Drilling industry focuses nowadays on process optimization and cost reduction. Unwanted events should be predicted and avoided to increase drilling efficiency, improve safety, and save costs. Development and application of mathematical models enable us to understand the dynamics of the drilling process, learn parameter interaction and regulate system behavior. It is also a way to reduce the risk of occurrence of such events or mitigate negative outcomes.

Challenges in two-dimensional modeling of drillstring vibrations include: (1) correct and precise interpretation of coupled two-dimensional motion, (2) use of sub-models, as down-hole weight on bit (WOB) model, downhole torque on bit (TOB) model and friction model, and (3) proper definition of associated boundary conditions. In this paper, we propose a two-dimensional axial-torsional model that considers these criteria. We present a new way to calculate downhole WOB, which can be used as an alternative to a constant WOB value. Dynamic boundary conditions are introduced to represent the respective phases of stick-slip. The model is formulated using the finite element method and intended for vertical wells. The main goal for developing this model is evaluation of the effect of surface rotational and axial velocities on the downhole drill bit dynamics. A dimensionless parameter, stick-slip severity, is used to represent the intensity of torsional oscillations.

The developed model is based on mathematical relations and defined boundary conditions, which describe the dynamics of drillstring during stick-slip events. The model allows to study the effect of up to twelve input parameters on stick-slip severity to determine a suitable range. Results presented in this paper show that axial velocity applied from the surface may cause initiation of stick-slip, which in turn provokes axial vibrations. Increase in surface axial velocity leads to higher amplitude of downhole torsional oscillations. To mitigate stick-slip, surface rotational velocity should be increased.

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