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S-Wave Attenuation Due to Fluid Acceleration

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Abstract

At high frequency, the time scale is very short and acceleration becomes important. There must be a pressure difference in the main pores to drive fluid acceleration, which as a side effect is capable of inducing a reverse squirt in the throat connecting two pores. Based on such a mechanism, we develop a novel model of an S wave in fluid-saturated rock, which yields phase velocity (Vs) and the quality factor (Qs) as functions of frequency. Applications of the new model to Berea sandstone and Boise sandstone yield throat permeability. The second porosity represented by throats appears to be 5% of the total porosity. Nonetheless, Qs is predicted as 106 at a frequency of 10 Hz, far higher than seismic Qs measured in the field. This may be because groundwater has softened the skeleton of sedimentary rocks and/or because internal reflections at multiple lithological interfaces attenuate seismic waves.

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Data Availability

The data yielded from the model are available with https://doi.org/10.6084/m9.figshare.14493768 at https://figshare.com/s/b35b063cf723574c902b.

Abbreviations

a :

Length of main pore

b :

Length of throat

\(c_{{\text{s}}}\) :

S-wave velocity in undrained rock (\(c_{{\text{s}}} = \sqrt {\frac{G}{\rho }}\))

\(c_{{{\text{s0}}}}\) :

S-wave velocity of skeleton (\(c_{{{\text{s0}}}} = \sqrt {\frac{G}{{\rho_{{\text{s}}} }}}\))

f :

Frequency

G :

Shear modulus of skeleton

k :

Wavenumber

k D :

Darcy permeability

k 1 :

Local permeability of S wave in main pore

k 2 :

Local permeability of S wave in throat

P p1 :

Fluid pressure in main pore

P p2 :

Fluid pressure in throat

q 1 :

Darcy flux rate in main pore

q 2 :

Darcy flux rate in throat

Q E :

Quality factor of FOM

Q p :

Quality factor of P wave

Q s :

Quality factor of S wave

t :

Time

u :

Shear displacement

v :

Lagrangian velocity of solid

V s :

S-wave velocity

x :

Direction of Lagrangian motion

y :

S-wave direction

\(\phi\) :

Total porosity

\(\phi_{1}\) :

Local porosity in main pore

\(\phi_{2}\) :

Local porosity in throat

\(\mu\) :

Fluid dynamic viscosity

\(\omega\) :

Angular frequency

\(\omega_{{\text{C}}}\) :

Characteristic angular frequency

\(\Omega\) :

Dimensionless angular frequency

\(\rho\) :

Total density (\(\rho = \rho_{{\text{s}}} + \phi \rho_{{\text{f}}}\))

\(\rho_{{\text{f}}}\) :

Fluid density

\(\rho_{{\text{s}}}\) :

Skeleton density

References

  • Batzle, M. L., Han, D. H., & Hofmann, R. (2006). Fluid mobility and frequency-dependent seismic velocity—Direct measurements. Geophysics, 71, N1–N9.

    Article  Google Scholar 

  • Bear, J. (1972). Dynamics of fluids in porous medium. Dover.

    Google Scholar 

  • Biot, M. A. (1956a). Theory of propagation of elastic waves in a fluid-saturated porous solid I. Lower frequency range. Journal of the Acoustical Society of America, 28, 168–178.

    Article  Google Scholar 

  • Biot, M. A. (1956b). Theory of propagation of elastic waves in a fluid-saturated porous solid II. Higher frequency range. Journal of the Acoustical Society of America, 28, 179–191.

    Article  Google Scholar 

  • Blair, D. P. (1990). A direction comparison between vibrational resonance and pulse transmission data for assessment of seismic attenuation in rock. Geophysics, 55(1), 55–60.

    Article  Google Scholar 

  • Chapman, S., Borgomano, J., Yin, H., Fortin, J., & Quintal, B. (2019). Forced oscillation measurements of seismic wave attenuation and stiffness moduli dispersion in glycerine-saturated Berea sandstone. Geophysical Prospecting, 67, 956–968.

    Article  Google Scholar 

  • Cheng, J. C. (2012). Principles in acoustics. Science Press of China.

    Google Scholar 

  • Cheung, C. S., Baud, P., & Wong, T. (2012). Effect of grain size distribution on the development of compaction localization in porous sandstone. Geophysical Research Letters, 39, L21302.

    Article  Google Scholar 

  • Domenico, P. A., & Schwartz, F. W. (1997). Physical and chemical hydrogeology. Wiley.

    Google Scholar 

  • Dvorkin, J., Mavko, G., & Nur, A. (1995). Squirt flow in fully saturated rocks. Geophysics, 60(1), 97–107.

    Article  Google Scholar 

  • Dziewonski, A. M., & Anderson, D. L. (1981). Preliminary reference Earth model. Physics of the Earth and Planetary Interiors, 25(4), 297–336.

    Article  Google Scholar 

  • Gassmann, F. (1951). Uber die Elasticität Poröser Medien [On the elasticity of porous media]. Vierteljahrsschrift Der Naturforschenden Gesllschaft in Zürich, 96, 1–23.

    Google Scholar 

  • Gregory, A. R. (1976). Fluid saturation effects on dynamic elastic properties of sedimentary rocks. Geophysics, 41, 895–921.

    Article  Google Scholar 

  • Hauksson, E., & Shearer, P. M. (2006). Attenuation models (QP and QS) in three dimensions of the southern California crust: Inferred fluid saturation at seismogenic depths. Journal of Geophysical Research, 111, B05302.

    Article  Google Scholar 

  • Jaeger, J. C., Cook, N., & Zimmerman, R. (2007). Fundamentals of rock mechanics (4th ed.). Wiley-Blackwell.

    Google Scholar 

  • Johnson, D. L., Koplik, J., & Dashen, R. (1987). Theory of dynamic permeability and tortuosity in fluid-saturated porous media. Journal of Fluid Mechanics, 176, 379–402.

    Article  Google Scholar 

  • Jones, T., & Nur, A. (1983). Velocity and attenuation in sandstone at elevated temperatures and pressures. Geophysical Research Letters, 10, 140–143.

    Article  Google Scholar 

  • Kundu, P. K. (1990). Fluid mechanics. Academic Press.

    Google Scholar 

  • Li, G. (2020a). S wave attenuation based on Stokes boundary layer. Geophysical Prospecting, 68, 910–917.

    Article  Google Scholar 

  • Li, G. (2020b). Velocity and attenuation of ultrasonic S-wave in Berea sandstone. Acta Geodaetica Et Geophysica, 55, 335–345.

    Article  Google Scholar 

  • Li, G., Liu, K., & Li, X. (2020). Comparison of fluid pressure wave between Biot theory and storativity equation. Geofluids. https://doi.org/10.1155/2020/8820296

    Article  Google Scholar 

  • Li, G., Mu, Y., & Xie, C. (2021). Unsymmetric compressibility matrix to model P wave attenuation. Acta Geodaetica Et Geophysica. https://doi.org/10.1007/s40328-021-00344-6

    Article  Google Scholar 

  • Mikhaltsevitch, V., Lebedev, M., & Gurevich, B. (2014). A laboratory study of low-frequency wave dispersion and attenuation in water saturated sandstones. Leading Edge, 33, 616–622.

    Article  Google Scholar 

  • Mochizuki, S. (1982). Attenuation in partially saturated rocks. Journal of Geophysical Research, 87, 8598–8604.

    Article  Google Scholar 

  • Murphy, W. F., Winkler, K. W., & Kleinberg, R. L. (1986). Acoustic relaxation in sedimentary rocks: Dependence on grain contacts and fluid saturation. Geophysics, 51, 757–766.

    Article  Google Scholar 

  • Øren, P., & Bakke, S. (2003). Reconstruction of Berea sandstone and pore-scale modelling of wettability effects. Journal of Petroleum Science and Engineering, 39, 177–199.

    Article  Google Scholar 

  • Pride, S. R., & Berryman, J. G. (2003a). Linear dynamics of double porosity and dual-permeability materials I. Governing equations and acoustic attenuation. Physical Review E, 68, 036603.

    Article  Google Scholar 

  • Pride, S. R., & Berryman, J. G. (2003b). Linear dynamics of double porosity and dual-permeability materials II. Fluid transport equations. Physical Review E, 68, 036604.

    Article  Google Scholar 

  • Pride, S. R., Berryman, J. G., & Harris, J. M. (2004). Seismic attenuation due to wave-induced flow. Journal of Geophysical Research, 109, B01201. https://doi.org/10.1029/2003JB002639

    Article  Google Scholar 

  • Ricker, N. (1977). Transient waves in visco-elastic media. Elsevier.

    Google Scholar 

  • Schlichting, H. (1968). Boundary layer theory (6th ed.). Springer.

    Google Scholar 

  • Subramaniyan, S., Quintal, B., Madonna, C., & Saenger, E. (2015). Laboratory-based seismic attenuation in Fontainebleau sandstone: Evidence of squirt flow. Journal of Geophysical Research, 120, 7526–7535.

    Article  Google Scholar 

  • Toksöz, M. N., Johnston, D. H., & Timur, A. (1979). Attenuation of seismic waves in dry and saturated rocks I. Laboratory Measurements. Geophysics, 44(4), 681–690.

    Article  Google Scholar 

  • Walsh, J. B. (1965). The effect of cracks on the compressibility of rock. J. Geophy. Res., 70(2), 381–389.

    Article  Google Scholar 

  • Wang, H. F. (2000). Theory of linear poroelasticity—With applications to geomechanics and hydrogeology. Princeton University Press.

    Google Scholar 

  • Waters, K. H. (1981). Reflection seismology—a tool for energy resource exploration. Wiley.

    Google Scholar 

  • White, J. E. (1975). Computed seismic speeds and attenuation in rocks with partial gas saturation. Geophysics, 40, 224–232.

    Article  Google Scholar 

  • Zhao, L., Cao, C., Yao, Q., Wang, Y., Li, H., Yuan, H., Geng, J., & Han, D. (2020). Gassmann consistency for different inclusion-based effective medium theories: Implications for elastic interactions and poroelasticity. Journal of Geophysical Research, 125(3), e2019JB018328.

    Google Scholar 

  • Zhao, L., Wang, Y., Yao, Q., Geng, J., Li, H., Yuan, H., & Han, D. (2021). Extended Gassmann equation with dynamic volumetric strain: Modeling wave dispersion and attenuation of heterogeneous porous rocks. Geophysics, 86(3), 149–164.

    Article  Google Scholar 

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Acknowledgements

The research was sponsored by the National Natural Science Foundation of China under Grants 42064006 and 41873075. The authors would like to thank sincerely the Editor and two anonymous reviewers for their positive comments and constructive suggestions.

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Authors and Affiliations

  1. Department of Geophysics, Yunnan University, Kunming, 650504, Yunnan, China

    Guangquan Li & Shiwei Liu

  2. Institute of Geochemistry, Chinese Academy of Sciences, 99 West Lincheng Road, Guiyang, 550081, Guizhou, China

    Yonggang Liu

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Li, G., Liu, Y. & Liu, S. S-Wave Attenuation Due to Fluid Acceleration. Pure Appl. Geophys. 179, 1159–1172 (2022). https://doi.org/10.1007/s00024-022-02989-y

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