## 4 Discussion
### TC Permeability Coefficient Prediction Model  
The variable head permeability test is the traditional method used to measure the permeability coefficients of clay-quartz mixtures. The method is time- and effort-consuming. NMR technology has  
Figure 11: The relationships between \(T_{2c}\) and the clay content (CC). The lines at the top and bottom indicate the relationships between the \(T_{2c}\) and CC of kaolinite-quartz (K-Q) and montmorillonite-quartz (M-Q) mixtures, respectively. The Equations in the two boxes are the Equations of the two lines. \(R^{2}\) indicates the correlation coefficient of the Equation. Figure 10: \(T_{2}\) cut-offs (\(T_{2c}\)) of the eight mixtures: (**a**) K40; (**b**) K60; (**c**) K80; (**d**) K100; (**e**) M40; (**f**) M60; (**g**) M80; (**h**) M100. The black curve with a solid square and red curve with solid circle indicate \(T_{2}\) spectra of the saturated mixture before evaporation and the mixture at the beginning of the falling rate stage, respectively. The curves with an open square and open circle indicate the cumulative signal amplitude of \(T_{2}\) spectra with the same color. The intersection of the black curve with the open square and the red curve with the open circle was enlarged and placed in the upper left corner of (**a–d**). gradually become a commonly used method for determining the permeability of porous media. For shales, coals, and sandstones, it has been proved that the Schlumberger Doll Research (SDR) model and Timur-Coates model (TC model) can be used to predict permeability. Because of the complexity of the structures, whether these two models are suitable for predicting the permeability of clay-quartz mixtures with different mineral compositions has not been previously verified. The SDR model can be expressed by Equation (19):  
\[k_{SDR}=a\varphi_{NMR}{}^{m}\Big{(}T_{2 gm}\Big{)}^{n} \tag{19}\]  
where \(k_{SDR}\) is the permeability predicted by the SDR model and \(\varphi_{NMR}\) is the porosity measured by the NMR test. The empirical constants \(a\), \(m\), and \(n\) are related to the basic properties of porous media. This model used the average pore size to predict permeability. However, because the surface relaxivity of mixtures with different mineral compositions varied greatly, the \(T_{2 gm}\) could not be used directly to compare the average pore size. It is unreasonable to use the SDR model to predict the permeability of clay-quartz mixtures with different mineral compositions. The TC model can be expressed by Equation (20):  
\[k_{TC}=a\varphi_{NMR}{}^{m}\Big{(}\frac{FFV}{BFV}\Big{)}^{n} \tag{20}\]  
where \(k_{TC}\) is the permeability predicted by the TC model, \(FFV\) is the volume of free fluid, and \(BFV\) is the volume of bound fluid. The TC model is also called the free fluid model. In NMR tests, \(FFV\) corresponded to the cumulative \(T_{2}\) signal amplitude between \(T_{2c}\) and \(T_{2 max}\) (\(A_{FF}\)), and \(BFV\) corresponded to the cumulative \(T_{2}\) signal amplitude between \(T_{2 min}\) and \(T_{2c}\) (\(A_{BF}\)). Equation (21) can be revised to form:  
\[k_{TC}=a\varphi_{NMR}{}^{m}\Big{(}\frac{A_{FF}}{A_{BF}}\Big{)}^{n} \tag{21}\]  
To verify the applicability of the TC model, variable head permeability tests were used to measure the permeability coefficients of the eight mixtures. What is worth noting is that the permeability coefficient (\(K\)) determined from variable head permeability tests is not the permeability (\(k\)) in Equation (21). The \(k\) reflects the inherent permeability of porous media, independent of fluid properties. The \(K\) is not only affected by the properties of porous media, but also by the properties of fluids. The relationship between \(K\) and \(k\) is shown in the following Equation:  
\[K=\frac{\rho\mathcal{g}}{\eta}k \tag{22}\]  
where \(\varphi\), \(\mathcal{g}\), and \(\eta\) are the density, gravitational acceleration, and dynamic viscosity coefficient of the fluid, respectively. For the distilled water used in this study, these three parameters were fixed values.