**4. Contrast Experiment of the Stress Shadow Effect between Different Borehole Spacings**

The study of the stress shadow effect between natural primary cracks or artificial cracks was initially carried out for fractures created during shale gas extraction [14,15]; in Yu et al. [16], an analytical equation was given for the redistribution of the induced stress components between the stress shadows [17–19]:

$$\begin{split} \Delta \sigma\_{xx,\text{induced}} &= \frac{\int\_{-A\_{\xi}}^{A\_{\xi}} (\sigma\_{xx,\text{induced}}l\_{1}^{2} + \sigma\_{yy,\text{induced}}m\_{1}^{2} + 2\pi\_{xy,\text{induced}}l\_{1}^{2}m\_{1}^{2})dx}{2A\_{\xi}}\\ \Delta \sigma\_{yy,\text{induced}} &= \frac{\int\_{-A\_{\xi}}^{A\_{\xi}} (\sigma\_{xx,\text{induced}}l\_{2}^{2} + \sigma\_{yy,\text{induced}}m\_{2}^{2} + 2\pi\_{xy,\text{induced}}l\_{2}^{2}m\_{2}^{2})dx}{2A\_{\xi}} \end{split} \tag{7}$$

where *x* is the average induced stress along the tangent direction and normal direction of the crack, *Ac* is the half length of the crack, taken as 0.5 m; *mi* is the distance between end cracks, m; *li* is the crack width, m; *σyy*,*induced*, *σxx*,*induced*, *τxy*,*induced* is the stress tensor, respectively, MPa; Δ*σyy*,*induced*, Δ*σxx*,*induced*, Δ*τxy*,*induced* is the increment in the stress tensor, respectively, MPa [20–23].

Figure 6a,b shows the influence range of the von Mises equivalent stress in the rock mass caused by 0.12 m borehole fracturing with 3.0 m spacing; Figure 6c,d shows the influence range of the von Mises equivalent stress in the rock mass caused by 0.12 m borehole fracturing with 3.5 m spacing, and Figure 6e,f shows the influence range of the von Mises equivalent stress in the rock mass caused by 0.12 m borehole fracturing with 4.0 m spacing. Through comparative analysis, with the increase in the borehole spacing, the damage range around the borehole also increased, especially the damage range at the orifice, and the overall average value of the equivalent stress around the borehole

also increased at the spacing of 4.0 m. Therefore, in order to avoid the influence of the stress shadow effect on the spatial shape of the fracture development and the expansion and the pressure relief in advancing during the actual hydraulic-fracturing rock-breaking construction, it is necessary to ensure that the influence range of the equivalent stress of the fracture is at the overlapping critical edge as far as possible.

**Figure 6.** Stress shadow effect represented by an equal-effect stress field with different hole spacings. (**a**) Equivalent stress field of 0.12 m borehole with 3 m spacing. (**b**) Equivalent stress field of 0.12 m borehole with 3 m spacing (details). (**c**) Equivalent stress field of 0.12 m borehole with 3.5 m spacing. (**d**) Equivalent stress field of 0.12 m borehole with 3.5 m spacing (details). (**e**) Equivalent stress field of 0.12 m borehole with 4 m spacing. (**f**) Equivalent stress field of 0.12 m borehole with 4 m spacing (details).

With the increase in the buried depth, the spatial shape trend of the fracture surface development and the expansion of the hydraulic-fracturing borehole wall changes significantly. Under the condition of shallow-buried in situ stress, where the horizontal maximum principal stress is the maximum principal stress, and the vertical stress is the minimum principal stress, the fracture surface tends to expand along the water square direction to weaken the thickness of the fractured rock stratum. Under the in situ stress condition of deep mines, where the vertical stress is the maximum principal stress, and the horizontal stress is the minimum principal stress, the fracture plane tends to shorten the fracture step of the fractured rock layer along the vertical direction. Therefore, it was concluded that under different in situ stress types and different purposes of cutting hard-rock layers, appropriate drilling orientations and induction methods should be selected to optimize and control the spatial morphology of the fracture plane.
