## 5 Correlation between Hydraulic Conductivity and PSD of Modified Soil Samples  
In soil, owing to the surface tension of water, the water in pores generated capillary negative pressure. The Young-Laplace equation can describe this negative pressure:  
\[\Delta p=\Upsilon\bigg{(}\frac{1}{r_{1}}+\frac{1}{r_{2}}\bigg{)} \tag{9}\]  
where \(\Delta p\) is the capillary pressure (Pa), \(\Upsilon\) is the surface tension of water (N/m), and \(r_{1}\) and \(r_{2}\) are the radii of curvature of pore bodies 1 and 2 (cm), respectively.  
Pores in soils can be considered as spherical pores [21], \(r_{1}=r_{2}=r\); thus, Equation (9) can also be written as follows:  
\[\Delta p=\frac{2\Upsilon}{r} \tag{10}\]  
When the hydraulic pressure (\(P\)) exceeded the capillary pressure, water may overcome the pressure of capillary water and permeate, hence hydraulic pressure (\(P\)) can be expressed as follows:  
\[P=\Delta p=\frac{2\Upsilon}{r} \tag{11}\]  
According to Equation (11), pores with a smaller radius had bigger capillary pressure, requiring larger hydraulic pressure for water to overcome the pressure of capillary water. Combined with the analysis of PSD obtained via NMR, this finding was probably due to the smaller pore radius of modified expansive soil than that of unmodified expansive soil (Figure 6), resulting in a higher average capillary negative pressure of the internal pores of modified expansive soil than that of unmodified expansive soil, and the internal pores were less permeable to water, so the hydraulic conductivity was smaller.  
In the saturated soil in the test, we supposed that the pores in saturated soil had only one state, namely permeable or impermeable. Under a certain water-head pressure, the capillary force of some pores cannot hinder the flow of water because of the large radius, so the state of these pores was defined as permeable, whereas the state of the remaining smaller pores was defined as impermeable. When the water-head pressure cannot overcome the pressure of the capillary water of the pore, the pore was defined as a water-proof state. In the hydraulic-conductivity test, \(\Upsilon\) is 7.2 \(\times\) 10\({}^{-3}\)N/m and \(P\) is 50 kPa. According to Equation (11), \(r\) was calculated as 0.288 \(\upmu\)m. To better analyse the results,  
Figure 14: Schematic diagram of water molecules in crystal layer of montmorillonite [33].  
Figure 15: The pore size distribution of sample before and after saturated.  
pores with a radius greater than the permeable pore radius were defined as permeable pores, whereas the remaining pores were defined as impermeable.  
The permeable pore volume of soil samples with different SiO\({}_{2}\) contents is shown in Figure 16. When the hydraulic pressure was 50 kPa, the permeable pore volume of samples with different SiO\({}_{2}\) contents (0%, 0.2%, 0.4%, 0.6%, 0.8%, and 1.0%) was 6.92, 5.38, 5.02, 3.68, 3.26, and 2.5 cm\({}^{3}\), respectively. With increasing SiO\({}_{2}\) content, the volume of the permeable pore in different samples gradually decreased. As shown in Figure 17, the relation between permeable pore volume and hydraulic conductivity was studied via linear regression analysis, and the coefficients of determination (R\({}^{2}\)) of the fitting curve was 0.98. This finding meant that permeable pore volume and hydraulic conductivity had a good linear relationship.