Fractal-Time-Dependent Fick Diffusion Model of Coal Particles Based on Desorption–Diffusion Experiments

The diffusion coefficient is a crucial parameter to manifest the diffusivity of methane in coal, which is obtained by solving mathematical and physical equations that characterize the diffusion process. The diffusion model established by combining the pore structure parameters and time-dependent characteristics in Euclidean space cannot reflect the natural heterogeneity of coal. There is a fundamental difference between the fractal structure and time-dependent diffusion characteristics in Euclidean space. Based on the classic homogeneous spherical Fick diffusion model of coal particles, the full-aperture comprehensive fractal dimension and time-dependent diffusion decay coefficient are introduced, forming the fractal-time-dependent Fick diffusion (FTFD) model of coal particles. To verify the new model, we obtained model parameters by mercury injection, low-temperature liquid nitrogen adsorption, and desorption–diffusion experiments, and the model fitting curve was drawn, which was compared with the desorption–diffusion rate experimental points. In addition, the calculation method of lost-gas content based on the √t approach was also corrected. The results indicated that the new model could describe the whole process of gas diffusion in the heterogeneous coal with good accuracy and stability. The new model also covered the traditional classical homogeneous spherical Fick diffusion and dynamic diffusion-coefficient models with more general theoretical significance. The lost-gas content calculated by the new model was consistent with the actual lost-gas content, improving the accuracy of the lost-gas calculation. The investigation results have important theoretical significance for a comprehensive understanding of the diffusion and transport of pore gas in coal seams.