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In the past decades, many permeability models have been proposed to characterize the coal permeability evolution in the elastic state. Considering that the coal near the mining working face is often in the plastic or post-peak failure state, it is significant to characterize the coal permeability evolution in the plastic and post-peak failure stages for safe mining. In this study, two permeability models are developed using a variable-order fractional derivative to characterize the coal permeability evolution during the whole process of elastic, plastic, and post-peak failure stages. The results indicate that both models have the ability to better describe the coal permeability evolution during the whole process in areas near the mining working face. In addition, the physical and mechanical interpretation of the variable-order function is given as an indicator of fracture development, indicating the sensitivity of stress variation to coal fracture development. Moreover, the relationships among the effective stress, damage variable, and post-peak permeability of coal are discussed.

Finding a fractional derivative that can more accurately describe the characteristics of a dynamic process is one of the objectives for fractional derivative modeling. In this paper, Atangana–Baleanu (AB) fractional order dashpots with constant and variable coefficients are constructed by using AB fractional derivatives with nonlocal and nonsingular kernel characteristics. Compared with Riemann–Liouville (RL) and conformable (KA) fractional order dashpot, AB fractional order dashpot can describe the viscoelastic properties of materials in full time. A new damage AB fractional derivative (DABFD) creep model is proposed by improving the Nishihara model with the help of the variable coefficient AB fractional dashpot. The parameters for the DABFD creep model are determined by fitting creep experimental data of Beishan granite. Compared with the Nishihara, damage RL fractional derivative (DRLFD) and damage KA fractional derivative (DKAFD) creep models, DABFD creep model has a better fitting and prediction accuracy. The parameter sensitivity for the DABFD creep model shows that different creep curves can be demonstrated with different DABFD creep model parameters. The model can better represent the viscoelastic properties and three stages of the creep process for rock.

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