Kinetic properties of coal gas desorption based on fractional order fractal diffusion equation in time

The precise characterization of gas transport mechanisms within the coal matrix was essential for ensuring the safety of coal seam mining and the effective utilization of coalbed methane. Traditional Fick's model, which assumes coal is isotropic and homogeneous, fails to accurately represent coal's properties. To address this, we introduced a fractal dimension into Fick's model, considering coal's non-homogeneity, complex topology, strong adsorption characteristics, and natural diffusion behaviors. We reformulated first-order differential equations into fractional partial differential equations, creating the time fractional order fractal diffusion (TFFD) model to better explain gas diffusion in heterogeneous coal. The model was analyzed using separation of variables, allowing us to determine gas diffusivity based on Caputo-type fractional order definitions and fractal theory. Results show that the TFFD model is particularly relevant during the rapid desorption phase, with the anomalous diffusion index mirroring the initial diffusion coefficient's variation patterns. The correlation between the fitted equations and experimental pressure data achieved a correlation coefficient exceeding 0.98, highlighting the TFFD model's enhanced relevance and applicability in characterizing the rapid gas desorption process within coal.