## Abstract

Drilling geometrically complex boreholes in the subsurface has been made possible with the development of downhole tools that steer the bit. This paper proposes a model of borehole propagation that can be used not only to predict the bit trajectory, here restricted to a vertical plane, but also to assess the conditions leading to abnormal situations. The model is formulated from considerations involving (a) a bit/rock interaction law that relates the force and moment on the bit to its penetration into the rock, (b) kinematic relationships that describe the local borehole geometry from the bit motion, and (c) a beam model of the bottom-hole assembly (BHA) that expresses the force and moment at the bit as functions of both the loads applied on the BHA and the geometrical constraints imposed by the stabilizers that center the BHA in the borehole. The coupling between the model components leads to the formulation of a delay differential equation that governs the evolution of the borehole inclination, with spatial delays corresponding to the positions of the stabilizers behind the bit. The drilling response at a length scale corresponding to these delays is affected by the geometric feedback provided by the stabilizers that force the BHA to conform locally to the borehole geometry. At a length scale sufficiently large compared to the delays, the propagating borehole follows a trajectory characterized by a quasi-stationary curvature, which ultimately evolves towards a constant inclination equilibrium solution if it exists. The trajectory is sometimes counterintuitive, with the borehole tending to steer opposite to what is intended with the steering tool. Moreover, under certain conditions, which depend on the BHA configuration, on the design and bluntness of the bit, and on the rock, perturbations in the borehole trajectory that are sensed by the stabilizers are amplified progressively at the bit, until a limit cycle characterized by finite oscillations is reached. Both the directional stability of the system and the counterintuitive behaviors can be related to a key dimensionless group that can be interpreted as the ratio of a pseudostiffness characterizing the bit/rock interaction and the bending stiffness of the BHA.

Original language | English (US) |
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Pages (from-to) | 1089-1114 |

Number of pages | 26 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 77 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

## Keywords

- Delay differential equation
- Directional drilling
- Self-excited oscillations