Establishment and Application of a Theoretical Model for Gas Extraction Considering the Influence of Borehole Cross-Section Instability

To resolve poor gas extraction efficiency and short service life due to borehole instability in deep coal seams, this study established a theoretical gas extraction model incorporating borehole cross-section deformation shapes via combined experimentation and numerical simulation, analyzing the impact mechanisms of instability. The results demonstrate that (1) Under increasing extraction negative pressure (10-55 kPa), the extraction flow and unit pressure drop of five instability models follow quadratic growth, with sudden deformation exhibiting maximal pressure loss (unit pressure drop ≤ 59.9 Pa/m), gradual deformation minimal loss (55% of sudden model), and irregular cross-section intermediate loss (72% of sudden model); (2) a modified negative pressure loss model quantifies the local effects of unstable cross-sections, revealing that the local loss coefficient (ξ) and Reynolds number (<i>Re</i>) obey ξ = <i>a</i> - <i>b</i> × ln (<i>Re</i> + <i>c</i>), where ξ declines significantly in low-<i>Re</i> ranges (gradual: <i>Re</i> = 0-100; sudden: <i>Re</i> = 0-125; irregular: <i>Re</i> = 0-150) before stabilizing; and (3) theoretical modeling of borehole instability effects enables the analysis of negative pressure distribution and extraction efficiency, with simulations showing intensified pressure loss under prolonged extraction or higher negative pressure, ordered as sudden instability > irregular > gradual deformation. These findings provide a theoretical basis for identifying instability zones/severity and enhancing deep-seam gas extraction efficiency.