Drill String Dynamics Research Articles

How to mathematically model a drill-string: lumped or continuous models?

This work compares the efficacy of lumped and continuous models in simulating drill-string dynamics. We aim to critically explore the inadequacy of lumped models and challenge their perpetuated use in the literature. Lumped models represent the drill-string with a pre-determined number of discrete masses, sometimes as few as one or two. This inherently restricts their ability to capture the distributed nature of drill-strings, compromising their accuracy. In this paper, these limitations are made evident by comparing simulations of lumped models with simulations of continuous models that increase in complexity. First, predictions of the axial behaviour obtained with two models, one lumped and another continuous, are analysed. Second, simulations involving a 3-D fully coupled continuous model are evaluated. The simulations demonstrate the inaccuracy of lumped models. Also, they refute the frequent assumption of representing the bottom hole assembly (BHA) as a single lumped mass. Beyond these results, theoretical arguments based on fundamental wave-reflection phenomena are employed to further support the inadequacy of lumped formulations for this purpose. Third, this paper also discusses the prevalent strategy of validating lumped models through comparisons with laboratory experiments. This approach suffers from being a makeshift validation, as similitude principles are often violated in laboratory setups, resulting in a practice that may have helped perpetuate lumped models in the literature. Consequently, using lumped models for general drill-string dynamics prediction is not appropriate. Recognising their limitations and exploring alternative approaches, such as continuous models, is crucial for accurate drill-string dynamics prediction.

Stick-slip vibration analysis of a coupling drilling-rock system with two delays and one delay-dependent coefficient

Drilling depths exceeding 10,000 m have recently been achieved, but the increasing depth poses challenges due to the flexible drill-string, making it more susceptible to stick-slip vibration. This vibration can lead to accidents such as excessive bit wear and tool falling into the well. Previous studies on modeling polycrystalline diamond compact (PDC) bits have focused on symmetrically distributed complete blade bits, which are not commonly used. To address this gap, this study introduces PDC bits that closely resemble the actual ones, with a structural configuration of both complete blades and part-blades symmetrically distributed. We also consider the impact of mud and rock chips on the drill bit by introducing a delay-dependent coefficient of response damping change. This study develops a stick-slip vibration model for a coupling drilling-rock system with two state-dependent delays tn1 and tn2, and one delay-dependent coefficient. Using the crossing curve method and numerical studies, we obtain different types of crossing curves reflecting the stability switching characteristics of the system in the dimensionless delay τ2τ1 space. During stick-slip vibration, τ1 mutates from 1.69 s to 2.51 s, and τ2 mutates from 1.3 s to 2.16 s, both within the stable region of the crossing curve. The critical steady state and the stable state typically cross the stable and unstable regions of the crossing curve in their respective times. Increasing the torsional damping within the range of 0.04 to 0.08 can accelerate system stabilization or delay the arrival of the critical steady state. Moreover, increasing the drive rotation speed within the range of 4 to 6 eliminates stick-slip vibration and accelerates the system towards a steady state. In conclusion, this study predicts the drill-string dynamics of a symmetrically distributed PDC bit with complete blades and part-blades. The insights gained can help minimize stick-slip vibration and provide recommendations for drilling operation parameters.

Steering ability rapid evaluation of the slide drilling system based on multi-body dynamics model

Efficiently and accurately predicting the steering ability of the bottom hole assembly (BHA) in slide drilling is a crucial yet challenging task. While the industry standard three-point geometry method is computationally efficient, it cannot capture the intricate mechanics involved in the drilling process. Conversely, utilizing the whole drill string dynamics model for simulating slide drilling provides a more comprehensive representation but involves time-consuming evaluations, making it difficult to balance precision and efficiency. This study addresses this challenge by proposing a rapid evaluation method that ensures both speed and accuracy by developing a multi-body dynamics rapid model of the slide drilling system. Based on the whole drill string model, this method incorporates two key techniques to achieve its primary objective. Firstly, it utilizes the equivalent torque model of the bit, which includes the fixed joint and the virtual rotational velocity, to replace the original revolute joint between the positive displacement motor (PDM) and the bit. Secondly, a segmentation model is established to decouple the simulation of the BHA drilling process from the drill string equilibrium process, thereby significantly reducing the slide drilling system’s degree of freedom (DOF). The proposed method offers a new approach to improving the efficiency of evaluating the steering ability of the slide drilling system while ensuring a high level of accuracy.

A Coupled Fluid-Structure Model for Estimation of Hydraulic Forces on the Drill-Pipes

Abstract Most models employed for fluid forces in drill-string dynamics are of reduced order. The simplified nature of these models often fails to describe the complex behavior of the fluid force, in particular, when the drill-string movement is non-trivial or even when non-Newtonian behavior is predominant. In this work, the fluid forces are estimated by modeling the dynamic of the drilling fluid for both Newtonian and non-Newtonian fluids, and compared with reduced-order models. To achieve it, the lattice-Boltzmann method is implemented to solve the fluid model, while prescribed movements are set for the tool-joint. With the obtained fields, the forces are calculated and compared with recurrent models. Finally, it is observed that some models are capable of describing the interaction as long as the dynamic of the tool-joint is simple. In the presence of other trajectories—e.g., bouncing and chaotic—the models fail to describe the details of the dynamic.

Investigation of drill string dynamics using probabilistic methods

The object of research is a drill string. The work is directed to the study of the drill string dynamics and its influence on the rock, which is drilled using probabilistic methods. It is shown that the nature of the vibrations of the top of the drill string and the downhole signals are different and have a stationary random character. It is proved that the process at the output of the drill string, taking into account random disturbances, is narrow-band (quasi-harmonic) or close to it. It is found that the energy composition of the spectrum sections and the appearance of resonance peaks and their displacement is directly determined by the physical and mechanical properties of the rocks being drilled, as one of the factors that determines the vibration state of the drill string. Graphs of the spectral density of the power of pitting vibrations are given, showing their amplification at frequencies of 30–40 Hz as the hardness of the rocks being drilled increases. The maximum of the power density bending curve shifts towards low frequencies. There is a «mutual convergence» of the maxima of the power spectral density of the oscillations of the top and bottom of the drill string. It is shown that an important feature of drill string vibrations in the process of drilling with ball bits is that the intensity of vibrations of the top of the drill string passes with frequencies close to low frequencies, and the vibrations of the bottom of the string – to high frequencies. It is determined that the process at the output of the drill string, taking into account random disturbances, is narrow-band (quasi-harmonic) or close to it. As a result, the assumption of a narrow band at the output of the system, which is the basis of theoretical research, is justified. It is proven that the drill string, as a system with liquid filling under pressure, is more stable compared to a hollow structure. The washing liquid, in this case, plays the role of a dynamic absorber.

MODELLING OF A DRILL STRING MOTION UNDER SEISMIC IMPACT

The article studies nonlinear dynamics of drill strings in a seismically active environment. The influence of seismic waves on the vibrations of drill strings for shallow drilling is considered. Deformations of drill strings are assumed to be finite. Based on the Ostrogradsky-Hamilton variational principle and the laws of soil motion, a nonlinear mathematical model of spatial deformation of drill strings is constructed. A numerical analysis of the model is performed using the Wolfram Mathematica symbolic computing package. Qualitative and quantitative changes in the dynamics of drill strings are observed, manifested in the form of beats due to the influence of seismic waves.

STUDY OF THE DYNAMICS OF ASSOCIATED DRILL STRINGS INTERACTING WITH THE WELL WALL

The depth of exploration and production wells during the development of oil and gas fields can exceed 5-10 km, and the insufficient strength and reliability of the drill string often limits the possibility of further increasing labour productivity. In addition, accidents at wells lead to significant material costs. Therefore, the design and operation of drill strings, as well as the implementation of work to eliminate complications during drilling, should be carried out on the basis of a science-based approach, taking into account the latest advances in the field of drill string dynamics and technologies related to work to eliminate stuck problems. Unsteady vibrations of inhomogeneous distributed systems is a very complex problem in the mechanics of a deformable solid and the theory of vibrations. In connection with the rapid development of the extractive industries, solving this problem is of particular importance. This is due to ensuring the strength of drill string structures with increasing power and speed of drilling units and mechanisms. The study of the problem revealed a number of little-studied problems, which include issues of nonlinear interaction of the column with the surrounding soil, accompanied by various types of complications (sticking, pipe ruptures, etc.), wave and oscillatory processes in the elements of the drilling dynamic system (DDS), finding the boundaries of the interaction areas of the column with the walls of the well using low-cost techniques. The purpose of the work is theoretical research of wave and oscillatory processes in the pipes of the drilling assembly and the dynamics of associated drill strings interacting with the well wall. In this work, a homogeneous rod was used as a drill string model. The drill string consists of a drill pipe string (DPC) and a bottom drill string assembly (BDSA), which includes a bit, a downhole motor, wellbore forming elements: calibrators, centralizers, sections of drill collars (DC), the main purpose of which is to creating an axial load on the bit. The DPC consists of sections of drill pipes that are identical in their characteristics (type, outer diameter, wall thickness, joints used, etc.). Therefore, in general, the drill string is a heterogeneous structure.

Investigation of field scenarios using a 4n degrees of freedom transient torque and drag model

A coupled drill string dynamics model presents an opportunity to understand, replicate and potentially control the vibrations occurring along a drill string in a deep well. This paper describes a 4n degrees of freedom (4nDOF) model of the axial, torsional and lateral dynamics, consisting of coupled wave equations and lumped elements, which include modeling of the contact between the drill string and wellbore with a stiff string assumption. The model accounts for the bit-rock interaction, structural damping, and hydraulic forces acting on the lumped elements. The 4nDOF model is compared with three field scenarios from a drilling operation in the North Sea: a three-stage top drive start-up procedure, a drilling sequence starting from off-bottom conditions, and a backreaming operation at low rotational speed. The selected cases include instances of stick–slip and higher modes of torsional vibrations, which persist despite the use of a surface vibration mitigation system. High-frequency data recorded by a downhole measurement sub placed near the drill bit are used to verify the simulated downhole rotational speed and triaxial accelerations. The model is useful for analyzing torsional stick–slip events, which are replicated with a similar period and amplitude as in the downhole measurements. Furthermore, the model accurately reproduces the transition between first and second mode torsional oscillations upon the activation of a tuned top drive vibration mitigation system. The model can also be used to analyze lateral deformations and whirling of the drill string at various depths. The field case simulations show that the lateral dynamics can undergo significant changes at certain points in the drill string during torsional stick–slip and higher-mode torsional oscillations. The proposed model can be used to run simulations in close to real-time on a standard personal workstation, thus making it a valuable tool for devising and testing new drill string vibration mitigation solutions.

Dynamic characteristic response of PDC bit vibration coupled with drill string dynamics

A thorough understanding of the rock breaking process is key to increasing the rate of penetration and reducing bit wear. Currently, many models of drill bit–rock interaction have been developed. However, these models inaccurately reflect the complex movement of polycrystalline diamond compact (PDC) bits in the well. In this study, the finite element method was used to discretize and simulate the rock removal process, and drill string nodes were taken as the boundary conditions of drill bit movement. A model combining drill string dynamics and drill bit–rock interaction was established to describe the complex dynamic behavior of a PDC bit in a well. The accuracy of the model was verified by a PDC bit drilling experiment. Then, the influences of rock type, weight on bit (WOB), rotational speed, and PDC bit blade number on PDC bit vibration characteristics were studied. Results show that drill bit vibration is stronger when drilling in rocks with high hardness and roughness, and the bit runs more smoothly with more blades. Therefore, drill bits with more blades should be used when drilling hard rocks. Bit vibration is enhanced by increasing rotational speed and WOB, but the effect of rotational speed on bit vibration is much smaller than that of WOB.

A high-fidelity model for nonlinear self-excited oscillations in rotary drilling systems

This paper presents an integrated model to analyze self-excited axial and torsional vibrations in rotary drilling systems using realistic PDC bits. The model combines a multi-degree-of-freedom (MDOF) drillstring representation with a detailed PDC bit–rock interaction model, accounting for cutter layouts and rock properties. The bit–rock interaction is described by rate-independent laws for reactive force and torque, encompassing cutting and frictional contact at the cutter face and wear flat. The regenerative rock cutting introduces state-dependent delays (up to 100) due to complex cutter layout of the bit, while the unilateral nature of frictional contact is formulated as a nonlinear boundary condition. The state-dependent delays are transformed into constant angular offsets prescribed by the bit design by introducing a bit trajectory function whose evolution is governed by a partial differential equation (PDE), which is coupled with the drillstring dynamics described by ordinary differential equations (ODEs). The Galerkin method with Chebyshev polynomials transforms the set of coupled PDE–ODEs into a system of ODEs. This approach bypasses the need to search for multiple delays at each time step, allowing implementation of this integrated rock-bit-drillstring model. Simulation results replicate field observed torque-on-bit (TOB) effects including velocity-weakening and hysteresis due to torsional stick–slips, attributing them to cutting depth variation rather than downhole friction. Preliminary findings regarding the influence of bit design on torsional stick–slips align with field drilling practices. The results of parametric study correspond to field test observations, enabling analysis of bit design, drillstring configuration, and drilling parameters on self-excited torsional stick–slips.

Distributed model for the drill-string system with multiple regenerative effects in the bit-rock interaction

In this paper, an integrated model is presented to study the axial–torsional dynamics of a distributed drill string. The coupled axial and torsional drill string dynamics typically lead to severe phenomena such as bit-bounce and stick–slip, which are the main causes of failure and efficiency reduction. The top-drive dynamics and the Bottom Hole Assembly (BHA) dynamics are interconnected through the drill pipes. In order to preserve the infinite-dimensional feature of the drill pipes in the mathematical modeling, a distributed model in terms of Delay Differential Algebraic Equations (DDAE) is proposed to represent the dynamics of the drill string. The bit-rock interaction, which consists of the cutting and the frictional components, is responsible for the coupling between the axial and torsional dynamics. A rate-independent bit-rock interaction law is employed for both cutting and frictional components. In order to capture the global bit motion, including multiple regenerative effects, the well-bottom depth function is used to determine the depth of cut. The well-bottom pattern evolution is formulated through algebraic equations rather than Partial Differential Equations (PDEs). A new formulation for the linearized drill string system in terms of Neutral-type Delay Differential Integral Equations (NDDIEs) is proposed to investigate the initiation of the oscillations around the nominal operation. By this formulation, the local stability of the distributed system is studied by determining the right-most eigenvalue of the linearized dynamics. A simulation-based case study reflecting real-life scenarios is presented to investigate the dynamical behavior of the system under different operating conditions.

Numerical analysis of whirl for drill string with aluminium alloy drill pipes in deep vertical well

In this study, a new finite element (FE) model using beam-shell composite elements is established. The model combines the advantage of computational speed for beam elements with the contact details of shell elements. The thermal instability of aluminium alloy materials and geothermal gradient are considered. Besides, kinematic contact model, Coulomb friction model, and initial imperfection are considered in this model. The drill string with aluminium alloy drill pipes of Songke-2 (SK-2) well is selected to simulation. The calculation results are consistent with the failure mode of the aluminium alloy drill pipes at engineering site. The effects of aluminium alloy drill pipes position, geothermal gradient, and rotational speed on the whirl characteristics of drill string with aluminium alloy drill pipes are analyzed. The results show that the closer the aluminium alloy drill pipes are to the drill bit, the more obvious the whirl phenomenon of the drill strings is. For deep wells with a maximum temperature of 170 °C, the temperature gradient has limited effect on the whirl state. The rotational speed has a significant impact on the whirl. When the rotational speed reaches 80 rpm, backward whirl occurs at the bottom of the borehole. Moreover, when the distance from the drill bit is different, the drill string will exhibit different whirl states. This work provides a new approach for modeling the dynamics of drill strings, selecting different element types based on the importance of different parts. Geothermal gradient is also considered in this model, which can help to calculate drill string dynamics for ultra deep drilling and geothermal drilling.

Analysis of the Influence of Downhole Drill String Vibration on Wellbore Stability

Most studies related to aspects of wellbore stability, such as wellbore breakage, block dropping, and wellbore expansion, revolve around the physicochemical interaction between drilling fluid and surrounding rock, but relevant studies show that drill string vibration during drilling also has a crucial and even decisive influence on wellbore stability. In order to thoroughly explore the influence mechanism of drill string vibration on wellbore stability, our research group established a finite element flexible simulation model of drill string dynamics and used a storage downhole vibration measurement device to collect downhole real drilling vibration data to verify the correctness of the simulation model. Then, based on the critical conditions of wellbore breakage, a wellbore stability evaluation method was established, and the wellbore stability under different drilling parameters and drilling tool combination conditions was evaluated and analyzed. The research results play an important role in revealing the influence mechanism of drill string vibration on wellbore stability and can provide theoretical guidance for engineering problems such as wellbore instability risk assessment.

Nonlinear analysis on the impact wear characteristics of casing based on drillstring dynamics

Casing integrity is significant for subsequent drilling and production and prediction of casing wear accurately helps to reduce casing damage. Thus, this study aims to represent the casing impact wear characteristics more intuitively and accurately during drilling. The vibrating drillstring when drilling will collide with casing frequently and increase the casing wear. The drillstring nonlinear dynamic model is established using beam finite element method in this work, and the drillstring dynamic characteristics are obtained using generalized-α method. Subsequently, the friction coefficient and wear factor are obtained through the casing wear test. On this basis, the casing impact wear characteristic combined with the casing wear efficiency model is described, and the residual strength of casing is analyzed. In addition, the effects of drilling depth, weight on bit (WOB), rotation speed and the eccentric distance of the drillstring on the casing wear are discussed. We hope that our research will aid in designing drilling parameters and reducing casing damage.

A Model-Based Intelligent Adjustment Method of Toolface for Bent-Housing Motor

The traditional toolface adjustment of a bent-housing motor is time-consuming and laborious. For the goal of rapid and accurate toolface adjustment, this paper presents an intelligent toolface adjustment method based on the drill string dynamics model and the BP (back propagation) neural network, optimized by a GA (genetic algorithm). Firstly, for the mode of rotating the drill string at the wellhead to change the downhole toolface, the drill string dynamics model is used to calculate the toolface change value. Then, the actual toolface change value is taken as the output data; the calculated toolface change value and the factors that have a significant impact on the change value of the toolface are taken as the input parameters; and the GA-BP neural network is adopted to fit the relationship between the input parameters and the output data. For the mode of changing the toolface by changing the WOB (weight on bit), the WOB change value and other parameters that cause the toolface to change were taken as the input data and the toolface change value was taken as the output data; the relationship between the WOB and the toolface was fitted with the GA-BP neural network. The results show that the prediction of absolute error under the mode of adjusting the toolface by rotating the drill string at the wellhead is within 10.9783°, and the prediction of absolute error under the mode of adjusting the toolface by changing the WOB is within 18.8833°. The intelligent models established under the two modes can meet the requirements of toolface control accuracy. Using the established intelligent prediction model, the toolface can be adjusted to the required value range at one time, which can improve drilling efficiency, and reduce labor intensity and the dependence on the experience of on-site personnel.

On Nonlinear Spatial Vibrations of Rotating Drill Strings under the Effect of a Fluid Flow

In this article, the development and subsequent numerical analysis of a nonlinear mathematical model of the drill-string dynamics taking into account the effect of a drilling fluid flow and the gravitational energy of the system is carried out. Spatial lateral vibrations of the drill string modeled as a rotating elastic rod are studied. The developed nonlinear model generalizes and refines the well-known linear models of rod vibrations with the considered effect. The obtained numerical results demonstrate the influence of geometric nonlinearity, the gravitational energy of the drilling system, additional Coriolis and centrifugal forces as well as the parameters of the fluid flow on spatial vibrations of the drill strings. It allows for giving some recommendations on the choice of the drilling system parameters for ensuring safe drilling operations.

A Novel Mesoscopic Drill Bit Model for Deep Drilling Applications

This paper deals with the development of a novel mesoscopic model of polycrystalline diamond compact (PDC) drill bits that can be implemented in complex drill string models for simulations to analyse the influence of rock inhomogeneities or the impact of anti-whirl bits on drill string dynamics. In contrast to existing modelling approaches, the model is developed at a mesoscopic level, where the basic bit–rock interaction is taken from the macroscopic bit model and the cutting characteristics are summarised at a microscopic cutting level into a simplified configuration via cutting blades. This model can therefore effectively describe asymmetries and thus interactions between the torsional and lateral dynamics of the drill bit, and is particularly suitable for investigating the effects of drilling into rock inhomogeneities and fault zones on drilling dynamics. By integration into a complex drill string model, simulation studies of drilling through a sandwich formation were carried out. The simulation results allow detailed stability statements and show the influence of formation properties and bit design on torsional and lateral drill string dynamics.

Modeling of drill string dynamics in directional wells for real-time simulation

Simple and computationally efficient drill string models running real-time describing motion in all axes in directional wells are important for the implementation of closed-loop control and assisted monitoring during drilling operations. This paper proposes a new simplified three-dimensional model based on a parametric curve and lumped-parameter modeling, where Kane’s method is used to establish the equations of motion. Validation of the steady-state motion and convergence for the lumped model in vertical and horizontal alignment was compared with a finite-element model. The configuration and restoring forces show good results compared with finite-element analysis. Hence, the model demonstrate the axial contraction as a function of the body restoring forces being oriented to the inertial frame, inherently producing nonlinear coupled axial tension forces. The qualitative response of the model is confirmed in simulation case studies, being showcased by a deviated J-well configuration. Traveling block velocity and top drive torque are included as actuated inputs to analyze off-bottom friction and contact along the wellbore. The model is proposed to act as a virtual sensor for drilling directional wells.

Spectral Analysis of the Infinite-Dimensional Sonic Drillstring Dynamics

By deploying sonic drilling for soil structure fracturing in the presence of consolidated/ unconsolidated formations, this technique greatly reduces the friction on the drillstring and bit by using energetic resonance, a bit-bouncing high-frequency axial vibration. While resonance must be avoided, to our knowledge, drilling is the only application area where resonance is necessary to break up the rocks. The problem is that the machine’s tool can encounter several different geological layers with many varieties of density. Hence, keeping the resonance of the tool plays an important role in drill processes, especially in tunnel or infrastructure shoring. In this paper, we analyze the sonic drillstring dynamics as an infinite-dimensional system from another viewpoint using the frequency domain approach. From the operator theory in defining the adequate function spaces, we show the system well-posedness. The hydraulic produced axial force that should preserve the resonant drillstring mode is defined from the spectrum study of the constructed linear operator guided by the ratio control from the top to tip boundary magnitudes.

Downhole vibration characteristics of 3D curved well under drillstring–bit–rock interaction

Understanding the characteristics of downhole vibration helps optimize the drilling parameters and improve the rate of penetration (ROP). This work establishes a nonlinear drillstring–bit–rock model to predict the downhole vibration behavior of 3D curved wells. The main novelty of this model is to consider the complex nonlinear bit–rock interaction and well trajectory. The model is solved by the Newmark method and verified by an indoor experiment and field data. The drillstring dynamics and effects of the main working parameters on the vibration characteristics are obtained and analyzed. Results show that, in the horizontal section, the drillstring motion and shape deformation are stable, and radial vibration is dominant. In the buildup section, the drillstring motion is irregular and messy, shape deformation becomes large, and radial vibration is significant but tangential vibration starts to become intense. In the vertical section, the drillstring motion is small in wellbore center, shape deformation is stable, and vibration severity reduces. The weight on bit (WOB) only affects drillstring vibration near the bit, and radial, tangential, and axial accelerations intensify as WOB increases. RPM has little effect on the drillstring vibration in 3D curved wells. In the buildup and horizontal sections, as the coefficient of friction (COF) increases, the radial and tangential accelerations rise while the axial vibration obviously weakens. When drilling in 3D curved wells, small WOB and large RPM can be adopted to reduce downhole vibration and increase ROP. The safety of drillstring and downhole tool also needs more attention when drilling in formations with large COF, such as gravel stone and sandstone. Large oil/water ratio or lubricant is recommended to improve the lubricity of drilling fluid. The research results could provide a valuable and useful guidance for drilling engineers to improve ROP and reduce downhole accidents.