# A Simple Water Retention Model Based on Grain Size Distribution
## 3 Proposed Model
### Primary Pore Filling
#### 3.2.3 Filling of Pores Associated with Silt Sized Particles  
For primary pores \(\mathrm{p_{i}}\) generated from silt sized particles (where \(\mathrm{d_{i,adjusted}}\) is between 2 and 60 \(\mathrm{\upmu m}\)), filling is assumed to occur if there are deemed to be large enough fractions of suitably smaller particles, and these occur in suitable proportions. In this context, the amounts of particles are not considered on a rigorous, global mass or volume balance; rather they are determined separately for each set of primary pores by ensuring that there are smaller particles present in the GSD such that a binary mixture of smaller particles can exist to fill the primary pores to a packing density consistent with those corresponding to the partial or maximal filling shown Figure 7b,c. Figure 7 provides values of packing densities for the Apollonian packing (0.844 for first level packing and 0.901 for second level packing) that can be adopted as values that would be achieved if different levels of primary pore filling were to occur. The obvious and direct criteria to consider would be whether there are enough particles \(\mathrm{d_{i1}}\) to fill the primary pores \(\mathrm{p_{i}}\) (refer to Figure 6), and enough \(\mathrm{d_{i2}}\) particles to fill the second level pores associated with \(\mathrm{d_{i1}}\). When this approach is implemented, the predicted WRCs are poor approximations to measured data, as the model fails to fill some pores that would improve the prediction accuracy. Upon assessment of the results from this direct Apollonian filling process, it is considered likely that its poor performance arises because the Apollonian gasket pore filling shown in Figure 7 is strictly a 2D representation of a 3D phenomenon. So, rather than considering pore filling directly using the 2D geometry of Figure 7, the data for packing density as a function of the size ratio (denoted SR) and proportions of particles in a 3D binary mixture in Figure 2 (reproduced below as Figure 8) will be used. Figure 7: Diagram showing the three possible states of a pore in the proposed model, along with the packing density (PD) associated with each state: (**a**) unfilled pores, where packing density is similar to random close packing density; (**b**) partial pore filling state where three particles fill the pore; (**c**) maximal pore filling state where six particles fill the pore. As an alternative approach, the pore filling criterion was modified to consider the size and proportion of mono-sized particles needed to fill the primary pores between particles \(\text{d}_{\text{i,adjusted}}\), to achieve the same packing density as partial or maximal filling in Figure 7. This was done by combining the packing density determined from Figure 7 with the literature data from Figure 2, to create Figure 8. Figure 8 shows the packing density achieved in binary particle mixtures with different smaller:coarse particle size ratios and fractions of coarser particles in the binary mixture. The unfilled pores condition in Figure 7 corresponds to an overall packing density that is a similar value to the random close packing density of spheres [56], which is consistent with the minimum void ratio of sands [57], aligns with the model's use of Equation (3), and is similar to the value for single sized particle packing in Figure 8 (when the coarse fraction is 0 or 1). Figure 8 indicates that in a binary particle mixture, a minimum SR of 6.5 is needed to achieve a packing density of 0.844 (i.e., partial filling Figure 7b), and the fraction of coarser particles (denoted FCP) must not be greater than about 0.72 (or else there are not enough smaller particles to fill the pores between the coarser particles to achieve the assumed packing density).