# A Simple Water Retention Model Based on Grain Size Distribution
## 2 Key Phenomena for Water Retention Curve Models
### Pore Dimension and Pore Filling  
The determination of the PSD from a GSD is arguably the most important aspect of the GSD-WRC approach, as it has the greatest impact on the prediction. Particle packing and determination of the resulting PSD is a complicated non-linear problem, thus it is common to simplify the particle packing for soils by assuming particles within a certain size range pack discretely and have pores of a similar size range [11; 12; 13; 14; 15; 16; 20; 21], with a two-dimensional framework typically adopted. An example of this assumption in a two-dimensional basis is mathematically described in Equation (1):  
\[\mathrm{p_{i}}=0.3\mathrm{d_{i}}, \tag{1}\]  
where \(\mathrm{p_{i}}\) is the pore diameter associated with particle size fraction \(\mathrm{i}\) of a soil, and \(\mathrm{d_{i}}\) is the mean diameter of the particles in size fraction \(\mathrm{i}\). The rationale behind Equation (1) estimating the pore diameter being 30% of the particle diameter is based on research of pore sizes between sand sized spherical particles of a uniform or similar size. It was found that particles tended to pack most commonly as shown in Figure 1, surrounded by either four (left) or six (right) other particles [22]. The value of 0.3 in Equation (1) is close to the average of 0.41 and 0.15 from Figure 1, and was shown to be useful for estimating water retention behavior of relatively uniform sized sand and glass beads [23]. Thus, the simplified particle packing approach Equation (1) is suitable for some materials, however, it is not suitable for many soils that contain particles of varying sizes. This is primarily because Equation (1) ignores the possibility of larger pores being filled with particles smaller than 0.3\(\mathrm{d_{i}}\). Pore filling creates more, smaller pores and increases the suction at which the residual pores drain. Predicting grain packing and pore distribution requires much more soil data, such as stress history, placement conditions, and particularly information on each particle's shape and size. The dependence of pore filling on the GSD is highlighted in several studies investigating the packing density (or total volume fraction of solids) of binary mixtures of spherical particles (i.e., two different sized particles present) [24; 25; 26]. For these mixtures, much higher densities (or pore filling) are achieved as the coarse:fine particle size ratio of the particles increases, as shown in Figure 2.  
In addition, Figure 2 shows that the ratio of coarse particle fraction to fine particle fraction also affects the packing density, with the highest density achieved at a coarse particle fraction of around 0.7 (for a size ratio around 19). This result suggests that a deficit of finer particles could result in the pores between coarse particles not being filled. The binary mixture studies discussed [24; 25; 26] have only used sand sized particles, thus the behavior observed may change for silt and clay sized particles that are much smaller, and possibly more platy. This is confirmed by research showing a reduction in packing density achieved for mono-sized particles smaller than around 60 \(\upmu\)m [27] (see Figure 3). The response between 3 and 60 \(\upmu\)m is due to the greater significance of frictional and adhesional forces for smaller particles, when compared to the particles self-weight [27]. Figure 1: Maximum pore diameter (p) created between particles for a mean diameter (\(\mathrm{d_{i}}\)), adapted from [22]. The particle shape is also important for determining the packing density or pore filling of particle mixtures; research shows that many different packing density trends are possible as the aspect ratio and geometry of particles change [28; 29; 30]. For many soils the change in aspect ratio with particle size can be significant, as values range from 1.3 for sand sized particles [31; 32] to several hundred for some clays [33].