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Article information
2014 , Volume 19, ¹ 6, p.54-64
Kurochkina E.P., Soboleva O.N.
Effective coefficients of a multiscale isotropic medium applied for the problem of propagation of acoustic waves
Wave propagation in heterogeneous media is a fundamental phenomenon of great scientific and practical interest. It is relevant to such important problems as detecting underground nuclear explosions, understanding the scale structure of oil, gas, and geothermal reservoirs. The spatial geometry of small-scale heterogeneities of medium is not exactly known. It has been shown that the irregularity of elastic parameters, density, permeability, porosity, increases as the scale of measurements decreases. It is customary to assume these parameters to be random fields characterized by the joint probability distribution functions. However, it is difficult to measure higher-order statistical moments for the geophysical parameters. At best, only the mean values and correlation functions of the second order are known. Hence, effective solutions cannot be constructed using the conventional perturbation theory. Geophysical parameters, for example, porosity, density, elastic modules can be well approximated by fractals and multiplicative hierarchical cascade models with lognormal distribution. As the first step towards the goal of finding effective coefficients in the problem of propagation of elastic waves in strongly heterogeneous solids, in this paper we study the propagation of acoustic waves in the same type of media in which local elastic parameters have essentially all variations of scales from a certain interval at each spatial point. The density of a medium and the elastic stiffness are approximated by a multiplicative continuous cascade. We use the subgrid modeling method and obtain effective coefficients for the estimation of the first statistical moment of the displacement in the acoustic equation if wavelength essentially exceeds then a maximum scale of heterogeneity. If a medium is assumed to satisfy the improved Kolmogorov similarity hypothesis, the effective coefficients take especially the simple form. Differential equations for obtaining effective coefficients are also derived for the media that do not satisfy the improved similarity hypothesis. The derived formulas are verified by direct 2D numerical modeling.
[full text]
Keywords: Propagation of waves, acoustic equation, effective coefficients, subgrid modeling, multiscale random media, multiplicative continuous cascade
Author(s):
Kurochkina Ekaterina Petrovna
PhD.
Position: Senior Research Scientist
Office: Institute of Thermophysics SB RAN
Address: 630090, Russia, Novosibirsk, pr. Lavrentieva, 1
Phone Office: (383)316 52 31
E-mail: kurochkina@itp.nsc.ru
Soboleva Olga Nikolaevna
Dr.
Position: Senior Research Scientist
Office: Institute of Computational Mathematics and Mathematical Geophysics, SB RAN
Address: 630090, Russia, Novosibirsk, pr. Lavrentieva, 1
Phone Office: (383)3306046
E-mail: olga@nmsf.sscc.ru
References:
[1] Bleistein N., Cohen J.K., Stockwell J.W. Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion. New York: Springer; 2001.
[2] Backus G. Long-wave elastic anisotropy produced by horizontal layering. Journal Geophysical Reserch. 1962; 67(11): 4427–4440.
[3] Molotkov L.A. On the equivalence between layer-periodic and transversal-isotropic media [Ob ekvivalentnosti sloisto-periodicheskiêh i transversal'no-izotropnyêh sred]. Matematicheskie voprosy teorii rasprostraneniya voln: Zapiski Nauchnych Seminarov POMI. Leningrad: Nauka, 1979. 89. 219–233. (In Russ.)
[4] Guillot L. Capdevile Y., and Marigo J.J. 2-D non periodic homogenization for the SH wave Equation. Geophysical Journal International. 2010; 182: 1438–1454.
[5] Capdeville Y. and Marigo J.J. A non-periodic two scale asymptotic method to take account of rough topographies for 2-D elastic wave propagation. Geophysical Journal International. 2013; 192: 163–189.
[6] Sahimi M. and Tajer S.E. Self-affine fractal distributions of bulk density, elastic moduli and seismic wave velocities of rock. Physical Rewiew. E. 2005; 71: 046301.
[7] Koohi lai Z., Vasheghani Farahani S., Jafari G.R. Non-Gaussianity effect of petrophysical quantities by using q-entropy and multi fractal random walk. Physica A: Statistical Mechanics and its Applications. 2013; 392(20): 5132–5137.
[8] Soboleva O.N., Kurochkina E. P. Effective coefficients of quasi-steady Maxwell's equations with multiscale isotropic random conductivity Physica A: Statistical Mechanics and its Applications. 2011; 390(2): 231–244.
[9] Kuz'min G.A., Soboleva O.N. Subgrid modeling of filtration in porous self-similar media. Journal Applied. Mechanics Techical Physics. 2002; 43: 583–592.
[10] Kolmogorov A.N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal Fluid Mechanics. 1962; 13: 82–85.
[11] Gnedenko B.V, Kolmogorov A.N. Limit distributions for sums of independent random variables. Cambridge (MA): Addison-Wesley; 1954.
[12] Ogorodnikov V.A., Prigarin S.M. Numerical Modeling of Random Processes and Fields: Algorithms and Applications, Utrecht, The Netherlands; 1996.
Bibliography link:
Kurochkina E.P., Soboleva O.N. Effective coefficients of a multiscale isotropic medium applied for the problem of propagation of acoustic waves // Computational technologies. 2014. V. 19. ¹ 6. P. 54-64
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