## 4 Conclusions  
The efficiency of CBM drainage systems is fundamentally controlled by the permeability of broken coal around drainage boreholes, with in the stress state acting as the primary determinant of permeability. Significantly, the effective stress serves as a holistic parameter to characterize the anisotropic stress distributions around drainage boreholes. Through systematic triaxial permeability testing on broken coal samples, the stage variation characteristics of permeability are revealed, and the quantitative relationship between the effective stress and permeability of broken coal is established. The following conclusions are drawn:  
1. A specific correlation exists between the permeability of broken coal and axial pressure. Axial pressure affects permeability by altering the pore structure of broken coal samples. During the loading process, as the axial pressure increases, the permeability of broken coal decreases exponentially. This decrease is accompanied by the compression of the pore structure, the gradual closure of permeability channels, and a consequent decrease in permeability. 2. A specific correlation also exists between the permeability of broken coal and the Talbot power exponent. As the Talbot power exponent (\(n\)) of the broken coal samples increases from 0.1 to 1.0, the permeability (\(k\)) shows a three-stage growth pattern. In the initial stage, the permeability increases slowly due to the predominance of smaller particle sizes. In the second stage, as the proportion of larger particles increases, the permeability increases linearly. In the third stage, with the dominance of larger particle sizes, the permeability exhibits exponential growth. 3. The seepage velocity may be influenced by the combined effect of the Talbot power exponent, pressure gradient, confining pressure, and axial pressure. Of these factors, the pressure gradient may have the most significant effect. The Talbot power exponent enhances the seepage velocity, while the axial and confining pressures inhibit it. 4. The permeability of broken coal with different Talbot power exponents typically exhibits an exponential decay as the effective stress increases. Both the parameters and the permeability-Talbot power exponent curve display three stages. As the Talbot power exponent increases, the power exponent (\(a\)) gradually decreases, while the coefficient (\(b\)) increases. This indicates that broken particles increase both the inhibitory effect of the effective stress and the amplitude effect. The stresses and broken state of the coal around the gas drainage borehole are complex and varied, resulting in the coexistence of multiple gas seepage channels. The permeability of each channel is interrelated, which greatly affects the gas migration path and drainage efficiency. In fact, we explored the serial seepage problem of the broken coal body around the drainage borehole, and due to the existence of different gas flow channels, the follow-up should focus on the parallel seepage modeling of coals with different degrees of brokenness and evaluate the comprehensive effects of parallel seepage mechanisms on coalbed methane drainage under different mining stages and stress environments. Conceptualization, methodology, and funding acquisition by [PERSON]; project administration and funding acquisition by [PERSON]; resources, visualization, supervision, funding acquisition, writing--review and editing by [PERSON]; validation, investigation, data curation, formal analysis, writing--original draft preparation, review and editing by S.Y. All authors have read and agreed to the published version of the manuscript. This research was funded by National Natural Science Foundation of China, grant number 52104216, 52374228, 52274226. Not applicable. Note applicable. Some or all data that support the findings of this study are available from the corresponding author upon reasonable request. DeepSeek R1 was used for grammar checking only during the manuscript revision. All academic content remains the responsibility of the authors.