Modeling and analysis of self-excited drill bit vibrations

The research reported in this thesis builds on a novel model developed atthe University of Minnesota to analyze the self-excited vibrations thatoccur when drilling with polycrystalline diamond cutter bits. The lumpedparameter model of the drilling system takes into consideration the axialand the torsional vibrations of the bit. These vibrations are coupledthrough a bit-rock interaction law. At the bit-rock interface, the cuttingprocess combined with the quasihelical motion of the bit leads to aregenerative effect that introduces a coupling between the axial andtorsional modes of vibrations and a state-dependent delay in the governingequations, while the frictional contact process is associated withdiscontinuities in the boundary conditions when the bit sticks in its axialand angular motion. The response of this complex system is characterized bya fast axial dynamics superposed to the slow torsional dynamics. A two time scales analysis that uses a combination of averaging methods anda singular perturbation approach is proposed to study the dynamical responseof the system. An approximate model of the decoupled axial dynamics permitsto derive a pseudo analytical expression of the solution of the axialequation. Its averaged behavior influences the slow torsional dynamics bygenerating an apparent velocity weakening friction law that has beenproposed empirically in earlier works. The analytical expression of thesolution of the axial dynamics is used to derive an approximate analyticalexpression of the velocity weakening friction law related to the physicalparameters of the system. This expression can be used to providerecommendations on the operating parameters and the drillstring or the bitdesign in order to reduce the amplitude of the torsional vibrations.Moreover, it is an appropriate candidate model to replace empirical frictionlaws encountered in torsional models used for control. In this thesis, we also analyze the axial and torsional vibrations by basingthe model on a continuum representation of the drillstring rather than onthe low dimensional lumped parameter model. The dynamic response of thedrilling structure is computed using the finite element method. While thegeneral tendencies of the system response predicted by the discrete modelare confirmed by this computational model (for example that the occurrenceof stick-slip vibrations as well as the risk of bit bouncing are enhancedwith an increase of the weight-on-bit or a decrease of the rotationalspeed), new features in the self-excited response of the drillstring aredetected. In particular, stick-slip vibrations are predicted to occur atnatural frequencies of the drillstring different from the fundamental one(as sometimes observed in field operations), depending on the operatingparameters. Finally, we describe the experimental strategy chosen for the validation ofthe model and discuss results of tests conducted with DIVA, an analogexperimental set-up of the lumpedparameter model. Some results of the experiments conducted in an artificialrock seem to validate the model studied here although the same experimentsobtained with natural rockswere unsuccessful. Different problems with the design of the experimentalsetup were identified. By using the outcome of the analysis of the uncoupleddynamics, we could provide critical recommendation to elaborate and todesign a simpler and stiffer analog experiment (TAZ) used to study the selfexcitation of the axial dynamics that ultimately lead to the excitation ofthe torsional dynamics.