A high-fidelity model for nonlinear self-excited oscillations in rotary drilling systems

This paper presents an integrated model to analyze self-excited axial and torsional vibrations in rotary drilling systems using realistic PDC bits. The model combines a multi-degree-of-freedom (MDOF) drillstring representation with a detailed PDC bit–rock interaction model, accounting for cutter layouts and rock properties. The bit–rock interaction is described by rate-independent laws for reactive force and torque, encompassing cutting and frictional contact at the cutter face and wear flat. The regenerative rock cutting introduces state-dependent delays (up to 100) due to complex cutter layout of the bit, while the unilateral nature of frictional contact is formulated as a nonlinear boundary condition. The state-dependent delays are transformed into constant angular offsets prescribed by the bit design by introducing a bit trajectory function whose evolution is governed by a partial differential equation (PDE), which is coupled with the drillstring dynamics described by ordinary differential equations (ODEs). The Galerkin method with Chebyshev polynomials transforms the set of coupled PDE–ODEs into a system of ODEs. This approach bypasses the need to search for multiple delays at each time step, allowing implementation of this integrated rock-bit-drillstring model. Simulation results replicate field observed torque-on-bit (TOB) effects including velocity-weakening and hysteresis due to torsional stick–slips, attributing them to cutting depth variation rather than downhole friction. Preliminary findings regarding the influence of bit design on torsional stick–slips align with field drilling practices. The results of parametric study correspond to field test observations, enabling analysis of bit design, drillstring configuration, and drilling parameters on self-excited torsional stick–slips.