Drill string vibrations have resulted in an increase in non-productive time, drilling cost, and a need for drill string system optimization in the oil and gas industry. Higher vibrations can lead to washout, blow-out phenomena, and a rapid wear and tear of drill string components. An in-depth understanding of the root causes of the drill string vibrations is of utmost importance. The effects of mass imbalance, bottom hole assembly (BHA) design, weight on bit, hole angle, hole size, RPM, drilling fluid viscosity, bit and formation type on drill string vibrations have been studied in the past. The main objective of this paper is to propose a novel drill string vibration model taking into consideration the effects of axial compression load, borehole clearance, and contact force in conjunction with Euler-Bernoulli, Rayleigh, and Shear beam theories. The proposed model studies the effects of axial compression load, borehole clearance, and stiffness on the natural frequency and frequency response of drill string. The drill string is considered as a cantilever beam with a circular cross-sectional area in a horizontal drilling condition, while the fluid (water) around it is considered as a spring and dashpot model. Drill string vibrations are modeled based on a distributed system approach. The forces acting on the cantilever beam element are axial compression load, shear forces, bending moment, fluid stiffness and damping forces. A fourth order partial differential equation is derived and the solution is obtained based on the assumed modes approach to obtain the characteristic equation and mode shapes. Natural frequency is determined using the eigen values obtained from the characteristic equation. The orthogonality conditions are further derived to decouple the derived equation. The drill string is initially assumed unbounded, hence the amount of contact force required to restrict the drill string within the borehole is determined. An iterative process is employed to restrict the drill string deflection within the borehole with a corresponding increase in the stiffness of the drill string system. This will help in determining the experienced down hole contact forces and the stiffness requirements of the bottom hole assembly. Dominant frequencies are identified, compared with field results, further guiding the operator to avoid rotating the drill string at particular frequencies. A detailed study of the effects of contact forces, borehole clearances, and axial compression load will provide comprehensive guidelines in BHA design of drill string assembly. To conclude, this paper proposed an accurate model to identify dominant frequencies and provides a thorough BHA design guidelines for the drill string design manufacturers and field operators leading to a reduction in drilling cost and unproductive time, and environmentally friendly drilling operations.

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