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A Nonlinear Viscoelastic Model for Wave Propagation in Porous Sandstones Under Uniaxial Loads

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Abstract

Wave propagation is widely used in geologic mining and geotechnical engineering to acquire the structure information of rock masses. As rocks, by their nature, are filled with cracks and pores, wave propagation in rocks shows its clear dependence on stress. To describe the stress dependence of wave propagation, a nonlinear viscoelastic model consisting of a modified Hooke Model and Kelvin Model was proposed based on experimental observations on sandstones under uniaxial loads. With this model, the nonlinear behavior of rocks at the early stage of compression is attributed to the deformation of voids with a variable tangent modulus. Thus, wave velocity and wave attenuation can be altered, which change with the closure of voids and act as a function of stress. The proposed model was validated by comparison with experimental measurements from ultrasonic transmission at varying uniaxial stresses. It is shown that the effects of applied stress on the stress–strain relationship, wave velocity and wave attenuation for P-waves are well describe by the developed viscoelastic model. Changes in wave attenuation are also analyzed in terms of viscosity-like parameter defined by this model, which is found to be negatively proportional to the uniaxial stress.

Highlights

  • The wave propagation in porous sandstones under uniaxial loads was investigated in experimental and analytical ways.

  • A viscoelastic model was proposed to incorporate the nonlinear mechanical and acoustic behaviors observed at the early stage of compression.

  • Both the nonlinear stress-strain relationship and the stress dependence of wave velocity and wave attenuation can be well described by the proposed model.

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Data will be made available on request.

Abbreviations

\(\gamma\) :

Attenuation coefficient (m–1)

\(\omega\) :

Angle frequency (rad/s)

\({\mathbb{F}}\) :

Application of fourier transform

\(M_{1}\) :

Transmitted wave spectrum from specimen at 67 mm length

\(M_{2}\) :

Transmitted wave spectrum from specimen at 100 mm length

\(x\) :

Propagation distance (m)

\(\varepsilon_{hs}\) :

Static elastic strain of rock matrix

\(\varepsilon_{vs}\) :

Static elastic strain of rock matrix

\(E_{1}\) :

Elastic modulus of rock matrix (GPa)

\(E_{2}\) :

Elastic modulus of rock void (GPa)

\(\sigma_{s}\) :

Static stress of whole rock (MPa)

\(\varepsilon_{s}\) :

Static strain of whole rock

\(E_{2}^{\prime}\) :

Tangent modulus of rock void (GPa)

\(E_{2d}\) :

Dynamic modulus of rock void (MPa)

\(\alpha_{1}\) :

Hardening coefficient of rock matrix

\(\alpha_{2}\) :

Hardening coefficient of rock void

\(\sigma_{d}\) :

Dynamic stress of whole rock (MPa)

\(\sigma_{vd}\) :

Dynamic stress of rock void (MPa)

\(\sigma_{hd}\) :

Dynamic elastic stress of rock matrix (MPa)

\(\sigma_{\eta }\) :

Viscous stress of rock matrix (MPa)

\(\varepsilon_{d}\) :

Dynamic strain of whole rock

\(\varepsilon_{vd}\) :

Dynamic strain of rock void

\(\varepsilon_{hd}\) :

Dynamic elastic strain of rock matrix

\(\varepsilon_{\eta }\) :

Viscous strain of rock matrix

\(\eta\) :

Viscosity-like parameter of rock matrix (Pa.s)

\(t\) :

Time(s)

\(\phi_{0}\) :

Initial porosity of compliant pores (%)

\(u_{d}\) :

Dynamic displacement of whole rock (m)

\(A_{0}\) :

Amplitude of dynamic displacement of whole rock

\(K\) :

Angular wavenumber (rad/m)

\(A_{1}\) :

Amplitude of dynamic viscous displacement of rock matrix

\(\rho\) :

Rock density (kg/m3)

\(k\) :

Real part of angular wavenumber (rad/m)

\(V_{ph}\) :

Phase velocity of P-wave (m/s)

\(E^{\prime}\) :

Tangent modulus of whole rock (GPa)

\(\eta_{dif}\) :

Viscosity coefficient of diffusion creep (Pa.s)

\(\dot{\varepsilon }_{dif}\) :

Strain rate of diffusion creep (s–1)

\(\sigma_{dev}\) :

Deviator stress (MPa)

\(\eta_{dis}\) :

Viscosity coefficient of dislocation creep (Pa.s)

\(\dot{\varepsilon }_{dis}\) :

Strain rate of dislocation creep (s–1)

\(b\) :

Length of Burgers vector (m)

\(\mu\) :

Shear modulus (GPa)

\(\upsilon\) :

Dislocation velocity (m/s)

\(\Delta u\) :

Discrepancy of displacement between hard pores and grains (m)

\(u_{d}^{hp}\) :

Dynamic displacement of hard pore (m)

\(u_{d}^{rg}\) :

Dynamic displacement of rock grain (m)

\(A_{hp}\) :

Amplitude of dynamic displacement of hard pore

\(A_{rg}\) :

Amplitude of dynamic displacement of rock grain

\(\Delta \dot{\varepsilon }\) :

Discrepancy of strain rate between hard pores and grains (s–1)

\(\dot{\varepsilon }_{d}^{hp}\) :

Strain rate of hard pore (s–1)

\(\dot{\varepsilon }_{d}^{rg}\) :

Strain rate of rock grain (s–1)

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (grant number U19A2098), the National Natural Science Foundation of China (grant number 42102320), the National Natural Science Foundation of China (grant number 42007272) and the Chinese Fundamental Research Funds for the Central Universities (grant number 2021SCU12038). We are very grateful to Prof. Jianfeng Liu and Dr. Huining Xu from the Key Laboratory of Geotechnical Engineering, Sichuan University for carrying out the ultrasonic measurement at varying uniaxial stress.

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Authors and Affiliations

  1. Institute for Disaster Management and Reconstruction, Sichuan University-The Hong Kong Polytechnic University, No. 24 South Section 1, Yihuan Road, Chengdu, 610065, China

    Shuang Yang

  2. College of Water Resource and Hydropower, Sichuan University, No. 24 South Section 1, Yihuan Road, Chengdu, 610065, China

    Jianhui Deng, Hua Li & Fei Wang

  3. College of Architecture and Civil Engineering, Chengdu University, Chengdu, 610106, China

    Fei Chen

  4. School of Mechanical Engineering, Chengdu University, Chengdu, 610106, China

    Xuankun Li

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  1. Shuang Yang
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Correspondence to Hua Li.

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Yang, S., Deng, J., Li, H. et al. A Nonlinear Viscoelastic Model for Wave Propagation in Porous Sandstones Under Uniaxial Loads. Rock Mech Rock Eng 56, 7639–7653 (2023). https://doi.org/10.1007/s00603-023-03459-0

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