Low-frequency scattering on a half-space filled with periodic fluid-solid medium with dipped layers

Authors

  • Yu. V. Roganov Tesseral Technologies Inc., Kyiv, Ukraine
  • V. Yu. Roganov Institute of Cybernetic of NAS of Ukraine, Kyiv, Ukraine

DOI:

https://doi.org/10.24028/gzh.0203-3100.v39i4.2017.107508

Keywords:

periodical solid-fluid medium, dispersion equation, scattering, reflection and refraction coefficients

Abstract

A low-frequency effective model has been developed for a medium with periodical liquid and solid layers with the slip between layers. It is shown that for an effective periodically n-layered medium with solid dipped layers with slip there exist n+1 plane waves with a fixed horizontal slowness that propagate downward. The boundary conditions are determined for low-frequency scattering at the boundary between a solid half-space and a half-space filled with an effective medium. These conditions depend on the dip angle of the layers and their filling. Based on the boundary conditions, linear systems of equations for the reflection and refraction coefficients are derived. Low-frequency scattering on a half-space with dipped solid layers with the slip is described by a system of n+3 equations with n+3 unknowns. In the presence of liquid layer, the number of equations and unknowns is equal to n+2. If the lower half-space consists of horizontal layers, the number of equations and unknowns is equal to 3. Explicit formulas for the roots of this system of equations are obtained for the case when the layers are horizontal. The theory is demonstrated on various examples of calculating the reflection and refraction coefficients.

References

Aki K., Richards P., 1983. Quantitative seismology. Theory and methods. Moscow: Mir, 520 p. (in Russian).

Molotkov L. A., 2001. The study of wave propagation in porous and fractured media based on effective models of BIO and layered media. St. Petersburg: Nauka, 348 p. (in Russian).

Molotkov L. A.,1979. Equivalence of periodically layered and transversally isotropic media. Zapiski nauchnykh seminarov LOMI 89, 219—233 (in Russian).

Molotkov L. A., 1994. On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces. Zapiski nauchnykh seminarov POMI 210, 192—212 (in Russian).

Molotkov L. A., 1991. New method for deriving equations of an effective average model of periodic media. Zapiski nauchnykh seminarov LOMI 195, 82—102 (in Russian).

Molotkov L. A., Bakulin A. V., 1995. The effective model of a stratified solid-fluid medium as a special case of the Biot model. Zapiski nauchnykh seminarov POMI 230, 172—195 (in Russian).

Molotkov L. A., Khilo A. E., 1984. Single-phase and multiphase effective models describing periodic media. Zapiski nauchnykh seminarov LOMI 140, 105—122 (in Russian).

Molotkov L. A., Perekareva M. N., 2006. Investigation of the wave field in an effective model of a layered elastic-fluid medium. Zapiski nauchnykh seminarov POMI 332, 175—192 (in Russian).

Roganov Yu. V., Roganov V. Yu., 2016. Wave propagation in periodic fluidsolid layered media. Geofizicheskiy zhurnal 38 (6), 101—117 (in Russian).

Backus G. E., 1962. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res. 67, 4427—4440. doi: 10.1029/JZ067i011p04427.

Corredor R., Santos J., Gauzellino P., Carcione J., 2016. Validation of the boundary conditions to model the seismic response of fractures. Geophys. Prosp. 64, 1149—1165. doi:10.1111/1365-2478.12375.

Deresiewicz H., Rice J. T., 1960. The effect of boundaries on wave propagation in liquid-filled porous solids: I. Reflection of plane waves at a true plane boundary. Bull. Seismol. Soc. Am. 50, 599—607.

Lovera O. M., 1987. Boundary conditions for a fluid-saturated porous solid. Geophysics, 52 (2), 174—178.

Molotkov L. A., 1982. Equivalence of periodically layered and transversally isotropic media. J. Soviet Math. 19(4), 1454—1466. doi: 10.1007/BF01085033.

Molotkov L. A., 1992. New method for deriving equations of an effective average model of periodic media. J. Soviet Math., 62(6), 3103—3107. doi: 10.1007/BF01095684.

Molotkov L. A., 1997. On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces. J. Math. Sci. 83(2), 288—301. doi: 10.1007/BF02405824.

Molotkov L. A., Bakulin A. V., 1998. The effective model of a stratified solid-fluid medium as a special case of the Biot model. J. Soviet Math. 91(2), 2812—2827. doi: 10.1007/BF02433997.

Molotkov L. A., Khilo A. E., 1986. Single-phase and multiphase effective models describing periodic media. J. Soviet Math., 32 (2), 173—185. doi: 10.1007/BF0108415.

Molotkov L. A., Perekareva M. N., 2007. Investigation of the wave field in an effective model of a layered elastic-fluid medium. J. Math. Sci. 142(6), 2620—2629. doi:10.1007/s10958-007-0150-1.

Nakagawa S., Schoenberg M., 2007. Poroelastic modeling of seismic boundary conditions across a fracture. J. Acoust. Soc. Am. 122(2), 831—847. doi: 10.1121/1.2747206.

Rajesh S., 2015. Reflection/refraction at the interface of an elastic solid and a partially saturated porous solid containing liquid filled bound pores and a connected pore space saturated by two-phase fluid. Lat. Am. J. Solids Struct. 12(10), 1870—1900. doi: 10.1590/1679-78251834.

Roganov Yu., Stovas A., 2012. Low-frequency wave propagation in periodically layered media. Geophys. Prosp. 60, 825—837.

Schoenberg M., 1983. Reflection of elastic waves from periodically stratified media with interfacial slip, Geophys. Prosp. 31, 265—292. doi: 10.1111/j.1365-2478.1983.tb01054.x.

Schoenberg M., 1984. Wave propagation in alternating solid and fluid layers. Wave Motion 6, 303—320. doi: 10.1016/0165-2125(84) 90033-7.

Schoenberg M., Muir F., 1989. A calculus for finely layered anisotropic media. Geophysics 54(5), 581—589. doi: 10.1190/1.1442685.

Published

2017-07-25

How to Cite

Roganov, Y. V., & Roganov, V. Y. (2017). Low-frequency scattering on a half-space filled with periodic fluid-solid medium with dipped layers. Geofizicheskiy Zhurnal, 39(4), 55–76. https://doi.org/10.24028/gzh.0203-3100.v39i4.2017.107508

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Articles