An Experimental and Theoretical Study of a Coupling Mechanism Between Longitudinal and Torsional Drillstring Vibrations at the Bit

Abstract

Summary. A mechanism that couples longitudinal and torsional drillstring vibrations at the bit was studied. Torsional vibrations are associated with dynamic variations of the rotational bit speed. When a roller bit runs over a multilobed pattern, these speed variations have been shown to affect the input of longitudinal vibrations. The theory for this coupling mechanism is verified experimentally by high-rate data of near-bit accelerations and torque recorded in a 1000-m [3,280-ft] -deep well. Introduction Several publications have dealt with vibrations in drillstrings. As discussed in the literature, vibrations influence the drilling situation and can result in a lower rate of penetration or may cause fatigue failures. On the other hand, a thorough understanding of the vibrations can enable predictions of downhole conditions on the basis of top measurements. The theory presented here is an attempt to explain some phenomena seen in experiments performed at Ullrigg, a full-scale offshore-type research drilling rig. These experiments comprise both real drilling and "drilling" with an exciter system, which induces a longitudinal, sinusoidal vibration with a constant amplitude at the bit. The experimental setup with data transmission system is thoroughly described in Ref. 1. Frequency spectra on drilling data gathered at Ullrigg often contain small side lobes near the most dominant frequency components. The sources for these side lobes are also discussed here. Theory It is assumed here that the axial motion of the bit can be described by an elevation function, s(s theta), where s theta is the angular displacement of the bit. Assuming also that the shape and orientation of the downhole patterns do not change or at least change very slowly with time implies that ,............................(1) where n is an integer. The axial force through the bit, or weight on bit (WOB), can formally be written as ,............................(2) where F denotes the mean force and F denotes the dynamic or fluctuating force. The torque on bit (TOB) may also be separated into a mean torque, tau, and a dynamic part that is represented by ,............................(3) Because of this dynamic torque, the rotation rate of the bit will fluctuate about a mean value omega: ,............................(4) where omega = omega(tau) may be regarded as a dynamic torsional response to the input torque tau. Now, turning to the special case where the downhole patterns are pure sinusoidal three-lobed patterns with amplitude s1, the axial bit motion can be written as .............................(5) Note that Eq. 5 is correct only with a three-cone roller bit and if the roller radius is much smaller than the pattern curvature radius. This requirement is reasonably well satisfied for the exciter system, where s1 = 2 mm [0.08 in.]. In the first linear approximation, it is assumed that the dynamic variations of force and bit speed are negligibly small--i.e., .............................................(6) and .............................................(7) The input for axial and torsional vibrations is then decoupled and may be written as .......................................(8) and ........................................(9) The negative sign appears because the torque input--i.e., the torque transfer from the formation to the bit--is minus the TOB. Hence, the dynamic torque input is positive when the bit is running downward on a negative slope: ds/ds theta less than 0. Eq. 8 has been used in several earlier publications on drillstring vibrations. Experiments have revealed, however, that the assumptions for this first linear approximation are poorly satisfiedi.e., the dynamic force, F, and the dynamic speed, omega, are not negligibly small. Hence the displacement, ,...................................(10) and the torsional excitation moment, ,............................(11) SPEDE