This paper deals with the theoretic simulation of a drill bit whirling under conditions of its contact interaction with the bore-hole bottom rock plane. The bit is considered to be an absolutely rigid ellipsoidal body with uneven surface. It is attached to the lower end of a rotating elastic drill string. In the perturbed state, the bit can roll without sliding on the bore-hole bottom, performing whirling vibrations (the model of dynamic equilibrium with pure rolling when maximum cohesive force does not exceed the ultimate Coulombic friction). To describe these motions, a nonholonomic dynamic model is proposed, constitutive partial differential equations are deduced. With their use, the whirling vibrations of oblong and oblate ellipsoidal bits are analyzed, the functions of cohesive (frictional) forces are calculated. It is shown that the system of elastic drill string and ellipsoidal bit can acquire stable or unstable whirl modes with approaching critical Eulerian values by the parameters of axial force, torque and angular velocity. The analogy of the found modes of motions with ones of the Celtic stones is established. It is shown that the ellipsoidal bits can stop their whirling vibrations and change directions of their circumferential motions in the same manner as the ellipsoidal Celtic stones do. As this takes place, the trajectories of the oblate ellipsoidal bits are characterized by more complicated paths and irregularities.
Schen, A.E., Snell, A.D. and Stanes, B.H. (2005) Optimization of Bit Drilling Performance Using a New Small Vibration Logging Tool. SPE/IADC Drilling Conference, Amsterdam, 23-25 February 2005.
Aarrestad, T.V. and Kyllingstad, A. (1988) An Experimental and Theoretical Study of a Coupling Mechanism between Longitudinal and Torsional Drillstring Vibrations at the Bit. SPE Drilling Engineering, 3, 12-18.
Chen, S.L., Blackwood, K. and Lamine, E. (2002) Field Investigation of the Effects of Stick-Slip, Lateral, and Whirl Vibrations on Roller-Cone Bit Performance. SPE Drilling & Completion, 17, 15-20.
Wu, X., Paez, L.C., Partin, U.T. and Agnihotri, M. (2010) Decoupling Stick/Slip and Whirl to Achieve Breakthrough in Drilling Performance. IADC/SPE Drilling Conference and Exhibition, New Orleans, 2-4 February 2010.
Ledgerwood, L.W., Jain, J.R., Hoffmann, O.J. and Spencer, R.W. (2013) Downhole Measurement and Monitoring Lead to an Enhanced Understanding of Drilling Vibrations and Polycrystalline Diamond Compact Bit Damage. SPE Drilling & Completion, 28, 254-262. http://dx.doi.org/10.2118/134488-PA
Ledgerwood, L.W., Hoffmann, O.J., Jain, J.R., El Hakam, C., Herbig, C., Spencer, R.W. and Hughes, B. (2010) Downhole Vibration Measurement, Monitoring, and Modeling Reveal Stick-Slip as a Primary Cause of PDC-Bit Damage in Today’s Applications. SPE Annual Technical Conference and Exhibition, Florence, 19-22 September 2010, 2652-2661. http://dx.doi.org/10.2118/134488-MS
Sowers, S.F., Dupriest, F.E., Bailey, J.R. and Wang, L. (2009) Roller Reamers Improve Drilling Performance in Wells Limited by Bit and Bottomhole Assembly Vibrations. SPE/IADC Drilling Conference and Exhibition, Amsterdam, 17-19 March 2009. http://dx.doi.org/10.2118/119375-ms
Ritto, T.G., Soize, C. and Sampaio, R. (2009) Non-Linear Dynamics of a Drill-String with Uncertain Model of the Bit-Rock Interaction. International Journal of Non-Linear Mechanics, 44, 865-876.
Samuel, R. (2010) Friction Factors: What Are They for Torque, Drag, Vibration, Bottom Hole Assembly and Transient Surge/Swab Analyses? Journal of Petroleum Science and Engineering, 73, 258-266.
Gulyayev, V.I. and Shevchuk, L.V. (2013) Nonholonomic Dynamics of Drill String Bit Whirling in a Deep Bore-Hole. Journal of Multi-Body Dynamics, 227, 234-244. http://dx.doi.org/10.1177/1464419313482658
Gulyayev, V.I. and Glushakova, O.V. (2011) Large-Scale and Small-Scale Self-Excited Torsional Vibrations of Homogeneous and Sectional Drill Strings. Interaction and Multiscale Mechanics, 4, 291-311.
Gulyayev, V.I., Hudoliy, S.N. and Glushakova, O.V. (2011) Simulation of Torsion Relaxation Auto-Oscillations of Drill String Bit with Viscous and Coulombic Friction Moment Models. Journal of Multi-Body Dynamics, 225, 139-152. http://dx.doi.org/10.1177/1464419311405571
Gulyayev, V.I. and Tolbatov, E.Yu. (2002) Forced and Self-Excited Vibrations of Pipes Containing Mobile Boiling Fluid Clots. Journal of Sound and Vibration, 257, 425-437. http://dx.doi.org/10.1006/jsvi.2002.5045
Gulyayev, V.I. and Tolbatov, E.Yu. (2004) Dynamics of Spiral Tubes Containing Internal Moving Masses of Boiling Liquid. Journal of Sound and Vibration, 274, 233-248. http://dx.doi.org/10.1016/j.jsv.2003.05.013
Gulyayev, V.I. and Borshch, O.I. (2011) Free Vibrations of Drill Strings in Hyper Deep Vertical Bore-Wells. Journal of Petroleum Science and Engineering, 78, 759-764. http://dx.doi.org/10.1016/j.petrol.2011.09.001