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Ultrasonic P-wave to ascertain the mean grain diameter of D’Euville limestone

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Abstract

In this paper, ultrasonic P-wave is used to ascertain the mean grain diameter of heterogeneous D’Euville limestone. A previous model of P-wave is used to yield phase velocity (Vp) and the quality factor due to viscous squirt (Qps) in the ultrasonic band, for the limestone saturated with water. As the limestone has grain diameter (d) ranging from 400 to 1000 μm, we set the mean d to be 800, 600 and 400 μm in turn. Vp is fitted precisely as possible between the model and ultrasonic measurement. The emission frequency was 500 kHz. With the mean d of 800 μm, the model yields Qps agreeing with the measurement not very accurately and the average frequency (between emission and receiving) is inverted as 251 kHz (which is too low). With the mean d of 600 μm, the model yields Qps agreeing with the measurement surprisingly well and the average frequency is inverted as 355 kHz (which is very reasonable). With the mean d of 400 μm, neither Vp nor Qps fits well between the model and measurement and the average frequency is inverted as 631 kHz (which is unreasonable). It is concluded that the mean d of D’Euville limestone is 600 μm.

Highlights

Phase velocity and the quality factor of ultrasonic P-wave in water-saturated D’Euville limestone are accurately simulated

The mean grain diameter of heterogeneous D’Euville limestone is 600 μm

Phase velocity and the quality factor of the limestone are featured with novel spikes in the frequency domain

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Data Availability

The data from the model are available under doi https://doi.org/10.6084/m9.figshare.22199461 at https://figshare.com/s/d5e12fcda17f82c9791c.

Abbreviations

\({a_{ij}}\) :

Asymmetric compressibility matrices

b :

Aperture distance of COG

d :

Diameter of grains

D :

Depth of the rock unit

f :

Frequency

G :

Shear modulus of skeleton (dry rock)

k :

Wavenumber

\({K_d}\) :

Bulk modulus of skeleton (\({K_d}=\frac{1}{{{\beta _d}}}\))

L :

Length of the rock unit

Q p :

The total quality factor of P-wave

Q ps :

The quality factor of P-wave due to viscous squirt

Q s :

The total quality factor of S-wave

V p :

Phase velocity of P-wave

V s :

Phase velocity of S-wave

\({\beta _d}\) :

Bulk compressibility of dry rock

\({\beta _f}\) :

Bulk compressibility of fluid

\({\beta _s}\) :

Compressibility of solid material

\(\varphi\) :

Permeability angle

\(\eta\) :

The total porosity (\(\eta ={\eta _1}+{\eta _2}\))

\({\eta _1}\) :

The first porosity

\({\eta _2}\) :

The second porosity

\(\mu\) :

Fluid dynamic viscosity

\(\nu\) :

Fluid kinematic viscosity (\(\nu =\frac{\mu }{{{\rho _f}}}\))

\(\omega\) :

Angular frequency

\(\gamma\) :

Squirt coefficient

\(\rho\) :

The total density (\(\rho ={\rho _s}+\eta {\rho _f}\))

\({\rho _f}\) :

Fluid density

\({\rho _s}\) :

Skeleton density

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Acknowledgements

The research was sponsored by National Natural Science Foundation of China (grant no. 42064006). We are deeply grateful to Editor-in-Chief (Gy?rgy Het?nyi), 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, Yunnan, 650504, China

    Guangquan Li & Xiang Li

  2. Department of Geology, Yunnan University, Kunming, Yunnan, 650504, China

    Yuchao Wang

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  1. Guangquan Li
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  2. Xiang Li
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Correspondence to Guangquan Li.

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Li, G., Li, X. & Wang, Y. Ultrasonic P-wave to ascertain the mean grain diameter of D’Euville limestone. Acta Geod Geophys 58, 451–463 (2023). https://doi.org/10.1007/s40328-023-00421-y

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