An elastic two-phase composite, with no restriction on the shape of the two phases, has stiffness bounds given by the Reuss and Voigt equations, and a narrower range determined by the Hashin-Shtrikman bounds. Averages are given by the Voigt-Reuss-Hill, Hashin-Shtrikman, Gassmann, Backus and Wyllie equations. To obtain stiffness bounds and averages, we invoke the correspondence principle to compute the solution of the viscoelastic problem from the corresponding elastic solution. Then, seismic velocities and attenuation are established for the above — physical and heuristic — models which account for general geometrical shapes, unlike the Backus average. The approach is relevant to the seismic characterization of solid composites such as hydrocarbon source rocks.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Backus G.E., 1962. Long-wave elastic anisotropy produced by horizontal layering. J. Geophys. Res., 67, 4427–4440.
Biwa S., Watanabe Y. and Ohno N., 2003. Analysis of wave attenuation in unidirectional viscoelastic composites by a differential scheme. Compos. Sci. Technol., 63, 237–247.
Carcione J.M., 1992. Anisotropic Q and velocity dispersion of finely layered media. Geophys. Prospect., 40, 761–783.
Carcione J.M., 2000. A model for seismic velocity and attenuation in petroleum source rocks. Geophysics, 65, 1080–1092.
Carcione J.M., 2014, Wave Fields in Real Media. Theory and Numerical Simulation of Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media. 3rd Edition, Elsevier, Amsterdam, The Netherlands.
Carcione J.M., Cavallini F. and Helbig K., 1998. Anisotropic attenuation and material symmetry. Acustica, 84, 495–502.
Carcione J.M. and Gurevich B., 2011. Differential form and numerical implementation of Biot’s poroelasticity equations with squirt dissipation. Geophysics, 76, N55–N64.
Carcione J.M., Helle H.B. and Avseth P., 2011. Source-rock seismic-velocity models, Gassmann versus Backus. Geophysics, 76, N37–N45.
Chen C.P. and Lakes R.S., 1993. Analysis of high loss viscoelastic composites. J. Mater. Sci., 28, 4299–4304.
Ciz R. and Shapiro S., 2007. Generalization of Gassmann equations for porous media saturated with a solid material. Geophysics, 72, A75–A79.
Eshelby J.D., 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London, 241, 376–396.
Gibiansky L.V. and Lakes R., 1997. Bounds on the complex bulk and shear moduli of a two-dimensional two-phase viscoelastic composite. Mech. Mater., 25, 79–95.
Glubokovskikh S. and Gurevich B., 2017. Optimal bounds for attenuation of elastic waves in porous fluid-saturated media. J. Acoust. Soc. Amer., 142, 3321–3329.
Graham G.A.C., 1968. The correspondence principle of linear viscoelasticity theory for mixed boundary value problems involving time-dependent boundary regions. Quart. Appl. Math., 26, 167–174.
Gurevich B. and Makarynska D., 2012. Rigorous bounds for seismic dispersion and attenuation due to wave-induced fluid flow in porous rocks. Geophysics, 77, L45–L51.
Hashin Z. and Shtrikman S., 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids, 11, 127–140.
Hill R., 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids, 11, 357–372.
Hilton H.H., 2009. The elusive and fickle viscoelastic Poisson’s ratio and its relation to the elasticviscoelastic correspondence principle. J. Mech. Mater. Struct., 4, 1341–1364.
Khazanovich L., 2008. The elastic-viscoelastic correspondence principle for non-homogeneous materials with time translation non-invariant properties. Int. J. Solids Struct., 45, 2–10.
Mainardi F., 2010. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London, U.K.
Mavko G. and Saxena N., 2016. Rock-physics models for heterogeneous creeping rocks and viscous fluids. Geophysics, 81, D427–D440.
Mavko G., Mukerji T. and Dvorkin J., 2009. The Rock Physics Handbook. Cambridge University Press, Cambridge, U.K.
Qadrouh A.N., Carcione J.M., Ba J. and Gei D., 2017. Backus and Wyllie averages for attenuation. Pure Appl. Geophys., 175, 165–170.
Roscoe R., 1969. Bounds for the real and imaginary parts of the dynamic moduli of composite viscoelastic systems. J. Mech. Phys. Solids, 17, 17–22.
Zhu Y., Tsvankin I. and Vasconcelos I., 2007. Effective attenuation anisotropy of thin-layered media. Geophysics, 72, D93–D106.
We are grateful to Petr Jílek and anonymous reviewers for a detailed review of our work. The authors are also grateful to the Jiangsu Innovation and Entrepreneurship Plan, Specially-Appointed Professor Plan of Jiangsu Province, and the National Natural Science Foundation of China (Grant No. 41974123).
About this article
Cite this article
Qadrouh, A.N., Carcione, J.M., Alajmi, M. et al. Bounds and averages of seismic quality factor Q. Stud Geophys Geod 64, 100–113 (2020). https://doi.org/10.1007/s11200-019-1247-y
- seismic attenuation
- Voigt and Reuss bounds
- Hashin-Shtrikman bounds
- Reuss-Voigt-Hill average
- Gassmann-Krief-Ciz-Shapiro average
- Backus and Wyllie average
- Q bounds