# Numerical Modeling of 3D Slopes with Weak Zones by Random Field and Finite Elements
## 2 Methodology
### Method of Generating 3D Random Field  
The stabilities of slope projects are strongly affected by the distribution of its internal weak zones. In the destruction of a slope composed of multiple zones, landslides are usually dominated by the spatial distribution of the weak zones. However, due to the constraints of project budget and construction schedules, it is almost impossible to obtain a complete distribution of zones within the slope through on-site surveys. In typical slope projects, only limited survey work is carried out. Therefore, the distribution characteristics of weak zones are only known in the survey operation area, and such information cannot be obtained at other locations.  
The spatial distribution of weak zones at the same site and the spatial distribution of the mechanical characteristics of the rock and weak zones should have similar characteristics. In the random field, the spatial correlation of the layer of weak zones in the layer of weak zones between the two-slope rock and weak zones elements can be expressed using the square exponential autocorrelation function \(R(x,y,z)\), as shown below:  
\[R(x,y,z)=\exp\left\{-\pi\left(\frac{\Delta x}{\theta_{x}}\right)^{2}-\pi \left(\frac{\Delta y}{\theta_{y}}\right)^{2}-\pi\left(\frac{\Delta z}{\theta_{ z}}\right)^{2}\right\} \tag{7}\]  
where \(R\) (\(x,y,z\)) represents the spatial correlation of the material unit; \(\theta_{x}\), \(\theta_{y}\) and \(\theta_{z}\) are the correlation lengths along \(x\), \(y\) and \(z\) directions, respectively.  
In this study, the anisotropic random field is used to characterize the inhomogeneity of the spatial distribution of the weak zones. When the spatial correlation of each unit in the simulated weak zone is relatively high, it can be regarded as the same weak zone. In addition, since the weak zones are usually continuously layered inside the slope, the simulation results obtained using this method are closer to the ground truth.  
For a 3D stationary Gaussian random field, a two-phase random media can be reconstructed for the rock and weak zones using the following formula:  
\[B(x,y,z)=\left\{\begin{array}{cc}\text{rock}&\Phi[G(x,y,z)]\geq p_{0}\\ \text{weak zones}&\Phi[G(x,y,z)]{<}p_{0}\end{array}\right. \tag{8}\]  
where \(B(x,y,z)\) is the random media of weak zones and rock; and \(p_{0}\) is the volume fraction of weak zones. Reconstructions of typical weak zones and rock random media are shown in Figure 1, where 0 represents the volume fraction of weak zones, and 1 represents the volume fraction of rock [33].  
The two-phase random media simulation method based on Gaussian random field is adopted in this research. This method can flexibly control the shape and orientation of the weak zone of the slope according to the actual geological survey situation, and can truly reflect the random distribution characteristics of the layer of weak zones in the slope. On the basis of the reconstruction of the typical two-phase random media shown in Figure 1, two typical samples of the two-phase random media are generated according to Equation (8), as shown in Figure 2, where the volume fraction of the weak zones is as assumed to be 20%.