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Analysis of the viscoelasticity in coal based on the fractal theory

Coal is a complex viscoelastic porous medium with fractal characteristics at different scales. To model the macroscale structure of coal, a fractal viscoelastic model is established, and the P-wave velocity dispersion and attenuation characteristics are discussed based on the complex modulus derived from this model. The numerical simulation results indicate that the fractional order α and relaxation time τ greatly affect the P-wave velocity dispersion and attenuation. The fractal viscoelastic model indicates a full-band velocity dispersion between 1 Hz and 104 Hz. Meanwhile, the P-wave velocity has a weaker dispersion with the fractal viscoelastic model than with the Kelvin-Voigt model and Zener model between 1 Hz and 104 Hz for the same relaxation time and elastic modulus, but the velocity at 1 Hz based on the fractal viscoelastic model is higher with the Kelvin-Voigt model and Zener model. Simultaneously, the velocities of five coal samples are tested, and the attenuation factor is calculated using a low-frequency system. The experimental results indicate a strong dispersion in coal in the range of 10–250 Hz. The classic Kelvin-Voigt model and Zener model cannot describe the dispersion characteristics of coal, but the fractal viscoelastic model can describe them well by using the appropriate fractional order and relaxation time.