## 3 Theoretical Study
### Permeability Model  
The permeability evolution model of gas-bearing coal considering changes in the loading and unloading rates was established based on some ideal assumptions. The basic assumptions were as follows: (1) Coal is a homogeneous continuous medium. (2) Coal contains only a single-phase gas fluid. The gas in coal exists in an adsorbed and free state, and the gas flow conforms to [PERSON]'s law. (3) The process of gas adsorption and desorption in coal accords with the Langmuir adsorption equilibrium equation, and both of them are reversible. The free gas conforms to the Ideal Gas Law. (4) The coal particles and gas are incompressible. (5) The whole process is isothermal regardless of the influence of temperature change. (6) The deformation of coal is much smaller than the size of the coal. (7) The Klinkenberg effect is not considered. According to the previous study, the permeability has a significant relationship with the porosity. The relationship between permeability and porosity can be expressed as follows [35; 36]:  
\[k\ =\ \frac{{d_{e}}^{2}\phi^{3}}{72{(1-\phi)}^{2}} \tag{1}\]  
where \(k\) is the permeability of coal, \(\phi\) is the porosity and \(d_{e}\) is the effective diameter of coal particle. From Equation (1), the relationship between permeability and initial permeability can be obtained as:  
\[\frac{k}{k_{0}}\ =\ \left(\frac{\phi}{\phi_{0}}\right)^{3}\!\!\left(\frac{1-\phi_{ 0}}{1-\phi}\right)^{2} \tag{2}\]  
where \(k_{0}\) and \(\phi_{0}\) present the initial permeability and initial porosity, respectively. Since the porosity is much less than 1, the second item on the right side of Equation (2) can be omitted. Then, the relationship between permeability and porosity can be written as:  
\[\frac{k}{k_{0}}\ =\ \left(\frac{\phi}{\phi_{0}}\right)^{3} \tag{3}\]  
Based on the definition, the porosity can be expressed as:  
\[\phi\ =\ 1-\frac{V_{s}}{V}\ =\ 1-\frac{V_{s0}(1+\Delta V_{s}/V_{s0})}{V_{0}+ \Delta V}\ =\ 1-\frac{\left(V_{0}-V_{p0}\right)\!\left(1+\Delta V_{s}/V_{s0}\right)}{V_{0}+ \Delta V}\ =\ 1-\frac{1-\phi_{0}}{1+\varepsilon_{s}}(1+\varepsilon_{s}) \tag{4}\]  
where \(V\) is the volume of coal, \(V_{s}\) is the volume of coal skeleton, \(V_{0}\) is the initial volume of coal, \(\Delta V\) is the volume change, \(V_{s0}\) is the initial volume of coal skeleton, \(\Delta V_{s}\) is the volume change of coal skeleton, \(\varepsilon_{v}\) is the volumetric strain of coal and \(\varepsilon_{s}\) is the volumetric strain of coal skeleton. The initial volume and volume of coal are:  
\[V_{0}\ =\ V_{p0}+V_{s0} \tag{5}\]  
\[V\ =\ V_{p}+V_{s} \tag{6}\]  
where \(V_{p}\) and \(V_{p0}\) present the pore volume and the initial pore volume, respectively. From Equation (5) and Equation (6), the volumetric strain can be expressed as:  
\[\varepsilon_{v}\ =\ \frac{V-V_{0}}{V_{0}}\ =\ \frac{V_{p}-V_{p0}+V_{s}-V_{s0}}{V_{0} }\ =\ \frac{V_{p}-V_{p0}}{V_{p0}}\frac{V_{p0}}{V_{0}}+\frac{V_{s}-V_{s0}}{V_{s0}} \frac{V_{s0}}{V_{0}} \tag{7}\]  
where \(V_{p0}/V_{0}\) presents the initial porosity and \(V_{s0}/V_{0}\) can be represented by \((1-\phi_{0})\). Rearranging Equation (7) leads to:  
\[\varepsilon_{v}\ =\ \varepsilon_{p}\phi_{0}+\varepsilon_{s}(1-\phi_{0}) \tag{8}\]where \(\varepsilon_{p}\) is the pore compression strain and it can be written as:  
\[\varepsilon_{p}\ =\ C_{p}(d\sigma-dp) \tag{9}\]  
where \(\sigma\) and \(p\) present the stress and gas pressure, respectively, \(C_{p}\) is the pore volume compressibility [37].