# A Simple Water Retention Model Based on Grain Size Distribution
## 3 Proposed Model  
The GSD-WRC model proposed in this paper had to meet several criteria. It should:  
* Achieve reasonably accurate predictions without requiring calibration to WRC data;
* Preferably use input parameters that are relatively easy to measure;
* Consider pore filling;  
Figure 4: A fractal shape called an Apollonian gasket, where the largest possible circle is progressively placed into each area that is not contained within a circle. * Be capable of considering variation in contact angle with respect to particle size. These criteria arose from the broader research goals and the limitations discussed in Sections 2.1 and 2.2. Note that the model does not attempt to capture volume change as this would necessarily compromise its simplicity. Thus, when using this model, and many others, limitations due to the lack of volume change must be kept in mind. To provide a high-level view of how the proposed model is implemented, a series of sequential steps are provided in the following list:
* The GSD of a material is used to calculate its clay, silt, and sand and larger fractions. * The sizes in the GSD are then adjusted using an aspect ratio, such that smaller particles (e.g., clay or silt) are reduced in size to a greater extent than larger particles. * A series of primary pore sizes are then created, by assuming the particles in each particle size bin of the adjusted GSD pack together to create a total volume of pores of a proportional size. * The volume of primary pores associated with each pore size is estimated by assuming the pore volume fraction is proportional to the mass fraction of the associated particle size bin. That is, for a particle size bin accounting for 10% of a soil's mass, the volume of pores associated with this bin account for 10% of its total pore volume, as determined from its adopted saturated volumetric water content. * A set of criteria are applied to each primary pore size to determine what degree of pore filling should occur within each pore size. Any required pore filling is done so to achieve the packing densities at different levels of Apollonian packing. * Whether pores are unfilled, partially filled, or filled is determined by considering whether there is a sufficient volume of smaller particle fractions that could fill each set of primary pores in a binary mixture to achieve the packing density corresponding to Apollonian packing. * Having determined the degree of filling for each set of primary pores, they are then redistributed (as required) into a larger number of smaller pores, according to the appropriate Apollonian geometry, and combined to give a final pore size distribution. * Each pore size in the final distribution is transformed to an equivalent area circular pore, with a drainage suction, creating the final pore size distribution. * The predicted WRC is the theoretical WRC of the pore size distribution, which is estimated by assuming a number of arbitrary suctions, and calculating the volumetric water content of the water filled or undrained pores. ### Inferring a Primary Pore Size Distribution from Grain Size Distribution (GSD)  
The main input of the proposed model is the GSD of a soil, which is used to estimate a PSD, which in turn is converted into a WRC using Jurin's law, for a given grain packing arrangement (porosity or void ratio). The particles corresponding to measured sizes in a GSD are often assumed to be spherical; however, research has shown that small particles (e.g., clays or silts) are actually non-spherical and that an aspect ratio should be considered to better capture particle size and shape. The smaller particles in a soil are usually measured with methods other than sieving such as sedimentation or laser diffraction, which are more likely to detect the larger dimension of non-spherical particles.