Global axial–torsional dynamics during rotary drilling

We have studied the global dynamics of the bottom hole assembly (BHA) during rotary drilling with a lumped parameter axial–torsional model for the drill-string and a linear cutting force model. Our approach accounts for bit-bounce and stick-slip along with the regenerative effect and is independent of the drill-string and the bit–rock interaction model. Regenerative axial dynamics due to variable depth of cut is incorporated through a functional description of the cut surface profile instead of a delay differential equation with a state-dependent delay. The evolution of the cut surface is governed by a nonlinear partial differential equation (PDE) which is coupled with the ordinary differential equations (ODEs) governing the longitudinal and angular dynamics of the BHA. The boundary condition for the PDE captures multiple regeneration in the event of bit-bounce. Interruption in the torsional dynamics is included by considering separate evolution equations for the various states during the stick period. Finite-dimensional approximation for our coupled PDE-ODE model has been obtained and validated by comparing our results against existing results. Bifurcation analysis of our system reveals a supercritical Hopf bifurcation leading to periodic vibrations without bit-bounce and stick-slip which is followed by solutions involving bit-bounce or stick-slip depending on the operating parameters. Further inroads into the unstable regime leads to a variety of complex behavior including co-existence of periodic and chaotic solutions involving both bit-bounce and stick-slip.