## 3 Results
### Embedding an Artificial Micro-fracture with Different Length  
To further investigate the impact of micro-fracture length on tight sandstone reservoirs, we incorporated a single fracture into the digital rock of the reservoir with an inclination of 0 degrees and an aperture of 10 voxels. The fracture length varied from 50 to 450 voxels in a sequential manner, resulting in a multi-scale digital rock as illustrated in Figure 9. The pore network model of the resulting tight sandstone reservoir multi-scale digital rock was then extracted using the maximal balls method with 0.6 h, as depicted in Figure 9.  
Figure 10 shows the pore structure distribution of the embedding of artificial fracture with different lengths, demonstrating the influence of micro-fracture length on the pore throat structure. As illustrated, the distribution of pore and throat radius shifts from a single-peaked Gaussian distribution to a double-peaked and multi-peaked distribution with increasing fracture length. Notably, this is particularly evident in the distribution of throat radius. The contribution of pore and throat radius to permeability is depicted in the relationship curves shown in Figure 11. These curves reveal that the range in which pore and throat radius contribute to the larger permeability value is relatively similar to that of the original digital rock. However, the contribution to permeability at 5 \(\upmu\)m becomes more  
Figure 8: The capillary pressure curve of embedding an artificial fracture with a different aperture in different directions. (**a**) X direction, (**b**) Y direction, (**c**) Z direction. significant with increasing fracture length which aligns with the proportion increase at 5 \(\upmu\)m illustrated in the figure. The effect of micro-fracture length on the permeability of a tight sandstone reservoir in different directions was summarized in Table 2. The time needed for each pore network  
Figure 11: The contribution for permeability of embedding an artificial fracture with different lengths. The pore’s contribution to permeability is (**a**), while the throat’s contribution to permeability is (**b**). Figure 9: The digital rocks embedding an artificial fracture with a different length and pore networks extracted using the maximal balls method. (**a**) shows the digital rock and pore network with the micro-fracture whose length is 50 \(\upmu\)m, (**b**) is the digital rock and pore network with the micro-fracture whose length is 200 \(\upmu\)m, (**c**) represents the digital rock and pore network with the micro-fracture whose length is 300 \(\upmu\)m, and (**d**) displays the digital rock and pore network with the micro-fracture whose length is 50 \(\upmu\)m.  
Figure 10: The pore structure distribution of the embedding artificial fracture with different lengths. (**a**) is the pore radius distribution, and (**b**) is the throat radius distribution. model is 0.1 h. Table 2 demonstrates that increasing the length of micro-fractures within the rock can lead to increased porosity. Increasing the length of micro-fractures within the rock can lead to increased permeability. Analyzing the permeability in different directions can provide valuable insights into how micro-fractures affect fluid flow. In this study, we investigate the effect of micro-fracture aperture on the permeability of a tight sandstone reservoir in different directions. Our results in Table 2 demonstrate that increasing the aperture of micro-fractures within the rock can lead to increased permeability. This is because micro-fractures significantly enhance the connectivity between pores which allows fluids to flow more freely, as illustrated in Figure 9. Additionally, as the fractures widen, the volume of open space within the digital rock increases, contributing to an overall increase in porosity. However, the relationship between porosity and micro-fracture length is close to linear. Analyzing the permeability in different directions can provide valuable insights into how micro-fractures affect fluid flow.