## 1 Introduction  
Darcy flow is the dominant flow in conventional gas reservoirs. However, in some unconventional gas reservoirs (e.g., tight sands and particularly in coals) where pore-throat radii as small as a few micro-nanometers are common, diffusion plays an important role and should be taken into account [1]. The common Darcy equation cannot fully capture the physics of flow in the micro-nanopore structure of coalbed methane (CBM) reservoirs. For CBM reservoirs, the viscous effects and other flow phenomenon such as diffusion and slippage effects should be considered. Therefore, a more rigorous approach is needed to accommodate submicron effects in micro-nanopores of some low-permeability CBM reservoirs.  
Gas near solid surface has a tendency to slip. Diffusion is one of the key processes controlling gas transport in porous media [2]. Diffusion may dominate gas transport when seepage velocity is \(\sim\)0.005 m per year [3]. Knudsen diffusion of gas molecules and their collision with solid walls is one of the most fundamental studies in unconventional gas reservoirs [4] For coals, previous studies indicate that gas diffusion correlated well with coal type and coal rank due to the meso and macropore quantity [5; 6; 7; 8]. However, in two coals with the same particle size and coal rank but different maceral composition, the previous research indicates that the coal with high interinites shows a fast diffusivity [9; 10; 11; 12; 13]. For gas diffusivity, the determination methods may include the particle method, the steady state method and the inverse diffusion method [14; 15; 16]. The effective diffusivity strongly depends on pore size when the average pore size is less than 1 \(\upmu\)m [17; 18; 19]. The effect of gas slippage in porous media can improve permeability [20; 21]. In general, Klinkenberg's effect becomes important when the mean free path of gas molecules is comparable to the pore-throat radius. Normally, Klinkenberg's effect increases in fine-grained, low-permeability porous media [22]. The effect of gas slippage is even more pronounced for rarefied gas flow where the Knudsen number is much higher (\(K_{n}>10\)). When the molecular mean free path is within two orders of magnitude of pore-throat diameter (0.1 \(<K_{n}<\) 1), gas molecules tend to slip on the pore surface. Consequently, permeability to gas results in higher values compared with permeability to liquid for a given porous medium including coals and shales. Many techniques can be used to measure the diffusivity and permeability in coals [14; 15; 16; 20; 21; 22]. However, some scholars proposed some theoretical models to evaluate the diffusivity and permeability in porous media from the perspective of pore structure in coals. The scanning electron microscope (SEM), low-temperature N\({}_{2}\) adsorption/desorption, mercury intrusion porosimetry (MIP), and nuclear magnetic resonance (NMR) techniques can be adopted to acquire the info of pore structure. In addition, then the effective diffusion coefficient and absolute permeability can be evaluated with the info of pore structure from these methods [23; 24; 25].  
Mercury intrusion porosimetry (MIP) is a widely used technique for determining pore structure in porous materials including conventional and unconventional reservoirs (e.g., tight sand, shales and coals) [26; 27; 28; 29]. Application of the MIP method to coals has been established for a long time as a routine way to assess the pore information including pore volume, porosity and pore size distribution. In this work, the info of pore structures for different rank coals was acquired by MIP data first. The theoretical models for effective gas diffusion coefficients and permeability in coals were derived. In addition, then the regular CH\({}_{4}\) diffusivity and permeability measurements are conducted to compare with the results of the derived CH\({}_{4}\) diffusivity and permeability with MIP data. Finally, the relationships among gas diffusivity and permeability were evaluated with the info of pore structures.