# 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
Thus, it is assumed partial pore filling in pore \(\text{p}_{\text{i}}\), requires particles at least 6.5 times smaller than \(\text{d}_{\text{i,adjusted}}\), and the mass fraction for \(\text{d}_{\text{i,adjusted}}\) must not be greater than 72% of the \"total\" mass fraction; here, the \"total\" mass fraction is the sum of mass fractions from \(\text{d}_{\text{i,adjusted}}\) and all adjusted particle size bins at least 6.5 times smaller. Similarly for maximal filling, a SR of 19.2 is required to achieve a packing density of 0.901 (Figure 7c), and the FCP value must not be greater than 0.75. From this information, the minimum possible SR of filling particles was adopted as 6.5 and 19.2 to achieve partial and maximal pore filling respectively (Figure 7b,c). Additionally, it was assumed there are enough finer particles to achieve partial or maximal pore filling if the FCP is less than 0.72 or 0.75 respectively. Hence, two decisions must be made to fill the primary pores in the silt fraction with mono-sized smaller particles:  
* What is the size ratio that defines a suitably smaller particle, to form a binary mixture with a particular degree of filling (packing density)? That is, what size particles (\(\text{d}_{\text{ss,i}}\)) are small enough to effectively fill the pores between the larger particles \(\text{d}_{\text{i,adjusted}}\)? Figure 8: Packing density measured for binary mixtures of spherical particles, showing the effect of size ratio (SR) and the fraction of coarser particles. The horizontal lines show the packing density (PD) of states (b) and (c) from Figure 7. The maximum FCP values that could accommodate these PD values are indicated. Data adapted from [25]. * What is the maximum fraction \(x_{i}\) of larger particles \(d_{i,adjusted}\) that still ensures that there are enough smaller particles (\(d_{ss,i}\)) to achieve the assumed packing density in the coarser particle fraction? That is, how to decide when there are too few smaller particles to effectively fill the pores between larger particles? Partial or maximal pore filling is assumed to occur if the mass fraction of particles sufficiently smaller than \(d_{i,adjusted}\), denoted here by \(x_{ss,i}\), satisfies Equation (6):  
\[FCP_{req}>\frac{x_{i}}{x_{i}+x_{ss,i}}=\frac{coarse\;fraction}{coarse\;fraction+ fine\;fraction} \tag{6}\]  
where \(x_{i}\) is the mass fraction of the i-th particle size bin, and \(FCP_{req}\) is 0.72 for partial pore filling or 0.75 for maximal pore filling. Equation (6) can also be re-expressed in terms of \(x_{ss,i}\) as shown in Equation (7). \[x_{ss,i}>\left(\frac{1}{FCP_{req}}-1\right)x_{i} \tag{7}\]  
In Equations (6) and (7) the sum of smaller particles, \(x_{ss,i}\), can be determined by subtracting from unity the mass fraction of particles larger than \(d_{ss,i}\), as shown in Equation (8):  
\[x_{ss,i}=1\;-\sum_{d_{m}>d_{ss,i}}x_{m} \tag{8}\]  
where \(d_{m}\) is the adjusted particle diameter associated with the mth particle size bin, \(x_{m}\) is the mass fraction of the mth adjusted particle size bin, and \(d_{ssi}\) is the adjusted particle size considered small enough to fill the pores between \(d_{i,adjusted}\) and is determined using Equation (9):  
\[d_{ss,i}=\frac{d_{i,adjusted}}{SR} \tag{9}\]  
where \(d_{i,adjusted}\) is the particle size as determined from Equation (2), and SR is the size ratio taken as 6.5 for partial or 19.2 for maximal pore filling. Note, if Equation (6) or (7) is not true using the SR and FCP values from both partial or maximal pore filling, then no filling is assumed to occur (Figure 7a). The pore filling conditions described so far for silt-sized particles are performed on a per bin basis; that is, each pore \(p_{i}\) is has its degree of pore filling determined. However, a per bin determination of the degree of pore filling is not always conducted. Improved prediction results occur when the largest particle texture class is considered as a whole (not on a per bin basis), and the smaller texture classes are considered on a per bin basis.