Vibration of drillstring has detrimental effects on drilling performance. This type of vibration is much more severe in air and gas drilling than in liquid drilling due to the low damping effect of air and gas. It is highly desirable for drilling engineers to have a reliable mathematical model to simulate the vibration and identify resonance conditions before and during the drilling operations. This paper fills the gap. Traditionally drillstring vibrations have been analyzed using bit-force as a boundary condition. Accuracy of such analyses is often questionable as the discrepancies have been observed between model-predictions and field data. Mathematical models of longitudinal vibration of drill strings excited by both bit-force and bit-displacement have been established in this study. Equations are solved using the method of separation of variables. Calculations with the solutions indicate that the resonance rotary speeds of drillstring predicted by the bit-force model are non-resonance rotary speeds predicted by the bit-displacement model, and vice versa. A field case study is presented in this paper. Data from a field operation with drillstring failure seems to support the bit-displacement model, not the bit-force model. The models are presented to drilling engineers to further verify their accuracies. Once fully validated, the mathematical models will provide drilling engineers an effective tool for designing drill string for air and gas drilling operations against drillstring failure.
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