Drillstring vibration is a dangerous dynamic phenomenon mostly happened. It may result in catastrophic and costly problems, such as failure of drillstring components and also leads to the wear of bit and decreased efficiency. Various studies were carried out to analyze the behavior of the drillstring; however, few of them took the bit-rock interaction into consideration. Therefore, a better understanding of the dynamic behavior of the drillstring in the presence of bits is necessary. In this paper, a new model is developed basing on Timoshenko beam theory to describe the dynamic behavior of drillstring. Firstly, several sources of excitation in vibration are analyzed in order to figure out their influences on the vibration. Then, boundary conditions and initial conditions are presented. Afterwards, the dynamic behavior of drillstring is considered as composed of axial-torsional vibration and lateral vibration, and governing equations are gotten by analyzing the equilibrium of tiny part of drillstring. These partial difference equations in space and in time are discretized by means of the finite element method and finite difference method, respectively. A computational code in Scilab is written to solve these equations and numerical simulations with this code are performed in several situations.


Drillstring vibration, especially in the form of torsional stick-slip motions, axial bit-bounce motions, and lateral whirl motions result in failure of drillstring components, deterioration of the well trajectory, etc (Ezzeddine 2013, Ghasemloonia et al. 2015). In order to avoid these failures, a better understanding of the dynamic behavior of the drillstring in the presence of bits should be obtained.

Static modeling, analytical dynamic modeling, and numerical modeling (e.g. finite element or finite difference) are approaches mostly used to analyze the behavior of the drillstring. The dynamic behavior of drillstring is a complex superposition of lateral, axial, and torsional-beam vibration, so these vibrations were separately studied at the beginning of research. Then the coupling of axial-lateral, axial-torsional, and torsional-lateral modes are investigated. But there are few studies on the coupling of three vibrations. Analytical dynamic modeling and numerical modeling could be classified into Frequency-domain method and Time-domain method. To simplify the proceeding of calculation, lumped parameter model or called a MDOF (multi-degree-of-freedom) model is developed. In this type of model, the drillstring is regarded as a linear system with series of rigid masses connected by massless elastic springs. The Newtonian approach and the energy variational approach (Hamilton's principle with the variational approach) are extensively used to derive the equations of motion of the drillstring.

In most of studies on the dynamic behavior of drilling, mass imbalance, drillstring-wellbore contact, misalignment, are considered as the source of excitations, but bit-rock interaction is neglected or just modeled as a simple boundary condition. For example, in many studies about axial vibration, a free boundary condition with a sinusoidal force excitation at the location of bit is implemented; an equivalent massspring-damper is also used to simulate bit-rock interaction. Previous studies suggest that bit-rock interaction is influenced by various factors such as rate of penetration (ROP), type of bit, type of rock, pressure of mud, temperature, etc. That means even if the surrounding environment condition is the same, the movement of drillstring influences the force between bit and rock, which inversely influences the movement of drillstring. Ritto (2010) combined the drillstring and bit with a probabilistic model of bit-rock interaction; however, unfortunately there are some errors in his derivation of model.

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