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.