Modeling and analysis of stick-slip and bit bounce in oil well drillstrings equipped with drag bits

Rotary drilling systems equipped with drag bits or fixed cutter bits (also called PDC), used for drilling deep boreholes for the production and the exploration of oil and natural gas, often suffer from severe vibrations. These vibrations are detrimental to the bit and the drillstring causing different failures of equipment (e.g., twist-off, abrasive wear of tubulars, bit damage), and inefficiencies in the drilling operation (reduction of the rate of penetration (ROP)). Despite extensive research conducted in the last several decades, there is still a need to develop a consistent model that adequately captures all phenomena related to drillstring vibrations such as nonlinear cutting and friction forces at the bit/rock formation interface, drive system characteristics and coupling between various motions. In this work, a physically consistent nonlinear model for the axial and torsional motions of a rotating drillstring equipped with a drag bit is proposed. A more realistic cutting and contact model is used to represent bit/rock formation interaction at the bit. The dynamics of both drive systems for rotary and translational motions of the drillstring, including the hoisting system are also considered. In this model, the rotational and translational motions of the bit are obtained as a result of the overall dynamic behavior rather than prescribed functions or constants. The dynamic behavior predicted by the proposed model qualitatively agree well with field observations and published theoretical results. The effects of various operational parameters on the dynamic behavior are investigated with the objective of achieving a smooth and efficient drilling. The results show that with proper choice of operational parameters, it may be possible to minimize the effects of stick-slip and bit-bounce and increase the ROP. Therefore, it is expected that the results will help reduce the time spent in drilling process and costs incurred due to severe vibrations and consequent damage to equipment.     

Instability regimes and self-excited vibrations in deep drilling systems

This paper analyzes the stability of the discrete model proposed by Richard et al. (2004 [1], 2007 [2]) to study the self-excited axial and torsional vibrations of deep drilling systems. This model, which relies on a rate-independent bit/rock interaction law, reduces to a coupled system of state-dependent delay differential equations governing the axial and angular perturbations to the stationary motion of the bit. A linear stability analysis indicates that, although the steady-state motion of the bit is always unstable, the nature of the instability depends on the nominal angular velocity Ω₀ of the drillstring imposed at the rig. On the one hand, if Ω₀ is larger than a critical velocity Ω_c, the angular dynamics is responsible for the instability. However, on the timescale of the resonance period of the drillstring viewed as a torsional pendulum, the system behaves like a marginally stable one, provided that exogenous perturbations are of limited magnitude. The instability then only appears on a much larger timescale, in the form of slowly growing oscillations that ultimately lead to an undesired drilling regime such as bit-bouncing or stick-slip vibrations. On the other hand, if Ω₀ is smaller than Ω_c, the instability manifests itself on the timescale of the bit motion due to a dominating unstable axial dynamics; perturbations to the steady-state motion then rapidly degenerate into stick-slip limit cycles or bit-bouncing. For typical deep drilling field conditions, the critical angular velocity Ω_c is virtually independent of the axial force acting on the bit and of the bit bluntness. It can be approximated by a power law monomial, a function of known parameters of the drilling system and of the intrinsic specific energy (a quantity characterizing the energy required to drill a particular rock). This approximation holds on account that the dissipation in the drilling structure is negligible with respect to that taking place through the bit/rock interaction, as is typically the case. These findings are further illustrated on an example of deep drilling and shown to match the trends observed in the field.     

Vibration analysis of drillstrings with self-excited stick-slip oscillations

Drillstring vibration is one of the major causes for a deteriorated drilling performance. Field experience revealed that it is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. Sick-slip oscillations due to contact between the drilling bit and formation is known to excite severe torsional and axial vibrations in the drillstring. A dynamic model of the drillstring including the drillpipes and drillcollars is formulated. The equation of motion of the rotating drillstring is derived using Lagrangian approach in conjunction with the finite element method. The model accounts for the torsional-bending inertia coupling and the axial-bending geometric nonlinear coupling. In addition, the model accounts for the gyroscopic effect, the effect of the gravitational force field, and the stick-slip interaction forces. Explicit expressions of the finite element coefficient matrices are derived using a consistent mass formulation. The generalized eigenvalue problem is solved to determine modal transformations, which are invoked to obtain the reduced-order modal form of the dynamic equations. The developed model is integrated into a computational scheme to calculate time-response of the drillstring system in the presence of stick-slip excitations.     

Non-linear control of torsional and bending vibrations of oilwell drillstrings

Drillstring dynamics is highly non-linear in nature and its model can only be described by a set of non-linear differential equations. In addition to this complexity, the drillstring dynamics are not linearly controllable and thus linear control methods are not suitable for suppressing the coupled torsional and lateral vibrations of a rotating drillstring. In this paper a non-linear dynamic inversion control design method is used to suppress the lateral and the torsional vibrations of a non-linear drillstring. It was found that the designed controller is effective in suppressing the torsional vibrations and reducing the lateral vibrations significantly.     

A simplified model to explore the root cause of stick-slip vibrations in drilling systems with drag bits

In this paper a study of the self-excited stick-slip oscillations of a rotary drilling system with a drag bit, using a discrete model that takes into consideration the axial and torsional vibration modes of the system, is described. Coupling between these two vibration modes takes place through a bit-rock interaction law, which accounts for both the frictional contact and the cutting processes. The cutting process introduces a delay in the equations of motion that is responsible for the existence of self-excited vibrations, which can degenerate into stick-slip oscillations and/or bit bouncing under certain conditions. From analysis of this new model it is concluded that the experimentally observed decrease of the reacting torque with the angular velocity is actually an expression of the system response, rather than an intrinsic rate dependence of the interface laws between the rock and the drill bit, as is commonly assumed.     

Contact-impact analysis of geared rotor systems

In this paper, a finite element formulation, used to analyze the contact-impact behavior of geared rotor systems coupled with the rotational, lateral, and axial vibrations between gears at high rotational speeds, has been developed. A gear impact element to model the contact-impact behavior between gears has been developed and its numerical method is discussed. A relative displacement measurement idea has been proposed to measure vibration parameter for contrast experiment in high rotational geared system. The equations of motion are derived and solved iteratively during each time increment until the unbalanced force decrease to an acceptable tolerance level. Based on the proposed method, an analysis program, GEARS, has been developed. The contact-impact behavior of geared rotor systems is analyzed especially under high rotational speed condition as numerical examples, which are demonstrated to show the effectiveness of the proposed method.     

Experimental investigations of the response of a cracked rotor to periodic axial excitation

In this paper, the coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses is investigated experimentally. The cracked rotor is excited axially using an electrodynamic exciter at a frequency equal to its bending natural frequency in both non-rotating and rotating conditions. The resulting time domain and frequency domain signals of the cracked rotor are studied. Spectral response of the cracked rotor with and without axial excitation is found to be distinctively different. When excited axially, it shows prominent presence of rotor bending natural frequency. However for an uncracked rotor, the response is similar with or without axial excitation. It is thus proposed that the response of the rotor to axial impulse excitation could be used for more reliable diagnosis of rotor cracks.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation

The free in-plane vibrations of circular rings with wheel-plates as generalised elastic foundations are studied using analytical methods and numerical simulations. The three-parameter Winkler elastic layer is proposed as a mathematical model of the foundation. The effects of rotary inertia and shear deformation are included in the analytical model of the system. The motion equations of systems are derived on the basis of the thin ring theory and Timoshenko?s theory. The separation of variables method is used to find general solutions to the free vibrations. Elaborated analytical models are used to determine the natural frequencies and the natural mode shapes of vibrations of an arbitrarily chosen set of simplified models of aviation gears and railway wheels. The eigenvalue problem is formulated and solved by using a finite element representation for each simplified model. The results for these models are discussed and compared. The proposed solutions are verified by experimental investigation. It is important to note that the solutions proposed here could be useful to engineers dealing with the dynamics of aviation gears, railway wheels and other circular ring systems.     

Modeling of the intensification of oil production by an acoustic action on the oil pool from the borehole

An improved model of physical processes that occur under an acoustic action (AA) is presented in the context of both thermal AA mechanism and its nonthermal alternatives. It is assumed that the fluid in an oil pool consists of light and heavy hydrocarbon phases, which are in thermodynamic equilibrium. External actions, such as filtration or AA, can shift the point of equilibrium of the fluid fractions in such a way that the heavy fraction may deposit on the pore walls or dissolve. In this case, the process of long-term mudding is governed by the inhomogeneity of the pressure field around the borehole and, correspondingly, by the change in the equilibrium concentration of the heavy impurity in the course of filtration. In the framework of the proposed model, the effect of an acoustic action can manifest itself both indirectly, as a heating of the surrounding medium because of sound absorption (which also changes the equilibrium concentration of impurity), and directly, as the dependence of the equilibrium concentration and relaxation time on the average density of acoustic energy of ultrasonic vibrations. The proposed model makes it possible to reproduce the characteristic features of the fluid filtration from the borehole before and after the AA, including the long-term intensification of petroleum recovery from the collector with a gradually decreasing recovery rate.     

Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping,via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system?s characteristics on this critical amplitude has also been discussed.     

An approach based on tool mode control for surface roughness reduction in high-frequency vibration cutting

The presented research work, aimed at deeper understanding of vibrational process during high-frequency vibration cutting, is accomplished by treating cutting tool as an elastic structure which is characterized by several modes of natural vibrations. An approach for surface quality improvement is proposed in this paper by taking into account that quality of machined surface is related to the intensity of tool-tip (cutting edge) vibrations. It is based on the excitation of a particular higher vibration mode of a turning tool, which leads to the reduction of deleterious vibrations in the machine-tool-workpiece system through intensification of internal energy dissipation in the tool material. The combined application of numerical analysis with accurate finite element model as well as different experimental methods during investigation of the vibration turning process allowed to determine that the most favorable is the second flexural vibration mode of the tool in the direction of vertical cutting force component. This mode is excited by means of piezoelectric transducer vibrating in axial tool direction at the corresponding natural frequency, thereby enabling minimization of surface roughness and tool wear.     

Vibrations of inclined bars with end constraint

Considered are vibrations of inclined bars with the bottom end fixed and the top end having a constraint that this end can move only in the vertical direction. During the vibrations of such bars, the axial and lateral motions are coupled. The frequencies of the coupled vibrations of small amplitudes of inclined bars with different end conditions are presented along with the normal functions. The application of the method of normalfunction expansion to the forced as well as transient vibrations of the inclined bars is also outlined.     

Vibration response of misaligned rotors

Misalignment is one of the common faults observed in rotors. Effect of misalignment on vibration response of coupled rotors is investigated in the present study. The coupled rotor system is modelled using Timoshenko beam elements with all six dof. An experimental approach is proposed for the first time for determination of magnitude and harmonic nature of the misalignment excitation. Misalignment effect at coupling location of rotor FE model is simulated using nodal force vector. The force vector is found using misalignment coupling stiffness matrix, derived from experimental data and applied misalignment between the two rotors. Steady-state vibration response is studied for sub-critical speeds. Effect of the types of misalignment (parallel and angular) on the vibration behaviour of the coupled rotor is examined. Along with lateral vibrations, axial and torsional vibrations are also investigated and nature of the vibration response is also examined. It has been found that the misalignment couples vibrations in bending, longitudinal and torsional modes. Some diagnostic features in the fast Fourier transform (FFT) of torsional and longitudinal response related to parallel and angular misalignment have been revealed. Full spectra and orbit plots are effectively used to reveal the unique nature of misalignment fault leading to reliable misalignment diagnostic information, not clearly brought out by earlier studies.     

Model of liquid-metal splashing in the cathode spot of a vacuum arc discharge

The formation of microjets is studied during the extrusion of a melted metal by the plasma pressure from craters formed on a cathode in a burning vacuum arc. An analytic model of liquid-metal splashing that includes two stages is proposed. At the first stage, the liquid motion has the axial symmetry and a liquid-metal wall surrounding the crater is formed. At the second stage, the axial symmetry is broken due to the development of the Plateau–Rayleigh instability in the upper part of the wall. The wall breakup process is shown to have a threshold. The minimal plasma pressure and the minimal electric current flowing through the crater required for obtaining the liquid-metal splashing regime are found. The basic spatial and temporal characteristics of the jet formation process are found using the analytic model.     

A new type of semi-active hydraulic engine mount using controllable area of inertia track

Automotive engine mounts function to constrain the engine shake motion resulting at low-frequencies, as well as to isolate noises and vibrations generated by the engine with unbalanced disturbances at the high frequencies. The property of the mount depends on vibration amplitude and excitation frequency. It means that the excitation amplitude is large in low excitation frequency range and small in high frequency range. In this paper, a new hydraulic engine mount with a controllable area of inertia track is proposed and investigated. Theoretical works with the mount model to isolate the engine-related vibrations were conducted by an optimal algorithm to control the area of the inertia track under shocks and multi-signal force excitations. This research clearly gives an analysis of the considerable changes in the mount dynamic properties according to the changes in the inertia track area. Consequently, when the inertia track area is tuned, the transmissibility of the mount is effectively reduced.     

A SIMPLE MODEL TO ESTIMATE THE IMPACT FORCE INDUCED BY PISTON SLAP

The dynamics of piston's secondary motion (lateral and rotational motion) across the clearance between piston and cylinder inner wall of reciprocating machines are analyzed. This paper presents an analytical model, which can predict the impact forces and vibratory response of engine block surface induced by the piston slap of an internal combustion engine. A piston is modelled on a three-degree-of-freedom system to represent its planar motion. When slap occurs, the impact point between piston skirt and cylinder inner wall is modelled on a two-degree-of-freedom vibratory system. The equivalent parameters such as mass, spring constant, and damping constant of piston and cylinder inner wall are estimated by using measured (driving) point mobility. Those parameters are used to calculate the impact force and for estimating the vibration level of engine block surfaces. The predicted results are compared with experimental results to verify the model.     

Nonlinear vibration of micromachined asymmetric resonators

In this paper, the nonlinear dynamics of a beam-type resonant structure due to stretching of the beam is addressed. The resonant beam is excited by attached electrostatic comb-drive actuators. This structure is modeled as a thin beam-lumped mass system, in which an initial axial force is exerted to the beam. This axial force may have different origins, e.g., residual stress due to micro-machining. The governing equations of motion are derived using the mode summation method, generalized orthogonality condition, and multiple scales method for both free and forced vibrations. The effects of the initial axial force, modal damping of the beam, the location, mass, and rotary inertia of the lumped mass on the free and forced vibration of the resonator are investigated. For the case of the forced vibration, the primary resonance of the first mode is investigated. It has been shown that there are certain combinations of the model parameters depicting a remarkable dynamic behavior, in which the second to first resonance frequencies ratio is close to three. These particular cases result in the internal resonance between the first and second modes. This phenomenon is investigated in detail.     

Transition to deformed shapes as a nuclear Jahn-Teller effect

The Jahn-Teller effect is well known in molecular physics as an interplay between degenerate electronic states and a molecular vibration, giving rise to a spontaneous breaking of a molecular symmetry. We translate the concept of the Jahn-Teller effect to nuclear physics and discuss nuclear collective motion and deformation from that point of view. It becomes transparent that the well-known surface quadrupole motion, with its tendency to stable deformations, can be understood as a nuclear Jahn-Teller mode emerging from the interaction between degenerate nucleon states and giant resonance vibrations. The axial symmetry may break down further to non-axial or to reflection asymmetry shapes by the same mechanism.     

Analysis of macroparticle charging in the near-electrode layer of a high-frequency capacitive discharge

Spatial variation of dust particle charges are estimated numerically for typical laboratory experiment conditions in a radio-frequency (rf) capacitive discharge. The surface potentials of macroparticles levitating in the upper part of the near-electrode layer of the rf discharge are measured. A model is proposed for the formation of irregular dust oscillations due to stochastic motion of dust in the bulk of a spatially inhomogeneous plasma (in the presence of a dust charge gradient). This mechanism is used for analyzing the results of measurements of the amplitude of vertical vibrations of dust particles in the near-electrode layer of the rf discharge. It is found that the dust charge gradient may be responsible for the development of such vibrations.