Binary-Classification Physical Fractal Models in Different Coal Structures

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, α, β, and γ, emerged. Among them, α mainly affects the attenuation at low frequencies; β controls the attenuation in the middle-frequency band; and γ 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.