Analysis of Energy Dissipation and Sealing Performance of the Premium Connection in Modal Vibrations

Abstract

Traditional static analysis cannot effectively explain the issue of the sealing performance of the premium connection being decreased due to the vibration of the tubing, leading to the failure of the connection sealing. In this paper, based on the energy dissipation theory and considering the influence of the micro contact slip of the sealing surface under the vibration of the tubing, a finite element model of the premium connection is established. The natural frequency and vibration mode are obtained through modal analysis experiments, and the accuracy of the finite element model is verified. The results show that the first five natural frequencies are mainly concentrated in the axial direction of the tubing, with the amplitude of the radial vibration mode being small. The vibration mode results are applied to the model as boundary conditions. It is found that an increase in the axial displacement amplitude leads to an increase in the energy dissipation of the sealing surface of the premium connection, which reduces the normal contact pressure and the effective length of the sealing surface, ultimately leading to a decrease in the sealing performance.

1. Introduction

Tubing and casing are connected through threaded connection, forming a string that is extended for thousands of meters, with the connection sections being regarded as the weakest part of the entire string [1]. The connection joining the tubing and casing are classified into two types: API threaded connections and premium connections. In Figure 1, schematics of the API and premium connection are shown. The premium connection is primarily composed of three parts: the connection threads, the primary sealing structure, and the torque shoulder [2]. The threads are typically designed in a trapezoidal shape to enhance the strength of the premium connection, serving both connection and load-bearing functions. The primary sealing structure is formed by a metal-to-metal interference fit, including cone-to-cone, sphere-to-cone, and cylinder-to-sphere contact surfaces, where the interference fit ensures gas tightness. The torque shoulder is generally designed with a negative angle, stopping the threads at the proper position, bearing compressive and torsional loads, and providing auxiliary sealing [3].

During large-displacement fracturing and high-gas-production operations, changes in the fluid pressure and flow rate inside the tubing cause string vibration, generating dynamic loads on the string [4]. Premium connections, as the connecting structures of tubing, are frequently subjected to sealing failure under complex load conditions, which can even result in accidents. According to field statistics from oilfields, connection failures are reported to account for 85–95% of all string failures [5,6]. Examples of the premium connection failures caused by string vibration are presented in Figure 2. In Figure 2a, the external wall of the coupling is shown to be worn due to the reciprocating friction between the premium connection and the wellbore, while in Figure 2b, the sealing surface is worn due to the reciprocating friction between the sealing surfaces of the premium connection.

In previous studies, the sealing performance of the premium connection has mainly been related to the equivalent stress, contact length and contact pressure of the premium connection. Cui et al. [7] established two kinds of premium connection with a trapezoidal thread and torque shoulder, and obtained the spatial variation caused by the stress distribution and deformation on the connection model. Yang et al. [8] proposed a theoretical model for evaluating the sealing performance of the premium connection based on the make-up torque, and calculated the elastic–plastic contact pressure distribution of the sealing interface. Xu et al. [9,10] proposed a quantitative model to directly calculate the air tightness of conical premium connections. The circumferential leakage width and radial average leakage height of the micro-leakage channel between the sealing surfaces were obtained. Galle et al. [11] compared different thread designs and evaluated their effects on the maximum allowable torque during tightening and the tension under service conditions. The influence of changes in the sealing area on the sealing performance and performance limit is discussed. Zhang et al. [12] studied the effects of the sealing surface contact stress, surface roughness and sealing surface contact length on the sealing performance of the premium connection. Shen et al. [13] analyzed the sealing pressure of the premium connection filled with threaded grease. Based on the gas state equation, the time at which the leakage gas broke through the sealing pressure of the thread grease and escaped from the coupling was analyzed. However, the above research is mainly based on a static analysis of the sealing performance of the premium connection, which can not effectively solve the problem of the decrease in the sealing performance of the premium connection caused by the vibration of the tubing.

Under vibration conditions, the sealing surface will experience gross slip and generate energy dissipation. With the passage of time, the energy dissipation of the connection will accumulate, resulting in fretting wear or the fatigue failure of the sealing surface [14], thus affecting its sealing performance. In the micro slip state, the energy dissipation is usually represented by the load–displacement hysteresis curve [15]. As shown in Figure 3, the load–displacement hysteresis curve is categorized into three types: linear, parallelogram, and elliptical [16]. The linear curve typically occurs under very small displacement amplitudes or large normal loads, where the contact surfaces experience only adhesion or microscopic slip. The parallelogram curve occurs under larger displacement amplitudes or smaller normal loads, where the macro-slip of the contact surfaces is observed. The elliptical curve mainly occurs in the later stages of the micro-slip cycle, where significant plastic deformation is usually present on the friction surfaces [17]. Masuko et al. [18,19] study the hysteresis characteristics of connection surfaces using analytical and experimental methods, defining the stiffness and slip characteristics of connection surfaces under preload, and investigate the horizontal displacement of connection surfaces under different loading and unloading conditions, plotting load–displacement curves. Zhang Ying [20] develops a contact stress model for the sealing surface and analyzes the impact of different sealing cone angles on energy dissipation in the premium connection under oscillating loads. YU Y et al. [21] establish a micro-slip model for the shear layer of the sealing surface and analyze the impact of the shear layer coefficient on the slip state and energy dissipation of the sealing surface under dynamic loads. Christian et al. [22] study how the coupling of the neglected normal and in-plane elastic responses affects the tribological properties when the Hertzian or random rough indenter slides over the elastomer. This coupling will affect the gap morphology and thus affect the leakage. These studies provide new insights into the analysis of sealing performance for the premium connection; however, the models established are based on certain simplifications and assumptions, and do not include vibration modal analysis or its impact on the sealing performance. During the service of the premium connection in high-temperature and high-pressure gas wells, vibration induced by changes in the internal flow rate and pressure can further exacerbate the risk of leakage at the natural frequency [23,24,25].

According to the above content, this paper considers the energy dissipation and sealing performance of the premium connection under vibration conditions, establishes a finite element model of the premium connection with a cone-to-cone surface, and carries out a modal analysis experiment on the premium connection to obtain the modal vibration mode and natural frequency. The accuracy of the finite element analysis results is verified by experiments. The modal vibration mode is transformed into the displacement amplitude and applied to the finite element model of the premium connection. The force–displacement hysteresis curve and the contact pressure of the sealing surface of the premium connection are analyzed, and the energy dissipation and sealing performance of the premium connection are analyzed.

2. Materials and Methods

2.1. Acquisition of Modal Shapes

In order to obtain the dynamic loads at the location of the premium connection under vibration conditions, a modal analysis of the string containing the connection is first conducted. Traditional modal analysis methods are based on a linear model, where the vibration equation of the string is given as follows [26].

$$[M]\left(\ddot{u}\right)+[C]\left(\dot{u}\right)+[K]\left(u\right)=f(t)$$

(1)

In the equation:

[M]—Mass matrix;

[K]—Stiffness matrix;

[C]—Damping matrix;

($\ddot{u}$)—Acceleration vector;

($\dot{u}$)—Velocity vector;

(u)—Displacement vector;

f(t)—Force vector.

From Equation (1), the equation of motion for undamped vibration can be obtained as follows:

$$[M]\left(\ddot{u}\right)+[K]\left(u\right)=0$$

(2)

Assuming the solution of Equation (2) is

$$\left(u\right)=\left(\phi \right)e^{j\omega t}$$

(3)

substituting Equation (3) into Equation (2) yields the characteristic equation, as follows:

$$\left([K]-{\omega }^2[M]\right)\left(\phi \right)=\left(0\right)$$

(4)

By solving the equation, the eigenvalues ω, also known as circular frequencies, and the eigenvectors (φ), also known as modal shapes, are obtained. The eigenvectors represent the relative amplitudes at different positions within the structure.

If a force (such as make-up torque) has already been applied to the sealing surface of the connection before the modal analysis, the contact state is also altered, and the stiffness matrix [K] varies with the load. In this case, Equation (1) is represented as follows:

$$[M]\left(\ddot{u}\right)+[K(u)]\left(u\right)=f(t)$$

(5)

where f(t) includes the static load f₁(t) and the dynamic load f₂(t).

$$f(t)=f_1(t)+f_2(t)$$

(6)

The amplitude of the dynamic load is generally smaller than that of the static load, so it does not disrupt the structural equilibrium established by the static load. After applying the static load, the stiffness matrix [K₁(u)] of the system is given by the following:

$$\left[K_1(u)\right]\left(u\right)=f_1(t)$$

(7)

When f₁(t) reaches its final value, the final values of the static load (u) and the dynamic load [K₁(u)] can be obtained. At this point, the system stiffness matrix [K₁(u)] can be considered as the initial stiffness matrix. Therefore, Equation (5) is as follows:

$$[M]\left(\ddot{u}\right)+[K_1(u)]\left(u\right)=f(t)$$

(8)

The eigenvalue equation is given by the following:

$$\left([K_1]-{\omega }^2[M]\right)\left(\phi \right)=\left(0\right)$$

(9)

By solving Equation (9), we can obtain the modal shape and modal frequency of the premium connection.

According to the above method, we can obtain the data of the modal shape of the premium connection and apply it to the premium connection by transforming the modal shape into cyclic displacement, and then analyze the energy dissipation at the sealing surface of the premium connection.

2.2. Establish a Finite Element Model of the Premium Connection

Considering that plastic deformation will occur during the analysis of finite element software, considering the material nonlinearity of the finite element analysis model, the plastic stress–strain of the material is set in Abaqus finite element software (v.6.6), as shown in Figure 4. According to the cone–cone Φ88.9 mm × 6.45 mm P110 premium connection shown in

Table 1. Parameters of a cone–cone premium connection.

Loading Flank Angle/°	Guide Surface Angle/°	Shoulder Angle/°	Thread Distance/mm	Thread Taper	Sealing Surface Taper
−3°	10°	−10°	4.234	1/16	1/2

, the finite element model is established as shown in Figure 5. The material parameters of the premium connection are shown in

Table 2. Material parameters of the premium connection.

Density/(kg/m3)	Elastic Modulus/GPa	Friction Coefficient	Sealing Surface Taper	Yield Strength/ MPa	Poisson Ratio
7850	206	0.02	1:2	758	0.3

.

The finite element analysis model of the premium connection incorporates the threaded structure but involves certain simplifications, specifically simplifying the connection between the crest and root of the internal and external threads. According to the relevant literature, the radial interference of the threads is set to 0.06–0.10 mm [27]. In the finite element software, the mesh is refined for the sealing surface and the torque shoulder of the premium connection, with a mesh size of 2 mm, using C3D8R elements. The overall tubing model consists of 140,371 elements. The contact pairs are configured as face-to-face contacts, with a total of 5 contact pairs created, including pairs for the sealing surface, the torque shoulder, and three pairs for the threaded sections of the premium connection. The interference between the sealing surfaces and the interference between the torque shoulders are set according to the parameters for an optimal make-up torque [28].

In the finite element simulation of the premium connection, the number and size of the mesh have a certain influence on the simulation data of the finite element. Through a large number of experiments on the finite element mesh density, the appropriate mesh density is selected so that it will not have a serious impact on the finite element simulation results. In the analysis of the premium connection, the C3D8R hexahedral element is used for meshing. Figure 6 shows the maximum von Mises stress distribution under different mesh sizes. By comparing the change in the stress curve under grid densities of 2 mm, 2.5 mm, 3 mm, 3.5 mm and 4 mm, it can be seen that when the mesh size is in the range of 2 mm~3 mm, the maximum stress of the sealing surface varies between 1.4% and 2.6%, so it has little effect on the analysis. Therefore, it is more appropriate to set the mesh size to 2 mm.

2.3. Modal Analysis Testing of the Premium Connection

To verify the accuracy of the finite element model of the premium connection, a modal analysis test is conducted. Figure 7 illustrates the setup of the modal testing for the premium connection. Since modal testing requires the test object to be in free boundary conditions, a rubber rope suspension method is used to simulate these conditions. A hammer is employed to provide excitation, with the excitation signals from the hammer and the acceleration signals from the three-axis acceleration sensor being collected by the MI-7004 signal collector (Econ Technologies, Hangzhou, China). The collected signals are processed in the software to complete the modal analysis and identification of the premium connection.

The test is conducted using a fixed-point excitation method with a single input and multiple output signals. In this method, the displacement of the three-axis acceleration sensor remains unchanged while the hammer impact points are varied. The measurement points for the premium connection are shown in Figure 8. Since the tubing can be simplified as a uniform cross-sectional cylinder, 8 tap points are set on the same cross-section, with cross-sections spaced 20 cm apart along the axis of the tubing. At the “protrusion” of the coupling, the center of the coupling is designated as an impact section, resulting in a total of 96 impact points. During the test, the force hammer’s impact direction is always maintained normal to the impact points.

2.4. Method of Energy Dissipation and Sealing Performance Analysis of the Premium Connection

2.4.1. Method of Energy Dissipation Analysis

As shown in Figure 9, under the influence of external cyclic loading, a closed hysteresis curve is formed by the force and displacement. The area enclosed by the hysteresis curve represents the energy dissipation value during one cycle of loading.

The force–displacement hysteresis curve includes the process of unloading and loading [29]. The following is its mathematical expression:

The unloading process of an elastic rod:

$${\displaystyle \frac{F_u-F_0}{2}}=-k_s\left({\displaystyle \frac{u{(L)}_0-u{(L)}_u}{2}}\right)+D_s$$

(10)

In the equation:

F₀ and u(L)₀—force and displacement at the beginning of unloading process;

F_u and u(L)u—force and displacement during unloading;

D_s—load of gross slip;

k_s—constant of interfacial adhesion stiffness.

The reloading process is described as follows:

$${\displaystyle \frac{F_u+F_0}{2}}=-k_s\left({\displaystyle \frac{u{(L)}_0+u{(L)}_u}{2}}\right)+D_s$$

(11)

The force–displacement hysteresis curve of the sealing surface can be obtained from Equations (10) and (11), and its area is the energy dissipation.

$$\Delta E_D=4{\displaystyle {\int }_{l_n}^L\mu p\left(x\right)}{u}_s\left(x\right)dx=4{\displaystyle {\int }_{l_n}^L\mu p\left(x\right)\left[u\left(x\right)-u_c\left(x\right)\right]}dx$$

(12)

In the equation:

u_c(x)—adhesive displacement;

u_s(x)—slip displacement;

μ—friction coefficient;

P(x)—interface pressure.

2.4.2. Sealing Performance Analysis Method

To quantitatively assess the sealing performance of the premium connection, the contact pressure over the effective sealing length of the seal surface is calculated and integrated. The ability of the metal-to-metal sealing structure to prevent fluid flow, based on this integral result, is referred to as the sealing performance evaluation index, W_a, defined by Equation (13) [30].

$$W_a={\displaystyle {\int }_{L_{yx}}{p}_c^x(L)}dL$$

(13)

In the equation:

W_a—The sealing performance evaluation indexmm·MPa;

L_yx—Effective sealing length, mm;

p_c—Normal contact pressure distribution on the sealing surface, MPa;

x—Sealing parameters.

Murtagian determined the sealing parameter x in Equation (13) through experimental methods. When thread lubricant is present in the metal–metal sealing structure, the sealing parameter x is 1.2; when thread lubricant is absent, the sealing parameter x is 1.4, as detailed in Equation (14) [31]:

$$W_a=\left\{\begin{array}{l}{\displaystyle {\int }_{L_{yx}}{p}_c^{1.2}(L)}dL(\mathrm{The}\mathrm{presence}\mathrm{of}\mathrm{thread}\mathrm{sealant},\mathrm{mm}·{\mathrm{MPa}}^{1.2})\\ {\displaystyle {\int }_{L_{yx}}{p}_c^{1.4}(L)}dL(\mathrm{The}\mathrm{absence}\mathrm{of}\mathrm{thread}\mathrm{sealant},\mathrm{mm}·{\mathrm{MPa}}^{1.4})\end{array}\right.$$

(14)

In the premium connection, the metal–metal sealing structure includes two types: the cone-to-cone sealing surface and the torque shoulder structure. Therefore, Equation (15) is revised in this paper to the following:

$$W_a={\displaystyle {\int }_{L_{seal}+L_{tor}}{p}_c^x(L)}dL$$

(15)

In the equation:

L_seal—Effective sealing length of the sealing surface;

L_tor—Effective sealing length of the torque shoulder.

When the maximum leakage rate of the metal-to-metal sealing structure is 15 min/0.025 cm³ [30], a critical sealing index, denoted as W_ac, is proposed, as shown in Equation (16). By comparing the sealing performance evaluation index W_a with the critical sealing index W_ac, the sealing performance can be assessed. When W_a < W_ac, it indicates that leakage may occur, and preventive measures should be taken; when W_a > W_ac, it indicates that the sealing surface is safe and there is no risk of leakage, ensuring a reliable sealing performance.

$$W_{a\mathrm{c}}=\left\{\begin{array}{l}1.843\times {\left({\displaystyle \frac{p_g}{p_a}}\right)}^{1.177}(\mathrm{The}\mathrm{presence}\mathrm{of}\mathrm{thread}\mathrm{sealant},\mathrm{mm}·{\mathrm{MPa}}^{1.2})\\ 103.6\times {\left({\displaystyle \frac{p_g}{p_a}}\right)}^{0.838}(\mathrm{The}\mathrm{absence}\mathrm{of}\mathrm{thread}\mathrm{sealant},\mathrm{mm}·{\mathrm{MPa}}^{1.4})\end{array}\right.$$

(16)

In the equation:

p_g—Sealing pressure, MPa;

p_a—Standard atmospheric pressure, MPa, is typically taken as 0.1 MPa in normal conditions.

Based on the analysis in Figure 10, it can be observed that under the same gas pressure, the leakage potential of the premium connection with thread sealant is significantly lower than that without thread sealant. This indicates the critical role of thread sealant in the gas sealing performance.

3. Result and Discussion

3.1. Modal Analysis and Model Validation of the Premium Connection

This section analyzes only the first five natural frequencies and mode shapes of the tubing with the premium connection. Therefore, only the first five natural frequencies are extracted from the finite element software, and higher-order natural frequencies obtained from the experiments are not discussed in this section. As shown in Figure 11, the modal assurance criterion (MAC) parameters from the post-processing of the software are used to assess the reliability of the experimental results. A higher MAC value indicates greater similarity between mode shapes, with a MAC value of 1 indicating identical mode shapes and a MAC value of 0 indicating significant differences between mode shapes. The analysis of the MAC parameters reveals some deviation in the mode shape analysis for the first natural frequency. However, the mode shape test results for the remaining natural frequencies are satisfactory, indicating that the modal test results are reliable and provide a high level of reference.

Considering the potential errors introduced during the test impact and the inherent uncertainties in the finite element analysis, this section compares and analyzes the modal results by integrating the finite element analysis results with the experimental results.

Table 3. Comparison of finite element analysis and experimental results.

	f1/Hz	f2/Hz	f3/Hz	f4/Hz	f5/Hz
Experimental Analysis	193.25	378.40	621.93	1127.64	1448.13
Finite Element Analysis	140	378	663	1127	1554
Error	27%	0.1%	6.83%	0.05%	6.85%

presents the natural frequencies of the tubing obtained through software post-processing analysis. It is observed that the first-order natural frequency obtained from the experiment differs significantly from the finite element analysis results, with an error of up to 27% when using experimental data as the baseline. This discrepancy is attributed to the hardness of the hammer used during the test, which results in better excitation effects for higher-order modes but poorer excitation effects for lower-order modes. Consequently, the error in the first-order modal analysis results is relatively large. Aside from the significant difference in the first-order natural frequency, the results for the other natural frequencies are generally satisfactory.

Figure 12 shows the frequency response function of the premium connection under the optimal makeup torque during the experiment. As shown in Figure 13, the experimental results indicate that the third natural frequency is 621.93 Hz, which is close to the fourth natural frequency at 795.21 Hz. However, the fourth natural frequency observed in the experimental results is not present in the finite element analysis, leading to doubts about its authenticity. To address this issue, the modal shapes from the fourth to the sixth natural frequencies were analyzed and compared, as shown in Figure 9. The comparison revealed that the third natural frequency (621.93 Hz) obtained from the experimental post-processing has three peaks, which is consistent with the results for a simply supported beam’s third mode shape. On the other hand, the mode shape for the fourth natural frequency (795.21 Hz) obtained from the experimental post-processing appears to be “incomplete”. Therefore, it is concluded that the frequency of 795.21 Hz falls between the third and fourth natural frequencies, and that the experimental result for the fourth natural frequency is likely due to testing errors.

Figure 14 shows the comparison between the experimental results of the tubing vibration mode under the optimal make-up torque for the premium connection with the finite element analysis (FEA) results. It is observed that the FEA results align well with the experimental analysis results (the magnification factor between the FEA and experimental results is 10). Due to limitations in post-processing software (Econ 4.2.39), the experimental post-processing results can only show the modal trend under the natural frequency and cannot display the specific amplitude of the tubing string’s vibration. By comparing the similarity of the vibration trends between the FEA and the experiment, as well as the accuracy of the natural frequency results previously analyzed, it is concluded that the FEA modal analysis results are relatively accurate and hold a certain degree of reference value. Therefore, the specific vibration amplitudes from the FEA results are adopted for further analysis in this paper.

3.2. Energy Dissipation Analysis of the Premium Connection Under Modal Shapes

Through the mutual verification of the modal test and finite element analysis, it can be concluded that the first five modal shapes of the tubing mainly appear in the x-axis direction, while the vibration amplitude in the y-axis direction is small. Based on the results of the finite element modal analysis, the amplitude of each vibration mode is extracted and used as the displacement amplitude applied to the finite element model to analyze the energy dissipation at the sealing surface of the premium connection. The constraint condition of the premium connection is shown in Figure 15. In the post-processing results of the finite element modal analysis of Section 3.1, the displacement amplitude of the first modal frequency along the y-axis of the tubing is 1.376 mm, the internal pressure is set to 80 MPa, and the friction coefficient of the sealing surface is 0.02.

Figure 16 shows the hysteresis curve of the first modal shape boundary condition. It can be seen from the diagram that the applied displacement amplitude is attenuated from 1.376 mm to 0.09 mm at the sealing surface. It can be seen from the analysis in Figure 16 that due to the large axial displacement load on the tubing, the contact length and equivalent stress of the sealing surface increase, resulting in a large gross slip between the sealing surfaces. The proportion of gross slip accounts for 93.9% of the displacement amplitude, and the micro slip only accounts for 6.1% of the displacement amplitude, resulting in the shape of the hysteresis curve between the sealing surfaces being a parallelogram. At this time, the energy dissipation value generated by the connection sealing surface interface is large.

The second mode shape of the tubing also mainly occurs in its x-axis direction, and the amplitude of the vibration mode in its y-axis direction is small. The displacement amplitude along the y-axis direction of the tubing under the second-order natural frequency is 2.258 mm, and an internal pressure of 80 MPa is added. The friction coefficient of the sealing surface is 0.02. The hysteresis curve obtained by applying the displacement amplitude to the finite element model of the premium connection is shown in Figure 17. In Figure 17, the proportion of gross slip accounts for 94.1% of the displacement amplitude, which is higher than the 93.9% of gross slip under the first modal shape. It is concluded that under the same boundary conditions, with the increase in the applied displacement amplitude (modal shape), the proportion of the gross slip region in the hysteresis curve increases, and the energy dissipation value also increases.

3.3. Sealing Performance Analysis of the Premium Connection Under Modal Shapes

Figure 18 illustrates the von Mises stress distribution for the premium connection. Considering the high-cycle displacement loading experienced by the tubing during extended service in the well, such displacement amplitudes can accelerate the formation of small cracks between the sealing surfaces. The prolonged reduction in the combined stiffness of the sealing surfaces inevitably leads to significant stress concentrations at the root of the bearing surfaces of the first and second threads of the threaded connection, resulting in substantial plastic deformation. As shown in Figure 18, the maximum von Mises stress at the thread of the premium connection is 850 MPa, while the yield strength of the premium connection in this paper is 758 MPa. The accumulation of plastic deformation damages the connection’s auxiliary sealing structure, thereby impairing the sealing performance of the premium connection.

Figure 19 shows the normal contact pressure distribution of the sealing surface of the premium connection before and after the first-order vibration mode loading. From the analysis of the figure, it is observed that the normal contact pressure on the sealing surface before the application of the loading displacement amplitude is higher than that after the application. The maximum normal contact pressure is reduced by approximately 30%, while the effective sealing length decreases from 1.4 mm to 1.1 mm, representing a reduction of about 21%. The parameters from Figure 17 are substituted into Equations (14) and (16) to calculate the variation in the sealing index values of the sealing surface before and after the application of the loading displacement amplitude, as shown in

Table 4. First-order mode shape sealing surface sealing index.

Internal Pressure/MPa	Sealing Index Before Loading/mm·MPa1.4	Sealing Index After Loading/mm·MPa1.4	Decrease Ratio/%
80	4670	1722	63.12

. The sealing index value is reduced by 2948 (mm·MPa1.4) from before to after loading, which constitutes a reduction of 63.12% relative to the sealing index before loading.

Figure 20 illustrates the distribution of normal contact pressure on the sealing surface of the premium connection both before and after the application of the second-order vibration mode loading. As shown in the figure, the maximum normal contact pressure decreases by approximately 31% after loading with the second-order vibration mode, and the effective sealing length is reduced by about 22%. According to the data presented in

Table 5. Second-order mode shape sealing surface sealing index.

Internal Pressure/MPa	Sealing Index Before Loading/mm·MPa1.4	Sealing Index After Loading/mm·MPa1.4	Decrease Ratio/%
80	4670	1704	63.51

, the sealing index of the sealing surface before and after loading decreased by 2966 (mm·MPa^1.4), accounting for 63.51% of the sealing index before loading. A comparison with the analysis in Figure 19 reveals that, as the applied displacement amplitude increases, the maximum normal contact force on the sealing surface remains almost unchanged, with only a slight reduction. Additionally, there is no significant change in the effective sealing length, and the effective sealing length under the first-order mode shape is essentially the same.

Based on the above analysis, it is concluded that after the application of the displacement amplitude, a significant reduction in the normal contact pressure on the sealing surface occurs, which leads to a decrease in the gas-tight sealing performance of the premium connection. The effective sealing length of the sealing surface initially increases with time and the displacement load, but subsequently remains unchanged. This behavior is attributed to the constraint effect of the front-end threads, which causes stress concentration at the front-end threads. In severe cases, this stress concentration can lead to damage to the sealing structure and, ultimately, sealing failure.

4. Conclusions

In this paper, the first five natural frequencies and modal shapes of the finite element model of the premium connection are obtained by using the time domain modal parameter identification method, and the results are verified by experiments. The tubing with modal vibration mode is analyzed, and the energy dissipation and sealing performance of the premium connection are studied. The main conclusions are as follows:

(1)

The finite element analysis shows that the vibration modes of the first five natural frequencies mainly occur in the x-axis direction of the tubing, and the amplitude of the vibration mode in the y-axis direction is small. The test verifies the accuracy of the finite element analysis results.

(2)

In the displacement cycle, the gross slip stage occupies a large proportion, resulting in the hysteresis curve being a “parallelogram”. Under the second-order mode shape, the gross slip ratio is 94.1%, which is slightly higher than the 93.9% of the first-order mode shape. As the displacement amplitude increases, the macroscopic slip region and energy dissipation in the hysteresis curve increase.

(3)

After applying the displacement amplitude, the effective sealing length of the sealing surface of the premium connection is reduced by about 21%, and the normal contact pressure is reduced by about 30%. As the displacement amplitude increases, the sealing performance tends to be stable, indicating that the vibration load will lead to a decrease in air tightness, but the sealing performance tends to be stable under a larger displacement amplitude.