Large-Scale Simulation of Acoustic Waves in Random Multiscale Media

  • Olga N. Soboleva
  • Ekaterina P. Kurochkina
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


The effective coefficients in the problem of the acoustic wave propagation have been calculated for a multiscale 3D medium by using a subgrid modeling approach. The density and the elastic stiffness have been represented by the Kolmogorov multiplicative cascades with a log-normal probability distribution. The wavelength is assumed to be large as compared with the scale of heterogeneities of the medium. We consider the regime in which the waves propagate over a distance of the typical wavelength in source. If a medium is assumed to satisfy the improved Kolmogorov similarity hypothesis, the term for the effective coefficient of the elastic stiffness coincides with the Landau-Lifshitz-Matheron formula. The theoretical results are compared with the results of a direct 3D numerical simulation.


Propagation of acoustic waves Subgrid modeling Multiplicative cascades 



The work was supported by the RFBR N15-01-01458.


  1. 1.
    Imomnazarov, Kh., Mikhailov A.A.: Application of a spectral method for numerical modeling of propagation of seismic waves in porous media for dissipative case. Sib. Zh. Vychisl. Mat. 17, 139–147 (2014)Google Scholar
  2. 2.
    Capdeville, Y., Guillot, L., Marigo, J.J.: Second order homogenization of the elastic wave equation for non-periodic layered media. Geophys. J. Int. 170, 823–838 (2007)CrossRefGoogle Scholar
  3. 3.
    Shelukhin, V., Igor, Yeltsov I., Paranichev, I.: The electrokinetic cross-coupling coefficient: two-scale homogenization approach. World J. Mech. 1, 127–136 (2011)CrossRefGoogle Scholar
  4. 4.
    Fouque, J.-P., Garnnier, J., Papanicolaou, G., Solna, K.: Wave Propagation and Time Reversal in Randomly Reversal in Randomly Layered Media. Stochastic Modelling and Applied Probability, vol. 56. Springer, Berlin (2007)Google Scholar
  5. 5.
    Sahimi, M.: Flow phenomena in rocks: from continuum models, to fractals, percolation, cellular automata, and simulated annealing. Rev Mod. Phys. 65, 1393–1534 (1993)CrossRefGoogle Scholar
  6. 6.
    Koohi lai, Z., Vasheghani, F.S., Jafari, G.R.: Non-Gaussianity effect of petrophysical quantities by using q-entropy and multi fractal random walk. Phys. A. 392 3039–3044 (2013)Google Scholar
  7. 7.
    Kolmogorov, A.N.: A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82–85 (1962)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kuz’min, G.A., Soboleva, O.N.: Subgrid modeling of filtration in porous self-similar media. App. Mech. Tech. Phys. 43, 583–592 (2002)Google Scholar
  9. 9.
    Molchan, G.M.: Turbulent cascades: limitations and a statistical test of the lognormal hypothesis. Phys. Fluids 9(8), 2387–2396 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Molchan, G.M.: Scaling exponents and multifractal dimensions for independed random cascades. Commun. Math. Phys. 179, 681–702 (1996)CrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Novosibisk State Technical UniversityNovosibirskRussia
  2. 2.The Novosibisk State University - Baker Hughes Joint Laboratory of The Multi-Scale Geophysics and MechanicsNovosibirsk 90Russia

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