## 4 Results Analysis and Discussions
### Reasons for the Changes in Permeability under Cyclic Loading and Unloading Conditions
This indicates that as long as the porosity of the broken coal sample is same during each loading and unloading (Equation (8) shows that strain and porosity are equivalent), the permeability of the broken coal sample is the same, independent of the stress state of the broken coal sample at that time. For example, it can be seen from Figures 11 and 12 that in each loading and unloading process, when the strain or porosity at the loading stage was equal to the strain or porosity at the unloading stage, the effective stress at the loading stage was greater than the effective stress at the unloading stage. At this time, the stress states of the broken coal samples at the loading stage and unloading stage were different,  
\[d_{1}=d_{0}(1-\varepsilon). \tag{7}\]1 Relation curve between permeability and strain under cyclic loading and unloading conditions of broken coal samples: (**a**) the first cyclic loading and unloading, (**b**) the second cyclic loading and unloading, (**c**) the third cyclic loading and unloading, (**d**) the fitting formulas between permeability and strain. but the permeability was the same. Therefore, the permeability of broken coal samples under cyclic loading and unloading conditions was determined by its porosity, which conforms to the cubic law. The relationship between permeability and porosity is as follows [35; 36]:  
\[d_{1}=d_{0}(1-\varepsilon). \tag{7}\]2  
where \(\alpha\) is the porosity sensitive exponent; \(K_{0}\) and \(\phi_{0}\) are the initial permeability and porosity, respectively. It can be seen from Figure 15 that the porosity sensitive exponent of the permeability and porosity fitting formula in three loading stages is 3, 2.82 and 2.69, respectively. The porosity sensitive exponent of the permeability and porosity fitting formula in three unloading stages was 2.87, 2.72 and 2.63, respectively. It indicates that both in loading and unloading stages, the porosity sensitive exponent of broken coal samples all gradually decreased with increased cyclic loading and unloading (loading stage: 3 \(>\) 2.82 \(>\) 2.69; unloading stage: 2.87 \(>\) 2.72 \(>\) 2.63). During the same loading and unloading cycle, the porosity sensitive exponent of coal samples in the loading stage was higher than that in the unloading stage (first loading and unloading: 3 \(>\) 2.87; second loading and unloading: 2.82 \(>\) 2.72; third loading and unloading: 2.69 \(>\) 2.63). Therefore, this further verifies the above analysis results: Both in loading and unloading stages, the stress sensitivities of permeability of broken coal samples all gradually decreased with increased cyclic loading and unloading. During the same loading and unloading cycle, the stress sensitivity of permeability of coal samples in the loading stage was far higher than that in the unloading stage. \[d_{1}=d_{0}(1-\varepsilon). \tag{7}\]3 Relation curve between permeability and porosity under cyclic loading and unloading conditions of broken coal samples: (**a**) the first cyclic loading and unloading, (**b**) the second cyclic loading and unloading, (**c**) the third cyclic loading and unloading, (**d**) the fitting formulas between permeability and porosity.