Stability of a rotary drilling system with bits consisting of full and partial blades

One approach to analyze the stick-slip vibrations of a rotary drilling system with a PDC bit is based on the RGD model (Richard et al., 2007), a discrete model that assumes a rate-independent bit/rock interaction law. This model considers both axial and torsional vibration modes, which are coupled through the depth of cut, a state variable determined by the current bit positions and its motion history. The bit geometry is further simplified as n continuous blades that are evenly distributed around the axis of rotation. The model is governed by a set of delay differential equations, with one priori unknown time delay corresponding to the constant angle between two blades. According to the RGD model, there exists a critical rotational speed separating two regimes of instability (Depouhon and Detournay, 2014), a function of the number of blades and a characteristic time of the system. In this work, we extend the RGD model by considering a class of symmetric bits with the same number of full or partial blades. This extension results in two angular delays and a corresponding set of coefficients which captures the contribution of each angular delay to the cutting component of weight-on-bit and torque-on-bit. A linear stability analysis is conducted that leads to explicit expressions for the critical rotational speed(s), which is a function of the number of blades and the relative length of the partial blades. The stability maps indicate that the bits can have a better stability compared to their symmetric counterparts. Compared to the original RGD model, it is also discovered that for certain bit geometries, there may exist multiple critical rotation speeds. Time simulation is performed to verify the results from the linear stability analyses.