## 5 Conclusions  
As two typical microstructure parameters of porous medium, the median pore size and \(V_{D<0.1\upmu\text{m}}\) are used to analyse the micro-pore characteristics of NS-modified Qinghai-Tibet Plateau silty clay. Four common distributions are employed to estimate the distribution characteristics of microstructure parameters. Furthermore, the information spread distribution is introduced to compare with the normal distribution. The main conclusions of this study can be summarized as:  
1. Nanoparticles fill the pore space between soil particles, making the soil pores smaller and the soil more compacted. The microstructure parameters of the silty clay samples  
Figure 15: The residual plot of (**a**) normal and (**b**) information spread distributions for the median pore diameter.  
exhibit apparent randomness and uncertainty. Furthermore, the lower the NS content, the more pronounced the stochastic effect.
2. The mean median pore diameters of NS-modified silty clay samples are smaller than those without addition, and decrease with increasing NS content. Nevertheless, the average cumulative volume of pore sizes smaller than 0.1\(\mu\)m shows the opposite trend.
3. The normal, lognormal, Weibull, and Gamma distributions can be potentially used to describe the microscopic features of NS silty clay, and three distributions satisfy hypothesis testing, except for the Weibull distribution. The normal distribution has the best described effect in comparison to the others.
4. The information spread estimation has a better fitting effect than the normal distribution, but it is complicated. Therefore, the normal distribution has the most excellent evaluation effect to describe the statistical characteristics of the microstructure parameters of NS-modified silty clay.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app13053074/s1](https://www.mdpi.com/article/10.3390/app13053074/s1)]([https://www.mdpi.com/article/10.3390/app13053074/s1](https://www.mdpi.com/article/10.3390/app13053074/s1)), Figure S1. The probability difference between the potential probability distributions and measured median pore diameters; Figure S2. The probability difference between the potential probability distributions and measured \(V_{D<0.1\ \mu\text{m}}\); Table S1. The statistical characteristics of probability differences between potential probability distribution and measured \(V_{D<0.1\ \mu\text{m}}\).  
Conceptualization, K.L. and C.W.; methodology, K.L. and C.W.; validation, J.L. and L.X.; data curation, C.W.; writing--original draft preparation, K.L. and C.W.; writing--review and editing, K.L., C.W. and Y.W.; supervision, J.L. and Y.W.; funding acquisition, C.W and S.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Distinguished Youth Foundation of Gansu Province (22J85 RA392), the National Natural Science Foundation of China (41902267), the Construction of Science and Technology Project of Gansu Province (JK2022-09), the Fundamental Research Foundations for the Central Universities (Izujbky-2020-2), and the Foundation of State Key Laboratory of Frozen Soil Engineering (SKLFSE201904).  
Not applicable.  
The authors declare no conflict of interest.