## 2 Methodology  
To calculate stresses near the well, we applied the ANSYS 19 simulation software, which contains ratios describing the behavior of elastic and poroelastic material. This software has been widely used for solving geomechanical problems in the development of oil fields [21; 22; 23]. In order to understand how solid bodies (casing and cement stone) change their shape, it is useful to compute the relation between the forces applied on the object and the corresponding change in shape. In simple terms, it relates the stresses and the strains in the material. Thus, to calculate stresses in the casing and cement stone, we used the following ratios, which describe a linear elastic material (where the relationship between stress and strain is linear):  
* Equations of motion (moments): \[\sum_{j}\frac{\partial\sigma_{ij}}{\partial x_{j}}+\rho f_{i}=0;\;i,j\;\;=1,2,3,\] (1) where \(\sigma_{ji}\) are the components of the stress tensor; \(\partial x_{j}\) is the derivative of the J-coordinate; \(\rho f_{i}\) are the mass forces;
* Geometric equations: \[\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right);\;i,j=1,2,3,\] (2) where \(\epsilon_{ij}\) is the components of the strain tensor; \(u\) is the components of the displacement vector;
* Physics relations (in this case, [PERSON]'s law of linear elasticity): \[\{\sigma\}=[D]\{\epsilon\},\] where \(\{\sigma\}\) is the stress tensor; \([D]\) is the matrix of elastic constants; \(\{\epsilon\}\) is the strain tensor. ANSYS also provides a special model of porous elasticity, which contains the following system of equations describing the behavior of poroelastic material:  
\[\left\{ \begin{array}{l}\
abla\cdot\sigma+f=0\text{ or }\
abla\cdot(\sigma^{ \prime}-\alpha pI)+f=0\\ \alpha\;\dot{\varepsilon}_{V}+\frac{1}{K_{m}}\dot{p}+\
abla\cdot q=S\end{array} \right. \tag{3}\]  
where \(\sigma\) is the stress tensor; \(\dot{\varepsilon}_{V}\) is derivative of volumetric deformations of the rock matrix; \(\dot{p}\) is the pore pressure derivative; \(\
abla\cdot\) is the divergence operator; \(\sigma^{\prime}\) is the effective stress tensor; \(\alpha\) is the Biot coefficient; \(p\) is the pore pressure; \(I\) is the second order unit tensor; \(f\) is the force vector; \(\varepsilon_{V}\) stands for the volumetric deformations of the rock matrix; \(K_{m}\) is the Biot modulus; \(q\) is the fluid flow vector; \(S\) is the flow source. Besides, we applied the equations for the inter-relations between stresses and deformations:  
\[\sigma^{\prime}=D\epsilon^{\epsilon}, \tag{4}\]  
where \(\epsilon^{\epsilon}\) is the strain tensor; \(D\) is the matrix of elastic constants. [PERSON]'s law was used to describe the fluid flow in a porous media:  
\[q=-k\
abla p/\mu, \tag{5}\]  
where \(k\) is the second order permeability tensor; \(\
abla\) is the gradient operator; \(\mu\) is the fluid viscosity. Figure 1 shows the finite element model of the stress-strain state near the well. We developed the model based on the well patterns commonly used in the oil fields of the Perm Region. Table 1 presents the geometric characteristics of the model and the elastic properties of the steel casing. Apart from perforation channels and productive rocks, this model also includes the main structural elements of the well: the casing and cement stone. For modeling an elastic medium, eight-node finite elements in the form of solid185 prisms were used, and for a poroelastic medium, cpt215, was also represented by eight-node prisms. In the ANSYS system, it is possible to replace eight-node finite elements with sixteen-node ones: cpt216 and solid186, respectively. However, this approach significantly increases the number of nodes in the model, which in turn, significantly increases the calculation time. Since we needed to perform calculations on a variety of options depending on the pressure in the well, we decided to save time and use eight-node finite elements, which are more efficient for our case.