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.