A Fractional-Order Damage-Based Permeability Model for Deep Coal Under Mining Disturbance
Abstract
Permeability models are essential for quantitatively describing coal permeability evolution and predicting gas migration during deep mining. Deep coal subjected to mining disturbance commonly exhibits pronounced nonlinear changes in permeability, limiting the applicability of conventional models. In this study, coal is idealized as a dual-component medium comprising the matrix and fractures, and deformation of both components induced by mining-related stress changes and gas adsorption is incorporated into the model. The conventional Weibull statistical damage variable is generalized to a fractional-order form using the Caputo derivative, yielding a Mittag–Leffler-type damage evolution law. By coupling this formulation with matrix–fracture deformation and an exponential damage–permeability term, a fractional-order damage-based permeability model is established to describe the complete evolution from elastic deformation through pre-peak damage to post-peak failure. The model parameters are calibrated separately using published datasets for protective-seam mining, top-coal caving, no-pillar mining, and a full-process loading case. The calibrated model yields coefficient of determination (R2) values of 0.9374, 0.9625, 0.9875, and 0.9980, respectively. The identified fractional order is λ = 1 for the three mining-disturbance datasets, whereas the full-process dataset yields λ = 0.7734. For the full-process dataset, the fractional-order model reduces root mean square error (RMSE) and mean absolute error (MAE) by approximately 31.4% and 34.7%, respectively, compared with its integer-order counterpart. Sensitivity analysis shows that λ, p, εd, and γ play distinct roles in permeability evolution. At an axial strain of 0.8%, increasing εd from 0.721% to 1.121% decreases k/k0 from 2.6919 to 1.3433, whereas increasing γ from 0 to 2.543 increases k/k0 from 1.0003 to 3.0334, indicating that εd and γ strongly affect the strain level and magnitude of permeability enhancement, respectively. The proposed model provides an effective tool for characterizing the nonlinear permeability evolution of deep coal under mining disturbance.