As one of the hot issues at the frontiers of science in the world, the multi-scale scientific question has occurred in the fields of natural science and engineering. The seepage in coal-rock, a branch of the multi-scale science, shows its multi-scale scientific question. Coal is a porous medium that contains multi-scale pores with the aperture from millimeter to nanometer. The pore size differential can reach one million orders of magnitude, which causes the multi-scale characteristics in space and time for coal permeability. Therefore, the research on the multi-scale permeability of coal is a critical scientific issue of the coal gas flow as well as an engineering extension of methane drainage. The unsteady diffusion-seepage experiment is conducted for CH4/He with and without stress using a cylindrical coal sample, accompanied by steady state seepage experiment. The experimental results show that the apparent diffusion coefficient of a cylindrical coal sample attenuates with time. This apparent diffusion coefficient shows two different multi-scale characteristics in time, the smooth and dynamic attenuation and the dynamic attenuation in a two-stage step. A dynamic model for the apparent diffusion coefficient is proposed, and it can accurately describe the complete unsteady flow process of gas in a cylindrical coal sample. The physical and mathematical models of the multi-scale pores in series are put forward. Then, the multi-scale structure of pore in series is validated by the mercury injection experiment. After that, the multi-scale permeability model is mathematically proved. Based on the Knudsen number (Kn), the continuous flow, slip flow, transition flow and free molecular flow are identified and introduced with the multi-scale pore size to build a multi-scale permeability model that reflects the effect of the effective stress and gas flow regime. The mechanism of the multi-scale seepage is revealed in this study. The size and the number of pores in series connection are the critical factors to influence the multi-scale permeability. The multi-scale effect can reach tens of thousands orders of magnitude within measurable range. The gas outflow firstly starts from the outside fractures, and then the inside small pores and finally the nano pores. With time goes on, the gradual increase in the number of pores in series connection leads to the gradual decrease in the equivalent pore size, which causes the equivalent pore aperture to get close to the minimum pore aperture. Therefore, the equivalent permeability quickly decreases with time, which is a reflection of the multi-scale space in coal. During the later stage of gas flow, the effect of slip and transition flow regime is larger than that of effective stress with Kn increasing and dominates the permeability. The new experimental observation and modelling of the multi-scale permeability provides an experimental solution for the research of the multi-scale seepage and overcomes the shortcoming of single tube theory. The diffusion and seepage are apparently unified, and the micro-level distinguishment and macro-level union of the multi-scale permeability are realized. © 2023 China Coal Society. All rights reserved.
