# A Simple Water Retention Model Based on Grain Size Distribution
## 3 Proposed Model
### Pore Size Distribution to Water Retention Curve  
Once each primary pore size \(\text{p}_{\text{i}}\) is filled with particles when deemed appropriate (according to Sections 3.2.1-3.2.4), each pore space associated with a particle size bin may be maximally filled and contain ten subpores as shown in Figure 10a, partially filled and contain four subpores (Figure 10b), or unfilled and only pore \(\text{p}_{\text{i}}\) is relevant. Each subpore is transformed to an equivalent area circular pore as shown in Figure 10. The equations to determine the area of a subpore depend upon the degree of pore filling, and for simplicity are given as expressions of the primary pore diameter \(\text{p}_{\text{i}}\) as shown in Equation (12); note a number of equations are shown, with the applicable subpore area being indicated by a pore color corresponding to what is shown in Figure 10.  
\[ \begin{split}\text{A}_{\text{ik}}&=8.163508\times 10^{ -3}\text{p}_{\text{i}}^{2}\hskip 42.679134 pt(\textit{central~{}pores})\\ \text{A}_{\text{ik}}&=8.436737\times 10^{-2}\text{p}_{ \text{i}}^{2}\hskip 42.679134 pt(\textit{green~{}pores})\\ \text{A}_{\text{ik}}&=6.566494\times 10^{-3}\text{p}_{ \text{i}}^{2}\hskip 42.679134 pt(\textit{cyan~{}pores})\\ \text{A}_{\text{ik}}&=4.640154\times 10^{-3}\text{p}_{ \text{i}}^{2}\hskip 42.679134 pt(\textit{red~{}pores})\end{split}  \tag{12}\]  
Figure 9: Flowchart detailing how the pore filling state is decided with the use of either Equation (10) or (11). In Equation (12), \(\rm A_{ik}\) is the area of the k-th subpore for the pore space of the i-th size bin, and \(\rm p_{i}\) is the primary pore diameter of the i-th particle size bin. Once the pores of area \(\rm A_{ik}\) have been converted to equivalent area circular pores, the capillary suction for each subpore (\(\rm s_{ik}\)) is estimated using the Young-Laplace equation shown in Equation (13):  
\[\rm s_{ik}=\frac{2\gamma\cos(\alpha_{i})}{r_{ik}} \tag{13}\]  
where \(\gamma\) is the surface tension of water, \(\rm r_{ik}\) is the circular radius of the k-th subpore within the pore space of the i-th size fraction, and \(\alpha_{i}\) is the receding contact angle of the water solid interface for the particles associated with the i-th size fraction. In the proposed model, the contact angle changes with particle size when the material tested is coal tailings, which are comprised of coal (maceral) and inorganic mineral particles. Indeed, coal macroals are known to be more hydrophobic than minerals [37; 38; 39; 40; 41; 42] and the contact angle of a given fraction depends on the relative proportion of macroals and minerals. In addition, it was found that those proportions vary with particle size [46]. For those materials containing coal particles, it is proposed to estimate the value of \(\alpha_{i}\) as a weighted average of the contact angle of macroals and minerals, as given by Equation (14):  
\[\alpha_{i}=\frac{\alpha_{c}(\rm V_{c,i+1}-V_{c,i})+\alpha_{m}(\rm V_{m,i+1}- V_{m,i})}{(\rm V_{c,i+1}-V_{c,i})+(\rm V_{m,i+1}-V_{m,i})} \tag{14}\]  
where \(\alpha_{c}\) is the contact angle measured for the coal particles, \(\rm V_{ci}\) is the volume fraction of coal particles for the i-th particle size bin, \(\alpha_{m}\) is the contact angle measured for the mineral  
Figure 10: Diagram showing: (**a**) the pore space associated with a maximally filled pore \(\rm p_{i}\), with the area \(\rm A_{ik}\) of one subpore \(\rm p_{ik}\) highlighted (with k ranging from 1 to 10); (**b**) the pore space associated with a partially filled pore with the area \(\rm A_{ik}\) of one subpore \(\rm p_{ik}\) highlighted (k ranging from 1 to 4). In both (**a**,**b**) each subpore \(\rm p_{ik}\) is transformed to a circular shape with an equivalent area, and subpores with the same color have identical areas. particles, and V\({}_{\rm mi}\) is the volume fraction of mineral particles for the i-th particle size bin.