Числовий метод обчислення критичного навантаження на долото при бурінні свердловин

Abstract

The article considers the influence 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, which in turn can lead to an accident. To determine the critical value of the axial load, was constructed a polygon of forces that acts at the lower end of the drill string, it can be found by a coordinate  that is exact of the maximum deflection of the drill string, and mathematical dependences were constructed in the form of a system of differential equations (Cauchy form). An algorithm for determining a given x coordinate was developed in the MatLab environment, which includes the solution of a system of differential equations by the Runge-Kutta method and the construction of an interpolation Lagrange polynomial. Graphical dependences of the change in the deflection of the drill string at different values of the bending moment are constructed. Based on the results of calculations, empirical models for determining the point of maximum deflection of a 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 unity, so it can be argued 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 of f (a, x), describes the data results with high accuracy.