## 4 Results and Discussions
### Possible Probability Distributions for Micro-Pore Parameters of NS-Modified Silty Clay  
In this study, the microscopic characteristics of NS-modified Qinghai-Tibet silty clay were described using two microporous parameters, including median pore diameter and \(V_{D<0.1~{}\upmu\text{m}}\). Figure 8 shows the probability differences between the potential probability distributions and the measured median pore radii at four NS contents (the additions with 0.25%, 1%, and 3% are in Figure S1). These probability differences have small values, indicating that the measured data are similar in shape to the four potential probability distribution curves. Nevertheless, 13.45% is the largest probability difference of the median pore size for the Weibull distribution without addition, which deviates from the measured results (Figure 8).  
Figure 7: SEM images of soil samples with various NS contents (from above to below, magnification times: \(\times\)35, \(\times\)500, \(\times\)2000, \(\times\)5000).  
From the large difference, it is indicated that the actual median pore size probability distribution cannot be described well by the Weibull distribution. The others are in good agreement with the experimental results. In particular, the predicted values from normal and lognormal distributions are approximately equal to the measured ones, with a maximum probability difference of 9.85%. Therefore, the three distributions can visually describe the probability distribution of the median pore size of NS Qinghai-Tibet silty clay.  
The measured mean \(V_{D<0.1\ \upmu\text{m}}\) curves at four NS contents are compared with the four distributions, and the probability differences between the probability distributions and experimental \(V_{D<0.1\ \upmu\text{m}}\) are shown in Figure 9. Similarly, the largest of the probability difference of \(V_{D<0.1\ \upmu\text{m}}\) for the Weibull distribution reaches 0.1827 with the addition of 0.25%, which deviates relatively from the measured results (Figure S2). The comparison results indicate that the probability law of mean \(V_{D<0.1\ \upmu\text{m}}\) cannot be reflected through the Weibull distribution. However, the other three types of distribution and measured results are in good agreement, the largest probability difference between the three distributions and measured results are smaller than 10.32%, except for the addition of 1%. The comparison results indicate that the randomness and variability are significant for the microstructure characteristics of modified Qinghai-Tibet silty clay. Therefore, the mean \(V_{D<0.1\ \upmu\text{m}}\) of NS-modified Qinghai-Tibet silty clay follows normal, lognormal, or Gamma distributions.  
The statistical characteristics (mean value, variance, and standard deviation) of the probability differences between these distributions and the experimental results at four NS contents are shown in Table 3 (for \(V_{D<0.1\ \upmu\text{m}}\) in Table S1). It can be found that the Weibull distribution has the maximum mean probability differences at four NS contents. For the median pore diameter, the maximum mean probability difference is 2.52% with the addition of 1%, and that is 2.89% with the addition of 3% for \(V_{D<0.1\ \upmu\text{m}}\). The statistical characteristics show that the predicted results from Weibull distribution deviate from the experimental ones. The mean and variance of the other three distributions are relatively small, indicating that the other three distributions perform better than the Weibull distribution on the prediction of the microstructure properties of the modified Qinghai-Tibet silty clay.  
Figure 8: The probability difference between distributions and experimental median pore diameters (\(F_{i}(x)\), i = 0, 1, 2, 3, and 4 represent the measured, normal, lognormal, Gamma, and Weibull distributions, respectively).