Dynamic Stability of Drillstrings Under Fluctuating Weight on Bit

Abstract

Summary This paper describes the theory and the underlying formulation behind the development of a drillstring-dynamics simulator that predicts rapidly growing lateral vibrations triggered by axially induced bit excitations. The analyses center on calculations of stable rotary speed ranges for a given set of drillstring parameters and are presented in vibration "severity" vs. rotary speed plots. The critical rotary speeds, which correspond to the rapidly growing lateral vibrations, are pinpointed by spikes on the severity plots. Some application, of pinpointed by spikes on the severity plots. Some application, of the drillstring-dynamics simulator are presented, and limitations and further model development are discussed. Introduction Field observations, in the form of downhole and surface vibration measurements, have clearly indicated that drill strings, particularly bottomhole assemblies (BHA's), generally are subjected to severe vibrations. These vibrations are induced primarily from two excitation sources: bit/formation and drillstring/borehole interactions. As a result of these excitations, the drillstring can vibrate in several ways: axially, laterally, torsionally, or more often, as combinations of these three basic modes (e.g., whirling). This makes the drillstring vibration problem fairly complex to investigate and makes full simulation of the dynamic events impractical. A more practical approach, therefore, would be to identify and investigate particular drillstring-vibration mechanisms individually. One such mechanism frequently observed in field vibration measurements is severe lateral vibrations induced as a result of axial excitations caused by bit/formation excitations, the subject matter of this paper. This drillstring-vibration mechanism is of particular significance because severe lateral vibration can induce accelerated fatigue failure in the drillstring and also can cause borehole enlargement and poor directional control. In addition, axial excitations also can be beneficial because they increase the rate of penetration (ROP) and reduce drag, so identification of damaging axial excitations becomes even more important. Drillstrings are subjected to axial loads with static and time-dependent components. The classic static-stability (buckling) theory of axially loaded drillstrings determines the critical value of static component of axial load, the maximum allowable static weight on bit (WOB) above which buckling will occur. The dynamic component of axial load is primarily caused by bit/formation interaction, which results in fluctuations in the WOB. When these fluctuations are taken into account, the loss of mechanical stability becomes evident as rapidly growing lateral drillstring vibrations (the dynamic counterpart of buckling). This occurs in much the same way as inducing a snaked motion in a vertically hanging rope by moving its end up and down at a particular frequency. This phenomenon, which is associated with particular frequency. This phenomenon, which is associated with specific axial fluctuations, is called parametric resonance. It can occur for much smaller WOB values than the critical WOB obtained from a static-stability (buckling) analysis.(Theoretically, dynamic instability may even occur at no WOB.) Therefore, under these circumstances, the static-stability analysis must be complemented by a dynamic-stability analysis that accounts for WOB fluctuations. Considerable work has been carried out on various aspects of drillstring vibrations. These have been mainly concerned either with the transient vibration modeling or with axial, lateral, and torsional natural excitation studies. The purpose of this paper is to complement these studies by establishing the conditions paper is to complement these studies by establishing the conditions under which the drillstring becomes laterally unstable as a result of axially induced vibrations (i.e., by establishing dynamic stability conditions). As outlined in this paper, such an analysis can be carried out by coupling axial and lateral vibration modes. The lateral vibration triggered by axial excitations causes the drillstring to precess around the wellbore. The complex problem of drillstring dynamics therefore is reduced to the determination of the onset of this drillstring precessional motion as a function of relevant drilling parameters (e.g., drillstring configuration, borehole inclination, rotary speed, WOB, and mud weight). Such an approach is mathematically and computationally attractive compared with more conventional transient vibration analysis. Lubkin and Stoker first studied dynamic instability for simply supported rods. Their work established the influence of axial vibration on the value of the static buckling load. Other researchers then carried out further work. Application of dynamic instability to the drillstring vibration problem was considered first in Refs. 16 through 18. This paper outlines the mechanism of the dynamic stability of drillstrings. The underlying formulation of a drillstring-dynamics model and its numerical implementation is presented. A number of model applications then are considered, and finally limitations and further developments are discussed. Dynamic Instability Mechanism The main aim of drillstring-dynamic-stability analysis is to determine under what conditions the axial vibrations (induced by WOB fluctuations) may trigger lateral vibrations that increase in amplitude. These conditions may occur when the energy associated with axial vibrations is diverted to lateral vibrations. This instability may occur when the frequency of WOB fluctuations is equal to twice a natural lateral frequency, w. This can be illustrated as follows. Fig.1a represents the motion of a point on the mid span of a simply supported column vibrating laterally in its lowest natural frequency. At a reference time t=t, the column is assumed to have an initial deflection w (which can be infinitesimally small). Let us now examine what would happen if the column were subjected to a fluctuating axial force, Ff = (F f0 sin 2wt) (Fig. 1c), which has a frequency 2w, twice that of the column lateral natural frequency. Fig. 1bshows the resulting lateral column deflection at various times. At time t1, the axial load, Ff, reaches its maximum, F f0 (and starts decreasing in magnitude), while the lateral column deflection continues to increase and Point A moves to the right to its maximum position. At time t2, when the lateral deflection reaches maximum, the value of Ff changes its sign, prompting Point A to move to the left. By time t4, Ff has completed one cycle, while the lateral motion has completes only one semicycle. At this moment, Ff reverses its sign again, prompting movement to the left after the column passes through the prompting movement to the left after the column passes through the neutral position. Byt8, Ff has completed two cycles while the lateral displacement has complete done cycle. Now, during periods t0 to t4, the axial load pumps the amount of energy U into the lateral vibration mode, which at t4 (when the column is passing its neutral position) manifests itself as excess kinetic energy. Likewise, during the next load cycle (periods t4 to t8), the axial load contributes a further amount of energy U that, when combined with the kinetic energy imparted during the previous load cycle (at t4), results in a lateral deflection with an amplitude greater than the previous half-cycle (i.e., the lateral vibration amplitude grows in magnitude during each load cycle). With further energy "packs," U, being pumped into lateral vibrations over each successive load cycle, the lateral motion amplitude grows infinitely. SPEDC P. 84