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.

Attenuation refers to any decrease in the power of a propagated signal through a medium. Attenuation measurement techniques include ultrasonic, resonance bar, and stress-strain methods. The stress-strain method measures elastic and viscoelastic properties in the seismic frequency range. The signals received via attenuation measurement systems using the stress-strain method can be considerably weak. Moreover, the noise in these signals causes errors when estimating the signal phase-angle difference between the sample signal and probe signal, thereby reducing system precision and measurement accuracy. Accurate measurements of such phase differences are essential to the measurement of attenuation. A seismic-wave attenuation with frequency-dependent measurement based on the stress-strain method and digital signal processing techniques has been performed. The system estimates the attenuation of a rock by measuring the phase shift in the stress-strain cycle. As a preprocessing method, the finite impulse response band-pass filters are designed to eliminate the influence of noise and direct current offset while ensuring that the phase difference of the measured signal remains unchanged. Three methods for phase difference estimation, i.e., crosscorrelation, fast Fourier transform (FFT), and Hilbert transform are compared, for different signal-to-noise ratios, sampling frequencies, data sample lengths, and true phase differences. The results find that the phase difference estimation based on FFT is the best among all three methods and can effectively improve the precision of the experimental results. Simulation and measurement results further indicate that the attenuation measurement system achieves stable and reliable attenuation measurements in the range of 3–2000 Hz.

Existing theoretical models of wave-induced flow face challenges in coal applications due to the scarcity of experimental data in the seismic-frequency band. Additionally, traditional viscoelastic combination models exhibit inherent limitations in accurately capturing the attenuation characteristics of rocks. To overcome these constraints, we propose a novel binary classification physical fractal model, which provides a more robust framework for analyzing wave dispersion and attenuation in complex coal. The fractal cell was regarded as an element to re-establish the viscoelastic constitutive equation. In the new constitutive equation, three key fractional orders, $\alpha$, $\beta$, and $\gamma$, emerged. Among them, $\alpha$ mainly affects the attenuation at low frequencies; $\beta$ controls the attenuation in the middle-frequency band; and $\gamma$ dominates the attenuation in the tail-frequency band. After fitting with the measured attenuation data of partially saturated coal samples under variable confining pressures and variable temperature conditions, the results show that this model can effectively represent the attenuation characteristics of elastic wave propagation in coals with different coal structures. It provides a new theoretical model and analysis ideas for the study of elastic wave attenuation in tectonic coals and is of great significance for an in-depth understanding of the physical properties of coals and related geophysical prospecting.

The mechanisms of frequency dispersion and attenuation of seismic waves in fluid-saturated coal are not yet fully understood, and traditional integer-order viscoelastic models using a single relaxation time struggle to accurately characterize the dynamic response across different frequencies. A new three-component physical fractal viscoelastic model, based on the self-similar fractal pore-fracture structure of saturated coal, decouples matrix and fluid dissipation through two independent Caputo fractional orders (α corresponding to the skeleton and β corresponding to the fluid), distinguishing it from the single-relaxation-time form of standard integer-order elements. Effective stress simultaneously modifies both the fractal orders (α, β) and the relaxation times (τ₂, τ₃), thereby increasing P-wave velocity and reducing the attenuation amplitude and its frequency dependence. In contrast, temperature and gas adsorption change only the relaxation time scale, leading to synchronous changes in wave velocity and attenuation in unison. The model provides petrophysical support for field-scale seismic dispersion inversion, fracture prediction, fluid identification, and sweet spot evaluation in coalbed methane reservoirs.