We consider a distributed axial-torsional drill-string model with a rate-independent bit-rock interaction law to study the occurrence and non-local characteristics of axial and torsional self-excited vibrations as caused by the regenerative effect. A first contribution of the paper is the derivation of a non-dimensional version of the full non-linear distributed drill-string–bit-rock interaction model and showing how it relates to the minimal set of characteristic quantities. Using this model the study shows how multiple axial modes of the drill-string are excited, or attenuated, depending on the bit rotation rate. This indicates that a lumped drill-string model approximation is insufficient for the general case. Then, a comprehensive simulation study is performed to create a stability map for the occurrence of stick-slip oscillations. In particular, the significance of the axial topside boundary condition, i.e., constant velocity vs. constant hook-load, is evaluated. A central finding is that increasing the axial loop gain (determined by the bit-rock parameters) tends to both increase the area of stable torsional dynamics and increase the rate of penetration for a constant imposed weight on bit. This also corresponds to a more severe axial instability.
- Distributed parameter systems
- Drill-string vibrations
- Hyperbolic systems
- Infinite dimensional systems