Skip to main content
SpringerLink
Log in

Stochastic Modeling of the Permeability of Randomly Generated Porous Media via the Lattice Boltzmann Method and Probabilistic Collocation Method

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

The permeability of natural porous media, such as soils and rocks, usually possesses uncertainties due to the randomness and spatial variation of microscopic pore structures. It is of great importance to develop an effective methodology to obtain statistical properties of permeability for porous media. In this work, an efficient approach is developed by combining the sphere packing algorithm, lattice Boltzmann method (LBM), and probabilistic collocation method (PCM). The porous media are generated by sphere packings of a specified size distribution, and the isotropy and representative elementary volume are verified by statistical analyses. Fluid flow in the complex pore structures is numerically resolved by LBM, with the permeability calculated by Darcy’s law. The uncertainty of permeability can be quantified by PCM with only several porosity samplings required at predetermined collocation points. In addition, the porosity–permeability relationships can be acquired efficiently. Numerical results indicate that, with the proposed approach, the computational efforts are reduced by more than two orders of magnitude compared to the Monte Carlo simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Adler, P.M., Jacquin, C.G., Quiblier, J.A.: Flow in simulated porous media. Int. J. Multiph. Flow 16(4), 691–712 (1990)

    Article  Google Scholar 

  • Bear, J.: Dynamics of Fluids in Porous Media, pp. 174–175. American Elsevier Pub. Co., New York (1972)

    Google Scholar 

  • Berryman, J.G.: Measurement of spatial correlation functions using image processing techniques. J. Appl. Phys. 57(7), 2374–2384 (1985)

    Article  Google Scholar 

  • Bosl, W.J., Dvorkin, J., Nur, A.: A study of porosity and permeability using a lattice Boltzmann simulation. Geophys. Res. Lett. 25(9), 1475–1478 (1998)

    Article  Google Scholar 

  • Carman, P.C.: Flow of Gases Through Porous Media. Butterworths Scientific, London (1956)

    Google Scholar 

  • Chen, H., Chen, S., Matthaeus, W.H.: Recovery of the Navier–Stokes equations using a lattice Boltzmann method. Phys. Rev. A 45, R5339 (1992)

    Article  Google Scholar 

  • Chen, S., Martínez, D., Mei, R.: On boundary conditions in lattice Boltzmann methods. Phys. Fluids 8(9), 2527–2536 (1998)

    Article  Google Scholar 

  • Coelho, D., Thovert, J.F., Adler, P.M.: Geometrical and transport properties of random packings of spheres and aspherical particles. Phys. Rev. E 55(2), 1959–1978 (1997)

    Article  Google Scholar 

  • Cundall, P.A., Strack, O.D.L.: A discrete numerical mode for granular assemblies. Géotechnique 29(1), 47–65 (1979)

    Article  Google Scholar 

  • Feng, Y.T., Han, K., Owen, D.R.J.: Filling domains with disks: an advancing front approach. Int. J. Numer. Methods Eng. 56(5), 699–713 (2003)

    Article  Google Scholar 

  • Ghanem, R., Spanos, P.D.: The Stochastic Finite Element Method: A Spectral Approach. Springer, New York (1991)

    Book  Google Scholar 

  • Guo, Z., Zheng, C., Shi, B.: Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 65(4 Pt 2B), 046308 (2002)

    Article  Google Scholar 

  • Han, K., Feng, Y.T., Owen, D.R.J.: Sphere packing with a geometric based compression algorithm. Powder Technol. 155(1), 33–41 (2005)

    Article  Google Scholar 

  • Jia, X., Williams, R.A.: A packing algorithm for particles of arbitrary shapes. Powder Technol. 120(3), 175–186 (2001)

    Article  Google Scholar 

  • Jiang, Z., Chen, W., Burkhart, C.: Efficient 3D porous microstructure reconstruction via Gaussian random field and hybrid optimization. J. Microsc. 252(2), 135–148 (2013)

    Article  Google Scholar 

  • Jin, G., Patzek, T., Silin, D.: Physics-based reconstruction of sedimentary rocks. SPE Presented at SPE Western Regional/AAPG Pacific (2003)

  • Joshi, M.: A Class of Stochastic Models for Porous Materials. University of Kansas, Lawrence (1974)

    Google Scholar 

  • Jude, J.S., Sarkar, S., Sameen, A.: Reconstruction of Porous Media Using Karhunen–Loève Expansion. Springer India, India (2013)

    Book  Google Scholar 

  • Kansal, A.R., Torquato, S., Stillinger, F.H.: Computer generation of dense polydisperse sphere packings. J. Chem. Phys. 117(18), 8212–8218 (2002)

    Article  Google Scholar 

  • Lerman, A.: Geochemical Processes: Water and Sediment Environments, p. 200. Wiley, Hoboken (1979)

    Google Scholar 

  • Li, Y., Ji, W.: Stability and convergence analysis of a dynamics-based collective method for random sphere packing. J. Comput. Phys. 250(10), 373–387 (2013)

    Article  Google Scholar 

  • Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: comparisons with other stochastic methods. Water Resour. Res. 43(9), 6627–6632 (2007)

    Article  Google Scholar 

  • Li, H., Zhang, D.: Efficient and Accurate Quantification of Uncertainty for Multiphase Flow with Probabilistic Collocation Method. SPE J. 14(4), 665–679 (2009)

    Article  Google Scholar 

  • Li, Y., Leboeuf, E.J., Basu, P.K., Mahadevan, S.: Stochastic modeling of the permeability of randomly generated porous media. Adv. Water Resour. 28(8), 835–844 (2005)

    Article  Google Scholar 

  • Liang, Z.R., Fernandes, C.P., Magnani, F.S., Philippi, P.C.: A reconstruction technique for three-dimensional porous media using image analysis and Fourier transforms. J. Petrol. Sci. Eng. 21(3–4), 273–283 (1998)

    Article  Google Scholar 

  • Maier, R.S., Kroll, D.M., Kutsovsky, Y.E., Davis, H.T., Bernard, R.S.: Simulation of flow through bead packs using the lattice Boltzmann method. Phys. Fluids 10(1), 60–74 (1998)

    Article  Google Scholar 

  • Mcnamara, G.R., Zanetti, G.: Use of the Boltzmann equation to simulate lattice gas automata. Phys. Rev. Lett. 61(20), 2332 (1988)

    Article  Google Scholar 

  • Mei, R., Wei, S., Yu, D., Luo, L.S.: Lattice Boltzmann method for 3-D flows with curved boundary. J. Comput. Phys. 161(2), 680–699 (2000)

    Article  Google Scholar 

  • Nabovati, A., Llewellin, E.W., Sousa, A.C.M.: A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method. Compos. A 40(6), 860–869 (2009)

    Article  Google Scholar 

  • Noble, D.R., Chen, S., Georgiadis, J.G., Buckius, R.O.: A consistent hydrodynamic boundary condition for the lattice Boltzmann method. Phys. Fluids 7(1), 203–209 (1995)

    Article  Google Scholar 

  • Osnes, H., Sundnes, J.: Uncertainty analysis of ventricular mechanics using the probabilistic collocation method. IEEE Trans. Bio-med. Eng. 59(8), 2171 (2012)

    Article  Google Scholar 

  • Palma, P.R.D., Guyennon, N., Heße, F., Romano, E.: Porous media flux sensitivity to pore-scale geostatistics: a bottom-up approach. Adv. Water Resour. 102, 99–110 (2017)

    Article  Google Scholar 

  • Pan, C., Hilpert, M., Miller, C.T.: Pore-scale modeling of saturated permeabilities in random sphere packings. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 64(2), 066702 (2001)

    Article  Google Scholar 

  • Pan, C., Luo, L.S., Miller, C.T.: An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Comput. Fluids 35(8), 898–909 (2006)

    Article  Google Scholar 

  • Pilotti, M.: Generation of realistic porous media by grains sedimentation. Transp. Porous Media 33(3), 257–278 (1998)

    Article  Google Scholar 

  • Qian, Y.H., D’Humières, D., Lallemand, P.: Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17(6BIS), 479 (2007)

    Google Scholar 

  • Quiblier, J.A.: A new three-dimensional modeling technique for studying porous media. J. Colloid Interface Sci. 98(1), 84–102 (1984)

    Article  Google Scholar 

  • Ren, J., Guo, P., Peng, S., Yang, C.: Investigation on permeability of shale matrix using the lattice Boltzmann method. J. Nat. Gas Sci. Eng. 29, 169–175 (2016)

    Article  Google Scholar 

  • Santiso, E., Müller, E.A.: Dense packing of binary and polydisperse hard spheres. Mol. Phys. 100(15), 2461–2469 (2002)

    Article  Google Scholar 

  • Skordos, P.A.: Initial and boundary conditions for the lattice Boltzmann method. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 48(6), 4823 (1993)

    Google Scholar 

  • Spanne, P., Thovert, J.F., Jacquin, C.J., Lindquist, W.B., Jones, K.W., Adler, P.M.: Synchrotron computed microtomography of porous media: topology and transports. Phys. Rev. Lett. 73(14), 2001–2004 (1994)

    Article  Google Scholar 

  • Sun, A.Y., Zeidouni, M., Nicot, J.P., Lu, Z., Zhang, D.: Assessing leakage detectability at geologic CO2 sequestration sites using the probabilistic collocation method. Adv. Water Resour. 56(2), 49–60 (2013)

    Article  Google Scholar 

  • Sutou, A., Dai, Y.: Global optimization approach to unequal sphere packing problems in 3D. J. Optim. Theory Appl. 114(3), 671–694 (2002)

    Article  Google Scholar 

  • Tatang, M.A., Pan, W., Prinn, R.G., Mcrae, G.J.: An efficient method for parametric uncertainty analysis of numerical geophysical models. J. Geophys. Res. Atmos. 102(D18), 21925–21932 (1997)

    Article  Google Scholar 

  • Valera, R.R., Morales, I.P., Vanmaercke, S., Morfa, C.R., Cortés, L.A., Casañas, D.G.: Modified algorithm for generating high volume fraction sphere packings. Comput. Particle Mech. 2(2), 161–172 (2015)

    Article  Google Scholar 

  • Wang, K., Li, G., Jiang, X.: Applying probabilistic collocation method to power flow analysis in networks with wind farms. Paper presented at Power and Energy Society General Meeting (2013)

  • Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938)

    Article  Google Scholar 

  • Wu, J., Shu, C.: Particulate flow simulation via a boundary condition-enforced immersed boundary-lattice Boltzmann scheme. Commun. Comput. Phys. 7(4), 793 (2010)

    Google Scholar 

  • Wu, B., Zheng, Y., Tian, Y., Wu, X., Yao, Y., Han, F., Liu, J., Zheng, C.: Systematic assessment of the uncertainty in integrated surface water-groundwater modeling based on the probabilistic collocation method. Water Resour. Res. 50(7), 5848–5865 (2015)

    Article  Google Scholar 

  • Yang, A., Miller, C.T., Turcoliver, L.D.: Simulation of correlated and uncorrelated packing of random size spheres. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 53(2), 1516 (1996)

    Google Scholar 

  • Yin, X., Zhang, J.: An improved bounce-back scheme for complex boundary conditions in lattice Boltzmann method. J. Comput. Phys. 231(11), 4295–4303 (2012)

    Article  Google Scholar 

  • Zauner, T.: Application of a force field algorithm for creating strongly correlated multiscale sphere packings. J. Comput. Phys. 313, 662–673 (2016)

    Article  Google Scholar 

  • Zhao, Y.L., Wang, Z.M., Ye, J.P., Sun, H.S., Gu, J.Y.: Lattice Boltzmann simulation of gas flow and permeability prediction in coal fracture networks. J. Nat. Gas Sci. Eng. 53, 153–162 (2018)

    Article  Google Scholar 

  • Zou, Q., He, X.: On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids 9(6), 1591–1598 (1996)

    Article  Google Scholar 

Download references

Acknowledgements

This work is partially funded by the National Science and Technology Major Project of China (Grant No. 2017ZX05039-005).

Author information

Authors and Affiliations

  1. Energy and Resources Engineering, College of Engineering, Peking University, Beijing, 100871, China

    Lei Zhao & Heng Li

Authors
  1. Lei Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Heng Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, L., Li, H. Stochastic Modeling of the Permeability of Randomly Generated Porous Media via the Lattice Boltzmann Method and Probabilistic Collocation Method. Transp Porous Med 128, 613–631 (2019). https://doi.org/10.1007/s11242-019-01261-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-019-01261-7

Keywords

Search

Navigation