A Coupled Axial-Torsional-Lateral Bit-Rock Interaction Model

The nonlinear interaction at the bit/rock interface is recognized as a root cause of drilling instabilities and thus plays a fundamental role in the modeling of the self-excited vibrations of rotary drilling systems. This paper introduces a general bit-rock interaction law relating the forces and torques on the drill bit to the motion of the bit and its penetration into the rock formation. The proposed model can be viewed as an extension of the bit-rock interaction law in the classical model proposed by Richard et al (T. Richard, C. Germay, E. Detournay, A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits, J. Sound Vib. 305 (3) (2007) 432-456). The latter model only considers the axial force and torque on the bit while not accounting for lateral bit forces, which is essential in the scope of lateral vibrations of drilling systems. The model presented in this paper extends the existing model from the following two perspectives. First, both axial and lateral drilling forces, as well as the torque on bit, are described following a unified modeling framework. Second, not only normal drilling modes, but also abnormal drilling modes, such as axial and torsional stick, bit bounce, and reverse rotation, are taken into consideration. The global drilling forces acting on the bit are described in terms of integrals of single-valued and set-valued functions. These non-smooth functions are introduced to describe the force laws associated with the multiple interaction modes of a local cutter on the bit with the rock, including normal cutting, local sticking, inverse motion, etc. It is shown that these normal and abnormal local cutting modes are determined by the geometry and the motion of the drill bit and can be further related to the normal and abnormal drilling modes of the bit as a whole. From this perspective, the global motion of the bit is indirectly connected to the global drilling force through the local cutting behavior. The proposed model leads to bit-rock interaction laws for axial-torsional, lateral, and coupled axial-torsional-lateral dynamics of a rotary drilling system. All these generated laws are also applicable to abnormal drilling modes and they are equivalent to published bit-rock interaction laws for the normal drilling mode.