## 4 Results and Discussion
### Insights into the Relationships among Coal Porosity, Gas Diffusivity and Permeability  
In this section, the gas diffusivity and permeability data are obtained from experimental measurement. Figure 5 shows that the \(\mathrm{R_{o,m}}\) exhibits no significant relationship with effective diffusion coefficient (\(D_{e}\)) and permeability [45]. However, there is a subtle exponential relationship with coefficient of determination (\(R^{2}\)) of 0.41 between permeability and diffusivity as shown in Figure 6. Although no exact function of permeability and diffusivity exists, it indeed demonstrates an increasing gas diffusivity trend with the increase of permeability. Coefficient of determination (\(R^{2}\)) can be calculated as follows:  
\[R^{2}=\frac{SSR}{SST}=\frac{SST-SSE}{SST}=1-\frac{SSE}{SST}=1-\frac{\sum_{i =1}^{n}{(y_{i}-\hat{y}_{i})^{2}}}{\sum_{i=1}^{n}{(y_{i}-\overline{y})^{2}}} \tag{15}\]  
where \(\hat{y}_{i}\) is the predicted value of \(y_{i}\); \(\overline{y}\) is the mean value of \(y\); SSE is the sum of squares due to error:  
\[\mathrm{SSE}=\sum_{i=1}^{n}{(y_{i}-\hat{y}_{i})^{2}} \tag{16}\]  
SSR is the sum of squares of the regression:  
\[\mathrm{SSR}=\sum_{i=1}^{n}{(\hat{y}_{i}-\overline{y})^{2}} \tag{17}\]  
SST is the total sum of squares:  
\[\mathrm{SST}=\sum_{i=1}^{n}{(y_{i}-\overline{y})^{2}} \tag{18}\]  
From the above equation, if using a good fitting model, \(R^{2}\) should vary between 0 and 1. A value closes to 1 indicates that the fit is good. Figure 7 shows the relationship of \(D^{\prime}\) and porosity constructed for the different rank coals and five other rocks. A distinct power function relationship exists between \(D^{\prime}\) and porosity for coals and other rocks. The cementation factor (m values) can be deduced by Equation (3). All twelve coals show relatively high m values (from 2.03 to 2.46), which tends to exhibit a larger m value for coals compared with other rocks. The results can be divided into two groups according to the different m values. The other rocks group includes dolomite, limestone, Indiana sandstone, Berea sandstone and red brick have the m value of 1.5, while for the different rank coals, the cementation factor is 2.3 due to the nature of organic rocks [2]. The coals exhibit an organized behavior and display an exponential relationship between \(D^{\prime}\) and average pore diameter (\(d_{a}\)) as shown in Figure 8. For the group of coals with m = 2.3, the relationship is expressed as follows with the coefficient of determination (\(R^{2}\)) of 0.78:  
\[D^{\prime}=0.00028*d_{a}^{1.217} \tag{19}\]  
where \(D^{\prime}\) denotes diffusivity (dimensionless) and \(d_{a}\) is the average pore diameter in \(\upmu\)m. For the group of other rocks with m = 1.5, the relationship can be summarized as follows with the coefficient of determination (\(R^{2}\)) of 0.87:  
\[D^{\prime}=0.00453*d_{a}^{0.422} \tag{20}\]  
The complicated pore structure of coals makes it difficult to make further conclusions. The relationship for permeability versus porosity of twelve coals with different ranks and five other rocks were plotted in Figure 9. The relationship between permeability and porosity has been well established by previous research [46; 47; 48]. In this study, the results show two different exponential relationships for these two group rocks with different m values. For the group of coals with m = 2.3,the relationship between permeability and porosity can be summarized as follows with the coefficient of determination (\(R^{2}\)) of 0.77:  
\[\mathrm{k}=583.8*\varphi_{a}^{2.95} \tag{21}\]  
where \(\mathrm{k}\) is permeability in \(\mathrm{\SIUnitSymbolMicro m}^{2}\) and \(\varphi_{a}\) is air-filled porosity. This study may provide a new perspective on the fluid flow evaluation of unconventional reservoirs with the classic mercury porosimetry technique.