1. Introduction
In high-pressure, high-temperature (HPHT) wells with pressures greater than 7 × 10
7 Pa and temperatures greater than 150 °C, premium connections are used to connect the tubing to several kilometers of string [
1], as shown in
Figure 1, forming a underground channel for oil and gas. The heavy downhole loads greatly threaten the sealability of premium connections. A premium connection, as shown in
Figure 2, contains threads, sealing surfaces and a torque shoulder. The sealability of a premium connection is guaranteed by two metal sealing surfaces with an interference fit. The sealing surfaces consist of a large number of asperities with different diameters. When in contact, the actual contact area between the two sealing surfaces is significantly lower than the nominal contact area [
2]. Additionally, the contact pressure on the surface also changes significantly, which affects the sealability of the premium connection.
Previous studies on the sealability of premium connections have mostly been macroscopic, assuming that the sealing surface is smooth in analyzing the stress, contact pressure, contact area or contact length. Considering the roughness of premium connections, Zhang Ying established a gas leakage rate model of a cone–cone premium connection, and investigated the impact of sealing surface roughness on the leakage rate of the premium connection [
3]. Wang and Shen et al. established a two-dimensional (2D) finite element model (FEM) of premium connections with two sealing structures, and verified the accuracy of the simplified symmetry model through experiments [
4,
5]. Sches et al. carried out tests on a full-size premium connection and determined the optimal structural size according to the wear resistance, internal pressure strength and outer pressure resistance [
6]. Man et al. established a three-dimensional (3D) FEM of premium connections [
7,
8] that consider the structure of the sealing surface, downhole conditions, and loads, and obtained the stress, contact pressure, and contact length. In addition, Murtatisu suggested a method to assess the sealing performance via the sealing strength [
9]. Through physical tests and numerical simulation methods, the sealing strength and critical sealing strength were derived via the functional relationship between sealability and contact pressure [
10,
11,
12,
13]. Previous research has neglected the effect of changes in the real contact area and contact pressure due to elastic and plastic deformation of asperities on the sealability of the sealing surface upon contact.
The metal sealing surface is composed of a great number of asperities of different sizes. The profile of the rough surface can be described using either a statistical model or a fractal model. The statistical model mainly depends on the statistical parameters of the rough surface. Additionally, the statistical parameters are affected by the instrument’s resolution and the sampling length, so they cannot accurately characterize all the features of the rough surface. However, this problem can be solved using a fractal model. The rough surface described by the fractal model is characterized by continuity, scale independence and self-similarity; thus, it objectively represents the characteristics of the rough surface [
14]. The concept of fractals was first proposed by Mandelbrot, who discovered fractal geometry, which applies to objects with natural forms, no specific proportions and sizes, and infinite detail. Since then, the Weierstrass–Mandelbrot fractal function (W-M function) has been widely used in the study of tribology [
15]. Majumdar and Bhushan applied the W-M function to analyze the contact behavior of two rough surfaces, established Majumdar–Bhushan (M-B) fractal contact theory, and obtained expressions for the contact load and contact area using fractal parameters [
16]. Jourani et al. established a 3D quantitative model utilizing the W-M function to examine the impact of 3D fractals on both contact area and surface roughness [
17]. Due to the complex structure of premium connections, the length of the sealing surface is very short. As it is affected by the resolution of the measuring instrument and the sampling length, it cannot accurately reflect all the characteristics of the rough surface.
Compared to the traditional method of assuming a smooth sealing surface for premium connection, utilizing fractal functions to describe the rough morphology on the sealing surface of premium connections can reveal the contact behavior of the sealing surface more realistically. First, a fractal rough surface, including the geometric parameters and the material parameters of the premium connections, is constructed with the W-M function. A fractal contact model for rough surfaces is developed by taking into account the variations in actual contact area resulting from elastic, elastoplastic, and fully plastic deformation of asperities. Then, the influence law of the different values of the fractal dimension D and the scale coefficient G on the contact area and the contact pressure is analyzed. Secondly, the surface profile of the fractal roughness is created on the sealing surface, and a fractal FEM of full-size premium connection is established. The distribution of the Von Mises stress, contact pressure and effective contact length under different values of the fractal dimension, axial tension and internal pressure are obtained. Finally, based on the theory of sealing strength, an analysis of the sealability of the premium connection is conducted under different loads. The findings of this paper could provide a new method of researching the sealability of premium connections at the microscale, and may serve as a reference for the design and application of premium connections. It may also provide a reference for the sealability analyses of metal sealing parts such as bolts and flanges.
4. Analysis of the Sealability of Premium Connections
For a premium connection with a metal-to-metal sealing structure, achieving sealability relies predominantly on the distribution of contact pressure across the sealing surface. The influencing factors mainly include the geometric structure of the sealing surface, the properties of the metal materials, the properties of the sealing surface, the service environment, the load conditions, and whether a thread compound is used. The traditional design concept of sealability is that if the mean contact stress on the sealing surface surpasses the fluid pressure in the pipe, it will meet the requirements of sealability. However, sealing structures designed in line with this concept still leak [
30]. Thus, this study utilized a model for sealing contact strength to assess the sealability of a premium connection.
On the basis of the theory of sealing contact energy, Murtagian established a weighted model of sealing contact strength to evaluate the sealability through experiments and a numerical simulation study [
9].
Here,
Pav is the average contact pressure (in Pa),
W is the sealing strength (in m·Pa
1.2),
l is the contact length of the sealing surface (in m) and
m is the correlation index. If a thread compound is used,
m = 1.2; if no thread compound is used,
m = 1.4. In this case, we selected
m = 1.4.
Here, is the critical sealing strength (in m·Pa1.2), Pa is the atmospheric pressure (in Pa, the standard atmospheric pressure here was 1 × 10−5 Pa) and Pg is the pressure of gas to be sealed (in Pa).
In order to effectively assess the sealing ability of a premium connection, Xie optimized Equation (24) and established a model for the critical sealing strength of a cone–cone premium connection [
31].
To ensure that a premium connection does not leak, it must meet the condition .
According to Equation (25), the critical sealing contact strength
of a premium connection is 3.12 × 10
6 m·Pa
1.4. According to the results of the above FEA and Equation (23), the sealing contact strength
W of a premium connection under different axial forces and internal pressures can be calculated (as shown in
Table 5). In the example shown in
Figure 15, the sealing contact strength
W under axial tension and internal pressure is greater than the critical sealing contact strength
, but there are still potential sealing failure points. For example, when the axial force is 1.2 × 10
6 N, the sealing contact strength
W is reduced by 30%. Although the sealing contact strength
W increases under the internal pressure, a large area of plastic deformation appears on the sealing surface when the internal pressure is 1 × 10
8 Pa (as shown in
Figure 13). The strength failure can also lead to sealing failure on the sealing surface. Through the above analysis, it can be concluded that the premium connection in this study has a higher risk of sealing failure when the axial tension is 1.2 × 10
6 N or the internal pressure is 1 × 10
8 Pa.
5. Conclusions
On the basis of fractal contact theory, the impact of fractal parameters on the contact behavior of a sealing surface was assessed. The rough surface profile and the deformation of asperities on the sealing surface of a premium connection were taken into account. Through the use of ABAQUS software and contact strength theory, the fractal FEM of a full-size premium connection was established, and the sealability of the premium connection under axial tension and internal pressure was analyzed. The following conclusions can be drawn from this study.
- (1)
Compared with the scale coefficient G, the influence of fractal dimension D on contact area and contact pressure is more significant. With the increase in fractal dimension, the number of asperities on the fractal surface increases, and the contact area exhibits exponential growth, while the contact pressure decreases exponentially. When the fractal dimension D is less than 2.5, the maximum Von Mises stress is 8.81 × 108 Pa and the maximum contact pressure on the sealing surface is 1.20 × 109 Pa, making it prone to gluing and ultimately leading to sealing failure. Conversely, when the fractal dimension D is greater than 2.7, the contact pressure distribution on the sealing surface is more uniform, which improves sealability.
- (2)
As the axial tension increases, stress concentration in the area gradually shifts from the sealing surface to the threaded portion, resulting in a reduction in the contact pressure and the effective contact length of the sealing surface. When the axial tension reaches 1.2 × 106 N, the sealing surface experiences significant displacement along the axial direction, the effective contact length is reduced from 2.72 × 10−3 m to 2.24 × 10−3 m, and the maximum contact pressure is reduced from 8.10 × 108 Pa to 6.39 × 108 Pa, which leads to a 30% decrease in sealing strength and therefore poses a high risk of sealing failure.
- (3)
As the internal pressure increases, the plastic deformation ratio of the fractal surface asperities significantly increases, leading to a proportional increase in the contact pressure and effective contact length of the sealing surface. When the internal pressure reaches 1 × 108 Pa, the effective contact length is increased from 9.08 × 10−3 m to 1.06 × 10−2 m, the maximum contact pressure is increased from 8.67 × 108 Pa to 1.37 × 109 Pa, and the sealing strength is increased by 23%. At the same time, the maximum Von Mises stress on the sealing surface reached 9 × 108 Pa, resulting in a significant stress concentration on the sealing surface.
In summary, the sealability of the premium connection is primarily affected by the axial tension. Within the range in which the structural integrity is not compromised by internal pressure, increasing the internal pressure and ensuring smoother sealing surfaces can both contribute to maintaining sealability. Therefore, to ensure the sealability of the premium connections, axial tension should be kept below 1.2 × 106 N, and internal pressure should be limited to below 1 × 108 Pa. Moreover, to avoid the sealing surfaces gluing during assembly, surface roughness should be minimized during manufacture. Compared to the traditional method of assuming a smooth sealing surface for premium connection, utilizing fractal functions to describe the rough morphology on the sealing surface of premium connections can reveal the contact behavior of the sealing surface more realistically. The premium connections’ fractal FEM established by this method is closer to the actual situation, considering the effect of the asperities’ elastic–plastic deformation on the sealability. Consequently, this research provides a new method for researching the sealability of premium connections at the microscale, and serves as a reference for the design and application of premium connections.
However, within the model in this paper, the influence of deformation behavior on the surface asperities and the interaction between them was ignored. Both of these factors can reduce the stiffness of the sealing surface, and the greater the load, the more significant the influence [
23]. Therefore, in future research, it is necessary to conduct rough surface testing experiments to reveal the relationship between fractal parameters and traditional surface roughness evaluation parameters, and to conduct rough surface loading experiments to verify and modify the fractal finite element model of the premium connection in order to improve the reliability of sealability research on premium connections.