# Hydraulic Fracture Propagation near the Cavity in a Poroelastic Media
## 2 Basic Equations of the Model  
Let us consider the problem of crack propagation in a homogeneous poroelastic medium, the deformations of which are determined by the following equations:  
\[\sigma_{ij}=2\mu\varepsilon_{ij}+\lambda\varepsilon_{0}\delta_{ij}-\alpha p \delta_{ij} \tag{1}\]  
where \(\sigma_{ij}\) is the solid body stress tensor; \(\varepsilon_{ij}=(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i} )/2\) is the solid body strain tensor; \(u_{i}\) are the body displacements; \(\lambda\), \(\mu\) are the Lame parameters of dry rock; \(\varepsilon_{0}\) is the volume strain; \(p\) is the fluid pressure within the poroelastic body; and \(\alpha\) is the Biot coefficient. The fluid flow in a porous medium is determined by the continuity equation:  
\[\frac{1}{M}\dot{p}+\alpha\dot{\varepsilon}_{kk}+v_{k,k}=0 \tag{2}\]  
where \(M\) is the Biot modulus, and \(v\) is the fluid flow velocity. The coefficients \(\alpha\) and \(M\) can be determined through the modulus of volumetric compression of the fluid \(K_{f}\), the modulus of volumetric compression of the solid phase (skeleton) \(K_{s}\), the modulus of volumetric compression of the porous medium \(K\) and porosity \(\gamma\):  
\[\frac{1}{M}=\frac{\gamma}{K_{f}}+\frac{\alpha-\gamma}{K_{s}},\quad\frac{1}{K_ {s}}=\frac{1-\alpha}{K} \tag{3}\]  
Filtration of the fluid through a porous medium is specified by [PERSON]'s law:  
\[v_{i}=-\frac{k}{\eta}p_{i} \tag{4}\]  
where \(k\) is the medium permeability, and \(\eta\) is the dynamic viscosity of the fluid. The fluid flow in the fracture was described by the continuity equation:  
\[\frac{\partial d}{\partial t}+\
abla\cdot\overset{\rightarrow}{q}+v_{t}+v_{ b}=0 \tag{5}\]where \(d\) is the crack opening; \(\overset{\rightarrow}{q}=(q_{1},q_{3},q_{2})\); \(q_{i}=\widetilde{dk}(\partial p/\partial x_{i});\widetilde{k}\) is the effective permeability of the crack in the \(i\)th direction; \(v_{t}\) and \(v_{b}\) are the rates of fluid leakage into the formation through the upper and lower crack surfaces. Within the problem under consideration we believe that the fluid flow inside the crack is Poiseuille flow, and the fluid is Newtonian. Thus, the effective fracture permeability can be defined as:  
\[\widetilde{k}=-\frac{d^{2}}{12\eta} \tag{6}\]  
In the process of modeling we assume that the fluid flowing in the fracture and the fluid filtered in the porous medium are the same. Crack propagation modeling is done using the Cohesive Zone Model (CZM) approach [48]. It is assumed that there is a softening zone of the material formed right before the fracture. In this zone, stresses \(\sigma\) occur, and they are associated with displacements \(u\) according to a certain law. Two parameters are required to determine the law, i.e., \(\sigma_{c}\) and \(G_{\text{IC}}\). The parameter \(\sigma_{c}\) is the critical stress, and \(G_{\text{IC}}\) is the critical energy release rate during fracturing, which is calculated as:  
\[G_{\text{IC}}=\int_{0}^{u_{f}}\sigma du \tag{7}\]  
where \(u_{f}\) is the displacement discontinuity that corresponds to the total destruction of the material. In the case in which the cohesive failure zone is small compared to the length of the crack, the stress intensity factor and the critical energy released in failing are related by the formula [49]:  
\[K_{\text{IC}}=\sqrt{G_{\text{IC}}\frac{E}{1-v^{2}}} \tag{8}\]  
where \(E\) is the elasticity modulus of the material. The destruction of the rock starts when the critical stress, \(\sigma_{c}\), is attained. Then, new values of the critical stress, \(\overline{\sigma}_{c}\), and the elastic modulus, \(\overline{E}\), are calculated, according to the specified nature of the fracture, which is determined by the fracture variable \(D(u)\), which depends on the deformations outside the linear elastic zone. Figure 1b shows an example of the function \(D(u)\), where \(D=0\) corresponds to the intact state, and \(D=1\) corresponds to complete damage.