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Analysis of Variance of Porosity and Heterogeneity of Permeability at the Pore Scale

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Abstract

We present a new statistical variance approach for characterizing heterogeneities related to pore spaces in reservoir rocks. Laboratory-based computer microtomography data for reservoir sandstone samples were acquired and processed using advanced image segmentation techniques. The samples were processed using a method based on the digital rock physics concept using the high-performance Navier–Stokes flow solver in the GeoDict commercial software package. The digitized structures were subjected to computational fluid dynamic simulations. The effects of structural matrix modifications caused by the precipitation of minerals on the porosity–permeability relationship and the characterization of the representative elementary volume were assessed. The variances of the digital flow fields were compared at the pore scale (6 µm). The algorithm for analysing variance was benchmarked using a synthetic dataset that provided artificial repetitive structural patterns at both low and high resolutions. This gave an estimate of the sensitivity of the proposed algorithm to minor inhomogeneities. Representative elementary volume variance analysis was performed by comparing the correlation coefficients for various pore–grain composition patterns with the variances of simulated mean flow velocities. Probability density functions indicate that the flow velocities and pore space geometries differed greatly for different samples. The normalized probability density functions of the mean flows shifted to higher velocities as the resolution decreased. We found that a representative elementary volume analysis was more reliably achieved by analysing the mean flow velocity variance than by analysing the pore microstructure alone.

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References

  1. Andrä, H., et al.: Digital rock physics benchmarks—part II: computing effective properties. Comput. Geosci. 50, 33–43 (2013)

  2. Amer Water Works Assn: Recommended Practice for Backflow: introduction to Chemistry and Control (Manual of Water Supply Practices), 3rd edn, p. 30. American Water Works Association, Denver (2004). ISBN 1583212884

  3. Armstrong, M.: Basic Linear Geostatistics. Springer, Berlin (1998). ISBN 978-3-642-58727-6

  4. Bachmat, Y., Bear, J.: Macroscopic modelling of transport phenomena in porous media. 1: the continuum approach. Transp. Porous Media 1, 213–240 (1986)

  5. Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1972)

  6. Biswal, B., et al.: Stochastic multiscale model for carbonate rocks. Phys. Rev. E 75, 061303 (2007)

  7. Biswal, B., et al.: Modeling of multiscale porous media. Image Anal. Stereol. 28, 23–34 (2009a)

  8. Biswal, B., et al.: Towards precise prediction of transport properties from synthetic computer tomography of reconstructed porous media. Phys. Rev. E 80, 041301 (2009b)

  9. Buades, A., et al.: Non-local means denoising. Image Process. Line 1, 208–212 (2011)

  10. Butz, T.: Fouriertransformation für Fußgänger. Teubner, pp. 72–73, (1998). ISBN 978-3-322-94867-0

  11. Chagneau, A., et al.: Mineral precipitation-induced porosity reduction and its effect on transport parameters in diffusion-controlled porous media. Geochem. Trans. 16, 13 (2015)

  12. Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn. Wiley, New York (2008). ISBN 978-81-265-1805-0

  13. Guan, K.M., et al.: Effects of image resolution on sandstone porosity and permeability as obtained from X-ray microscopy. Transp. Porous Media 128, 233–245 (2018)

  14. Guibert, R., et al.: Computational permeability determination from pore-scale imaging: sample size, mesh and method sensitivities. Transp. Porous Media 107, 641–656 (2015)

  15. Hilfer, R., Lemmer, A.: Differential porosimetry and permeametry for random porous media. Phys. Rev. E 92, 013305 (2015)

  16. Hilfer, R., Zauner, T.: High-precision synthetic computed tomography of reconstructed porous media. Phys. Rev. E 84, 062301 (2011)

  17. Krige, D.G.: A statistical approach to some mine valuations and allied problems at the Witwatersrand. J. Chem. Metall. Min. Soc. S. Afr. 52, 119–139 (1951)

  18. Hinz, C., et al.: Pore scale modelling of calcite cement dissolution in a reservoir sandstone matrix. In: E3S Web of Conferences, vol. 98, p. 05010 (2019)

  19. Landau, L.D., Lifshitz, E.M.: Fluid Mechanics—Course of Theoretical Physics. Pergamon Press, Oxford (1966)

  20. Latief, F.D.E., Biswal, B., Fauzi, U., Hilfer, R.: Continuum reconstruction of the pore scale microstructure for Fontainebleau sandstone. Physica A 389, 1607–1618 (2009)

  21. Linden, S.: The LIR space partitioning system applied to Stokes equations. PhD Thesis, TU Kaiserslautern (2014)

  22. Matheron, G.: Principles of geostatistics. Econ. Geol. 58, 1246–1266 (1963)

  23. Menke, H.P., Bijeljic, B., Blunt, M.J.: Reservoir condition imaging of reactive transport in heterogeneous carbonates using fast synchrotron tomography—effect of initial pore structure and flow conditions. Chem. Geol. 428, 15–26 (2016)

  24. Pyrcz, M.J., Deutsch, C.: The whole story on the hole effect. Centre for computational Geostatistics Department of Civil and Environmental Engineering, University of Alberta, Edmonton, Alberta (2003)

  25. Rossi, M.E., Deutsch, C.V.: Mineral Resource Estimation. Springer (2013). ISBN 978-14020-5716-8

  26. Saxena, N., et al.: References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks. Adv Water Resour 109, 211–235 (2017)

  27. Saxena, N., et al.: Imaging and computational considerations for image computed permeability: operating envelope of digital rock physics. Adv. Water Resour. 116, 127–144 (2018)

  28. Shevlyakov, G.L., Oja, H.: Robust Correlation: Theory and Applications. Wiley, pp.10–23, (2016). ISBN 978-1-118-49345-8

  29. von Hagen, W.: The Definitive Guide to GCC, 2nd edn. Apress, New York (2006). ISBN 978-1-590-59585-5

  30. Webster, R., Oliver, M.A.: Geostatistics for Environmental Scientists, 2nd edn. Wiley, New York (2007). ISBN 978-0-470-51726-0

  31. Weinberg, S.L., Abramowitz, S.K.: Statistics using SPSS: An interactive approach, second edition. Cambridge University Press, p. 397 (2008). ISBN 978-0-521-89922-2

  32. Wolf, M.H.: Visualisierung und Quantifizierung der Fluiddynamik in Bohrkernen aus dem Salinar und Deckgebirge des Raumes Staßfurt mittels Positronen-Emissions-Tomographie. PhD thesis, University Leipzig (2011). http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-77160 (in German)

  33. Yun, H.-S., et al.: Determination of representative elementary volume of fault core materials by particle distribution analysis. Geosci. J. 22, 104–119 (2018)

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Acknowledgements

This work was supported by the German Federal Ministry of Education and Research (BMBF) ‘Geological Research for Sustainability (GEO:N)’ program, which is part of the BMBF ‘Research for Sustainable Development (FONA3)’ framework program. It is part of the project ResKin (Reaction kinetics in reservoir rocks, 03G0871E). We would like to thank our team at the JGU Mainz for considerable support during the difficult process of writing and debugging the automatized C++ code and while writing this manuscript.

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Correspondence to Arne Jacob.

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Jacob, A., Enzmann, F., Hinz, C. et al. Analysis of Variance of Porosity and Heterogeneity of Permeability at the Pore Scale. Transp Porous Med 130, 867–887 (2019). https://doi.org/10.1007/s11242-019-01342-7

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Keywords

  • Pore-scale variance analysis
  • REV estimation
  • Porous media
  • Digital rock physics