# NUMERICAL METHOD OF CALCULATING THE CRITICAL LOAD ON THE CHILL AT DRILLING WELLS

## Authors

• M. I. Horbiichuk Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska Street Ivano-Frankivsk Ukraine, 76019
• D. R. Kropyvnytskyi Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska Street Ivano-Frankivsk Ukraine, 76019

## Keywords:

bit load, drill string, Runge-Kutta method, least squares method, empirical model.

## Abstract

The paper considers the effect of axial load on the bit on the shape of the drill string. It is determined that increasing this parameter to a critical value will lead to deformation of the drill string, as a result it  can cause an accident. To determine the critical value of the axial load, a polygon of forces acting at the lower end of the drill string was constructed, determined by the  coordinate that is the point of maximum deflection of the drill string, and mathematical dependences were made as a system of differential equations (Cauchy form). An algorithm for determining a given x coordinate has been developed in the MatLab environment, which includes solving a system of differential equations using the Runge-Kutta method and building an interpolation Lagrange polynomial. Graphical dependences of drill string deflection change at different values ​​of bending moment are developed. It has been shown that increasing load on the bit makes the torque to increase, as a result of which the intensity of the curvature of the drill string increases.

Based on the results of calculations, empirical models for determining the point of maximum deflection of the drill string in the form of a polynomial of the 2nd order were synthesized, the parameters of which were calculated by the method of least squares. The adequacy of the models was checked using a correlation coefficient. The calculated values ​​of the correlation coefficient are close to 1, so it can be  claimed that the proposed empirical model adequately describes the "experimental" data.

A series of machine experiments was performed at different values ​​of the maximum degree of the polynomial and it was determined that for the degree of the polynomial equal to 3, the empirical model, which is a function f (a, x) describes the results with high accuracy and the number of polynomial members is 20.

It was determined that when drilling a well, the critical load on the bit can be calculated by two factors - the point of maximum deflection of the drill string and the length, which is determined by the difference between the maximum deflection point and the neutral section point.

## References

Osnovy naftohazovoyi spravy / Biletskyy V. S., Orlovskyy V. M., Dmytrenko V. I., Pokhylko V. M. Poltava: PoltNTU, Kyyiv: FOP Khalikov R. KH., 2017. 312 s.

Vuds H., Lubynskyy A. Yskrevlenye skvazhyn pry burenyy / per. s anhl. M.: Hostekhyzdat, 1960. 161 s.

Tymoshenko S. P., Here Dzh. Mekhanyka materyalov: uchebnyk, 2-e yzd. SPb: Lan, 2002. 672 s.

Sultanov B. Z. Upravlenye ustoychyvostyu y dynamykoy burylnoy kolonny. M.: Nedra, 1991. 208 s.

Horbiychuk M. I., Pistun YE. P. Chyslovi metody i modelyuvannya na EOM: navchalʹnyy posibnyk. Ivano-Frakvivsʹk, 2010. 409 s.

Ermakov S. M., Ermakov S. M., Zhyhlyavskyy A. A. Matematycheskaya teoryya optymalnoho éksperymenta. M.: Nauka, 1987. 320 s.

Mikhail I. Gorbiychuk, Taras V. Humenyuk Synthesis Method of Empirical Models Optimal by Complexity under Uncertainty Conditions Journal of Automation and Information Sciences. – vol. 48, is. 9. – P. 64 -74.

Horbiychuk M. I., Kohutyak M. I., Zayachuk YA. I. Induktyvnyy metod pobudovy matematychnykh modeley hazoperekachuvalnykh ahrehativ pryrodnoho hazu // Naftova i hazova promyslovist. 2008. № 5. S. 32 – 35.

2023-02-20

## How to Cite

Horbiichuk, M. I., & Kropyvnytskyi, D. R. (2023). NUMERICAL METHOD OF CALCULATING THE CRITICAL LOAD ON THE CHILL AT DRILLING WELLS. METHODS AND DEVICES OF QUALITY CONTROL, (1(48). https://doi.org/10.31471/1993-9981-2022-1(48)-115-126

## Section

MATHEMATICAL MODELLING FOR THE UNDESTROYED CONTROL PROBLEMS