Whirl and Chaotic Motion of Stabilized Drill Collars
- J.D. Jansen (Koninklijke/Shell E and P Laboratorium)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling Engineering
- Publication Date
- June 1992
- Document Type
- Journal Paper
- 107 - 114
- 1992. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 1.11 Drilling Fluids and Materials, 1.6.1 Drilling Operation Management, 4.1.5 Processing Equipment, 1.6 Drilling Operations, 1.2.5 Drilling vibration management, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 1.12.1 Measurement While Drilling, 1.5 Drill Bits, 4.3.4 Scale, 1.10 Drilling Equipment
- 3 in the last 30 days
- 662 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 5.00|
|SPE Non-Member Price:||USD 35.00|
Stabilized bottomhole assemblies (BHA's) often exhibit complicated lateral vibrations. This paper aims to clarify this behavior with the theory of rotor dynamics. The motion of the collars varies from simple whirling motion, like that of an unbalanced centrifuge, to highly irregular motion caused by nonlinear effects of fluid forces, stabilizer clearance, and borehole-wall contact. In the extreme case, the vibrations can be classified mathematically as chaotic. The results presented in this paper are important in the interpretation of field measurements and indicate the limits of large-scale computer simulations for the prediction of directional tendencies.
Lateral (bending) vibration of the stabilized part of a BHA often results in large side forces on the bit, and, thus, may influence the directional tendency of the assembly. Furthermore, lateral vibration may cause premature failure of downhole components and damage to the borehole wall. To control the lateral vibration of an assembly, the nature of the motion involved must be understood on the basis of both experiment and theoretical analysis. This paper uses the theory of rotor dynamics (a well-developed area in mechanical engineering) as a basis for qualitative description of the dynamics of stabilized drill collars.
Drilling with a slightly bent drill collar or an unbalanced measurement-while-drilling tool can result in violent lateral vibrations analogous to the circular (whirling) motion of an unbalanced centrifuge. This phenomenon is illustrated by considering a single span of vibrating drill collars between two stabilizers (Fig. 1). For low vibrational frequencies, the deflection of the collars resembles a simple sinusoid. The motion of the collars then can be described completely by the displacements (x , x ) of the collar axis in a plane midway between the stabilizers (Figs. 2 and 3). If a constant rotary speed is assumed, and if the influences of gravity, fluid forces, stabilizer clearance, and friction are neglected, the equations of motion of the system (Appendix A, Refs. 7 and 8) are as follows.
where m =equivalent mass of the drill collars, k= equivalent bending stiffness of the collars, e =eccentricity of the center of mass, = rotary speed, t = time, and the dots above the variables indicate differentiation with respect to time. The equivalent mass and stiffness are derived from the properties of the drill collars with a virtual-work approach. Eqs. 1a and 1b each describe a plane lateral vibration with terms that represent an inertial force, an elastic restoring force, and a harmonic excitation. The excitation force can be recognized as the centrifugal force arising from the rotation of unbalanced collars-i.e., collars having a center of mass that does not coincide with the geometric center. The steady-state solution of Eqs. 1a and 1b is
where w= and represents the first natural angular frequency for lateral vibration of the span of drill collars. Eqs. 2a and 2b represent two lateral vibrations that are 90 out of phase. Together, they describe a forward whirling motion of the phase. Together, they describe a forward whirling motion of the collars around the center of the borehole, with an angular frequency (whirl speed) equal to and an amplitude that depends on the ratio of the whirl speed to the natural angular frequency. When = (i.e., when the collars rotate at their critical speed), resonance occurs and the amplitude becomes unbounded. If damping is taken into account, the maximum amplitude remains finite and the critical speed is defined as the rotary speed at which the amplitude is maximum. The critical speed is influenced by the effects of drilling fluid, stabilizer clearance, and stabilizer friction. Eqs. B-4a and B-4b in Appendix B are explicit expressions for the critical speed and the corresponding whirl amplitude. The following trends may be derived from these expressions: added fluid mass and increased stabilizer clearance reduce the critical speed and fluid damping reduces the whirl amplitude. It is not possible to establish the effect of stabilizer friction on the critical speed with a simple expression. An implicit formula for the amplitude of whirling collars under the influences of friction, clearance, and fluid forces is given as Eq. B-3. Note that the equations in the appendices have been made dimensionless by scaling them according to Eq. A-8. An example of forward whirl caused by mass eccentricity is depicted in Fig. 4, which was obtained by numerical integration of the equations of motion given in Appendix A. After the transients have been damped out, the numerical result matches the analytical value of the steady-state response obtained from Eq. B-3. Note that all figures are plotted with scaled variables. Table 1 gives the parameters used in Figs. 4 through 9, and Appendix C gives a numerical example that indicates the order of magnitude of these parameters. The collar displacements (y , y ) in Fig. 4A and the parameters. The collar displacements (y , y ) in Fig. 4A and the whirl amplitude, r, in Fig. 4B are scaled with respect to the collar clearance; an amplitude r= 1 implies contact between the collars and the borehole wall. The dimensionless time, , in Fig. 4B is scaled with respect to the natural angular frequency, . If the parameter values from the example of Appendix C are used, it follows parameter values from the example of Appendix C are used, it follows that = 250 corresponds to t 32 sec.
|File Size||681 KB||Number of Pages||8|