# Advanced Scanning Technology for Volume Change Measurement of Residual Soil
## 3 Mathematical Equations  
In the SWCC tests, the change in the mass of a soil specimen is usually recorded, but the total volume change in the specimen is not measured. Therefore, the gravimetric water content versus soil suction of a soil specimen is plotted in the SWCC. However, according to [PERSON] et al. [11] and [PERSON] et al. [48], the volume change in a soil specimen may be  
Figure 1: Locations of the soil samples on a Singapore geological map.  
significant and important to the data interpretation of the SWCC. Hence, the shrinkage curve is established by measuring the volume change in a soil specimen as it progressively loses moisture during the drying process. A relationship between the void ratio and gravimetric water content of a soil specimen is then formed and used in conjunction with the SWCC. A shrinkage equation that fits a wide variety of soils was proposed by [PERSON] et al. [11], as can be seen in Equation (1):  
\[e\left(w\right)=a_{sh}\left[\frac{\left(w\right)^{C_{sh}}}{\left(b_{sh}\right) ^{C_{sh}}}+1\right]^{\frac{1}{C_{sh}}} \tag{1}\]  
where \(e\left(w\right)\) is the void ratio of the soil specimen; \(a_{sh}\) is the minimum void ratio of the soil specimen; \(b_{sh}\) is the slope of the tangency; \(C_{sh}\) is the curvature of the shrinkage curve; and \(w\) is the gravimetric water content of the soil specimen.  
The SWCCs, in terms of the volumetric water content and degree of saturation of a soil specimen versus soil suction, are plotted to determine the correct air-entry value and compute the unsaturated soil property functions of the soil specimen [11]. [PERSON] and [PERSON] [49] developed a general SWCC model (Equation (2)) to best fit the unimodal SWCC based on the pore-size distribution of soil, and it has been observed to fit well with experimental data for a suction range from 0 to 106 kPa. A later study conducted by [PERSON] and [PERSON] [50] also concluded that [PERSON] and [PERSON]'s [49] equation better fits the SWCC among all the various equations.  
\[\theta_{w} =\left[1-\frac{ln\left(1+\frac{\psi}{c_{r}}\right)}{ln\left(1+ \frac{10^{6}}{c_{r}}\right)}\right]\frac{\theta_{s}}{\left\{\frac{ln\left[e+ \left(\frac{\psi}{a}\right)^{n}\right]}\right\}^{m}} \tag{2}\]  
where \(\theta_{s}\) is the saturated volumetric water content; \(\psi\) is the matric suction (kPa); \(C_{r}\) is the matric suction corresponding to the residual water content (kPa); \(a\) is the fitting parameter of SWCC and is closely related to the air-entry value of the soil; \(n\) is the fitting parameter of SWCC and controls the slope at the inflection point of SWCC; and \(m\) is the fitting parameter of SWCC and is closely related to the residual water content.  
[PERSON] et al. [51] adopted (Equation (3)) as the best-fitting SWCC with a dual sub-curve (bimodal SWCC). This equation needs the correction factor \(\left(C_{(\psi)}\right)\) to guarantee that the water content is negligible at 1 GPa of soil suction.  
\[\theta_{w}=C_{(\psi)}\begin{bmatrix}\theta_{r}+\left(\theta_{s1}-\theta_{s2} \right)\left(1-erfc\frac{ln\left(\frac{\theta_{s1}-\psi}{\theta_{s1}-\psi_{s1} }\right)}{s_{1}}\right)+\left(\theta_{s2}-\theta_{r}\right)\\ \left(1-erfc\frac{ln\left(\frac{\theta_{s2}-\psi}{\theta_{s2}-\psi_{s2}} \right)}{s_{2}}\right)\end{bmatrix} \tag{3}\]  
where \(\theta_{w}\) is volumetric water content at any suction; \(\theta_{s}\) is saturated volumetric water content; \(\psi\) is soil suction of the soil (kPa); \(a\) is soil parameter that is primarily determined by the soil's air-entry value (AEV) (kPa); \(n\) is a function of the rate of soil water extraction when the AEV has been reached; \(m\) is a function of the residual water content; \(\psi_{m}\) is soil suction at the inflection point (kPa); \(\psi_{a}\) is the air-entry value of soil (kPa); and \(s\) is the geometric standard deviation of SWCC.