A Study of Slip/Stick Motion of the Bit

Abstract

Summary. This paper deals with torsional oscillations caused by slip/stick motion of the drill-collar section. This phenomenon is associated with a large-amplitude, saw-tooth-like variation in the applied torque. "Slip/stick motion" refers to the belief that the amplitude of the torsional oscillations becomes so large that the drill-collar section periodically comes to a complete stop and does not come free until enough torque is built up in the drillstring to overcome the static friction. A mathematical model of slip/stick motion is presented. This model includes parameters describing downhole friction effects and a simplified description of the drillstring. The effects of damping. finite rotary-table inertia, and the rotary-speed control system are discussed. Theoretical predictions are compared with measured torque signals recorded during field drilling. This kind of drilling performance is likely to be less effective than normal drilling and may also lead to fatigue problems. This paper discusses ways to avoid severe torsional oscillations by using a more sophisticated feedback system to control the rotary Introduction Vibrations within a drillstring are associated with drilling problems. The influence on drilling performance by torsional vibrations has not been extensively discussed in the literature, but recent measurements with a rotary-table torquemeter have shown that low-frequency torsional oscillations can he severe. Torsional vibrations in the drillstring have been the subject of recent papers. Halsey et al. conclude that torsional resonance frequencies in the drillstring are very nearly independent of drilling parameters such as weight on bit and rotary speed, so long as the drillstring rotates freely. They observed that only the intensities of the observed torsional resonances are affected by downhole conditions, at least in normal drilling. Here, slip/stick motion was considered as an exception to normal drilling, with the onset of slip/stick motion indicated by the breakup of the normal torsional resonance pattern. Dawson et al. treat the special case of slip/stick motion. Their formulation begins with a simple torsional pendulum. but models the frictional torque as a function of the rotational velocity in an attempt to model more realistically the friction between the drill-string and wellbore. One important conclusion from their work is that slip/stick motion is lost above a certain critical rotary rate. This agrees with field observations. In the mathematical model presented here, slip/stick motion is assumed to occur. That is, the model does not predict whether slip/stick motion will or will not occur under a given set of conditions. The simple torsional pendulum is taken as a reasonable approximation of the drillstring, in torsional oscillations. This, of course, describes only the lowest torsional resonance mode of the drillstring. Input to the simple model includes the torque necessary to rotate the drill collars and the additional torque necessary to overcome the static friction between the drill collars and the well-bore walls. The model predicts the slip/stick period. the fraction of time the bit is at rest, and the increase in dynamic torque resulting from slip/stick motion. Pendulum Model It is assumed here that the drillstring behaves like a simple torsional pendulum. This implies that the twisting of the relatively stiff bottomhole assembly (BHA) is neglected and that the round-trip time for torsional waves is negligibly small compared to the natural oscillation period. It is also assumed that the rotary-table speed, denoted here by, is constant. These assumptions also are adopted by Dawson et al. The validity of these assumptions will be discussed below. Because the BHA is treated as a lumped flywheel. the equation of motion for the bit is ............................... (1) where denotes the angle of rotation. J is the effective moment of inertia. and S is the torsional stiffness of the drillpipe section. Finally. r denotes the frictional torque acting on the BHA. For a drillstring composed of a drillpipe section of length L1 and a uniform drill-collar section of length L2, a good approximation for the inertia is.................................... (2) where p is the density and I1 and I2 are the cross-sectional polar moments for the two sections. The torsional stiffness is just .................................. (3) where G is the shear modulus of the drillstring material. The frictional torque T is modeled by .................................... (4a) and ...................... (4b) Here Tc is a constant Coulomb friction torque and is the extra torque needed to start the BHA or to break it loose. Viscous friction is neglected in this model for mathematical convenience and because it is assumed to be small. The effect of an additional viscous damping term will be discussed below. Without loss of generality, it is now assumed that the bit comes loose at time t=0 as indicated in the second part of Fig. 1. Because the torque level in the string must be equal to the maximum friction torque when this occurs, the appropriate initial conditions are ............................. (5) and ........................................ (6) SPEDE P. 369^