Natural Frequency Analysis of a Stepped Drill String in Vertical Oil Wells Subjected to Coupled Axial–Torsional–Lateral Vibrations

Drilling oil and gas wells is a complex process that requires a combination of several parameters to dig into the ground. Inappropriate drilling parameter settings and reaction forces can lead to unwanted vibrations, which can negatively impact the drill string and cause damage to drill bits. To reduce unwanted oscillations, drilling vibration modeling is the first approach used to determine the behavior of the drill string under various conditions. Natural frequencies, one of the modal characteristics of a vibrating drill string, can be estimated by analytical or numerical models. However, as the field conditions become more complicated, analytical models become increasingly difficult to use, and alternative approaches must be adopted. The main objective of this paper is to investigate the natural frequencies of drill strings with real geometry under coupled vibration modes using both analytical and finite element methods. This study bridges the literature gap in modeling stepped drill string geometries, which are usually represented as uniform beams. This paper used analytical and finite element models to determine the drill string’s lateral, axial, and torsional natural frequencies under varying lengths of drill pipes and drill collars. To assess the reliability of finite element models under complex geometry, the drill string was approximated as a stepped beam rather than a uniform beam. Then, a comparison was made with analytical models. The results showed that the length of drill pipes has a pronounced effect on the natural frequencies of the overall drill string for the three vibrational modes, while drill collar length only has a notable impact on the torsional mode. These findings contribute to drilling systems’ reliability and efficiency in the oil and gas energy sector.