## 2 Relationship between the Porosity and Target Particle Number  
Suppose a cuboid container is filled with sand; the volume of the container is \(V=BS\), where \(S\) is the vertical sectional area perpendicular to the width direction, as shown in the left figure in Figure 1. If the volume of sand particles in the container is \(V_{s}\), the pore volume is \(V_{v}=V-V_{s}\). The basic definition of sand porosity \(n_{3}\) is as follows:  
\[n_{3}=\frac{V_{v}}{V}=1-\frac{V_{s}}{V} \tag{1}\]  
where the subscript '\(3\)' of \(n_{3}\) indicates the actual porosity of the three-dimensional sand to make it easy to distinguish. The volume of all sand particles can be expressed as the sum of the product of the individual sand particle volume \(v_{si}\) of each particle size and the corresponding particle number. \[V_{s}=\sum_{i=1}^{m}N_{i}v_{si} \tag{2}\]  
where \(m\) is the number of types of sand particle size. If a sand particle is taken as the target particle and we let b be the mass percentage of the target particle mass to the total mass of the sand, then the target particle volume expression is obtained on the basis of the unchanged target particle mass, that is, \(\rho_{b}V_{sb}=\overline{\rho}V_{s}b\). \[V_{sb}=C_{b}V_{s}b \tag{3}\]  
\[C_{b}=\frac{\overline{\rho}}{\rho_{b}} \tag{4}\]  
In which \(\overline{\rho}\) is the average particle density and \(\rho_{b}\) is the target particle density of sand. Suppose the sand particle density of each particle size is the same, which is acceptable; then, \(C_{b}=1\). Similar to Equation (2), the target particle volume can be expressed as  
\[V_{sb}=N_{b}v_{sb} \tag{5}\]  
where \(v_{sb}\) is the individual target particle volume and \(N_{b}\) is the number of target particles. The total volume of sand particles expressed with the number of target particles was obtained by plugging Equation (3) into Equation (5), namely,  
\[V_{s}=\frac{v_{sb}}{C_{b}b}N_{b} \tag{6}\]  
This was obtained as follows by plugging into Equation (1). \[n_{3}=1-\frac{v_{sb}}{C_{b}bV}N_{b} \tag{7}\]  
For a particular type of sand, \(C_{b}\) and \(b\) are usually unchanged. The above equation shows that the porosity of sand can be characterized by counting the number of target particles in sand. Due to the difficulty of counting target particles inside the sand pile, it is only possible to observe target particles on the surface of the sand. Figure 1 shows the case in which the sand was placed in a rectangular container with a width of \(B\) and a cross-sectional area of \(S\). At an observation depth of \(H\), this includes sand particles in contact or not with the container wall. Some of the sand particles that are not in contact with the container wall can be partially observed and are partially obscured. Therefore, the sand in the container could be divided into two parts, visible sand and invisible sand, and the range [0, \(B\)] of the sand body was divided into the sum of the range of sand surface [0, \(H\)] and the interior of sand [\(H\), \(B\)]. If all of the sand particles in  
Figure 1: Sketch of the surface layer of sand. the container were uniformly distributed, the observed porosity \(n_{3}^{f}\) of sand, which can be understood as the porosity obtained from surface observation, was the same as \(n_{3}\). Using \(N_{b}^{f}\) to represent the number of target particles in the range of the sand surface \([0,H]\), the number of target particles at the sand surface is given by  
\[N_{b}=N_{b}^{f}\frac{B}{H} \tag{8}\]  
This was obtained as follows by plugging into Equation (7). \[n_{3}=\frac{V_{v}}{V}=1-\frac{V_{s}}{V} \tag{1}\]0  
From the above equation, it was concluded that the porosity of the sand \(n_{3}\) could be evaluated by the number of target particles on the surface of the sand \(N_{b}^{f}\), assuming that the sand particles are uniformly distributed along the direction of observation.