## 1 Introduction  
Expansive soils can cause deformation and failure of civil engineering structure and geological environment engineering, such as pavements and slopes [1, 2]. Hydraulic conductivity, as one of the most essential engineering characteristics of expansive soil, is closely correlated with the strength and deformation of expansive soil and significantly influences engineering construction [3]. Expansive soil slopes and subgrades often fail repeatedly under rainfall-infiltration conditions. Modifying and replacing expansive soil is often considered an economic solution to this problem [4]. Researchers believe that the replacement soil has to be of low hydraulic conductivity to prevent a bathtub effect and inhibit the moistening of the underlying expansive soil [5]. Thus, the modified materials and the hydraulic conductivity of expansive soil are essential for geotechnical engineering. At present, lime is commonly used as an expansive soil modified material. Line is a strong alkaline material. And if its added amount is high, it will destroy the acid-base balance of the soil and cause soil hardening. Several studies have shown that the addition of nanomaterials to soil can reduce hydraulic conductivity [6, 7, 8, 9]. [PERSON] [6] found that the hydraulic conductivity of raw clay was reduced by adding 4% nanoclay. [PERSON] and [PERSON] [7] studied that 2% Al\({}_{2}\)O\({}_{3}\) and nano-CuO can decrease the hydraulic conductivity of clay by 30% and 45%, respectively. [PERSON] and [PERSON] [8] showed that nanomaterials (nano-copper and nano-alumina) can reduce the hydraulic conductivity of compacted soil-bentonite mixture with different plasticity indexes. [PERSON] and [PERSON] [9] found that silica nanomaterials can increase soil cohesion/adhesiveness and reduce soil hydraulic conductivity. The modifier used in the current research was superhydrophobic nano-SiO\({}_{2}\). As an inorganic micro-/nanofiller, nano-SiO\({}_{2}\) has excellent mechanical properties and is an important raw material for preparing large-scale industrial applications of superhydrophobic materials. Given its hydrophobic properties, it is gradually being used to treat expansive soils. Nano-SiO\({}_{2}\) is an environmentally friendly material, which is consistent with the main component in soil. Micro-addition does not change the PH value of the soil, and does not cause soil hardening. However, using superhydrophobic nano-SiO\({}_{2}\) as a soil modifier is not common. Few studies have been conducted on using hydrophobic silica to modify expansive soil. The mechanism of superhydrophobic nano-SiO\({}_{2}\) changing the hydraulic conductivity of expansive soil remains unknown. According to the relevant literature, the hydraulic conductivity of clays depends, in part, on their mineralogy, cation valence, and cation exchange capacity, as well as their structure through clay layer orientation at the atomic scale and through particle and pore distribution at the aggregate scale [10]. For clayey soils, pore size distribution (PSD) is the main characterisation of soil physical form and structure and is associated with the hydraulic conductivity of soil [11; 12]. The microstructure characteristics of soil are the basis for theoretical research and provide reasonable explanations for many macroscopic phenomena [13]. However, given the special properties of expansive soil, such as shrinkage and swelling, research on the hydraulic conductivity of expansive soil is more complicated than those on other clays. Moreover, analysing expansive soil from a microscopic point of view is difficult. The common test methods to obtain soil PSD include mercury injection porosimetry (MIP) and nuclear magnetic resonance (NMR) analyses. The principle of the MIP method is to press mercury into the pores and measure the volume of mercury entering under different pressures to calculate pore radius. Although this method is theoretically feasible, pressing mercury into extremely fine pores and blind ends is difficult, and certain test errors are inevitable.