3.1 Risk assessment method based on quantitative vibration intensity

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.

Figure 2Analysis process of drillstring vibration intensity.

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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 n_c1, and the high critical speeds are expressed as n_c2, if the design speed is satisfied $\mathrm{1.3}{n}_{\mathrm{c}\mathrm{1}}\le n\le \mathrm{0.75}{n}_{\mathrm{c}\mathrm{2}}$, no resonance will occur and the point control requirements will be satisfied. If the above conditions are not satisfied, the recommended speed value is given as the intermediate value of the two-stage critical speed, that is $n^{\prime }=\mathrm{0.5}(n_{\mathrm{c}\mathrm{1}}+n_{\mathrm{c}\mathrm{2}})$.

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 σ_b, yield limit σ_s, symmetrical cyclic fatigue limit σ₋₁ and allowable stress amplitude σ_a which considering the effects of mean stress. According to the dynamic model, the vibration stress will be obtained, and the time series of the equivalent stress will be obtained, so the transient stress σ, the equivalent stress amplitude σ_da and the equivalent average stress σ_dn can be obtained. And by contrasting allowable stress amplitude σ_a, the fatigue limit σ₋₁, and yield limit σ_s, the condition is defined as “safety” when equivalent stress amplitude σ_da is low, less than the revised allowable fatigue stress amplitude σ_a; The condition is defined as “warning” when equivalent stress amplitude σ_da exceeds revised allowable fatigue stress amplitude σ_a, but less than the fatigue limit σ₋₁, and should be handled carefully, it is not allowed to occur that the equivalent stress amplitude exceeds the fatigue limit σ₋₁ or the instantaneous stress σ exceeds the yield strength σ_s. Therefore, it is necessary for the construction personnel to make quick treatment and control the vibration intensity.

3.2 Calculation method and judgment principle

3.3 Secondary development software

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⁻³.

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⁻¹, and then meshing, at last in the task volume input file name and the number of CPU, and then click OK, begin to automatically calculate the dynamic stress of drillstring assembly. In the column of material mechanics parameters, input material yield strength and tensile strength, in the column of calculation results, input maximum instantaneous stress, the maximum and the minimum fluctuation stress, and then click OK, appear evaluation results of assembly dynamic stress.

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.

Figure 3Modal analysis calculation process.

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4.1 Initial conditions for calculation

BHA design parameters: Drill bit (φ215.9 mm), rotary steerable tool (φ197 mm), drillstring (φ127 mm), undersized centralizer (φ210 mm), flexible sub (φ172 mm), drill collar (φ172 mm, length 6 m), non-magnetic drill pipe (φ127).

The elastic modulus of drill assembly is 206 GPa, Poisson's ratio is 0.3, drillstring material density is 7850 kg m⁻³, and drilling fluid density is 1150 kg m⁻³. Well bore diameter is 216 mm. The rotary table torque applied to the drillstring is 10 kN m, weight on bit is 80 kN, and excitation load is applied to the pushing block. The whole drillstring is affected by gravity, buoyancy and damping force of drilling fluid.

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.

4.2 Dynamic Stress Calculation

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.

Figure 4Time series curve of typical principal stress.

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For the first principal stress, the maximum stress ${\mathit{\sigma }}_{\mathrm{1}max}=\mathrm{52.8}$ (MPa), and the minimum stress ${\mathit{\sigma }}_{\mathrm{1}min}=-\mathrm{9.06}$ (MPa), so the main stress amplitude will be calculated as follows.

The average stress in the direction of the principal stress amplitude can be calculated:

Similarly, the same formula is used to calculate the second and third principal stress, as shown in Table 1. The equivalent stress amplitude σ_da is calculated:

equivalent mean stress calculated:

Relative to drillstring materials, σ_b=827 (MPa), σ_s=758 (MPa), other coefficients are selected in accordance with Sect. 4.1. We can obtain,

wherein, σ_s is yield strength, MPa; σ_b is tensile strength, MPa.

Then the allowable stress under symmetric circulation is,

According to the above calculation data, the allowable stress amplitude σ_a considering the influence of average stress is,

Then, σ is obviously less than σ_s, and σ_da<σ_a, therefore, meets the fatigue strength requirement.

Table 1Calculation data of three principal stresses (MPa).

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4.3 Natural vibration characteristic

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.

Table 2Natural frequency of the BHA.

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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⁻¹. In the work under the rotational speed of drilling tools, resonance is inclined to be caused, and then lead to drilling tool failure fracture. In drilling process, rotary speed needs to stay away from these dangerous values.

4.4 Dynamic analysis

When the speed is 60, 100 and 140 r min⁻¹ respectively, the changes of the maximum stress point in different cycles are shown in Fig. 5. Stress changing period and maximum stress differ with different rotating speed. When the speed is 140 r min⁻¹, the stress peak is the largest, up to 60 MPa. When the speed is 100 r min⁻¹, the stress peak is the minimum.

Figure 5Dynamic stress time series of each node at different rotary speeds.

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Additionally, when the rotating speed is 60, 100 and 140 r min⁻¹ respectively, the drill bit lateral force changes in different periods are shown in Fig. 6. When the speed is 140 r min⁻¹, the lateral force reaches a maximum of 40 kN at 6.95 s. At different speeds, the lateral forces tend to stabilize with time. When the rotating speed is 140 r min⁻¹, displacement of assembly at various points in a cycle is shown in Fig. 7a, due to the constraints on both ends of the assembly, there is no displacement in X and Z directions, the other drillstring point has larger displacement changes. When the speed is 140 r min⁻¹, as shown in Fig. 7b, the stress at each node of the drillstring constantly changes within a period. The node with maximum stress is in the middle of the drillstring as indicated by the arrow. The maximum stress reached about 78 MPa at 5.863 s.

Figure 6Lateral force time series of each node.

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Figure 7The variation of the displacement and the stress of each node in the drill assembly with time.

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4.5 Safety evaluation

When the rotary speed is 140 r min⁻¹, the dangerous stress point is 14.62 m away from the bottom of the drillstring. The change of stress over time is shown in Fig. 8.

As shown in the Fig. 8, the maximum stress ${\mathit{\sigma }}_{max}=\mathrm{60.8264}$ MPa, the minimum stress ${\mathit{\sigma }}_{min}=\mathrm{0.4207}$ MPa, and equivalent stress amplitude is ${\mathit{\sigma }}_{\mathrm{da}}=\frac{{\mathit{\sigma }}_{max}-{\mathit{\sigma }}_{min}}{\mathrm{2}}=\frac{\mathrm{60.8264}-\mathrm{0.4207}}{\mathrm{2}}=\mathrm{30.20}$ (MPa). Since the σ is obviously less than σ_s and σ_da<σ_a, it meets the fatigue strength requirement.

Figure 8Curve of dynamic stress over time of each node at rotary speed is 140 r min⁻¹.

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4.6 Experimental vibration data analysis

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). a_x, a_y, a_z are defined as survey signals of triaxial accelerometers on the x, y, z axis respectively. m_x, m_y, m_z are defined as survey signals of triaxial magnetometers on the x, y, z axis respectively, the sampling frequency f_s is 1 kHz. Assume that the Earth's magnetic field strength as M. Obviously, $M=\sqrt{m_x^{\mathrm{2}}+m_y^{\mathrm{2}}+m_z^{\mathrm{2}}}$. Under certain sample frequency, measuring signal is time series and can be expressed as time function. We can use $a_h=\pm \sqrt{a_x^{\mathrm{2}}+a_y^{\mathrm{2}}}$ define the lateral vibration of the BHA and use a_z express the longitudinal vibration.

Figure 9The time series of drillstring vibration and the spectrogram of x and y signals of accelerometers, in the case of safety drilling.

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Figure 10The actual photo of the drillstring damage (a) and the vibration signal measured in the early stage of the drillstring damage (b).

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As shown in the Fig. 9, the time series of lateral vibration and longitudinal vibration of drillstring were illustrated, and the spectrogram of x and y signals of accelerometers were illustrated subsequently. The vibration data is measured in the case of safety drilling. In the secondary development software as mentioned above, the σ is less than σ_s and σ_da<σ_a, indicates that the working condition is safe. In the other case, if the vibration state of the drillstring is unsafe, the measured vibration signals level will increase, as shown in the Fig. 10, the vibration signal gradually increases until the drillstring is damaged. Simultaneously, we can calculate by the software we developed in this paper, σ≥σ_s or ${\mathit{\sigma }}_{\mathrm{da}}\le {\mathit{\sigma }}_{-\mathrm{1}}$, reveal the vibration intensity of drillstring is too high to safe.