A theoretical framework for development and intensity analysis of coal and gas outbursts based on dual energy constraints

The “source-sink” energy conservation model has long underpinned quantitative coal and gas outburst research, yet its oversimplified constraints limit precise energy pathway characterization, relegating analyses to semi-quantitative comparisons. This study addresses two essential processes for outburst initiation: the pre-release of coal deformation energy and the induced elevation of gas expansion energy release. Building on this foundation, we propose a gas expansion energy threshold formula derived from critical tensile failure stress and porosity conditions, and establish discriminant criteria for outburst-triggering energy that integrate working face stress distribution and gas spatial heterogeneity. Both coal seam energy and outburst thresholds exhibit nonlinear increases with distance from the working face. Notably, the threshold variation demonstrates higher-order mathematical characteristics, intersecting the energy profile from below to determine coal ejection zones. By adopting the energy conservation equation as a discriminant criterion, two computational models are developed: a coal transport work model for distinct outburst stages based on the deformation equivalence principle, and a coal fragmentation work model grounded in the minimum incremental area principle. These advances collectively refine energy constraints governing outburst progression. Under the hypothesis of chain-reaction development, an energy chain-release analysis method constrained by temporally sequential energy criteria is proposed. Numerical simulations reveal that when energy distribution ahead of the working face simultaneously satisfies initiation and propagation constraints, the system enters cyclical outburst phases. Outburst evolution transitions from initiation-dominated constraints in early stages to propagation-dominated constraints in later phases. Calibrating energy conversion efficiency (30.5%-66.1%, mean 52.3%±13.1%) aligns simulated intensities with recorded outburst intensity data from actual accidents. Crucially, tectonically deformed coal (f < 0.2) exhibits outburst potential even under low gas pressure ( < 0.52 MPa) combined with high in situ stress ( > 35 MPa), underscoring geomechanical stress as a prioritized prevention indicator. This study establishes a unified theoretical framework bridging energy quantification and outburst dynamics, advancing mechanistic understanding through multi-constraint energy-chain modeling.