# Model Modification of the Soil-Water Characteristic Curve of Unsaturated Weak Expansive Soil
## 6 Correction of the Soil-Water Characteristic Curve by the Mercury Intrusion Test
### SWCC Three-Parameter Model Fitting
#### 6.1.2 Three-Parameter Model Fitting Analysis by the Mercury Intrusion Test  
Based on the known pore structure of the soil in the mercury intrusion test, the suction force of the soil was calculated by the capillary principle in Equation (5). The fit analysis of the results was carried out by the three-parameter model of Equation (10), and the resulting fitting information of each parameter under different dry densities is shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\rho_{d}\)/(**\(\beta\)\(\cdot\)cm\({}^{-1}\)**)**} & \multicolumn{4}{c}{**Fitting Parameters**} & \multirow{2}{*}{**R\({}^{2}\)**} \\ \cline{2-2} \cline{4-5}  & \(\alpha\) & \(\beta\) & \(\gamma\) & \\ \hline
1.4 & 45.517 & \(-\)4.182 & \(-\)0.412 & 0.980 \\
1.5 & 46.665 & \(-\)4.124 & 0.524 & 0.993 \\
1.6 & 43.783 & \(-\)3.912 & 0.729 & 0.992 \\
1.7 & 43.347 & \(-\)3.860 & 3.887 & 0.995 \\
1.8 & 39.131 & \(-\)3.423 & 4.197 & 0.993 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Model fitting results.  
Figure 7: Profiles of the fitting parameters on dry density.  
The three-parameter logarithmic model has a fitting coefficient (R\({}^{2}\)) that ranges between 0.980 and 0.995, with a good fitting effect. The linear fitting analysis of the fitting parameters \(\alpha\), \(\beta\), and \(\gamma\) for the dry density \(\rho_{\rm d}\) was performed, and the results are shown in Figure 8.  
The relationships between \(\alpha\), \(\beta\), \(\gamma\), and \(\rho_{\rm d}\) of the three-parameter model of the mercury intrusion test are as follows:  
\[\alpha=69.433-16.09\rho_{d} \tag{14}\]  
\[\beta=-6.751+1.782\rho_{d} \tag{15}\]  
\[\gamma=-18.345+12.581\rho_{d} \tag{16}\]  
Substituting Equations (14)-(16) into Equation (9) results in a mathematical model of the soil-water characteristic curve containing only the dry density of the expanding soil, as shown in Equation (17). Combined Equation (2) can be based on the known dry density and pore diameter of the soil, and the suction-volume moisture content curve of the soil matrix can be predicted and calculated as follows:  
\[\theta_{2}=69.433-16.09\rho_{d}-(6.751-1.782\rho_{d})\ln(\psi_{2}-18.345+12.581 \rho_{d}) \tag{17}\]  
where \(\theta_{2}\) is the volumetric moisture content of the mercury intrusion test (%), and \(\psi_{2}\) is the matric suction calculated based on the capillary principle for the mercury intrusion test (kPa).  
It can be seen from Equation (17) that the soil-water characteristic curve of expansive soil is not a simple linear relationship with dry density, but the three parameters in the three-parameter model can be expressed by \(\rho_{d}\). The correlation is high, indicating that the distribution of pore structure and pore size in expansive soil are closely related to the soil-water characteristic curve. Under the same volume of water content, the greater the dry density of mudstone, the more internal small pores, and the greater the matrix suction because of the presence of capillary force.