Abstract
The effective coefficients for Maxwell’s equations in the frequency domain for a multiscale isotropic medium by using a subgrid modeling approach are calculated. The correlated fields of conductivity and permittivity are approximated by the Kolmogorov multiplicative continuous cascades with a lognormal probability distribution. The wavelength is assumed to be large when compared with the scale of heterogeneities of the medium. The equations for effective coefficients are obtained in the first order terms of \(\omega \varepsilon (\mathbf {x})/\sigma (\mathbf {x})\), where \(\varepsilon \left( \mathbf {x}\right) \) is the permittivity, \(\sigma \left( \mathbf {x}\right) \) is the electric conductivity and \(\omega \) is the cyclic frequency. The obtained effective parameters are frequency-independent and therefore it follows that they are also the effective parameters in the time domain. The theoretical results are compared with the results from direct 3D numerical simulations. The permittivity under certain conditions can influence a measured signal in a quasi-steady case if the parameters \(\sigma \) and \(\varepsilon \) are weakly correlated.
Similar content being viewed by others
References
Bekele A, Hudnall HW, Daigle JJ, Prudente A, Wolcott M (2005) Scale dependent variability of soil electrical conductivity by indirect measures of soil properties. J Terramech 42:339–351
Davydycheva S, Drushkin V, Habashy T (2003) Scale dependent variability of soil electrical conductivity by indirect measures of soil properties. Geophysics 68:1525–1536
Kolmogorov AN (1962) 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
Kurochkina EP, Soboleva ON (2013) The Subgrid modeling for Maxwell’s equations with multiscale isotropic random conductivity and permittivity. Prog Electromagn Res B 49:197–213
Kuz’min GA, Soboleva ON (2002) Subgrid modeling of filtration in porous self-similar media. J Appl Mech Tech Phys 43:583–592
Modersitzki J, Sleijpen G, Van der Vorst H (2000) Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems. Matrix Anal Appl 22:726–751
Ogorodnikov VA, Prigarin SM (1996) Numerical modeling of random processes and fields: algorithms and applications. VSP, Utrecht
Sahimi M (1993) Flow phenomena in rocks: from continuum models, to fractals, percolation, cellular automata, and simulated annealing. Rev Modern Phys 65:1393–1534
Shelukhin VV, Terentev SA (2009) Frequency dispertion of dialectric permitivity and electric conductivity of rocks via two-scale homogenization of the Maxwell equations. Progr Electromagn Res B 14:175–202
Wilson J, Schertzer D, Lovejoy S (1991) Continuous multiplicative cascade models of rain and clouds. In: Schertzer D, Lovejoy S (eds) Nonlinear variability in geophysics. Kluwer Academic Publishers, Norwell, pp 185–207
Acknowledgements
The Siberian Branch of the Russian Academy of Sciences (SB RAS) Siberian Supercomputer Center is gratefully acknowledged for providing supercomputer facilities. The work was supported by Program of Fundamental Scientific Research of State Academies of Sciences N 0315-2015-0002, Russia.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Soboleva, O.N., Epov, M.I. & Kurochkina, E.P. Effective coefficients in the electromagnetic logging problem with log-normal distribution, multiscale conductivity and permittivity. Stat Papers 59, 1339–1350 (2018). https://doi.org/10.1007/s00362-018-1035-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-018-1035-8