The ability to predict the response of a drill bit to the topside axial and rotational velocities of the drill-string is a prerequisite for any system aimed at automatically controlling the drilling parameters to optimize the rate of penetration and the overall quality of the well construction process. When drilling with a Polycrystalline Diamond Compact (PDC) bit, even the steady-state response can exhibit complex behavior, characterized by the presence of (at least) three different regimes whose range and parameters depend upon the bit characteristics and the mechanical properties of the formations being drilled. Transient effects significantly complicate the situation, especially when vibrations (axial, rotational or lateral) disturb the drilling process. Often, the root cause of these vibrations lies in the bit-rock interaction itself, while the drill string, through its elasticity and interaction with the borehole wall, may amplify or attenuate these vibrations. Therefore, continuous calibration of the drill string and bit-rock parameters from available surface and downhole measurements is critical for any automated control system relying on dynamic models of the drilling process.

We present a calibration procedure whose goal is two-fold: first, to identify the time-varying parameters involved in the bit-rock interaction, and second, to provide a low-order, transfer function model approximation of the drill string axial and rotational dynamics. Our approach is based on particle filter techniques and a refined instrumental variable method for transfer function model estimation, and allows for real-time estimation of the various model parameters. We illustrate its behavior against recorded drilling data, where the proposed methods are shown to capture the different dynamics in place. We explain, in addition, how the calibrated drill string and bit-rock interaction models can be integrated in a framework to identify drilling parameter regions prone to axial or rotational vibrations.

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