A Conservation Law for Evolving Borehole Geometry in Drilling

Accurately modeling the evolution of the borehole surface around the bit plays a crucial role in understanding the mechanism of self-excited vibrations in drilling processes. This paper studies the evolution of the bottom-hole surface under given axial-torsional-lateral bit motions. It is shown that the evolution of the bottom-hole surface geometry is governed by a partial differential equation (PDE) in the form of a conservation law with source terms. Scaling of the PDE is conducted to determine the dominant parameters influencing the evolution of the borehole surface. The scaled PDE is solved numerically using a finite-volume algorithm. A study of typical drilling modes demonstrates that the PDE model not only predicts the depth of cut ahead of cutting blades more accurately than classical time delay models but also reconstructs wellbore surface comparable to published simulation and experimental results. This PDE formulation shows a promising potential for the modeling of drill bit vibrations, especially bit whirl.