A non-linear Cosserat rod model for drill-string dynamics in arbitrary borehole geometries with contact and friction

Abstract

A model for drill-string dynamics in arbitrary borehole geometries based on the Cosserat rods theory is presented. The objective of this work is to exploit the capabilities of the Cosserat rods to capture the full range of possible dynamics. The continuous model is implemented in a finite element environment and verified with related benchmarks reported in the literature. Non-linearities of the drill-string problem occur due to geometrical factors such as finite displacements and rotations as well as to contact and friction at the borehole wall. A contact and friction model is introduced. An application is solved and compared against an approach where the friction effect is introduced through an external load (soft-string model). All possible translations and rotations in the space are considered in the Cosserat formulation of the 1-D continuum. Singularities that may arise due to the rotations description are successfully dealt with using a formulation based on quaternions. Thus, the advantage of capturing the whole dynamics with an intrinsic contact and friction model is apparent. The solution herein obtained can address more complex phenomena, which results in an improvement over the commonly used linear beam-like models where only torsional or axial-torsional effects are considered. Moreover, the beam-like models are not appropriate for non straight borehole geometries, whereas the Cosserat formulation overcomes this limitation. To sum up, a dynamic, continuous and non-linear model that considers the torsional, axial and bending effects in arbitrary borehole geometry is proposed. The formulation intrinsically includes contact and friction at the wellbore wall and bottom. A close agreement between the solution for the present Cosserat rod model and a soft-string model is found for the case under study.