Abstract
Moment tensor analyses suggest that shear microfractures dominate the compressive characteristics of most rock materials. However, the microfracture sizes estimated by the traditional shear model are unrealistically large. This paper presents a novel shear model for microfracture size estimation in sandstone specimens based on AE measurements. The sizes of 342 microfractures were estimated via the proposed model; most are found to range from 0.0022 to 1.80 mm with an average microfracture size slightly smaller than the average grain size. The microfracture size distribution characteristics observed here are similar to those observed by microscope for stressed sandstone, suggesting that more realistic microfracture sizes can be obtained with the proposed model than the traditional shear model.
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Abbreviations
- r, a, d :
-
Distance between the point and the center of the microfracture plane, shear microfracture radius, and diameter
- τ 0, τ a, τ f, τ e, Δτ, σ n :
-
Initial shear stress pre-fracture, ultimate shear stress after fracture, friction stress, effective stress, stress drop, and normal stress on the microfracture plane
- u 0, u a, \(\overline{\Delta u}^{a}\) :
-
Initial sliding displacement, final sliding displacement, and average sliding displacement between microfracture sides
- µ f, G, ν, K II, E :
-
Coefficient of friction, shear stiffness, Poisson's ratio, fracture toughness of type II, elasticity modulus
- U w, U s, U k, U f :
-
Total work done by the stress at the microfracture plane, surface energy, kinetic energy, frictional dissipation energy
- ρ :
-
Rock density
- R C, 〈R C〉:
-
Radiation pattern coefficient of P or S waves, average radiation pattern coefficients
- V C, J C :
-
P- or S-wave velocity, radiation energy flux of P or S wave
- L :
-
Distance between the AE source (microfracture) and AE sensor
- u :
-
Far field first motion displacement of the P wave
- C s :
-
Magnitude of the sensor response including the material constants
- R e (t,r) :
-
Reflection coefficient
- \(\mathop{t}\limits^{\rightharpoonup}\) :
-
Direction vector of AE sensor
- \(\mathop{r}\limits^{\rightharpoonup}\) :
-
Direction vector from the source to the sensor
- M 1, M 2, M 3 :
-
Three eigenvalues of the moment tensor
- e 1, e 2, e 3 :
-
Eigenvectors of the eigenvalues M1, M2, and M3
- l, n :
-
Normal vector of two node planes
- γ :
-
Tensile angle
- σ 1, σ 2, σ 3 :
-
Maximum, medium, and minimum principal stress
- l 1, l 2, l 3 :
-
Cosines of the normal direction of the given plane with respect to the principal stress axes σ1, σ2, and σ3
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Acknowledgements
The work presented in this paper is financially supported by the National Key Research Project (2016YFC0801607), and the National Natural Science Foundation of China (U1710253, 51974062, and 41672301), and the Fundamental Research Funds for the Central Universities of China (N180101028, N2001032). The authors are thankful to the referees and editors for their valuable comments and suggestions devoted to improving the quality of our manuscript.
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Zhang, P., Liu, H., Guan, K. et al. A Shear Model for Rock Microfracture Size Estimation Based on AE Measurement. Rock Mech Rock Eng 54, 2533–2546 (2021). https://doi.org/10.1007/s00603-021-02388-0
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DOI: https://doi.org/10.1007/s00603-021-02388-0