Skip to main content

Computations of Absolute Permeability on Micro-CT Images

Mathematical Geosciences volume 45, pages 103–125 (2013)Cite this article

Abstract

We apply an accurate numerical scheme to solve for Stokes flow directly on binarized three-dimensional rock images, such as those obtained by micro-CT imaging. The method imposes no-flow conditions exactly at the solid boundaries and employs an algebraic multigrid method to solve for the resultant set of linear equations. We compute the permeability of a range of consolidated and unconsolidated porous rocks; the results are comparable with those obtained using the lattice Boltzmann method and agree with experimental measurements on larger core samples. We show that the Kozeny–Carman equation can over-estimate permeability by a factor of 10 or more, particularly for the more heterogeneous systems studied. We study the existence and size of the representative elementary volume (REV) at lamina scale. We demonstrate that the REV for permeability is larger than for static properties—porosity and specific surface area—since it needs to account for the tortuosity and connectedness of the flow paths. For the carbonate samples, the REV appeared to be larger than the image size. We also study the anisotropy of permeability at the pore scale. We show that the permeability of sandpacks varies by less than 10 % in different directions. For sandstones, permeability changes by 25 % on average. However, the anisotropy of permeability in carbonates can be up to 50 %, indicating the existence of connected pores in one direction which are not connected in another.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

References

  • Adler PM, Jacquin CG, Quiblier JA (1990) Flow in simulated porous media. Int J Multiph Flow 16:691–712

    Article  Google Scholar 

  • Adler PM (1992) Porous media: geometry transports. Butterworth-Heinemann, London

    Google Scholar 

  • Al-Raoush R, Papadopoulos A (2010) Representative elementary volume analysis of porous media using X-ray computed tomography. Powder Technol 200:69–77

    Article  Google Scholar 

  • Bear J (1988) Dynamics of fluids in porous media. Elsevier, New York

    Google Scholar 

  • Bijeljic B, Mostaghimi P, Blunt MJ (2011) The signature of non-Fickian transport in highly heterogeneous media. Phys Rev Lett 107:204502

    Article  Google Scholar 

  • Blunt MJ (2001) Flow in porous media—pore-network models multiphase flow. Curr Opin Colloid Interface Sci 6:197–207

    Article  Google Scholar 

  • Blunt MJ, Bijeljic B, Dong H, Gharbi O, Iglauer S, Mostaghimi P, Paluszny A, Pentland C (2012) Pore-scale imaging and modelling. Adv Water Resour. doi:10.1016/j.advwatres.2012.03.003

    Google Scholar 

  • Boisvert JB, Manchuk JG, Neufeld C, Niven EB, Deutsch CV (2012) Micro-modeling for enhanced small scale porosity-permeability relationships. In: Abrahamsen P, Hauge R, Kolbjornsen O (eds) Geostatistics, Oslo, 2012. Springer, Dordrecht

    Google Scholar 

  • Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems 1. J Comput Phys 2:12–26

    Article  Google Scholar 

  • Chung TJ (2002) Computational fluid dynamics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Clavaud JB, Maineult A, Zamora M, Rasolofosaon P, Schlitter C (2008) Permeability anisotropy and its relations with porous medium structure. J Geophys Res 113:B01202

    Article  Google Scholar 

  • Coenen J, Tchouparova E, Jing X (2004) Measurement parameters resolution aspects of micro X-ray tomography for advanced core analysis. In: Proceedings of the international symposium of the society of core analysts, UAE. (SCA 2004-36)

  • Deutsch CV (2010) Estimation of vertical permeability in the McMurray formation. J Can Pet Technol 49(12):10–18

    Google Scholar 

  • d’Humières D, Ginzburg I (2009) Viscosity independent numerical errors for Lattice Boltzmann models: from recurrence equations to magic collision numbers. Comput Math Appl 58:823–840

    Article  Google Scholar 

  • Dong H, Blunt MJ (2009) Pore-network extraction from micro-computerized-tomography images. Phys Rev E 80:36307–36318

    Article  Google Scholar 

  • Flannery BP, Deckman HW, Roberge WG, D’Amico KL (1987) Three-dimensional X-ray microtomography. Science 237:1439–1444

    Article  Google Scholar 

  • Jin G, Patzek TW, Silin DB (2004) Direct prediction of the absolute permeability of unconsolidated consolidated reservoir rock. In: Proceedings of the SPE annual technical conference exhibition, Houston, TX. (SPE 90084)

  • Kainourgiakis ME, Kikkinides ES, Galani A, Charalambopoulou GC, Stubos AK (2005) Digitally reconstructed porous media: transport sorption properties. Transp Porous Media 58:43–62

    Article  Google Scholar 

  • Kang Q, Lichtner PC, Zhang D (2006) Lattice Boltzmann pore-scale model for multicomponent reactive transport in porous media. J Geophys Res 111:B05203

    Article  Google Scholar 

  • King MJ (1995) Application and analysis of a new method for calculating tensor permeability. In: De Haan HJ (ed) New developments in improved oil recovery. Geol Soc Spec Publ, vol 84. Geol Soc, London, pp 19–27

    Google Scholar 

  • Knackstedt MA, Arns CH, Limaye A, Sakellariou A, Senden TJ, Sheppard AP, Sock RM, Pinczewski WV, Bunn GF (2004) Digital core laboratory: properties of reservoir core derived from 3D images. In: Proceedings of the SPE Asia Pacific conference on integrated modelling for asset management, Kuala Lumpur, Malaysia. (SPE 87009)

  • Lemaitre R, Adler PM (1990) Fractal porous media IV: three-dimensional Stokes flow through random media and regular fractals. Transp Porous Media 5:325–340

    Article  Google Scholar 

  • Manwart C, Aaltosalmi U, Koponen A, Hilfer R, Timonen J (2002) Lattice-Boltzmann finite-difference simulations for the permeability for three-dimensional porous media. Phys Rev E 66:16702–16713

    Article  Google Scholar 

  • Mostaghimi P, Bijeljic B, Blunt MJ (2012) Simulation of flow and dispersion on pore-space images. SPE J. doi:10.2118/135261-PA

    Google Scholar 

  • Mourzenko V, Thovert JF, Vizika O, Adler PM (2008) Geometrical transport properties of random packings of polydisperse spheres. Phys Rev E 77:66306–66320

    Article  Google Scholar 

  • Noble DR, Chen S, Georgiadis JG, Buckius RO (1995) A consistent hydrodynamic boundary condition for the lattice Boltzmann method. Phys Fluids 7:203–210

    Article  Google Scholar 

  • Norris RJ, Lewis JJM (1991) The geological modeling of effective permeability in complex heterolithic facies. In: Proceedings of the SPE annual technical conference exhibition, Dallas, TX. (SPE 22692)

  • Okabe H, Blunt MJ (2004) Prediction of permeability for porous media reconstructed using multiple-point statistics. Phys Rev E 70:66135

    Article  Google Scholar 

  • Okabe H, Oseto K (2006) Pore-scale heterogeneity assessed by the lattice-Boltzmann method. In: Proceedings of the international symposium of the society of core analysts, Trondheim, Norway. (SCA 2006-44)

  • Øren PE, Bakke S (2002) Process based reconstruction of sandstones prediction of transport properties. Transp Porous Media 46:311–343

    Article  Google Scholar 

  • Øren PE, Bakke S, Arntzen OJ (1998) Extending predictive capabilities to network models. SPE J 3:324–336

    Google Scholar 

  • Øren PE, Antonsen F, Rueslatten HG, Bakke S (2002) Numerical simulations of NMR responses for improved interpretations of NMR measurements in reservoir rocks. In: Proceedings of the SPE annual technical conference exhibition, San Antonio, TX. (SPE 77398)

  • Ovaysi S, Piri M (2010) Direct pore-level modeling of incompressible fluid flow in porous media. J Comput Phys 229:56–76

    Article  Google Scholar 

  • Pan C, Hilpert M, Miller CT (2001) Pore-scale modeling of saturated permeabilities in random sphere packings. Phys Rev E 64:66702–66711

    Article  Google Scholar 

  • Pan C, Hilpert M, Miller CT (2004) Lattice-Boltzmann simulation of two-phase flow in porous media. Water Resour Res 40:W01501

    Article  Google Scholar 

  • Patankar SV (1980) Numerical heat transfer fluid flow. Hemisphere Publication, Washington

    Google Scholar 

  • Philipse AP, Pathmamanoharan C (1993) Liquid permeation (sedimentation) of dense colloidal hard-sphere packings. J Colloid Interface Sci 159:96–107

    Article  Google Scholar 

  • Piller M, Schena G, Nolich M, Favretto S, Radaelli F, Rossi E (2009) Analysis of hydraulic permeability in porous media: from high resolution X-ray tomography to direct numerical simulation. Transp Porous Media 80:57–78

    Article  Google Scholar 

  • Roberts AP (1997) Statistical reconstruction of three-dimensional porous media from two-dimensional images. Phys Rev E 56:3203–3212

    Article  Google Scholar 

  • Shabro V, Torres-Verdin C, Javadpour F, Sepehrnoori K (2012) Finite-difference approximation for fluid-flow simulation and calculation of permeability in porous media. Transp Porous Media 94:775–793

    Article  Google Scholar 

  • Silin DB, Patzek TW (2009) Predicting relative-permeability curves directly from rock images. In: Proceedings of the SPE annual technical conference exhibition, New Orleans, LA. (SPE 124974)

  • Stüben K (2001) A review of algebraic multigrid. J Comput Appl Math 128:281–309

    Article  Google Scholar 

  • Talabi O, Blunt MJ (2010) Pore-scale network simulation of NMR response in two-phase flow. J Pet Sci Eng 72:1–9

    Article  Google Scholar 

  • Tamamidis P, Zhang G, Assanis DN (1996) Comparison of pressure-based artificial compressibility methods for solving 3D steady incompressible viscous flows. J Comput Phys 124:1–13

    Article  Google Scholar 

  • Thies-Weesie DME, Philipse AP (1994) Liquid permeation of bidisperse colloidal hard-sphere packings the Kozeny-Carman scaling relation. J Colloid Interface Sci 162:470–480

    Article  Google Scholar 

  • Thovert JF, Salles J, Adler PM (1993) Computerized characterization of the geometry of real porous media: their discretization, analysis interpretation. J Microsc 170:65–79

    Article  Google Scholar 

  • van Genabeek O, Rothman DH (1996) Macroscopic manifestations of microscopic flows through porous media: phenomenology from simulation. Annu Rev Earth Planet Sci 24:63–87

    Article  Google Scholar 

  • White FM (1999) Fluid mechanics. McGraw-Hill, New York

    Google Scholar 

  • Wright HMN, Roberts JJ, Cashman KV (2006) Permeability of anisotropic tube pumice: model calculations and measurements. Geophys Res Lett 33:L17316

    Article  Google Scholar 

  • Zhang D, Zhang R, Chen S, Soll WE (2000) Pore scale study of flow in porous media: scale dependency, REV, statistical REV. Geophys Res Lett 27:1195–1198

    Article  Google Scholar 

  • Zinszner B, Meynot C (1982) Visualisation des propriétés capillaires des roches réservoirs. Rev Inst Fr Pét 37:337–361

    Google Scholar 

Download references

Acknowledgements

We would like to thank the sponsors of the Imperial College Consortium on Pore-scale Modelling for funding this work.

Author information

Authors and Affiliations

  1. Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ, UK

    Peyman Mostaghimi, Martin J. Blunt & Branko Bijeljic

Authors
  1. Peyman Mostaghimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin J. Blunt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Branko Bijeljic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peyman Mostaghimi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mostaghimi, P., Blunt, M.J. & Bijeljic, B. Computations of Absolute Permeability on Micro-CT Images. Math Geosci 45, 103–125 (2013). https://doi.org/10.1007/s11004-012-9431-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11004-012-9431-4

Keywords

  • Permeability
  • Micro-CT imaging
  • Direct simulation
  • REV
  • Anisotropy