## 2 Methodology
Figure 1: Finite element scheme used for calculations. In the ANSYS software package, to determine the convergence criterion, the calculation accuracy was set for stresses, which was \(1\times 10^{-8}\), and for pressure at \(1\times 10^{-10}\). At lower accuracy values, for example \(1\times 10^{-9}\) for stresses, the numerical calculation stopped converging. The following commands were used to set up the calculation:  
1. ANTYPE, STATIC: this command was used for static analysis, which is used in calculating stresses in a linearly elastic medium and pressure for a poroelastic medium. 2. NROPT, full: This command configures the Newton-Raphson method for a static or full transient analysis. The \"full\" parameter means using the full Newton-Raphson method with nonsymmetric element matrices, if such an option is available. A contact element represents a simulation component used to model the contact behavior between materials with different deformation properties. In our case, the finite element model applies contact elements between casing-cement stone and cement stone-rock. Thus, the contact elements simulate the slip of cement stone along the casing and the rocks. On the contrary, if we were to consider the model with the continuous contact elements, the stresses would not be redistributed accurately. When vertical stresses are applied to the upper boundary of the finite element model, stress concentrations occur within the casing-cement stone and cement stone-rock points of contact due to the different values of their elastic characteristics. Therefore, the stress distribution will not correspond to reality. To avoid this effect, researchers set a condition in the form of displacements fixed along the normal to the surface at the upper boundary of the model. This approach is possible when stresses are calculated near the open trunk of a vertical well or a cased well without perforation channels. However, if the model includes perforations, the vertical stress component will not be calculated accurately with the same boundary conditions. To test the model, calculations were first performed without taking into account the perforation channels, cement stone, and production column, that is, assuming an open vertical borehole. Filtration rates were calculated, and well productivity was determined. The results of the numerical calculation of the well productivity were compared with the analytical calculation using the Dupuy formula. The discrepancies between the numerical and analytical solutions amounted to 3.8%, which is considered quite acceptable. In the future, the authors plan to compare the calculation model with data from field studies of the productivity of wells, developing this productive facility. At the same time, if we use contact elements for obtaining the slip of material, the stress concentration will not occur, and radial stresses between the rock, cement stone, and casing will be transmitted naturally. The chosen approach allows us to calculate the stress distribution (including the vertical component) in the near-well zone more accurately. The following assumptions were made during the calculations:  
1. A linear elastic model was used to describe the behavior of a steel production column, since it adequately describes the properties of durable steel in the range of design stresses. 2. The plastic properties of the reservoir rock have not been studied in the laboratory, so it was decided to consider it as a poroelastic medium. It is worth noting that during the operation of the well, the near-well zone and the rock areas around the perforation \begin{table}
\begin{tabular}{c c c} \hline \(\boldsymbol{\aleph}\) & **Characteristic** & **Value** \\ \hline
1 & Outer diameter of the casing, mm & 168.3 \\
2 & Casing thickness, mm & 7.3 \\
3 & Drilling size, mm & 215.9 \\
4 & Radius of the model, m & 5 \\
5 & [PERSON]’s modulus of the casing, GPa & 200 \\
6 & Poisson’s ratio of the casing, d.f. & 0.2 \\
7 & Yield strength of casing steel, MPa & 372 \\ \hline \end{tabular}
\end{table}