Mathematical model for estimating the longitudinal resonance of the drill string for drilling directional sections of wells

Abstract

The article is devoted to the urgent problem of ensuring the dynamic stability of drill strings during the construction of oil and gas wells with a complex spatial trajectory of the shaft. The work substantiates that modern technologies for deepening inclined and horizontal sections are accompanied by significant friction and adhesion forces, which cause the occurrence of harmful phenomena of intermittent slip (stick-slip) and longitudinal friction, which prevent the effective transmission of axial load to the bit. The main method of overcoming these resistances is the introduction of hydraulic axial vibration generators (HGV) into the bottom of the drill string (BHS) assemblies. However, it has been established that additional vibrations generated by oscillators, at certain frequency regimes, can provoke the occurrence of longitudinal resonance, which leads to a critical increase in stress amplitudes, fatigue fracture of the metal and emergency tool breakage. The scientific novelty of the study lies in the development and theoretical substantiation of a refined mathematical model of longitudinal vibrations of a two-stage drill string, which takes into account the specifics of curved sections of the well. The authors used the apparatus of the classical wave equation of an elastic rod with the integration of viscous damping parameters and nonlinear Coulomb friction forces. The model describes the dynamic interaction of two stages of the column (the upper part and the BHA), which have different geometric dimensions and physical and mechanical properties. Particular attention is paid to the boundary conditions at the junction of the stages, where stress concentration occurs due to reflection and amplification of waves. The influence of the zenith angle, the intensity of the barrel curvature, the density of the drilling fluid and the characteristics of the GOK was taken into account. During the study, modeling was carried out for four types of layouts, including options using steel pipes and titanium colubing sections. The results of the analysis of the amplitude-frequency characteristics (AFC) showed that for shorter and stiffer steel columns, the resonance peaks are most pronounced, while in long columns (over 4000 m) there is a “blurring” of the resonance due to high energy dissipation. It was found that the use of light alloys, such as titanium, significantly shifts the spectrum of natural frequencies downward, which creates a risk of resonance at higher harmonics at high oscillator operating frequencies (up to 20 Hz). The practical significance of the work lies in deriving a system of inequalities to determine the boundaries of longitudinal resonance zones. The resulting analytical apparatus allows drilling companies to calculate critical frequencies at the design stage and choose such bit rotation modes and flushing fluid flow rates that would guarantee the operation of the equipment outside the limits of dangerous vibrations. This will make it possible to increase the operational reliability of the drilling tool and reduce the cost of building deep directional wells.