Advertisement

Computational Geosciences

, Volume 23, Issue 5, pp 881–893 | Cite as

Image-based simulations of absolute permeability with massively parallel pseudo-compressible stabilised finite element solver

  • Liang Yang
  • Jianhui YangEmail author
  • Edo Boek
  • Mikio Sakai
  • Christopher Pain
Original Paper

Abstract

We apply an accurate parallel stabilised finite element method to solve for Navier-Stokes equations directly on a binarised three-dimensional rock image, obtained by micro-CT imaging. The proposed algorithm has several advantages. First, the linear equal-order finite element space for velocity and pressure is ideal for presenting the pixel images. Second, the algorithm is fully explicit and versatile for describing complex boundary conditions. Third, the fully explicit matrix–free finite element implementation is ideal for parallelism on high-performance computers, similar to lattice Boltzmann. In the last, the memory usage is low compared with lattice Boltzmann or implicit finite volume. We compute the permeability of a range of rock images. The stabilisation parameter may affect the velocity, and an optimal parameter is chosen from the numerical tests. The steady state results are comparable with lattice Boltzmann method and implicit finite volume. The transient behaviour of pseudo-compressible stabilised finite element and lattice Boltzmann method is very similar. Our analysis shows that the stabilised finite element is an accurate and efficient method with low memory cost for the image- based simulations of flow in the pore scale up to 1 billion voxels on 128-GB ram workstation and on distributed clusters.

Keywords

Stabilised finite element Pore scale Micro-CT image permeability Comparative study Parallel computing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

Liang Yang received financial support from EPSRC grant EP/P013198/1 and Imperial College Research Computing Service [12].

References

  1. 1.
    Imperial College London: Pore Scale Modelling group: Micro-CT images and networks. http://www.imperial.ac.uk/earth-science/research/research-groups/perm/research/pore-scale-modelling/micro-ct-images-and-networks/ (2017)
  2. 2.
    Akanji, L.T., Matthai, S.K.: Finite element-based characterization of pore-scale geometry and its impact on fluid flow. Transp. Porous Media 81(2), 241–259 (2010)CrossRefGoogle Scholar
  3. 3.
    Arns, C.H., Bauget, F., Limaye, A., Sakellariou, A., Senden, T., Sheppard, A., Sok, R.M., Pinczewski, V., Bakke, S., Berge, L.I., et al.: Pore scale characterization of carbonates using X-ray microtomography. SPE J. 10(04), 475–484 (2005)CrossRefGoogle Scholar
  4. 4.
    Badia, S., Codina, R.: Algebraic pressure segregation methods for the incompressible navier-Stokes equations. Arch. Comput. Meth. Eng. 15(3), 1–52 (2007)CrossRefGoogle Scholar
  5. 5.
    Blunt, M.J.: Multiphase Flow in Permeable Media: A Pore-scale Perspective. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
  6. 6.
    Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Hoboken (2003)CrossRefGoogle Scholar
  7. 7.
    Dong, H., Blunt, M.J.: Pore-network extraction from micro-computerized-tomography images. Phys. Rev. E 80(3), 036307 (2009)CrossRefGoogle Scholar
  8. 8.
    Ergun, S.: Fluid flow through packed columns. Chem. Eng. Prog. 48, 89–94 (1952)Google Scholar
  9. 9.
    Ferrari, A., Lunati, I.: Direct numerical simulations of interface dynamics to link capillary pressure and total surface energy. Adv. Water Resour. 57, 19–31 (2013)CrossRefGoogle Scholar
  10. 10.
    Ghassemi, A., Pak, A.: Pore scale study of permeability and tortuosity for flow through particulate media using Lattice Boltzmann method. Int. J. Numer. Anal. Methods Geomech. 35(8), 886–901 (2011)CrossRefGoogle Scholar
  11. 11.
    Ginzburg, I.: Variably saturated flow described with the anisotropic lattice Boltzmann methods. Comput. Fluids 35(8-9), 831–848 (2006)CrossRefGoogle Scholar
  12. 12.
    Harvey, M.: Imperial College Research Computing Service.  https://doi.org/10.14469/HPC/2232. https://data.hpc.imperial.ac.uk/resolve/?doi=2232 (2017)
  13. 13.
    He, X., Doolen, G.D., Clark, T.: Comparison of the lattice Boltzmann method and the artificial compressibility method for navier–Stokes equations. J. Comput. Phys. 179(2), 439–451 (2002)CrossRefGoogle Scholar
  14. 14.
    Hughes, T.J., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics: V. Circumventing the babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)CrossRefGoogle Scholar
  15. 15.
    Jettestuen, E., Helland, J.O., Prodanović, M.: A level set method for simulating capillary-controlled displacements at the pore scale with nonzero contact angles. Water Resour. Res. 49(8), 4645–4661 (2013)CrossRefGoogle Scholar
  16. 16.
    Keehm, Y., Mukerji, T., Nur, A.: Permeability prediction from thin sections: 3D reconstruction and lattice-Boltzmann flow simulation. Geophysical Research Letters 31(4) (2004)Google Scholar
  17. 17.
    Lallemand, P., Luo, L.S.: Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, galilean invariance, and stability. Phys. Rev. E. 61(6), 6546 (2000)CrossRefGoogle Scholar
  18. 18.
    Macdonald, I., El-Sayed, M., Mow, K., Dullien, F.: Flow through porous media-the ergun equation revisited. Ind. Eng. Chem. Fundam. 18(3), 199–208 (1979)CrossRefGoogle Scholar
  19. 19.
    Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-CT images. Math. Geosci. 45(1), 103–125 (2013)CrossRefGoogle Scholar
  20. 20.
    Nithiarasu, P.: An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows. Int. J. Numer. Methods Eng. 56(13), 1815–1845 (2003)CrossRefGoogle Scholar
  21. 21.
    Phelan, F.R.Jr, Wise, G.: Analysis of transverse flow in aligned fibrous porous media. Compos. A: Appl. Sci. Manuf. 27(1), 25–34 (1996)CrossRefGoogle Scholar
  22. 22.
    Raeini, A.Q., Bijeljic, B., Blunt, M.J.: Generalized network modeling: Network extraction as a coarse-scale discretization of the void space of porous media. Phys. Rev. E 96(1), 013312 (2017)CrossRefGoogle Scholar
  23. 23.
    Raeini, A.Q., Blunt, M.J., Bijeljic, B.: Modelling two-phase flow in porous media at the pore scale using the volume-of-fluid method. J. Comput. Phys. 231(17), 5653–5668 (2012)CrossRefGoogle Scholar
  24. 24.
    Tamamidis, P., Zhang, G., Assanis, D.N.: Comparison of pressure-based and artificial compressibility methods for solving 3D steady incompressible viscous flows. J. Comput. Phys. 124(1), 1–13 (1996)CrossRefGoogle Scholar
  25. 25.
    Thauvin, F., Mohanty, K.: Network modeling of non-darcy flow through porous media. Transp. Porous Media 31(1), 19–37 (1998)CrossRefGoogle Scholar
  26. 26.
    Yang, J., Crawshaw, J., Boek, E.S.: Quantitative determination of molecular propagator distributions for solute transport in homogeneous and heterogeneous porous media using lattice Boltzmann simulations. Water Resour. Res. 49(12), 8531–8538 (2013)CrossRefGoogle Scholar
  27. 27.
    Yang, L., Badia, S., Codina, R.: A pseudo-compressible variational multiscale solver for turbulent incompressible flows. Comput. Mech. 58(6), 1051–1069 (2016)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Water, Energy and Environment (SWEE)Cranfield UniversityBedfordUK
  2. 2.Department of Earth Science and EngineeringImperial College LondonSouth KensingtonUK
  3. 3.Department of Chemical EngineeringImperial College LondonSouth KensingtonUK
  4. 4.Geoscience Research CentreTOTAL E & P UK LimitedWesthillUK
  5. 5.Division of Chemical Engineering & Renewable Energy, School of Engineering and Materials ScienceQueen Mary University of LondonBethnal GreenUK
  6. 6.Resilience Engineering Research Centre, School of EngineeringThe University of TokyoBunkyoJapan

Personalised recommendations