## 2 Background
### Incorporating Fines Cohesive Packing Behavior  
We present here a brief summary of the formulation for the packing model developed by [PERSON] et al. [41] for binary mixtures of glass beads, while more details can be found in the original paper. According to the packing model presented by [PERSON] et al. [41], two main mechanisms could explain the increase in porosity associated with non-ideal packing for any binary mixture: the loosening and the wall effects. Regarding the loosening effect, it impacts packing when the large particles are dominant (\(f_{f}<f_{f}^{*}\)) where fine particles occupy the spaces between large particles and loosen their packing. On the other hand, the wall effect occurs when the fine particles dominate the packing (\(f_{f}>f_{f}^{*}\)); in such case the large particles disrupt the original packing of the fine particles by introducing larger pores at the large particles wall boundaries. The impact of the above two mechanisms on porosity is accounted for through the use of empirical functions which are incorporated into the ideal packing model. The binary mixture porosity in the large particle dominated (\(n_{ml}\)) and fine particles dominated (\(n_{mf}\)) case, can be found using Equations (1) and (2), while the corresponding correction functions for the loosening (\(L\)) and wall (\(W\)) effect can be estimated using Equations (3) and (4) respectively:  
\[\frac{1}{1-n_{ml}}=\left(\frac{f_{f}}{1-n_{f}}+\frac{f_{l}}{1-n_{l}}\right)- \left(1-L(r)\right)\frac{f_{f}}{1-n_{f}},\quad\text{when }f_{f}<f_{f}^{*} \tag{1}\]  
\[\frac{1}{1-n_{mf}}=\left(\frac{f_{f}}{1-n_{f}}+\frac{f_{l}}{1-n_{l}}\right)- \left(1-W(r)\right)n_{l}\frac{f_{l}}{1-n_{l}},\quad\text{when }f_{f}>f_{f}^{*} \tag{2}\]  
\[L\left(r\right) =1-\left(1-r\right)^{3.3}-2.8\ r\left(1-r\right)^{2.7} \tag{3}\] \[W\left(r\right) =1-\left(1-r\right)^{2.0}-0.4\ r\left(1-r\right)^{3.7}. \tag{4}\]  
In the above expressions: \(f_{f}\) is the mass fraction of fine particles; \(f_{l}\) is the mass fraction of large particles (\(f_{l}=1-f_{f}\)); \(n_{f}\) and \(n_{l}\) are the porosities of the two end-members (i.e., pure unmixed fine and pure unmixed large particles respectively). The subscripts \(f\) and \(l\) refer to fine (i.e., clay or silt) and large (i.e., sand) particles, respectively throughout this paper. The quantity \(r\) is the effective size ratio, which for binary mixtures of non-cohesive spheres will be equal to the actual fine-to-large particle size ratio, \(R\)[41]. Nevertheless, the authors found that below a critical fine particle size value (\(d_{cri}\)), the porosity of fine glass beads increases drastically as the particle size decreases due to the effect of inter-particle forces (i.e., cohesive forces). The \"cohesive degree\" describes how \(n\) varies with particle size below \(d_{cri}\) which separates the cohesive and non-cohesive behavior. In the presence of cohesive behavior, the effective size ratio (\(r\)) will be larger than \(R\) due to fines aggregation and can be found using Equation (5):  
\[r=(1-x)R^{p}+x\ R. \tag{5}\]  
In Equation (5), \(p\) is a material constant that quantifies cohesive packing degree of the fine particles [41]. The degree of cohesive packing increases as \(p\) decreases, which leads to forming larger cohesive aggregates (i.e., \(d_{f}\) increases) and thus, \(r\) increases. The parameter \(x\) describes how the fines cohesive packing contributes to the packing disturbance of large particles in the binary mixture and it is given by [41]:  
\[x=1-1.543 exp\left(-0.697\frac{d_{l}}{d_{cri}}\right), \tag{6}\]  
based on Equation (6), it is evident that \(x\) decrease as \(d_{l}\) decreases and becomes more similar to \(d_{cri}\) indicating that fine particles significantly disturb the packing of larger particles in the mixture. In this case, \(r\) increases for a given \(p\) according to Equation (5), suggesting larger loosening and wall effects [Equations (3) and (4)] and causing porosity to increase at a given \(f_{f}\).