Measuring and Controlling Torsional Vibrations and Stick-Slip in a Viscous-Damped Drillstring Model
Authors: Omojuwa E , Osisanya S. , Ahmed R. 
Published: March 2015

Abstract
Failure of drillstring components due to torsional stresses are caused by torsional vibrations. This paper investigates the response of the drillstring to torsional vibrations through analysis on an elemental stepped-shaft drillstring model in a damped medium. The paper directs its analysis to the stick-slip motion which is the most common form of torsional vibrations in drillstrings; stick-slip motion is defined as the continuous stopping and release of the bit/BHA due to the irregular downhole rotation prompted by the existing relationship between the frictional torque and the torque applied from the surface to free the bit.
The interaction between the wellbore and the drill-bit is identified in this paper as the source of excitation, that is, as the downhole vibratory source; hence, the exciting torque or couple is derived from the resolution of forces and reactions at wellbore-drill-bit contact. The response equations obtained from theories of mechanical vibrations and mathematical principles in this paper are in two forms: the transient state and steady-state responses, which makes this research work different from other similar literatures where only the steady-state response is presented. Computer programs are written and tested with suitable drilling data to solve the response equations, display parameters required for vibration analysis and the graphics of the responses to different forms of excitations are also displayed.
In order to achieve further reduction of the oscillations of the drillstring model, a vibration control system is designed, which is a torsional vibrations absorber and its influence on the oscillations of the drillstring was also verified. The results and simulations from computer programs are also presented. In conclusion, recommendations are given based on practical solutions to the problems of vibrations and from the findings of this paper.

Introduction
Torsional or "stick-slip" vibrations are often regarded as one of the most damaging modes of vibration when drilling with low rotary speeds. "Stick-slip" is caused by irregular downhole rotation that fatigues drill collar connections, damages the bit and reduces rate of penetration (ROP). For a typical drillstring of length 15,000-ft (5000-m), torsional disturbance consists of a travelling torsional pulse that bounces back and forth between the top rotary and the drill bit every few seconds periodically forcing the drill bit to "stick" and "slip" for extended periods at the rock surface. The amplitude of this torsional excitation can be two to four times the target or average angular speed (typically between 30 and 150 RPM) set by the top-drive and this can give rise to enormously destructive fluctuating torques in the drillstring that, once out of control, invariably cause damage to the bit or drill string [1]. Torsional vibrations are recognized at the drill-floor by fluctuations in the power needed to maintain a constant rate of surface rotation [2]. The fatigues (cyclic stresses) imposed on the drillstring lead to the sudden failure of the drillstring components and a few other effects, such as: torsional cycles noted at the surface, cutters with impact damage, reduction in life of the bit [3]. Figure 1 shows a damaged BHA component due to shearing [3], an after-effect of torsional stresses.