# A Simple Water Retention Model Based on Grain Size Distribution
## 3 Proposed Model
If the sizes reported in the GSD are assumed to represent diameters of spherical clay and silt sized particles, which are in fact platy, then the mean size of the particles will be overestimated, and this will affect the inferred packing of these smaller particles, resulting in an underestimation of the number of smaller pores. Hence, a more realistic grain packing will be achieved by reducing the sizes of small particles in the GSD to reflect their non-sphericity. Figure 5 shows data derived from [31; 47; 48; 49; 50; 51] which can be used as a basis to correlate AR to particle size. In the proposed model, the sizes of the particles in each size fraction of the GSD (d\({}_{\rm i}\)), smaller than sands (<60 \(\mu\)m), are adjusted by factoring them by an aspect ratio (AR\({}_{\rm i}\)), determined by the function fitted to the data in Figure 5, according to Equation (2). \[ \begin{array}{ll}\text{d}_{\text{i,adjusted}}\ =\frac{\text{d}_{\text{i}}}{\Delta\text{R}_{ \text{i}}}&\text{d}_{\text{i}}\ \leq\ 60\ \mu\text{m}\\ \text{d}_{\text{i,adjusted}}\ =\text{d}_{\text{i}}&\text{d}_{\text{i}}>\ 60\ \mu\text{m}\end{array}  \tag{2}\]  
In Equation (2), \(\text{d}_{\text{i,adjusted}}\) is the adjusted maximum particle size in the i-th bin of the GSD, \(\text{d}_{\text{i}}\) is the unaltered maximum particle size in the i-th bin of the GSD, and \(\text{AR}_{\text{i}}\) is the aspect ratio determined from the fitting equation shown in Figure 5.  
It should be noted that some clays are reported to have aspect ratios on the order of hundreds [33], however these extreme values were not considered in this model, which could result in limitations for some materials. Note, this large variability in aspect ratio between clay particles may be due to their mineralogy, which was ignored in this model for the purpose of using simple to measure soil properties. For the first step in determining a primary packing, particles within each GSD size bin are assumed to pack together to create circular pores, according to one of the arrangements shown in Figure 1. Pores formed between three particles of diameter \(\text{d}_{\text{i,adjusted}}\) can accommodate smaller particles up to 0.15d\({}_{\text{i,adjusted}}\); pores formed between four particles of diameter \(\text{d}_{\text{i,adjusted}}\) can accommodate particles up to 0.41 \(\text{d}_{\text{i,adjusted}}\). Assuming pores are similarly likely to form from three or four particle arrangements, the average pore diameter of the two arrangements in Figure 1 (taken as 0.3 \(\text{d}_{\text{i,adjusted}}\)) is used as the basis to relate particle size to pore size according to Equation (3), which is similar to Equation (1) that has been found to be accurate for sands [23]. \[\text{p}_{\text{i}}=0.3\text{d}_{\text{i,adjusted}} \tag{3}\]  
In Equation (3) \(\text{p}_{\text{i}}\) is the primary pore diameter associated with the i-th particle size bin, and \(\text{d}_{\text{i,adjusted}}\) is the particle size as determined from Equation (2). After a series of primary pore sizes are obtained, the volume of primary pores with diameter \(\text{p}_{\text{i}}\) is determined by assuming the total pore volume associated with particle size  
Figure 5: Collated values of the average aspect ratio found for different particle sizes from the literature, measured using optical, electron, or atomic microscopy. A fitting equation and function (solid line) is shown, which describes the general trend in the aspect ratio values observed, as a function of particle size diameter \(\text{d}_{\text{i}}\). Data adapted from [31; 47; 48; 49; 50; 51]. bin i is proportional to the mass fraction of size bin i. Thus, the saturated volumetric water content associated with only the i-th particle size bin \(\theta_{\rm i}\), is given by Equation (4) as:  
\[\theta_{\rm i}=x_{\rm i}\theta_{\rm s} \tag{4}\]  
where \(x_{\rm i}\) is the mass fraction of particle size bin i, and \(\theta_{\rm s}\) is the saturated volumetric water content of the entire soil or material for the conditions under which the WRC is being determined.