## 3 Multi-Scale Numerical Modeling of Rock Failure Process under Compression
### Microscale Simulation
#### 3.1.1 Microscale Numerical Model  
FIB-SEM has been widely used to characterize shale's nanoscale structural features [41; 42]. Based on the images obtained by FIB-SEM, this section reconstructed an actual 3D multi-mineral component shale numerical model to study the influence of mineral distribution on the shale fracture process at the microscopic scale. In this way, the interaction between cracks and different mineral components can be observed, and the propagation process of cracks at the microscopic scale can be presented. The shale samples in this section were collected from the Yanchang Formation in the Ordos Basin, Gansu, China. The experimental equipment used in this section was the FIB-SEM HELIOS NANOLAB650 from the Research Institute of Petroleum Exploration and Development of China, and the acquisition pixel was about 10 nm.  
Figure 4a shows shale sample 3D FIB-SEM images, and the single image resolution is 1500 pixels by 1500 pixels, with a pixel size of 10 nm \(\times\) 10 nm. The interval of longitudinal cutting was 10 nm for each layer, and 750 layers were cut. Therefore, the size of the 3D FIB-SEM image of the shale sample obtained was about 15 \(\times\) 15 \(\times\) 7.5 \(\upmu\)m (Figure 4a(I)). In our research, the FIB-SEM image analysis results show that the white particles are pyrite, the light gray minerals are silicate minerals, the dark gray minerals are carbonate minerals, the gray-black minerals are organic matter, and the black is fractures (Figure 4a,b).  
The objective of this section (Figure 4a(II)) was to select the region of interest from Figure 4a(I)), the size of which was about 15 \(\times\) 7.5 \(\times\) 7.5 \(\upmu\)m. In an attempt to increase the calculation speed without affecting the calculation accuracy of the model, when constructing the numerical model, the pixels of the image were reduced five-fold, and one image was taken at intervals (the size of the element is 50 \(\times\) 50 \(\times\) 50 nm). The method introduced in Section 2.3 reconstructs the three-dimensional numerical model of shale at the microscopic scale, as shown in Figure 5c,d ((c) front view, (d) back view), represented by the elastic modulus diagram. Comparing Figure 5a,b and Figure 5c,d, it can be found that the reconstructed numerical model (Figure 5c,d) reproduces the real microstructure of the shale well, and the introduction of inhomogeneous coefficients characterizes the inhomogeneity of single minerals more realistically, which makes the reconstructed numerical model more natural and realistic. The total number of finite element elements in the specimen section is 6.75 million. The blue mineral is pyrite, the yellow mineral is the carbonate matrix dominated by calcite, and the red mineral is organic matter.  
In contrast, the transparent mineral is fractured, with contents of 1.09, 8.54, 18.18, and 0.19%. The shale numerical specimen was then built using the mechanical input parameters in Table 1[43; 44; 45]. The uniaxial compressive simulation was carried out with an external displacement at a constant rate of 2.25 \(\times\) 10\({}^{-4}\)\(\upmu\)m/step in the vertical direction.