# WGAN-Based Realization Process of Gravel Soil for Hydraulic Property Simulation
## 6 Discussion  
Our study demonstrates the effectiveness of our proposed WGAN-GP approach. Figure 3 provides a visual comparison between the real soil prototype and the generated realizations. Upon examination of both 3D models, it is challenging to distinguish between the generated microstructure and the real microstructure, indicating a high degree of similarity. Through fluid-flow simulations, we quantitatively assessed the hydraulic conductivity of the generated model and found them to closely resemble those of the original soil prototype. The pore distribution characteristics of the generated model follow a normal distribution, while the permeability distribution characteristics adhere to a log-normal distribution. This observation suggests a power-law relationship between pore and permeability distributions, which is consistent with the general principles of porous media (as referenced in Figure 7). This relationship was further confirmed by the literature of [PERSON] et al. (2022) ([PERSON] et al., 2022) in which a slightly different type of GAN and actor-critic (AC) reinforcement learning are used for 3D generation of porous systems. These findings indicate that our generator effectively captures the essential morphological features of natural soil. It is capable of reproducing the intricate micro-geometry of soil, encompassing a variety of realistic shapes of voids and matrix structures in the generated microstructures (as depicted in Figures 8-10). An important limitation of our approach is that the 64 \(\times\) 64 \(\times\) 64-voxel patches from the soil model are too small to capture the geometric characteristics of coarse grains. Figure 10: Lineal-path function (LF) of locating the same phase (pore in here) at different 3D distances within three groups of gravel soil models. The LF is calculated for each faces along the \(d_{XY}\) of the diagonal direction. The black curves represent the CF of the three groups of the training set, which is 64 \(\times\) 64 \(\times\) 64 in size and derived from primitive soil prototypes. Meanwhile, the red curves show the mean CF obtained from 14 WGAN-GP realizations of size 64 \(\times\) 64 \(\times\) 64. The light green curve is the LF of each realization. Despite fine grains dominating the hydraulic properties of the gap-grading gravel soil, the hydraulic effect of coarse particles cannot be overlooked. To address this, we propose gradually expanding the 3D patch size, such as using 128 \(\times\) 128 \(\times\) 128-voxel, 192 \(\times\) 192 \(\times\) 192-voxel patches, creating a series of pyramid representations, which is similar to the progressive training workflow [15; 27]. This approach aims to overcome the issue of small-scale patches lacking sufficient detail in modeling gravel soil prototypes. However, this will increase the demand on training hardware and software. For instance, on a GPU like the GeForce RTX 3090, training could take between 9 to 12 h, while training on CPUs would likely take substantially longer. A feasible strategy is to initially learn the basic soil prototype using 2D faces along one direction with larger patches, such as 700 \(\times\) 700 pixels, and then estimate voxel values in the other directions based on geostatistical variogram functions. This method could optimize training time, akin to techniques used in existing literature [28; 29; 30]. [PERSON] et al. (2017, 2018) [9; 25] introduced an intriguing method for rapidly generating a series of 2D or 3D unconditional realizations from a channelized aquifer training image using the Spatial Generative Adversarial Network (SGAN) they proposed. A key characteristic of the proposed SGAN is its definition of a (very) low-dimensional parameterization, which facilitates efficient probabilistic inversion using state-of-the-art Markov chain Monte Carlo (MCMC) methods. Essentially, it represents a machine learning strategy rooted in a multi-point geostatistical (MPS) method for inversion parameters.