Spatial-temporal dynamics of a drill string with complex time-delay effects: Bit bounce and stick-slip oscillations

In this article, an integrated model with spatially distributed inertia and general boundary conditions for the axial-torsional dynamics of a drill string is developed and studied. Based on the cutting geometry of the bit-rock interface, a notion called a complex delay is introduced. In the proposed complex-delay model, both the state-dependent delay from the torsional vibration and the multiple regenerative effects due to the loss of contact of drill bit are taken into account. Focusing on the system dynamics with the time delay, an efficient algorithm is developed to discretize and calculate the complex time delay by using a recursion method. The drill-string structure is spatially discretized by using the finite-element method. Numerical studies are conducted and several types of self-excited vibrations are illustrated, including stick-slip motions and bit bounce. The consideration of the complex delay in the drilling process suggests that the maximum number of regeneration periods can exceed ten and that multiple regenerative effects dominate the time-delay effects when the drill bit bounces away from the rock surface and experiences loss of contact. Finally, comparisons are made between the results obtained with the traditional state-dependent-delay model and the proposed complex-delay model. The results reveal that the complex-delay model is volume conservative in terms of the total cut material during the drilling process, while the state-dependent-delay model is not volume conservative. With just the state-dependent-delay model, due to the violation of cutting volume conservation, the result is an unrealistic torque on bit and a higher energy consumption of the drill bit, which all lead to an inaccurate picture of the system dynamics. By contrast, with the proposed complex-delay model, one is able to better capture the cutting dynamics and energy consumption of the drill bit. The findings also reveal that the time-delay effects are weakened, when loss of contact is experienced by the system.