Optimization and Experimental Study of the Subsea Retractable Connector Rubber Packer Based on Mooney-Rivlin Constitutive Model

Abstract

The sealing performance of the rubber packer is of vital importance for the subsea retractable connector, and the cross-sectional shape of the rubber packer is one of the most important factors affecting it. The compression distance of the rubber packer is increased by 19.54% utilizing the established two-dimensional numerical model. In addition, a new parameter called the anti-shoulder extrusion variable was defined in this paper. Shoulder extrusion will not occur when using this variable as a constraint during simulation. In general, the upper end and the lower end of a rubber packer are subject to different constraints, and the structural parameters of the rubber packer affect each other in terms of sealing performance. Therefore, the importance and originality of this study are exploring the optimization of the thickness and chamfer angles of the upper and lower ends of the rubber packer by use of a combination of the response surface optimization method and the multi-objective genetic algorithm, taking the thickness and chamfer angles of the upper and lower ends as design variables, and the stress on the inner side of the casing wall and the axial force of the compressed rubber packer as optimization objectives. Besides that, the anti-shoulder extrusion variables are also introduced as constraints to prevent shoulder extrusion. Ultimately, the cross-sectional shape of the rubber packer with a smaller-thickness and larger-angle upper end, and a larger-thickness and smaller-angle lower end can be obtained. The result to emerge from the test in this paper is that the pipe pressure that can be sealed by the optimized rubber packer structure is 25.61% higher than that before optimization. The anti-shoulder extrusion variable and the asymmetric cross-sectional shape of the rubber packer proposed in this paper shed new light on the finite element simulation of rubber and the research on similar seals.

1. Introduction

Subsea oil and gas pipelines are formed by connecting a large number of pipes and equipment [1], whose connections play a critical role in subsea oil and gas pipelines. Although relatively higher pipe pressure can be sealed by metal seal, due to the axial error between the oil and gas pipelines is rather sensitive, the pipe is required to boast a high accuracy of merging precision [2]. However, because of the errors of manufacturing and underwater installation, sometimes the axial error of the two connected pipes is comparatively large. In addition, the low visibility and high pressure under water is particularly unfavorable to the measurement of axial error by ROV or marine diver. The use of subsea retractable connectors to connect underwater production facilities can solve these problems and greatly reduce installation time and improve work efficiency, thereby shortening the construction period and reducing capital investment. Therefore, the subsea retractable connector is of great practical significance for the connection of subsea production facility. The core sealing component of the metal seal is the metal sealing ring, which can seal the high internal pressure of oil and gas pipelines [3], while the core sealing component of the subsea retractable connector is the rubber packer, which sealing performance is relatively poor. Therefore, it is of vital importance to improve the sealing performance of the rubber packer so that it can withstand the higher internal pressure of the oil and gas pipeline.

Patel et al. [4] attempted to point out that elastomer is a critical part in sealing components; its failure to seal will give rise to serious consequences, such as impacts on health, safety and environment. Therefore, the sealing performance of the rubber packer is of extreme importance to the subsea retractable connector. There are a number of factors affecting the sealing performance of rubber packer, ranging from the compression distance, its material, and the key dimensions of rubber packer, including its height, thickness, and chamfer angle. For instance, Huang et al. [5] pointed out that in the case where the shield tunnel in Shanghai was interrupted due to an accidental extreme overload, the rubber packer only lost its waterproof function due to a release of 6 mm compression distance.

Many researchers have studied the deformation and sealing performance of the elastomer from the aspects of theoretical calculation, the structure of the rubber packer and its material. In terms of theoretical calculation, Banks et al. [6], Gent et al. [7,8] and Horton et al. [9] attempted to study the bonded elastomer with small deformation and provide a relevant theoretical analysis; Suh et al. [10] focused on the elastomer with one side bonded to the rigid surface while the other side in friction contact, and compared the finite element analysis result with the experimental result so as to verify the analysis result; Konstantinidis et al. [11] analyzed the behavior of unbonded rubber layers under compression theoretically by use of the pressure solution, and its accuracy was verified; Al-Hiddabi et al. [12] analyzed the deformation of elastomer between rigid pipe and rigid casing. The analysis model of elastomer was established, which can be used to improve the sealing performance of elastomer; Constantinou et al. [13] proposed the expressions of compression modulus and maximum shear strain of bearings made of rubber layer and steel plate; Pinarbasi et al. [14,15] put forward the analytical solution for the bonded elastic layer of hollow and solid disks under uniform compression, and the expressions of compression modulus and displacement/stress distribution for the linear analysis of bonded elastic layer were derived. Zhang et al. [16] developed a theoretical model for the analysis of deformation based on the structure and working conditions of the rubber packer, and derived the distributions of contact pressure and shear stress on the sealing surface of the rubber packer by use of pressure methods; Wang et al. [17] proposed a method to study the sealing performance of elastomers based on the properties and geometric parameters of rubber materials, and which was verified by the experiment; Patel et al. [18] attempted to evaluate the sealing performance of elastomers by finite element simulation, and which was verified through analytical models. In terms of the structure and material of the rubber packer, Wang et al. [19] optimized the shape of the rubber packer and added the structure of shoulder protection by finite element analysis of the rubber packer, which was verified through the experiment; Lan et al. [20] compared three types of rubber packer structures, and ultimately achieved the sealing under high temperature and high pressure by the combination of the structure of a single rubber barrel with expanding back-up rings and a new type of fluorine rubber, called AFLAS; Zheng et al. [21] studied the key dimensions of the rubber packer and determined a set of optimal dimensions, which was verified by experiment; Liu et al. [22] carried out a tensile test on rubber as well as the finite element simulation and optimization on key dimensions of the rubber packer, covering its height, thickness and angle; Ma et al. [23] revealed the influence of friction coefficient between rubber packer and pipe on sealing performance, and proposed that reducing friction coefficient would contribute to increase the range of maximum contact pressure; Hu et al. [24] claimed that improper selection of rubber materials would give rise to seal failure, and the optimization of the constitutive models of the original three rubber materials was carried out, and eventually, material B75 was determined as the most suitable one; Dong et al. [25] tried to use the finite element model to apply a temperature load to the rubber packer and the influence was analyzed, and the gap between the rubber packer and the casing, angle of contact surface of the rubber packer and optimal value interval of initial setting load were determined; Liu et al. [26] designed a new sealing structure of the packer and its sealing performance and pressure resistance under high temperature and high pressure were verified. In addition, they also proposed that the upper packer plays a critical role in the sealing process, and the optimal size interval of the rubber packer was given; Hu et al. [27] attempted to carry out constitutive experiment and friction experiment on rubber materials, and the simulation model was verified by an experimental device with a transparent shell. It is concluded that the compression deformation and sealing performance increase with the increase in axial load.

The above studies on the optimization of the parameters of the rubber packer structure, however, only focus on a single variable under the condition of other variables remain unchanged. Therefore, the final structural size of the rubber packer is identified by determining the optimal value interval of a single variable sequentially. This kind of approach lacks consideration of the interaction between the structural parameters of the rubber packer; therefore, the optimal size of the rubber packer may fail to be within the optimal value interval of each structural parameter. In addition, these studies assume that the structure of the rubber packer is top-down symmetrical. The difference between the moving end and the fixed end when compressing the rubber packer is beyond their consideration, which may lead to the rubber packer failing to exert its maximum sealing potential. This paper takes the influence of the variations of structural parameters of the rubber packer on the sealing performance into consideration simultaneously and the structure of the rubber packer is optimized in order to obtain the optimal sealing performance by combining with the different constraint conditions at the two ends of the rubber packer.

In this paper, a numerical model that integrates the thickness of the upper and lower ends with the chamfer angle of the rubber packer is established. Under the condition that the response surface optimization method is available, the compression distance of the rubber packer is enhanced by 19.54% than the theoretical calculation by finite element simulation. In addition, the maximum equivalent stress on the inner side of the casing wall and the axial force of the compressed rubber packer are taken as the optimization objectives, and the anti-shoulder extrusion variable is introduced as the constraints to make sure no shoulder extrusion would occur on the rubber packer. A combination of the response surface optimization method and the multi-objective genetic algorithm was adopted to optimize the structural parameters of the rubber packer.

2. The Structure of Connector

The subsea retractable connector, as shown in Figure 1, is mainly applicable for shallow water, where the connection is assisted by divers. The principal task of this connector is to connect oil and gas pipelines as soon as possible on the seabed where the underwater depth does not exceed 100 m.

Hubs are welded on the sides of the central pipe and of the casing wall of the subsea retractable connector so as to connect with the hub of oil and gas pipeline; the gaskets on both sides of the rubber packer are designed with metal protective covers; central pipe can move in the casing before the actuating ring compressing the rubber packer; the left end of the actuating ring is designed with a hydraulic port, which can form a hydraulic chamber in cooperation with the central pipe; the left end of the actuating ring is connected with the end cover 2 by bolts, the end cover 2 is used to limit the displacement of the compressed rubber packer.

When working underwater, first, lower the connector between the oil and gas pipelines which are prepared to be connected on the seabed, then link the hub on the casing side of the connector with the hub of the oil and gas pipeline and then seal, adjust the length of the central pipe in casing by the diver stretching the central pipe, in order to realize the docking between the hub on the central pipe side of the connector and the hub of the oil and gas pipeline and then seal. Finally, press through the hydraulic port, move the actuating ring axially and compress the rubber packer until the end cover 2 contacts the central pipe, this marks the end of compression distance, so as to realize the sealing of oil and gas pipeline. The extensible length range of the central pipe in casing is 0–150 mm; the displacement of the axial movement of the actuating ring depends on the structural size of the end cover 2; therefore, the displacement of the connector is fixed after its installation is completed.

It can be found that the sealing performance of the rubber packer determines that of the connector. Therefore, it is of vital importance to conduct the optimization of the rubber packer so as to improve its sealing performance.

3. Constitutive Model of the Rubber Material

The core sealing component of the subsea retractable connector is the rubber packer, which is made of rubber or filled with rubber with other materials as the framework. This paper only considers the case that the material of packer is rubber, therefore, the rubber packer is an elastomer, which Poisson’s ratio is approximately equal to 0.5, it is almost incompressible in volume [1,28]. The incompressibility of the volume is reflected in the rubber packer through its radial expansion to compensate for the reduction in height [28]. In addition, the rubber packer features isotropy, geometric nonlinearity, material nonlinearity and contact nonlinearity [1,29].

The hyperelastic constitutive models of elastomers range from Neo Hookean model [30], Gent model [31], Yeoh model [32], to Mooney–Rivlin model [33,34], etc. Among them, Mooney–Rivlin two-parameter constitutive model is relatively simple and can realize the simulation of small or medium strain of rubber [24,35]. Even for an ideal material that is of high elasticity, incompressibility and remains isotropic before deformation, the Mooney–Rivlin two-parameter model is still valid for large deformation when the linear relationship between the shear force of the material and the amount of simple shear is achieved. Therefore, Mooney–Rivlin two-parameter model is used to simulate the deformation process of rubber in this paper.

Rubber is an isotropic material with high elasticity before it is compressed or stretched; therefore, the expression of deformation energy, W, can be obtained:

$$W=(I_1,I_2,I_3),$$

(1)

where $I_1$, $I_2$, $I_3$ are strain invariants, which can be expressed by the principal stretch ratio, ${\lambda }_i$

$$\left\{\begin{array}{l}{I}_1={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2\\ I_2={\lambda }_1^2{\lambda }_2^2+{\lambda }_2^2{\lambda }_3^2+{\lambda }_3^2{\lambda }_1^2\\ I_3={\lambda }_1^2{\lambda }_2^2{\lambda }_3^2\end{array}\right.$$

(2)

For homogeneous and incompressible material, it can be obtained as follows:

$${\lambda }_1{\lambda }_2{\lambda }_3=1,$$

(3)

Therefore $I_3=1$, it can be obtained that the deformation energy, W, is merely a function of $I_1$, $I_2$.When the material is not deformed,

$${\lambda }_1={\lambda }_2={\lambda }_3=1,$$

(4)

$I_1=I_2=3$ can be obtained, therefore, the expression of the deformation energy, W, can be transformed into

$$W={\displaystyle \sum _{i=0,j=0}^{\infty }{C}_{ij}{\left(I_1-3\right)}^i{\left(I_2-3\right)}^j},$$

(5)

where $C_{00}=0$. $C_{ij}$ is the physical constant of the material, which can be obtained from the experimental data of uniaxial stretch test [36]. Under the premise of satisfying the small deformation of simulated rubber, Equation (5) can be simplified to a two-parameter model:

$$W=C_1\left(I_1-3\right)C_2\left(I_2-3\right),$$

(6)

where $C_1=C_{10}$, $C_2=C_{01}$, and the following can be obtained

$$W=C_{10}\left(I_1-3\right)C_{01}\left(I_2-3\right),$$

(7)

$C_{10}$ and $C_{01}$ are the physical constants of the material of Mooney–Rivlin constitutive model.

In the case of small deformation of rubber, the relation between shear modulus $G$ and Young’s modulus, $E_0$, can be expressed as

$$G=\frac{E_0}{3}=2\left(C_{10}+C_{01}\right),$$

(8)

The relation between Young’s modulus, $E_0$, and hardness can be expressed as:

$$\lg E_0=0.0198H_r-0.5432,$$

(9)

By introducing Equation (8) into Equation (9), the relation between material constant and rubber hardness can be obtained, which can be expressed as

$$\lg \left[6C_{10}\left(1+\frac{C_{01}}{C_{10}}\right)\right]=0.0198H_r-0.5432,$$

(10)

In this paper, the value of $C_{01}/C_{10}$ is taken as 0.5. When the rubber with an international hardness of 90 is used, $C_{10}=1.92556$, $C_{01}=0.96278$.

4. Simulation and Optimization

In this paper, the simulation and optimization of the thicknesses of end faces and chamfer angles of the rubber packer can be realized by use of the finite element software ANSYS. The purpose of optimization is to identify the value of structural parameters of rubber packer with the best sealing performance under the same compression distance.

When the finite element simulation is carried out, the axisymmetric model can realize the significant reduction of the number of grids required for calculation, so as to effectively decrease the computing time. This paper only analyzes the sealing part of the retractable connector, and the model can be simplified to a two-dimensional axisymmetric numerical one, because the overall force of the sealing part is uniform, and the constraints and load are symmetrical along the axis. As shown in Figure 2a, two-dimensional axisymmetric numerical model is established for the central pipe, actuating ring, fixed ring, rubber packer and casing wall by the 3D software UG, and the values of structural parameters are shown in

Table 1. Structural parameters of numerical model.

Parameter	Value
Inner radius of casing wall, $R_{ti}$(mm)	142
Outer radius of rubber packer, $R_{ro}$(mm)	140
Inner radius of rubber packer, $R_{ri}$(mm)	115
Outer radius of central pipe, $R_{fo}$(mm)	115
Height of rubber packer, $H$(mm)	50
Thickness of rubber packer, $T$(mm)	25
Thickness of the upper end of the rubber packer, $t_u$(mm)	20
Thickness of the lower end of the rubber packer, $t_d$(mm)	20
Angle of the upper end of the rubber packer, $\alpha$(°)	45
Angle of the lower end of the rubber packer, $\beta$(°)	45

; Figure 2b shows the cross-sectional view of the rubber packer. The parameterized structural parameters cover thicknesses of end faces, $t_u$, $t_d$, and chamfer angles, $\alpha$, $\beta$.

The outer radius of the central pipe, $R_{fo}$, is relatively large because the end cap of this connector needs to be fixed on the end surface of the central pipe with bolts; the dimensions of rubber packer are measured according to the rubber packer structure before optimization.

4.1. Parameter Setting

In this paper, the material of the central pipe, actuating ring, fixed ring and casing wall are steel alloy F22, which mechanical properties are shown in

Table 2. Mechanical properties of the material.

Name	Poisson’s Ratio	Yield Strength (MPa)	Elastic Modulus (MPa)
steel alloy F22	0.3	515	2.1 × 105

. The material of the rubber packer is hydrogenated nitrile-butadiene rubber with an international hardness of 90. The friction coefficient between the actuating ring, the fixed ring and the central pipe is set to 0.1, and that between the rubber packer and the central pipe, actuating ring, fixed ring and casing wall is set to 0.2. Fixed constraints are applied on the central pipe, fixed ring and casing wall; for the structural limitation of the connector in this paper, the compression distance is fixed while the axial force is variable, therefore, axial displacement load is applied to the actuating ring.

Figure 3 shows the finite element simulation by adopting an axisymmetric model; Figure 3a shows the grid division of the sheet, in which the unit size of the rubber packer is 0.3 mm, the unit size of the actuating ring and fixed ring is 0.5 mm, and the unit size of the central pipe and casing is 1 mm. The default element is appointed to the whole model. As shown in Figure 3b, the central pipe, fixed ring and casing wall are completely constrained, and the actuating ring applies displacement load according to the compression distance calculated in Section 4.2.

When using the response surface optimization method, it is necessary to set the optimization interval of parameters, optimization objectives as well as the constraints.

Table 3. Value interval of structural parameters of the rubber packer.

$\mathbf{Thickness} \mathbf{of} \mathbf{the} \mathbf{Upper} \mathbf{End} {\mathit{t}}_{\mathit{u}}\left(\mathbf{mm}\right)$	$\mathbf{Thickness} \mathbf{of} \mathbf{the} \mathbf{Lower} \mathbf{End} {\mathit{t}}_{\mathit{d}}\left(\mathbf{mm}\right)$	$\mathbf{Chamfer} \mathbf{Angle} \mathbf{of} \mathbf{the} \mathbf{Upper} \mathbf{End} \mathit{\alpha }$	$\mathbf{Chamfer} \mathbf{Angle} \mathbf{of} \mathbf{the} \mathbf{Lower} \mathbf{End} \mathit{\beta }$
15–22	15–22	25–60	25–60

shows the value interval of the four structural parameters of the rubber packer.

Among them, the thicknesses of both end faces of the rubber packer, $t_u$, $t_d$, shall not be too small, otherwise the chamfer on both sides of the rubber packer will be too large, and the contact surface between the actuating ring and the rubber packer will be too small, which will lead to destabilization, resulting in the failure of the rubber packer to achieve the most ideal sealing state; similarly, it should not be too large, otherwise in the finite element calculation, rubber extrusion will easily occur, and the calculation results will fail to describe the real sealing state. Likewise, there is a limitation on the value interval of chamfer angle, $\alpha$, $\beta$. It is necessary to ensure that the combination of the value of the chamfer and the thickness of the end face can neither be too small to lose the function of the chamfer, nor can it be too large to cause the compressed rubber packer to lose stability. Based on simulation and engineering experience, the value interval of the thickness of end face is 15–22mm, and the value interval of chamfer angle is 25°–60°.

4.2. Compression Distance of the Rubber Packer

The compression distance of the rubber packer can be determined according to the product of the relative axial deformation and the length of the rubber packer. According to reference [20], the relative axial deformation of the rubber packer, ${\epsilon }_z$, can be obtained.

$${\epsilon }_{\mathrm{z}}=\frac{2R_{ti}\left(R_{ti}-R_{ro}\right)}{\left(R_{ti}^2-R_{ri}^2\right)}=0.0819,$$

(11)

where $R_{ti}$ is the inner radius of the casing wall; $R_{ro}$ is the outer radius of the rubber packer; $R_{ri}$ is the inner radius of the rubber packer. Therefore, the axial distance for compressing the rubber packer, $\Delta h_1$, is

$$\Delta h_1=H{\epsilon }_z=4.095,$$

(12)

where $H$ is the height of rubber packer. Figure 4a shows the simulation when the compression distance of the rubber packer is 4.095 mm.

The simulation results show that, there is still a large gap between the edge of the rubber packer and the casing, and there is no extrusion of the rubber packer, that is shoulder extrusion, the rubber packer can still be further compressed. Therefore, the rubber packer cannot achieve its most ideal sealing state by compressing with theoretical calculation values.

The main reason for the error between the simulation results and the theoretical calculation values lies in the default shape of rubber packer in the theoretical calculation is a regular rectangle. In engineering applications, however, both end faces of the rubber packer are designed with chamfers in order to prevent shoulder extrusion when compressed.

Therefore, in order to simulate the situation in engineering application, the compression distance of the rubber packer from the initial state to the final sealing state should be slightly larger than the theoretical calculation value. In this paper, a parameter called supplementary distance, $\Delta h_2$, is introduced, so as to compensate the error between the compression distance of the rubber packer applied in the engineering and that in the theoretical calculation on the premise of the response surface optimization method can be used.

$$\Delta h_2=H{\epsilon }_z^{\prime },$$

(13)

where ${{\epsilon }^{\prime }}_{\mathrm{z}}$ is the coefficient of supplementary distance.

Therefore, the calculation formula for the axial distance of the rubber packer, $\Delta h$, is

$$\Delta h=\Delta h_1+\Delta h_2,$$

(14)

Since the response surface optimization analysis is required in the following part of this paper, there will be chamfers of different sizes for combination and calculation. Therefore, the given axial distance of the rubber packer, $\Delta h$, should be as large as possible without grid distortion when calculating. To ensure that the response surface optimization method can be carried out, it is only necessary to find out the values of the structural parameters of the rubber packer that are most prone to grid distortion. When the structural parameter value does not lead to grid distortion under the compression distance, $\Delta h$, the chamfer of other sizes will also not cause grid distortion. As shown in

Table 4. Parameters most prone to calculation errors.

${\mathit{t}}_{\mathit{u}}\left(\mathbf{mm}\right)$	${\mathit{t}}_{\mathit{d}}\left(\mathbf{mm}\right)$	$\mathit{\alpha }$	$\mathit{\beta }$
22	22	25	25

, the parameter values most prone to grid distortion when using finite element simulation are the maximum value in the thickness interval and the minimum value in the chamfer interval of the end surface of the rubber packer.

Table 5. Finite element simulation results.

Distance (mm)	4.85	4.9	4.95	5
Whether a calculation error	No	No	No	Yes
Whether a shoulder extrusion	Slightly	Slightly	Seriously	-
$\Delta h_2$	0.755	0.805	0.855	0.905
${{\epsilon }^{\prime }}_{\mathrm{z}}$	0.0151	0.0161	0.0171	0.0181

can be obtained from finite element simulation. When the compression distance is 4.95 mm, there is a slight shoulder extrusion, which is acceptable and no computational error occurred. Therefore, the maximum compression distance, $\Delta h$, should be around 4.9 mm. To ensure the optimization can be carried out, in this paper, the maximum compression distance, $\Delta h$, is slightly reduced and the coefficient of the supplementary distance is rounded, ${\epsilon }_z^{\prime }$ = 0.016, $\Delta h_2=0.016H$ = 0.8 mm. Finally, $\Delta h$= 4.895 can be obtained. The simulation result at this time is the closest to the real situation.

Figure 4b is the diagram of simulation when the rubber packer is compressed by 4.895 mm, and Figure 4c,d are the diagrams of equivalent stress comparison of the inner side of the casing wall along path 1 to 2 under the compression distance of 4.095 mm and 4.895 mm, respectively. It can be seen that the optimized axial distance, $\Delta h_1$, is 19.54% higher than the original theoretical calculation value, $\Delta h_1$, resulting in a higher equivalent stress on the inner side of the rubber packer after compression distance optimization than that before optimization, which can seal higher pipe pressure.

4.3. Conditions under Which Shoulder Extrusion Does Not Occur

In the calculation process using finite element software combined with multi-objective genetic algorithm, the best candidate points are selected based on the value interval of parameters. However, by use of the structural parameter values of the candidate points during the finite element simulation, shoulder extrusion may occur. In order to prevent this from occurring in candidate points, conditions to prevent shoulder extrusion should be added to the optimization process.

As shown in Figure 5, ${\epsilon }_{tu}$ is the maximum equivalent strain at the upper end face of the rubber packer after compression; ${\epsilon }_{\alpha }$ is the maximum equivalent strain of chamfer at the upper end of the rubber packer after compression; ${\epsilon }_{td}$ is the maximum equivalent strain at the lower end face of the rubber packer after compression; ${\epsilon }_{\beta }$ is the maximum equivalent strain of chamfer at the lower end of the rubber packer after compression.

Figure 6 reveals the finite element simulation of conventional rubber packer structure carried out under different compression distance, and then the relation between four equivalent strain parameters, ${\epsilon }_{tu}$, ${\epsilon }_{\alpha }$, ${\epsilon }_{td}$, ${\epsilon }_{\beta }$ and the shoulder extrusion can be obtained.

In Figure 6a,b, when the compression distance of the rubber packer is 5.1 mm and 5.2 mm, there is no shoulder extrusion at the upper or lower ends after the compression, meanwhile, in Figure 6c,d, at the tenth second, ${\epsilon }_{tu}\ge {\epsilon }_{\alpha }$, ${\epsilon }_{td}\ge {\epsilon }_{\beta }$; in Figure 6e, when the compression distance of the rubber packer is 5.3 mm, a slight shoulder extrusion occurs at the upper end and no shoulder extrusion occurs at the lower end after the compression, meanwhile, in Figure 6g, at the tenth second, ${\epsilon }_{tu}\le {\epsilon }_{\alpha }$ while ${\epsilon }_{td}\ge {\epsilon }_{\beta }$; in Figure 6f, when the compression distance of the rubber packer is 5.4 mm, shoulder extrusion occurs at the upper end and slight shoulder extrusion occurs at the lower end after the compression, meanwhile, in Figure 6h, at the tenth second, ${\epsilon }_{tu}\le {\epsilon }_{\alpha }$ and ${\epsilon }_{td}\le {\epsilon }_{\beta }$; in Figure 6i,j, when the compression distance of the rubber packer exceeds 5.4 mm, the shoulder extrusion is gradually obvious after compression, at this point, ${\epsilon }_{tu}\ll {\epsilon }_{\alpha }$, ${\epsilon }_{td}\ll {\epsilon }_{\beta }$.

Figure 6. Diagram of comparison of compression distance and maximum equivalent strain. (a) 5.1 mm compression distance; (b) 5.2 mm compression distance; (c) Equivalent strain under 5.1 mm compression distance; (d) Equivalent strain under 5.2 mm compression distance; (e) 5.3 mm compression distance; (f) 5.4 mm compression distance; (g) Equivalent strain under 5.3 mm compression distance; (h) Equivalent strain under 5.4 mm compression distance; (i) 5.5 mm compression distance; (j) 5.6 mm compression distance; (k) Equivalent strain under 5.5 mm compression distance; (l) Equivalent strain under 5.6 mm compression distance.

Figure 6. Diagram of comparison of compression distance and maximum equivalent strain. (a) 5.1 mm compression distance; (b) 5.2 mm compression distance; (c) Equivalent strain under 5.1 mm compression distance; (d) Equivalent strain under 5.2 mm compression distance; (e) 5.3 mm compression distance; (f) 5.4 mm compression distance; (g) Equivalent strain under 5.3 mm compression distance; (h) Equivalent strain under 5.4 mm compression distance; (i) 5.5 mm compression distance; (j) 5.6 mm compression distance; (k) Equivalent strain under 5.5 mm compression distance; (l) Equivalent strain under 5.6 mm compression distance.

It can therefore be concluded that: at the end of the compression distance of the rubber packer, when ${\epsilon }_{tu}-{\epsilon }_{\alpha }\ge 0$, no shoulder extrusion would occur at the upper end of the rubber packer; when ${\epsilon }_{td}-{\epsilon }_{\beta }\ge 0$, no shoulder extrusion would occur at the lower end of the rubber packer.

In this paper, the anti-shoulder extrusion variables ${\epsilon }_1$, ${\epsilon }_2$ are introduced to prevent shoulder extrusion in the optimization process. The anti-shoulder extrusion variables, ${\epsilon }_1$, ${\epsilon }_2$ are defined as maximum equivalent strain value of the end face thickness minus the maximum equivalent strain value of the chamfer after compression.

$$\left\{\begin{array}{l}{\epsilon }_1={\epsilon }_{tu}-{\epsilon }_{\alpha }\ge 0\\ {\epsilon }_2={\epsilon }_{td}-{\epsilon }_{\beta }\ge 0\end{array}\right.,$$

(15)

where ${\epsilon }_1$ is the anti-shoulder extrusion constraint variable at the upper end of the rubber packer; ${\epsilon }_2$ is the anti-shoulder extrusion constraint variable at the lower end of the rubber packer.

4.4. Response Analysis and Optimization Result

Table 6 shows the objectives and constraints of optimization. As shown in Figure 7, in this paper, the equivalent stress on the inner side of the casing wall from path 1 to 2 can be viewed as an index to measure the sealing performance of the rubber packer. The higher the maximum equivalent stress and the larger the range, the better the sealing performance of the rubber packer is represented. The maximum equivalent stress of the path from 1 to 2 is greater than that before optimization. Meanwhile, the maximum equivalent stress is increased by 100% and rounded to the target of maximum equivalent stress optimization; the greater the axial force used to compress rubber packer, the better. However, due to the limitation of working conditions, the maximum axial force can be provided by the hydraulic system is 300 KN; there is no shoulder extrusion when compressing the rubber packer.

Table 6. The objectives and constraints of optimization.

Name	Type	Constraints
Maximum equivalent stress along path 1 to 2 ${\sigma }_{\mathrm{c}}$(MPa)	Target function	$3\le {\sigma }_{\mathrm{c}}\le 7$
Axial force $F$(N)	Target function	$1.3\times {10}^5\le F\le 3\times {10}^5$
Anti-shoulder extrusion variable ${\epsilon }_1$(mm/mm)	Constraint variable	$\ge 0$
Anti-shoulder extrusion variable ${\epsilon }_2$(mm/mm)	Constraint variable	$\ge 0$

Table 6. The objectives and constraints of optimization.

Name	Type	Constraints
Maximum equivalent stress along path 1 to 2 ${\sigma }_{\mathrm{c}}$(MPa)	Target function	$3\le {\sigma }_{\mathrm{c}}\le 7$
Axial force $F$(N)	Target function	$1.3\times {10}^5\le F\le 3\times {10}^5$
Anti-shoulder extrusion variable ${\epsilon }_1$(mm/mm)	Constraint variable	$\ge 0$
Anti-shoulder extrusion variable ${\epsilon }_2$(mm/mm)	Constraint variable	$\ge 0$

Figure 7. Diagram of target function.

Figure 7. Diagram of target function.

When performing response surface optimization calculation, the following assumptions were made in order to simplify the calculation process and, at the same time, to ignore factors that have a small effect on the deformation of the rubber packer: (1) Ignore the effect of the end face thickness and chamfer of the rubber packer on the compression distance within the value interval; (2) neglect the effect of gravity on the compression process of the rubber packer; (3) the center of the rubber packer always lies on the central axis when the compressed rubber packer is deformed; (4) the external seawater pressure and the oil and gas pressure inside the pipe are out of consideration.

The relationship between maximum equivalent stress, ${\sigma }_{\mathrm{c}}$, from path 1 to 2 and the four structural parameters to be optimized is shown below to demonstrate that the method of conducting independent analysis of individual structural parameters and deriving the optimal value interval is not reliable. From Figure 8a, when $t_u=18.5$, $t_d=18.5$, $\alpha =42.5$, $\beta =42.5$, the end thickness, $t_u$, $t_d$, and the chamfer angle, $\beta$ are approximately first-order linearly related to the maximum equivalent stress ${\sigma }_{\mathrm{c}}$ along the path 1 to 2. As shown in Figure 8b, when $t_u=18.5$, $t_d=15$, $\alpha =42.5$, $\beta =42.5$, the maximum equivalent stress, ${\sigma }_{\mathrm{c}}$, along the path from 1 to 2 varies drastically in proportion to the thickness of the upper end, $t_u$, and the chamfer angle of the upper end, $\alpha$.

Therefore, when only one structural parameter of the rubber packer is modified, it is appropriate to take the value according to the relation between maximum equivalent stress and a single structural parameter. However, the structural parameters would interact with each other when modifying multiple structural parameters, and the optimal size for the fit may not be the optimal value under a single structural parameter.

Therefore, the response surface optimization method combined with the MOGA algorithm are adopted in this paper so as to optimize the analysis of the numerical model.

Table 7. Combination of optimal solutions for four parameters.

Parameter	${\mathit{t}}_{\mathit{u}}\left(\mathbf{mm}\right)$	${\mathit{t}}_{\mathit{d}}\left(\mathbf{mm}\right)$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{\sigma }}_{\mathbf{c}}\left(\mathbf{MPa}\right)$	$\mathit{F}$$(\times {10}^5\mathbf{N})$	${\mathit{\epsilon }}_1$$(\times {10}^{-5})$	${\mathit{\epsilon }}_2$$(\times {10}^{-5})$
Candidate point 1	21.011	21.952	55.374	51.607	6.513	2.9813	29.87	177.57
Candidate point 2	20.952	21.951	58.38	49.17	6.3195	2.8564	297.37	23.066
Candidate point 3	20.953	21.969	56.483	51.394	6.2964	2.8272	87.349	224.92
Candidate point 4	20.945	21.963	55.515	51.478	6.2805	2.8334	424.22	125.05
Candidate point 5	20.973	21.92	55.319	51.471	6.2608	2.8395	302.86	106.55

shows the five best candidates obtained by using the optimal solution in the response surface optimization method. The optimization objectives meet the requirements of the preset objectives; the maximum equivalent stress along the path 1 to 2 and the axial force are less than the maximum axial force that can be provided, and no shoulder extrusion occurs during the compression process.

In addition, it can be seen that the structural parameters of the five candidate points are all $t_u<t_d$, $\alpha >\beta$, and the final cross-sectional shape of the rubber packer with small thickness and large angle at the upper end and large thickness and small angle at the lower end can be obtained.

The reasons for obtaining this cross-sectional shape are: (1) In the process of the rubber packer being compressed, the upper end of the packer is more prone to shoulder extrusion than the lower end due to the action of friction on the inside of the casing wall, therefore, the thickness of the upper end of the rubber packer is smaller than that of the lower end, while the angle of the upper end is larger than that of the lower end; (2) the structure with large thickness and small angle at the lower end can ensure that the axial force required to compress the rubber packer can be enhanced under the same compression distance, which will result in greater contact stress between the rubber packer and the casing wall contact surface, so that a better sealing performance can be achieved.

Select candidate point 1 as the best reference size, round the value of parameter, and finally set: $t_u$ = 21 mm, $t_d$ = 22 mm, $\alpha$ = 55°, $\beta$ = 52°.

4.5. Performance Comparison before and after Optimization

The dimensions of the rubber packer before optimization are $t_u$ = 20, $t_d$ = 20, $\alpha$ = 45°, $\beta$ = 45°, and the dimensions after optimization are $t_u$ = 21 mm, $t_d$ = 22 mm, $\alpha$ = 55°, $\beta$ = 52°. In the following part, the models of the rubber packer structures with two sets of dimensions are established and simulated for comparative analysis.

4.5.1. Comparison of Deformation

As can be seen in Figure 9a,b, the rubber packer before optimization still has a certain gap at the upper and lower ends, while the rubber packer after optimization has a better compression state under the same compression distance and there is no shoulder extrusion at the upper end or lower end.

At this point, Equation (16) can be obtained according to Figure 10.

$$\left\{\begin{array}{l}{\epsilon }_1={\epsilon }_{tu}-{\epsilon }_{\alpha }=0.01453\\ {\epsilon }_2={\epsilon }_{td}-{\epsilon }_{\beta }=0.01688\end{array}\right.,$$

(16)

According to Equation (16), it can be found that the rubber packer is in the state before a shoulder extrusion will occur, and the reason for not reaching a more desirable critical state of shoulder extrusion lies in the values of the structural parameters obtained after the optimization are rounded off in this paper. Thus, the most obvious finding to emerge from this part is that, the anti-shoulder extrusion variable introduced in this paper can reflect the deformation state of the rubber packer effectively.

In addition, it is evident from the partial enlargements C and D of Figure 9b that the rubber packer after optimization is approximately symmetrical at the upper and lower ends of the packer at the end of the compression distance. As for the rubber packer before optimization, the gap at the upper end of the rubber packer is smaller and the gap at the lower end is larger after compression, which means that if the rubber packer is further compressed, shoulder extrusion will occur at the upper end of the rubber packer, while the lower end is still in a state that has not reached the optimal compression state. It can also be found from the partial enlargement A in Figure 6e that a slight shoulder extrusion occurs at the upper end of the rubber packer while no shoulder extrusion occurs at the lower end. It is obvious from the partial enlargement B and C in Figure 6f that the shoulder extrusion is more serious at the upper end than that at the lower end for the top-down symmetrical cross-sectional structure of the rubber packer.

These results have confirmed that the cross-sectional structure of the rubber packer with small thickness and large angle at the upper end and large thickness and small angle at the lower end finally obtained in this paper is better than the conventional top-down symmetrical rubber packer cross-sectional structure, which allows for greater sealing potential of the rubber packer.

4.5.2. Equivalent Stress Comparison along Path 1 to 2

As shown in Figure 11, the maximum equivalent stress of the inner side of the casing wall from path 1 to 2 is 3.2474 MPa before optimization and 5.7463 MPa after optimization, which is increased by 76.95%.

It can be found from Figure 11 that the maximum equivalent stress on the inner side of the casing wall only occurs at the upper end of the rubber packer, therefore, the upper end of the rubber packer plays a decisive role in the sealing performance.

In addition, it can be obtained from Figure 11c that the equivalent stress on the inner side of the casing wall only rise significantly when the rubber packer before optimization is between 20–30 mm along path 1 to 2; while the rubber packer after optimization shows a near-vertical rise when it is between 10–20 mm along path 1 to 2, and it can be observed that the slope of the rise is greater in the rubber packer after optimization than that before optimization. It can be concluded from the analysis that although the maximum equivalent stress along path 1 to 2 occurs at the upper end of the rubber packer, the structure of the lower end of the rubber packer, provides a lifting effect on the equivalent stress value on the inner side of the casing wall; therefore, it is equally important.

Therefore, the study above strengthens the idea that the optimized rubber packer with thick-lower end, small-angle, and top-down asymmetrical structure is more advantageous than the top-down symmetrical structure.

5. Experimental Validation

The rated working pressure of this connector is 2000 psi. One of the most crucial tests in API RP 17R, API SPEC 6A and API SPEC 17D for connector performance testing is hydrostatic pressure test. Therefore, this paper conducts hydrostatic pressure test to compare the sealing performance of the rubber packer before and after optimization.

According to the requirements for hydrostatic pressure test in API, the hydrostatic pressure test pressure shall be of 1.5 times the rated working pressure. The acceptance criterion for hydrostatic pressure test shall be no visible leakage during the hold period, and an acceptable pressure settling rate should not exceed 5% of the test pressure per hour.

As shown in Figure 12a, a full-scale test device is designed for hydrostatic pressure test. The test device mainly consists of the actuating ring, casing, central pipe and pedestal. The press applies a downward displacement to the actuating ring so as to compress the rubber packer. At the end of the compression distance, pressure is applied to the test device through the hydraulic port on the outer side of the packer. Figure 12b shows the hydrostatic pressure test for the rubber packer before and after optimization.

The specific processes of hydrostatic pressure test are as follows:

(1)

Prepare the test device, hydraulic pump, press and other related equipment and clean them;

(2)

Place the test device on the press and fix it;

(3)

Start the press, set the axial distance to 4.9 mm, and start pressing after compression distance is finished;

(4)

When the pressure gauge value reaches 5 MPa, 10 MPa, 15 MPa and 21 MPa, stop pressing, and hold the pressure for 15 min after the pressure is stable, at the end of the pressure-holding period, check whether there is leakage in the test device and record the pressure gauge indication;

(5)

Decompression step by step and disassembly of the test device.

Figure 13 shows the curve of pressure variation with time in the hydrostatic pressure test. It can be seen from the partial enlargement A that the pressure settling of the rubber packer before optimization is relatively large within 58–72 min. The pressure dropped by 0.6 MPa during the hold period and the pressure settling rate exceeded 5% of the test pressure per hour. Finally, the pressure is maintained at 16.4 MPa; while the pressure settling rate of the optimized packer is within the range allowed by API standard, and finally the pressure is maintained at 20.6 MPa.

The results of two hydrostatic tests showed that the optimized rubber packer can seal higher pipe pressure under the same compression distance, and it met the API requirements for hydrostatic pressure test.

6. Conclusions

In this paper, based on the Mooney–Rivlin constitutive model and the finite element software, the influence of structural parameters of the upper and lower end surfaces of the subsea retractable connector rubber packer on its sealing performance has been studied, and the experimental comparison has been conducted. Therefore, obvious findings to emerge from this study are as follows:

(1)

The compression distance of the rubber packer was optimized on the basis of the structural parameters of the rubber packer end face could be optimized by use of the finite element response surface optimization method. The optimized compression distance, $\Delta h$, is 19.54% higher than the original theoretical calculated value, $\Delta h_1$, resulting in a higher equivalent stress along compression path and a better sealing performance of the rubber packer can be realized.

(2)

In the finite element simulation, the anti-shoulder extrusion variable, $\epsilon$, proposed in this paper can effectively reflect whether the shoulder extrusion occurs on the rubber packer during the compression. By means of introducing the anti-shoulder extrusion variable as constraints in the optimization process, the structural parameters of the rubber packer that lead to shoulder extrusion can be avoided in an effective way.

(3)

Under the same compression distance, the optimized rubber packer could meet the API requirements for the connector design validation, and it enjoys a better sealing performance. It helps to illustrate that the rubber packer cross-sectional structure with small thickness and large angle at the upper end and large thickness and small angle at the lower end obtained by use of the response surface optimization method of finite element software is more advantageous than the conventional one in terms of sealing performance. Meanwhile, this new structure allows similar deformation of the upper and lower ends of the compressed rubber packer, avoiding the situation where the upper end of the rubber packer has a shoulder extrusion while the lower end is not fully compressed.