## 4 Discussion
### Impact on Permeability  
The aim in this discussion is to investigate how incorporating the notion of fines cohesive packing affects the prediction of permeability, \(K\), for granular materials. Porosity, which can be significantly affected by cohesive packing as mentioned earlier, is a major factor that impacts \(K\). The Kozeny-Carman formula is frequently used for estimating \(K\) for mixed soils and granular materials [2; 4; 8; 33; 35; 74; 75; 76], where \(K\) is expressed as a function of porosity according to the following Equation (8):  
\[K=\left(\frac{d_{rep}^{2}n^{3}}{180{(1-n)}^{2}}\right), \tag{8}\]  
where \(d_{rep}\) is the representative particle size. For the pure unmixed sand or fines samples, representative particle size can be estimated from Equation (8) using the measured \(K\) and \(n\) values. For sand-fines mixtures, [PERSON] et al. [35] suggested to use a power-averaging method to determine \(d_{rep}\) of the mixture at any given \(f_{f}\) as follows:  
\[d_{rep}=\left(f_{l}\;d_{repL}^{m}+f_{f}\;d_{repF}^{m}\right)^{1/m}, \tag{9}\]  
where \(d_{repL}\) and \(d_{repF}\) are the representative particle size of the large and fine particles, respectively. The coefficient \(m\) varies sigmoidally from \(-1\) to \(0\) as \(f_{f}\) decreases from \(1\) to \(0\) and it can be estimated empirically [35] through Equation (10):  
\[m=\frac{1}{1+\exp[a\big{(}f_{l}^{*}-f_{l}\big{)}]}-1, \tag{10}\]  
where \(f_{l}^{*}\) is the critical large particle fraction (\(f_{l}^{*}=1-f_{f}^{*}\)) above which \(K\) varies significantly with increasing \(f_{l}\); and \(a\) is an empirical factor that controls the steepness of the \(a-f_{f}\) curve. In order to find \(a\), the \(d_{rep}\) for sand-fines mixtures was calculated using the measured \(n\) and \(K\) in Equation (8) and then Equations (9) and (10) were used to find the best fit factor \(a\), that can reproduce the \(d_{rep}\) of the mixture at variable fines fraction. An example is given in Figure 8 showing the calculated \(d_{rep}\) for mixture M1 and the best fit curve using Equations (9) and (10). We found that \(a=10\) provides the best fit for \(d_{rep}\) data calculated based on measured \(K\) and \(n\) for the different mixtures tested in the study (where \(K\) data is available). Once the estimating equation for \(d_{rep}\) is determined as function of \(f_{f}\), \(K\) can be calculated as function of \(f_{f}\) given that \(n\) is calculated using the current model [Equations (1)-(7)], where \(p=0.15\) and \(d_{cri}=0.15\) mm. In this case, \(K\) is predicted at any given \(f_{f}\), provided that the \(K\) and \(n\) of pure end-members are measured. The prediction of the current model shows good match with experimental data (Figure 9). The prediction of \(K\) assuming ideal packing for \(n\) is also shown using dashed curve in Figure 9 for comparison. The use of the current packing model improves the prediction of \(K\) in mixture M1 compared to the ideal packing model which underestimates \(n\) (Figure 3) and consequently underestimates \(K\) (Figure 9). On the other hand, both current model and ideal packing model predicts very similar \(K\) (Figure 9b) for mixture M11 which represents a near-ideal packing case (Figure 4b) as discussed earlier. That is, the current model can improve the prediction of \(K\) for mixtures where packing deviates from ideal case. The main discrepancy between the two models is again observed around critical fines fraction (Figure 9a), at which the maximum discrepancy in predicted \(n\) occurs. The maximum difference between the \(K\) predicted by incorporating the current model and that assuming ideal packing is 70%, for mixture M1. Note that the difference does not seem as significant in Figure 9a only because \(K\) is plotted in log scale. Another important observation is that the minimum \(K\) can be observed around \(f_{f}^{*}\) only if the mixture follows the ideal packing condition (Figure 9b). Otherwise, \(K\) at \(f_{f}=1\) is smaller than \(K\) at \(f_{f}^{*}\) (Figure 9a).