Nonlinear dynamic modeling and analysis of borehole propagation for directional drilling

Boreholes with complex trajectories are drilled with the help of downhole rotary steerable systems. These robotic actuators, which are embedded in the drillstring, are used to steer the bit in the desired direction. This paper presents a dynamic non-smooth borehole propagation model for planar directional drilling. Essential nonlinearities, induced by the saturation of the bit tilt and by non-ideal (undergauged) stabilizers, are modeled using complementarity conditions, leading to a closed-form analytical description of the model in terms of a so-called delay complementarity system. The analytical form of the model allows for a comprehensive dynamic and parametric analysis. Firstly, (quasi-)stationary solutions generated by constant actuator forces are analyzed parametrically as a function of the actuation force. Secondly, an analysis of the local stability of these solutions shows the coexistence of multiple (stable and unstable) solutions and their dependency on key system parameters, such as the weight-on-bit and bit characteristics. Thirdly, a numerical simulation study shows the existence of steady-state oscillations, which are a consequence of the non-smooth characteristics of the bit tilt saturation and the stabilizers. Such limit cycles represent borehole rippling, which is the planar equivalent of the highly detrimental borehole spiraling observed in practice. The constructed model and the pursued analysis provide essential insights in the effects causing undesired borehole rippling. Herewith, the presented results can be used to support improved directional drilling system design and to form the basis for further work on automation techniques for the downhole robotic actuator to mitigate spiraled boreholes.