A stuck drill pipe has been recognized as one of the most costly and non-productive challenges in drilling operations. Fishing jars are routinely used to un-lock or loosen the stuck (jammed) pipes which in many cases are expensive and the time taken to complete the job can reach several days of continuous jarring. The use of surface mounted vibratory systems has offered an alternative cost effective means to free the stuck pipes. Almost all of these systems are based on eccentric-weight oscillators which impart simple harmonic vertical forces that are transmitted down the pipe via elastic standing waves through the pipe material. A more recent development also uses a suspended oscillator but imparts a sinusoidal oscillatory displacement (rather than force) to the drill pipe at the top surface end, which again is transmitted down the pipe via elastic standing waves. This paper provides a generalized technique for solving the governing equations describing this top oscillatory system and the transmission of the elastic waves along the drill pipe. The transfer matrix technique is used to describe the travelling/standing waves along the pipe, the connecting couplings and the top suspended drive system. Effects of damping are introduced in the complex wave number and at the coupling locations. Examples of drill pipe scenarios are presented to elucidate the usefulness of the technique to determine the resonance condition, that is, the excitation frequencies for maximum retrieving forces at the stuck end, for any given drill pipe geometry. The resulting force amplitudes at the top driver end and the resulting retrieving forces imparted at the stuck end are quantified for any given imposed displacement amplitude at the drive end. A more complex system involving a drill pipe, spear and an elastic liner is also described where the transfer matrix technique is demonstrated to be an effective means to determine the overall system dynamics and resonance conditions.
Key words: Drilling, drill pipe, spear, liners, solid elastic dynamic, elastic waves.