# Model Modification of the Soil-Water Characteristic Curve of Unsaturated Weak Expansive Soil
## 5 Analysis of Soil-Water Characteristic Curves  
Considering the known pore structure of the soil, the relationship between the matric suction and the pore diameter D can be obtained by the capillary principle, as shown in Equation (2). Then, the soil-water characteristic curve of rock and soil can be derived. \[\psi=\frac{4 Tcos\theta}{D} \tag{2}\]  
where \(\psi\) is the matric suction of the soil (kPa), T is the unit of surface tension of the water (72.25 \(\times\) 10\({}^{-3}\) N\(\cdot\)m\({}^{-1}\)), \(\theta\) is the contact angle between water and soil, and D is the pore diameter (nm). The filter paper method test uses domestic \"double-circle\" brand NO. 203-type slow filter paper to measure substring suction and total suction force. This can be achieved using the filter curve equation in Equations (3) and (4). The matric suction filter equation is as follows:  
\[lg\psi=\left\{ \begin{array}{l}-0.0767w+5.493,w\leq 47\%\\ -0.0120w+2.470,w>47\%\end{array} \right\} \tag{3}\]  
The total suction filter equation is as follows:  
\[lgs=\left\{ \begin{array}{l}-0.0700w+5.257,w\leq 41\%\\ -1.1940w+51.321,w>41\%\end{array} \right\} \tag{4}\]  
where s and \(w\) are the total suction force of the soil body (kPa) and the moisture content of the filter paper quality (%), respectively. Considering that the results of the filter paper method are obviously affected by the environment in the low suction range, the suction value of 0-0.5 MPa is measured by the pressure plate method as a supplement. The characteristic curve of expansive soil water, obtained by the test method, is shown in Figure 6.  
Figure 5: Influence of compaction on the pore content of expansive soil. Based on the characteristic curve of the expansive soil and soil water (Figure 6), it is evident that the soil matric suction exhibits a non-linear decrease, with varying degrees of increase with increasing dry density under the same volumetric water content conditions. When combined with the PSD curve, it can be considered that the increase in the initial dry density of the soil sample changes the pore structure and its distribution law, and the pore structure of the sample determines the suction force and is reflected in the soil-water characteristic curve. The increase in the sample's initial dry density results in a corresponding increase in the number and proportion of small pores. Notably, the sample as a whole exhibits a higher suction value, reflecting the pore structure characteristics of the soil. In order to further verify the accuracy of the test results, three widely accepted typical SWCC models, namely, the Vangenuchten function (V-G), the Fredlund and Xing model (F-X), and the Gardner model, were used to fit the soil-water characteristic curve obtained by the test method. Among them, the Fredlund and Xing fitting coefficient R\({}^{2}\) is above 0.980, which is not only better than both van Genuchten and Gardner models but is also more suitable for suction fit analysis of weakly expansive soil. The fitting parameters are shown in Table 2.  
(a) The Fredlund and Xing model is as follows:  
\[\theta=\mathrm{C}(\psi)\frac{\theta_{\mathrm{s}}}{\left\{\ln\left[\mathrm{e}+ \left(\psi/\mathrm{a}\right)^{n}\right]\right\}^{\mathrm{m}}} \tag{5}\]  
where \(\theta\) denotes the actual volume water content (%), \(\theta_{\mathrm{s}}\) is the saturated volumetric water content (%), \(a\) represents the fitting parameter related to the air-entry value of the soil (kPa), \(n\) is the fitting parameter related to the slope of the SWCC, \(m\) denotes the fitting parameter related to the residual water content of the soil, and \(e\) is the Euler number, 2.71828. \[C(\psi)=\left[1-\frac{ln(1+\psi/C_{r})}{ln(1+10^{6}/C_{r})}\right] \tag{6}\]  
where \(C(\psi)\) is a correction factor, and \(C_{r}\) is the residual matric suction (kPa). (b) [PERSON] model is as follows:  
\[\theta=\frac{\theta_{\mathrm{s}}-\theta_{r}}{\left[\left(\frac{s}{b}\right)^{ n}+1\right]^{\mathrm{m}}}+\theta_{r} \tag{7}\]  
where \(b,\ n\), and \(m\) are fitting parameters.