# Hydraulic Fracture Propagation near the Cavity in a Poroelastic Media
## 3 Numerical Test Results and Discussion
### Influence of \(s\) and \(P\) on the Hydraulic Fracturing Parameters  
In the following numerical experiments, we investigate the effect of the hydrostatic stress in the medium and the distance between the initial crack and the cavity on the fluid pressure and the opening of the mouth of the crack. For simplicity, here, we will consider the uncoupled problem. In the first experiment, the crack was located at a distance equal to half of the radius of the cavity (\(s=0.5\ R\)), and the hydrostatic pressure in the medium was changed. Figure 8a,b show the pressure curves that were obtained as a function of injection time and the half-length of the crack for \(P=0\), \(P=1.5\) MPa, and \(P=3\) MPa. The  
Figure 7: **(a)** The direction of crack propagation, \(\beta\), as function of the angle, \(\varphi\), between the horizontal line and the segment that connects the center of the cavity and the tip of the crack; (b) image of angles \(\beta\) and \(\varphi\); (c,d) pressure on the mouth of the crack as a function of injection time and the half-length of the crack for different values of \(\alpha\); (e,f) opening of the mouth of the crack as a function of injection time and the half-length of the crack for different values of \(\alpha\); (a,c-f) were obtained for a hydrostatic stress in the medium equal to \(1.5\) MPa and an initial crack located at the distance of \(s=0.5\ R\) from the cavity (half of the radius of the cavity). graphs show that the fracture pressure increases as the hydrostatic stress increases. At the same time, the minimum on the pressure curves becomes more pronounced (which can be seen better in Figure 8b) when plotting the pressure versus the half-length of the crack). In the absence of hydrostatic stress, the minimum pressure disappears. Figure 8c,d show the opening of the mouth of the crack. As expected, the opening of the mouth of the crack increases as the pressure of the fluid increases. If the opening of the mouth of the crack is drawn as a function of the half-length of the crack (Figure 8d), it can be seen that, for a hydrostatic stress of 3 MPa, the opening pressure changes insignificantly when a certain length is reached. In the next experiment, the hydrostatic stress in the medium is 1.5 MPa, and the distance from the initial crack to the cavity changed and is equal to 0.25 \(R\), 0.5 \(R\), and 1.0 \(R\). Figure 8e,f show the fluid pressure at the mouth of the crack. It can be seen that, when the initial crack gets closer to the cavity, the fluid pressure at the mouth for the initial stages of crack propagation decreases, and the minimum becomes more pronounced. Figure 8g,h show graphs of the opening of the mouth of the crack. Figure 8h shows that, when the initial crack is close to the cavity, the opening quickly reaches a certain value, after which it changes insignificantly. At the same time, a decrease of the distance \(s\) leads to an increase in the values of the opening of the mouth at the initial stages of the propagation of the crack. Thus, we have shown that the value of the minimum on the pressure curve is sensitive to the stress state in the medium and the position of the initial crack. Based on these facts, in the future, we plan to develop a method for determining the stress state of a rock mass near mines. The method will consist of performing a series of hydraulic fracturing treatments near the mine in order to record the fluid pressure curves in the crack. In addition, the inverse problem of determining the stress state will be solved. The results of the provided numerical experiments comport with the existing research. Figures 3-6 show that, with an increase in material strength and elastic modulus and decrease in permeability, fracture trajectories are less curved, which is consistent with the conclusions in [41; 50; 51; 52]. Additionally, in the work, the Biot coefficient characterizing the degree of problem coupling was used to study the effect of the medium's poroelastic model on hydraulic fracturing (Figures 5a and 7).