# Hydraulic Fracture Propagation near the Cavity in a Poroelastic Media
## 3 Numerical Test Results and Discussion
### Influence of Poroelastic Parameters on Hydraulic Fracturing
Figure 5: Path of a crack near a cavity: (**a**) hydrostatic compression of the medium is 1.5 MPa, permeability \(k=1\)\(\mu\)m\({}^{2}\), and the different numbers indicate the curves obtained for different values of the Biot coefficients: 1—\(\alpha=0\); 2—\(\alpha=0.5\); 3—\(\alpha=0.75\); 4—\(\alpha=0.9\); (**b**) hydrostatic compression of the medium is 1.5 MPa, the Biot coefficients, \(\alpha\), is 0.75, and different numbers indicate the curves obtained for different values of the permeability: 1—\(k=10^{5}\)\(\mu\)m\({}^{2}\); 2—\(k=1000\)\(\mu\)m\({}^{2}\); 3—\(k=100\)\(\mu\)m\({}^{2}\); 4—\(k=1\)\(\mu\)m\({}^{2}\). Next, we consider the features of the behavior of the hydraulic fracturing parameters near a circular cavity. The formulation of the problem is used as shown in Figure 2a. Figure 7a shows the direction of crack propagation, \(\beta\), as a function of the angle \(\varphi\), which is the angle between the horizontal line and the segment that connects the center of the cavity and the tip of the crack (Figure 7b). The curves in Figure 7a were obtained for the initial crack located at a distance \(s=0.5\,R\) from the cavity. The hydrostatic stress in the medium is \(1.5\) MPa. The dotted line represents a curve in case the hydraulic fracture is propagated circumferentially around the cavity. When the problem coupling increases (\(\alpha\) increases), it is apparent that the presence of a cavity has less effect on the trajectory of the crack. The curves for high \(\alpha\) move away from the dotted line faster, and the direction of the propagation of the crack does not change from the angle \(\varphi\). Paths differ slightly up to angle values of 40-45 degrees. The fluid pressure inside the fracture has specific features during the propagation of hydraulic fracturing near the cavity. Figure 7c,d show the fluid pressure on the mouth of the crack, depending on the time of injection and the half-length of the crack. The presence of a minimum on the curves is clearly visible. Such a minimum is not observed during the development of a crack in a homogeneous area. This minimum is associated with the peculiarities of the stress state arising around the cavity under the hydrostatic stress of the medium. In addition, the graphs show that when the Biot coefficient, \(\alpha\), increases, the fluid pressure (including the value of the minimum) also increases. Figure 7e,f show the opening of the mouth of the fracture. Figure 7e shows that no features are observed for the time dependence. As expected, when the problem coupling increases, the opening of the fracture mouth and the fluid pressure increase. If the opening of the fracture mouth is plotted as a function of the fracture half-length (Figure 7e), the inflection points on the curves can be seen. Figure 6: Results of modeling the hydraulic fracture propagation for an uncoupled (curves 1) and a coupled (curves 2) problem at various hydrostatic stresses: (**a**) hydrostatic stress is \(0.75\) MPa; (**b**) hydrostatic stress is \(3\) MPa.