Vibration and high shock are major factors in the failure of downhole tools. It is important to study the causes of vibration and shock formation to prevent failure of the drillstring and bottom hole assembly (BHA). At present, it is generally recognized that the vibration of drillstring is the main reason for the failure, especially the lateral vibration. In this paper, the bottom tool of Rotary Steering Drilling System (RSS) calculation model was established based on the secondary development of ABAQUS software. Starting from the initial configuration of drilling tool, considering the contact impact of drilling tool and borehole wall, the dynamic excitation of guide mechanism and the drilling pressure, torque, rotational speed, gravity, buoyancy, drilling fluid damping. The dynamic characteristics of the inherent frequency and dynamic stress of the bottom hole assembly (BHA) were calculated and analyzed, and risk assessment method based on the quantitative vibration intensity was established. The reliability of typical drilling tool is evaluated, which provides a reference for the optimization design of BHA of Rotary Steering Drilling System.

Drillstring and downhole tool failure usually results from failing to control one or more of the vibration mechanisms. The drillstring motion state is affected by several factors and interaction among them, like drilling load, drillstring itself, drilling fluid, well wall and bottom rock (Wang et al., 2015). The vibration of the drillstring can be divided into three primary modes: axial, lateral (also referred to as transverse, or bending), and torsional modes, with different destructive natures (Ritto et al., 2013; Liao et al., 2012). In the axial mode, the vibration is a longitudinal movement along the drillstring. In the transverse mode, the vibration is a lateral movement that causes bending or bending of the component, resulting in a stress reversal in which one side is in a different tension state than the other side. In the torsional mode, the vibration is resistance to the rotation resulting in twisting as torque is applied to overcome resistance (Nave and Dvorkin, 2015; Ghasemloonia et al., 2014; Ritto et al., 2013).

Wilson and Heisig (2015) found that lateral vibration is more severe than vertical vibration, because the bending stress is high while vertical stress caused by axial vibration is not severe. Lateral vibration is very easy to happen in side tracking, and the lowest resonant frequency depends on properties of drillstring material and drilling fluid. Qu (1995) discussed lateral vibration under the effect of drilling fluid inside and outside by using solid-liquid coupling vibration theory, he concludes that drilling fluid can be equivalent to generalized distributive mass, and the effect of drilling fluid cannot be neglected. Shor et al. (2015) utilizing finite element method get conclusion that mass unbalance has a great influence on lateral vibration, and when the drillstring vibrates at the inherent frequency, the friction between the drillstring and well wall becomes more severe. Drillstring dynamics study are mainly by method of theoretical computation due to the difficulty of experiments. Millheim et al. (1978) lay the foundation of using finite element in dynamics analysis. The finite element computer program to analyze the behavior of bottom-hole assemblies of drill collars, stabilizers and drill bit is discussed. Wilson and Heisig (2015), Schmalhorst and Neubert (2003), Jogi et al. (2002) and Yigit et al. (1996) utilized both differential equation solution and finite element method to analysis the drillstring dynamics, and some of their work has formed commercial software. In most cases analytical models are used to get a fast overview of natural axial and torsional frequencies of drillstrings in straight boreholes. To develop practical and commercial computing software about drillstring dynamics, the model established must take lithology, bit type, drill assembly, damping and contact into consideration, meanwhile through a lot of experiments and experiment data analysis, comparison with theoretical computing results, various parameters or boundary conditions in theoretical calculation can be corrected. Therefore, it takes a lot of hard work to develop software for drillstring dynamics. Based on this situation, secondary development based on commercial finite element software is one feasible technical approach, which reduces large amount of work for calculation method and program verification.

Several studies have been devoted to better understand the full dynamics of the rotary drilling systems (Yigit and Christoforou, 1996, 1998; Khulief and Al-Naser, 2005; Sampaio et al., 2007; Kapitaniak et al., 2015). Gupta and Wahi (2016) developed a global dynamic of the coupled axial-torsional vibrations incorporating the possibility of bit-bounce. Huang et al. (2015) established a generalized quasi-static model of drillstring system, the forward model and inversion model are further combined into a unity. Lian et al. (2015) established a nonlinear dynamics model to investigate the motion behavior of drillstring in gas drilling of horizontal wells, a finite element model was also established and the buckling and contact of drillstring was analyzed. However, none of these studies have reported the dynamical motion state of the bottom hole assembly (BHA) in Rotary Steerable Drilling System (RSS) (Jones et al., 2016), the whole drillstring is rotated from the surface by a hydraulically driven top drive. The RSS has push-the-bit tools (Warren, 1998) which using pads on the outside of the tool which press against the well bore thereby causing the bit to press on the opposite side causing a direction change. The pads of the implementing agencies in RSS constantly pushed against the borehole wall, making bottom hole by a cycle of nonlinear damping force, which is lead to the bottom drilling tool movement of chaos and disorder (Wang et al., 2014; Xue et al., 2015, 2019).

Statistics study (Lin et al., 2016) shows that drillstring failure and cross-over sub failure account for approximately 54 % of drilling tool failures. Among the drillstring thread fracture incidents, 70 % of them occurred at box thread and 30 % occurred at pin thread. The causes for drillstring failure encountered during drilling practice have been systematically analyzed based on the statistical data (Wang et al., 2011; Abdo, 2006; Abdo and Abouelsoud, 2011). Fault tree of drillstring failure has been subsequently constructed (Wang et al., 2011).

In this paper, the bottom tool of RSS calculation model was established based on the secondary development of ABAQUS software. Starting from the initial configuration of drilling tool, considering the contact impact of drilling tool and borehole wall, the dynamic excitation of guide mechanism and the drilling pressure, torque, rotational speed, gravity, buoyancy, drilling fluid damping. The dynamic characteristics of the inherent frequency and dynamic stress of the BHA were calculated and analyzed, and risk assessment method based on the quantitative vibration intensity was established. The reliability of typical drilling tool is evaluated, which provides a reference for the optimization design of drilling tools.

Drillstring actual working condition is very complex, and it's hard to simulate and analyze the actual working condition precisely. Due to the complexity of the drilling operation, the mathematical calculation model cannot effectively explain the fatigue failure of the drillstring caused by the cyclic stress due to vibration and high shock. Since the dynamics of the drillstring involve wide vibration profiles including axial, lateral and torsional modes, the mathematical modeling of such long rotating components is highly nonlinearity.

Therefore, secondary development based on commercial finite element software
is a feasible technical solution. In this paper, the drillstring be
simplified by basic assumptions when establish a vibration analysis model in
ABAQUS software. The BHA calculation model of the RSS is shown in Fig. 1,
making the following assumptions in the calculation process of the entire
dynamic system.

The borehole section is circular and the well wall is rigid;

The axis of drillstring is at the same line with well axis before drillstring moves, and the drillstring node has three-dimensional freedom;

The bottom drillstring is considered as a beam, whose one end is hinging support and the other end is movable hinge support. The drill bit and the stabilizer are movable hinge support, push contact is one-way movable support;

The drillstring is considered as homogeneous, small deformation and elastic beam, the joint is neglected, and drillstring displacement is restrained by wellbore;

The drilling fluid influence on drillstring vibration characteristics is considered.

Dynamics model of Rotary Steering Drilling System (RSS).

Dynamic equilibrium equation of drilling dynamics system can be expressed
as:

Considering that the inside of the drill is filled with liquid and the outer
annulus also has drilling fluid, the unit of drillstring could have the
ability to withstand greater stress and deflection. The stiffness matrix of
the unit is shown in Eq. (2) (the equation details can be found in the Supplement).

The mass matrix of the unit is shown in Eq. (10) (the equation details can be found in the Supplement), and its mass matrix is
similar to the mass matrix of ordinary beam elements. Only some elements need
to be revised, multiplied by the coefficient

Then we can define

The

However, most engineering problems still involve damping, although the
damping may be small. The relationship of natural frequency between the
damped and the undamped is:

Based on the above analysis and Hamilton's principle, the dynamic model of
the drillstring system can be derived as follows:

The drillstring failure and cross-over sub failure account for approximately 54 % of drilling tool failures. Drillstring and downhole tool failure usually results from failing to control one or more of the vibration mechanisms. We established a risk assessment method based on quantitative vibration intensity, the evaluation process is shown in Fig. 2.

Analysis process of drillstring vibration intensity.

As show in Fig. 2, according to the initial designed speed and drilling tool
combination, the calculation model is established. Firstly, the critical
speed is calculated, and the range of relevant frequencies is taken to
determine the danger control point. The drilling tool is not allowed to be
attached to the critical speed. For a specific speed, it may be between the
two critical speeds, the low critical speeds are expressed as

From the calculation results above, the length of drillstring is large, the natural frequency is small, and sometimes the recommended value is difficult to stay away from the resonance speed. Of course, because of the existence of the wellbore, even if the drillstring speed is close to the critical speed, there will be no infinite amplitude, but the collision between the drillstring and the wellbore will be intensified. When the rotating speed is close to the natural frequency, under the action of periodic excitation load, the drillstring is usually in some violent vibration state, and the drillstring has a certain dynamic stress level. Further stress level judgment is needed to determine the reliability of the recommended speed.

The mechanical properties of the drillstring need to be determined when
determine stress intensity, that include the tensile strength

When the fatigue strength check with known three-way stress, the multi-direction stress is often converted into equivalent stress in engineering, and then the fatigue check is carried out according to the stress state. According to a large number of experiments, the deformation energy intensity theory combines the multidirectional stress state with the unidirectional stress state, which is more in line with the practical theory. According to the deformation energy intensity theory, the multidirectional stress will transform into a unidirectional force.

Then the equivalent stress amplitude

The equivalent mean stress

If the time series of equivalent stress is obtained by measurement, its
equivalent stress

where equivalent stress amplitude expressed
as:

Equivalent mean stress expressed as:

To simplified the calculation model, combined with the fatigue endurance
limit test of drillstring, the approximate calculation formula of the cycle
endurance limit of drillstring under the condition of underground corrosion
was adopted. The fatigue limit

The fatigue allowable stress amplitude formula for under symmetrical
circulation expressed as:

It is necessary to modify the allowable fatigue stress value when considering
the effect of average stress according to the previous calculation. Goodman
line is often used for modification in engineering, which is relatively
simple and the calculation results tend to be safe, so it is widely used
(Budynas and Nisbett, 2014). The Goodman's line calculation formula can be
expressed as:

The judgment principle of vibration stress intensity is defined in this
paper as following:

In this paper, we established calculation program of the RSS based on the
secondary development of Abaqus software. The modeling starts from the drill
bit, input the length, outer diameter and inner diameter of each section in
sequence according to the different cross-sections of drilling tool
combination. In the column of material properties, the elastic modulus,
Poisson's ratio, and density of materials are respectively input, where the
unit of elastic modulus is MPa and the unit of density is t mm

Load column input weight on bit, rotary torque, slap backup force and impact
force, according to the actual situation input weight on bit, rotary table
torque, slap pushing and collision force, the naming principles of section
number is the number under push position. At the same time input the
drillstring rotary speed in the working process, the unit is r min

Calculation process in the software is shown as in Fig. 3. Firstly, by using Abaqus model input, establish drillstring dynamics model, set analysis steps, define contact and impose static load, and simulate quasi-static deformation process of the contract model between drillstring and wellbore which is under axial pressure, and obtain drillstring initial configuration. Analyze the buckling results, the initial displacement is obtained and the deformation model is imported for further dynamic analysis. On the one hand, the collision calculation is carried out to obtain the collision contact force function. On the other hand, the natural frequency of the model is calculated to determine the damping coefficient of the system. The calculated collision contact force function and damping coefficient are introduced into the model, and other load functions such as speed, torque, gravity, buoyancy and pushing force are applied. Submit the work for dynamic calculation. During the calculation, the boundary changes are monitored in real time, and the boundary corrections are made in case of collision. The time history of dynamic stress under various excitation loads and the deformation, velocity and acceleration of drillstring are obtained.

Modal analysis calculation process.

BHA design parameters: Drill bit (

The elastic modulus of drill assembly is 206 GPa, Poisson's ratio is 0.3,
drillstring material density is 7850 kg m

When different angular velocity is applied to the drillstring, the corresponding period will be changed. The drill assembly has one end hinge branch, the other end movable hinge, the centralizer movable hinge, the shaft wall solid branch.

The principal stresses in three directions are shown in Fig. 4. The principal stress time series curve satisfies the condition of periodic cycle, but there is error in symmetry.

Time series curve of typical principal stress.

For the first principal stress, the maximum stress

Then the allowable stress under symmetric circulation is,

Calculation data of three principal stresses (MPa).

Firstly, the model of drillstring combination is established. Apply boundary conditions to the model, drillstring has one end hinge branch, the other end movable hinge, the centralizer movable hinge. Take the previous order eigenvalues, the natural frequency of the BHA will be obtained as shown in Table 2.

Natural frequency of the BHA.

As shown in Table 2, for a drillstring with a large length, the natural
frequency or critical speed is relatively dense, and it is difficult to
completely avoid the critical speed. At this time, it is necessary to avoid
the critical speed as much as possible, for example, the intermediate value
of the two-step critical speed can be taken. In the actual drilling process,
the drilling tool speed should be far away from the critical speed. The
assembly dangerous speed for natural frequency of 1, 2, 3 and 4 order are
corresponding to the rotational speed, i.e. 30, 73, 128
and 196 r min

When the speed is 60, 100 and 140 r min

Dynamic stress time series of each node at different rotary speeds.

Additionally, when the rotating speed is 60, 100 and 140 r min

Lateral force time series of each node.

The variation of the displacement and the stress of each node in the drill assembly with time.

When the rotary speed is 140 r min

As shown in the Fig. 8, the maximum stress

Curve of dynamic stress over time of each node at rotary speed is
140 r min

Direct measurement of stress values is difficult, besides the theoretical research, the measurements of drillstring vibration in in field were carried out by drilling engineer. To understand the intensity of drillstring vibration, Alley and Sutherland (1991) and Tian (2016) used Measurement While Drilling (MWD) to measure vibration parameters, the results show that the drillstring vibrates violently while drilling. The latest measurements were measured by the micro vibration logging tool, which was directly mounted on the drill bit to avoid interference of other downhole tools, so the vibration characteristics of drillstring and the fluctuation of WOB will be truly reflected by the measurements.

The rotary steerable system (RSS) with push-the-bit mode is installed near the bit for directional drilling, the whole drillstring of the drilling system is rotated from the surface by a top drive, which is typically a hydraulically driven mechanism. During directional drilling using RSS, specialized downhole equipment is employed to replace conventional directional tools such as mud motors. For RSS using push-the-bit mode, its push pads in implementing agency alternately in turn pushed against the borehole wall, making the bottom hole constantly under a cycle of nonlinear damping force, by which the movement of the RSS would be transformed to chaos and disorder. Thus, it is necessary and urgent for us to do some thorough research about the dynamic state of RSS. High-sample-rate downhole dynamics sensors were placed at 0.46 m above the bit in the BHA.

We develop a strap-down measurement while drilling (MWD) surveying system
that incorporates three-axis magnetometers and three-axis accelerometers
arranged in three mutually orthogonal directions (Russel and Russel, 1979).
The sensors are installed inside nonmagnetic drill collar that can avoid the
external magnetic interferences (Rehm and Garcia, 1989).

The time series of drillstring vibration and the spectrogram of

The actual photo of the drillstring damage

As shown in the Fig. 9, the time series of lateral vibration and longitudinal
vibration of drillstring were illustrated, and the spectrogram of

The vibrations induced in the drillstring are the main reasons for the drillstring failure. To develop practical and commercial computing software about drillstring dynamics is difficult, the model established must take lithology, bit type, drill assembly, damping and contact into consideration, meanwhile through a lot of experiments and experiment data analysis. In this paper, we developed a secondary development software based on commercial finite element software, which reduces large amount of work for calculation method and program verification. The dynamic characteristics of the inherent frequency and dynamic stress of the bottom hole assembly (BHA) were calculated and analyzed, and risk assessment method based on the quantitative vibration intensity was established. Considering the contact and collision between drillstring and borehole wall, taking the whole drillstring assembly system as the research object, the dynamic equation is solved by explicit central difference method in Abaqus, and the dynamic response of drillstring system is obtained. Then a risk assessment method based on quantitative vibration intensity was established and dynamic characteristics and safety of typical drillstring were analyzed. Using the secondary development function of Abaqus to develop specific graphical user interface and command flow to simplify, it is simpler and more effective to analyze specific engineering problems.

The “danger assessment method based on quantitative vibration intensity” established in this paper firstly requires avoiding the critical speed and selecting the ideal recommended speed; if it is not fully realized, the dynamic stress is evaluated, and the vibration intensity is determined according to the fatigue allowable stress and the yield limit of the drillstring. Divided into three grades to assess the reliability of the drill assembly with dynamic stress levels. This method combines the point control to avoid the resonance speed and the interval control based on the dynamic stress level, and forms the quantitative evaluation principle of the vibration risk of the drillstring.

The vibration and internal stress of the drillstring have a positive correlation, downhole measurement can only get the vibration signals and cannot get the magnitude of the stress value, simultaneously, internal stress is easily obtained by theoretical calculation. In this paper we developed a software for secondary development that can quickly get stress values. The combination of the two methods, ability to provide an optimal solution for the safety assessment of drilling tools. In addition, the effectiveness of our reliability assessment method is also proved by experimental vibration data.

The data used to support the findings of this study are included within the article.

The supplement related to this article is available online at:

QX, LaH, BL and CY analyzed the data and developed the model; QX and LeH performed the experiments; QX, RW and LeH prepared the figures. QX, LeH and CY wrote and edited the manuscript.

The authors declare that they have no conflict of interest.

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Key R&D Program of China (2016YFE0202200) and Natural Science Foundation of China (51704264). Edited by: Guimin Chen Reviewed by: three anonymous referees