## 3 Results
### Embedding an Artificial Micro-fracture with Different Aperture  
Based on the digital rock of the tight sandstone reservoir obtained from XCT technology, the artificial micro-fracture model is added with a dip of 0 degrees and a length of 500 voxels (penetrating), followed by a voxel ranging from 2 voxels to 50 voxels with a resolution of 1.0 \(\upmu\)m/voxel to form a multi-scale digital rock as shown in Figure 5. Due to the artificial micro-fracture surface's roughness, some spots have closed micro-fractures while others have larger micro-fracture apertures than desired. However, the average aperture of the micro-fracture is the set value. The micro-fracture is close to the real situation. Figure 4 also shows the pore network model for multi-scale digital rock extraction of tight sandstone reservoirs using the maximal balls method with 0.6 h. The micro-fracture was treated as a series of interconnected pores and throats. In order to further analyze the effect of micro-fracture on the pore structure for tight sandstone reservoirs, the pore radius, and throat radius distribution were plotted, see Figures 6a and 6b, respectively. In Figure 6, different colors represent different models for where the original model obtained from XCT technology is red, while the green, blue, and pink hues embody models with micro-fracture apertures of 10 \(\upmu\)m, 30 \(\upmu\)m, and 50 \(\upmu\)m, respectively. All curves for pore radius and throat radius distribution have a similar trend. For pore radius distribution, the size of the pore expands as the micro-fracture aperture increases, especially if the micro-fracture aperture is more than 10 \(\upmu\)m. For throat radius distribution, the size of the throat expands as the micro-fracture aperture increases due to the existence of the micro-fracture. The presence of the micro-fracture significantly changed the pore structure of tight sandstone reservoirs. Altering the radius of the pores and throats can substantially enhance the fluid flow conditions. This factor also underscores the significance of hydraulic fracturing in enhancing recovery. To further assess the significance of quantitatively evaluating the micro-fracture, the impact of varying pore and throat radii on permeability was computed and graphed. The formula used to calculate the influence of distinct pore and throat radii on permeability is expressed as follows:  
\[\Delta K_{i}=\frac{r_{i}^{2}*\alpha_{i}}{\sum r_{i}^{2}*\alpha_{i}}. \tag{4}\]  
For the \(i\) th interval, the permeability contribution value is denoted by \(\Delta K_{i}\), the corresponding pore or throat radius value is represented by \(r_{i}\), and its frequency is given by \(\alpha_{i}\). The impact of varying pore and throat radii on permeability was plotted in Figure 6a,b. In Figure 7, different colors represent different models. In comparison with the initial digital rock model, the curves of the digital rock with a different aperture for pore and throat are bimodal distributions. In Figure 7a,b, as the micro-fracture aperture increases, a new peak emerges in the curve, indicating that the fracture is beginning to impact the permeability. Figure 5: The digital rocks embedding an artificial fracture with a different aperture and pore networks extracted using the maximal balls method. (**a**) shows the digital rock and pore network with the micro-fracture whose aperture is 2 \(\upmu\)m, (**b**) is the digital rock and pore network with the micro-fracture whose aperture is 10 \(\upmu\)m, (**c**) represents the digital rock and pore network with the micro-fracture whose aperture is 30 \(\upmu\)m, and (**d**) displays the digital rock and pore network with the micro-fracture whose aperture is 50 \(\upmu\)m.  
Figure 6: The pore structure distribution of the embedding an artificial fracture with different aperture. (**a**) is the pore radius distribution, and (**b**) is the throat radius distribution.