# WGAN-Based Realization Process of Gravel Soil for Hydraulic Property Simulation
## 5 Results Analysis  
### Validation Analysis of Hydraulic Parameters  
Based on Table 2, a series of proposed hypothesis tests aimed to determine which distribution best satisfies the permeability distribution for each group of realizations. A Box-normal plot on porosity and permeability for all realizations was constructed to assess the goodness-of-fit of the distribution for 14 separate realizations. The results of this study indicate that the distribution of permeability values generated by the GAN model shows a closer alignment with a log-normal distribution. In contrast, the distribution of porosity values, as illustrated in Figure 7, more closely resembles a normal distribution. These observations are consistent with established knowledge regarding the relationship between permeability and porosity, as outlined in the Kozney-Carman formula, shown as follows:  
\[k=\frac{d_{h}^{2}}{180}\frac{{\varepsilon_{p}}^{3}}{\left(1-{\varepsilon_{p}} \right)^{2}} \tag{9}\]  
where \(d_{h}\) is the equivalent grain size of porous materials. This formula describes a power-law relationship where porosity influences permeability, and our findings underscore this relationship through the statistical characteristics observed in the GAN-generated data. As shown in Figure 7, we included scatter points representing the porosity and permeability values of the prototype for comparison with the generated results. Our analysis reveals that these prototype values closely approximate the mean of the generated data and largely fall within the interquartile range (IQR), indicating a confidence interval of approximately 75%. Notably, the porosity value of the prototype from Group 1 stands out as an exception. We also observe that the porosity distribution of the generated models follows a normal distribution, while the permeability adheres to a log-normal distribution. These two parameters exhibit a power-law relationship, as described by the Kozeny-Carman equation. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{2}{c}{\(k_{\text{xx}}\) **[cm/s]**} & \multicolumn{2}{c}{\(k_{\text{hy}}\) **[cm/s]**} & \multicolumn{2}{c}{\(k_{\text{hy}}\) **[cm/s]**} \\ \cline{2-7}  & **Prototype** & **Avg. Realizations** & **Prototype** & **Avg. Realizations** & **Prototype** & **Avg. Realizations** \\ \hline Group 1 & \(0.98\times 10^{-2}\) & \(1.32\times 10^{-2}\) & \(0.91\times 10^{-2}\) & \(1.55\times 10^{-2}\) & \(1.15\times 10^{-2}\) & \(1.43\times 10^{-2}\) \\ Group 2 & \(5.44\times 10^{-2}\) & \(4.40\times 10^{-2}\) & \(4.48\times 10^{-2}\) & \(4.44\times 10^{-2}\) & \(2.02\times 10^{-2}\) & \(2.29\times 10^{-2}\) \\ Group 3 & \(2.59\times 10^{-2}\) & \(3.00\times 10^{-2}\) & \(2.71\times 10^{-2}\) & \(2.91\times 10^{-2}\) & \(1.15\times 10^{-2}\) & \(1.30\times 10^{-2}\) \\ \hline \hline \end{tabular} Note. The average values were calculated from 14 realization values for each group based on Equation (8).  
\end{table} 
Table 3: Statistics of hydraulic conductivity for each group of soil model realizations. The hydraulic permeability of each group, as detailed in Table 3, indicates a flow rate on the order of \(\times 10^{-2}\) cm/s, which is consistent with classical criteria and engineering norms. Figure 8 introduces an alternative visualization method using a skeleton model to illustrate the relationship between connected pores and the resulting flow field. While porosity indicates a medium's capacity to store fluids, the Euler characteristic assesses its connectivity, which is crucial for fluid flow. Therefore, reconstructions of porous media should accurately reflect these functional aspects to model relevant physical processes at the pore-scale effectively. Additionally, Euler numbers corresponding to these structures are provided, offering a more intuitive and practically valuable approach compared to solely relying on Euler characteristics for assessing connectivity.