# Hydraulic Fracture Propagation near the Cavity in a Poroelastic Media
## 3 Numerical Test Results and Discussion
### Influence of Poroelastic Parameters on Hydraulic Fracturing  
Next, we consider the influence of the poroelastic parameters on the crack propagation path using the following example. Let the initial crack be oriented parallel to the surface of the circular cavity (Figure 5a) and located at a distance equal to half the radius of the cavity (\(s=0.5\)\(R\)). In the first numerical experiment, the influence of the Biot coefficient, \(\alpha\), on the crack propagation has been investigated at a constant medium permeability, i.e., \(k=1\)\(\upmu\)m\({}^{2}\), and a constant hydrostatic stress, \(P\), equal to \(1.5\) MPa. The value of \(\alpha\) determines the degree of mutual influence of the stress-strain state of the solid phase and the pore pressure of the fluid. The value \(\alpha=0\) corresponds to a fully uncoupled problem (i.e., the pore pressure  
Figure 4: Crack path at hydrostatic stress \(P=1.5\) MPa: (**a**) various values of the critical stress \(\sigma_{c}\) (1—\(\sigma_{c}=7.5\) MPa; 2—\(\sigma_{c}=0.75\) MPa; 3—\(\sigma_{c}=0.075\) MPa); (**b**) various values of the elastic modulus, \(E\) (1—\(E=30\) GPa; 2—\(E=3\) GPa; 3—\(E=0.3\) GPa). Figure 3: Path of hydraulic fractures at different positions of the initial crack and different hydrostatic stresses, \(P\). Curve numbers: 1—\(P=0\) MPa; 2—\(P=1.5\) MPa; 3—\(P=6\) MPa. and the stressed state of the medium do not affect each other), and \(\alpha=1\) corresponds to a fully coupled problem in which the influence of the stress-strain state and pore pressure on each other is maximal. Figure 5a shows the path of cracks near a circular cavity at \(\alpha=0\); 0.5; 0.75, and 0.9. At the initial stage of crack growth, the stress state around the cavity has the main influence on the direction of its propagation. In this case, as can be seen in Figure 5a, there are no differences between the paths of the cracks at different values of the Biot coefficient. They appear as the lengths of the cracks increase. Curve 1 (Figure 5a) corresponds to the uncoupled problem and coincides with curve 1, which was obtained by modeling hydraulic fracturing in an elastic formulation. As the value of \(\alpha\) increases, the effect introduced by pore pressure intensifies, and the path of the fracture deviates more from curve 1. The second numerical experiment presents the recorded value, i.e., \(\alpha=0.75\), and the changed permeability of the medium. The hydrostatic compression of the medium is 1.5 MPa. Figure 5b shows the curves for the permeability of the medium that are equal to \(10^{5}\), 1000, 100, and 1 \(\mu\)m\({}^{2}\). When the permeability is high (Figure 5b, curve 1), the fracture path is the same as in the uncoupled problem from the first experiment (Figure 5a, curve 1). This occurs because there is no significant increase in pore pressure during the deformation of the medium, and, therefore, it does not affect the growth of the crack significantly. As the permeability of the medium decreases, the effect of pore pressure increases, and the fracture paths begin to differ. With the chosen parameters of the model, the position of curve 4 is the limiting case, i.e., any additional decrease in permeability does not result in significant changes. Thus, when the problem coupling increases (\(\alpha\) increases) and the permeability decreases, then the fractures are less curved. This fact also can be demonstrated in the following example. Let's consider a horizontally-oriented, initial crack located at some distance from the cavity (Figure 6). Figure 6a,b show the path of hydraulic fracturing for hydrostatic stress in the medium equal to 0.75 MPa and 3 MPa, respectively. Curve 1 denotes a hydraulic fracture for an uncoupled problem (\(\alpha=0\)). Curve 2 denotes a hydraulic fracture obtained for a coupled problem with the Biot parameter \(\alpha=0.75\) and permeability \(k=1\)\(\mu\)m\({}^{2}\). In the case of a coupled problem, it is apparent that the cracks are less curved and propagation occurs closer to the cavity.