
  • A. Guadagnini
    • 1
    • 3
  • C. L. Winter
    • 3
    • 4
  • P. K. Smolarkiewicz
    • 5
  • P. Gouze
    • 6
  • S. Sadhukhan
    • 6
  • F. Inzoli
    • 7
  • G. Guédon
    • 7
  • E. Colombo
    • 7
  1. 1.Dipartimento di Ingegneria Civile e AmbientalePolitecnico di MilanoMilanItaly
  2. 2.Earth and Environmental Sciences Division and Theoretical DivisionLos Alamos National LaboratoryLos AlamosUSA
  3. 3.Department of Hydrology and Water ResourcesUniversity of ArizonaTucsonUSA
  4. 4.Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  5. 5.European Centre for Medium-Range Weather ForecastsReadingUK
  6. 6.GéosciencesUniversité Montpellier - CNRSMontpellierFrance
  7. 7.Dipartimento di Energia, Politecnico di MilanoMilanItaly

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