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International Summer School-Conference “Advanced Problems in Mechanics”

APM 2020: Advanced Problem in Mechanics II pp 533–539Cite as

Representative Elementary Volume via Averaged Scalar Minkowski Functionals

Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

Representative Elementary Volume (REV) at which the material properties do not vary with change in volume is an important quantity for making measurements or simulations which represent the whole. We discuss the geometrical method to evaluation of REV based on the quantities coming in the Steiner formula from convex geometry. For bodies in three-dimensional space this formula gives us four scalar functionals known as scalar Minkowski functionals. We demonstrate on certain samples that the values of such averaged functionals almost stabilize for cells for which the length of edges are greater than certain threshold value R. Therefore, from this point of view, it is reasonable to consider cubes of volume \(R^3\) as representative elementary volumes for certain physical parameters of porous medium.

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References

  1. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963). https://doi.org/10.1016/0022-5096(63)90036-x

    CrossRef  MATH  Google Scholar 

  2. Drugan, W.J., Willis, J.R.: A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J. Mech. Phys. Solids 44(4), 497–524 (1996). https://doi.org/10.1016/0022-5096(96)00007-5

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Zhang, D., Zhang, R., Chen, S., Soll, W.E.: Pore scale study of flow in porous media: scale dependency, REV, and statistical REV. Geophys. Res. Lett. 27(8), 1195–1198 (2000). https://doi.org/10.1029/1999GL011101

    CrossRef  Google Scholar 

  4. Mostaghimi, P., Blunt, M.J., Bijeljic, B.: Computations of absolute permeability on micro-CT images. Math. Geosci. 45(1), 103–125 (2013). https://doi.org/10.1007/s11004-012-9431-4

    CrossRef  MathSciNet  Google Scholar 

  5. Du, X., Ostoja-Starzewski, M.: On the size of representative volume element for Darcy law in random media. Proc. R. Soc. Lond. A 462, 2949–2963 (2006). https://doi.org/10.1098/rspa.2006.1704

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Sun, W.C., Andrade, J.E., Rudnicki, J.W.: Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int. J. Numer. Meth. Eng. 88, 1260–1279 (2011). https://doi.org/10.1002/nme.3220

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Bear, J.: Dynamics of fluids in porous media. Courier Corporation Chelmsford 120, 162–163 (2013). https://doi.org/10.1097/00010694-197508000-00022

    CrossRef  Google Scholar 

  8. Scholz, C., et al.: Direct relations between morphology and transport in Boolean models. Phys. Rev. E 92, 043023 (2015). https://doi.org/10.1103/PhysRevE.92.043023

    CrossRef  Google Scholar 

  9. Armstrong, R.T., et al.: Porous media characterization using Minkowski functionals: theories, applications and future directions. Transp. Porous Media 130, 305–335 (2018). https://doi.org/10.1007/s11242-01-1201-4

    CrossRef  MathSciNet  Google Scholar 

  10. Slotte, P.A., Berg, C.F., Khanamiri, H.H.: Predicting resistivity and permeability of porous media using Minkowski functionals. Transp. Porous Media 131(2), 705–722 (2019). https://doi.org/10.1007/s11242-019-01363-2

    CrossRef  MathSciNet  Google Scholar 

  11. Novikov, S.P., Taimanov, I.A.: Modern Geometrical Structures and Fields. American Mathematical Society, Providence (2006). https://doi.org/10.1090/gsm/071

  12. Schröder-Turk, G.E., et al.: Minkowski tensors of anisotropic spatial curvature. New J. Phys. 15, 083028 (2013). https://doi.org/10.1088/1367-2630/15/8/083028

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Gilmanov, R.R., Kalyuzhnyuk, A.V., Taimanov, I.A., Yakovlev, A.A.: Topological characteristics of digital oil reservoir models at different scales. In: Proceedings of the 20th Annual Conference of the International Association for Mathematical Geosciences, IAMG 2019, State College, Pennsylvania, USA, 10–16 August 2019, pp. 94–99 (2019)

    Google Scholar 

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Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems of SB RAS, 630090, Novosibirsk, Russia

    M. V. Andreeva & N. E. Russkikh

  2. Peter the Great St. Petersburg Polytechnic University, 195251, St. Petersburg, Russia

    A. V. Kalyuzhnyuk

  3. Gazpromneft Science and Technology Center, 190000, St. Petersburg, Russia

    V. V. Krutko

  4. Sobolev Institute of Mathematics of SB RAS, and Novosibirsk State University, 630090, Novosibirsk, Russia

    I. A. Taimanov

Authors
  1. M. V. Andreeva
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  2. A. V. Kalyuzhnyuk
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  3. V. V. Krutko
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  4. N. E. Russkikh
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  5. I. A. Taimanov
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Corresponding author

Correspondence to A. V. Kalyuzhnyuk .

Editor information

Editors and Affiliations

  1. IPME RAS, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    D. A. Indeitsev

  2. IPME RAS, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

    A. M. Krivtsov

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Andreeva, M.V., Kalyuzhnyuk, A.V., Krutko, V.V., Russkikh, N.E., Taimanov, I.A. (2022). Representative Elementary Volume via Averaged Scalar Minkowski Functionals. In: Indeitsev, D.A., Krivtsov, A.M. (eds) Advanced Problem in Mechanics II. APM 2020. Lecture Notes in Mechanical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-92144-6_40

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  • DOI: https://doi.org/10.1007/978-3-030-92144-6_40

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-92143-9

  • Online ISBN: 978-3-030-92144-6

  • eBook Packages: EngineeringEngineering (R0)

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