A New Method for Calculating the Setting Stress of Downhole Packer’s Inner Sleeve

Abstract

There is no analytical solution to the stress, strain and displacement changes in the inner sleeve of the downhole packer during service. In this paper, the inner sleeve structure is simplified based on the shell theory model, and the geometric equation and physical equation suitable for the inner-sleeve structure are established based on the control differential equation of the cylindrical shell derived by Flügge. Finally, the analytical solution calculation program of the radial displacement, and strain and stress value of each node of the cylindrical shell under the external load condition is compiled by using MATLAB R2024a software. The analytical solution is compared with the numerical solution of each parameter under the same conditions, and the root mean square error between the numerical solution and the analytical solution is evaluated. The results show that the analytical formulas of the stress, strain and displacement of the inner sleeve structure of the downhole packer established in this paper can accurately obtain the above parameters. The root mean square errors between the analytical formulas and the numerical solutions are 0.083, 0.074 and 0.086, indicating that the fitting degree between the two is good, which verifies the effectiveness of the theoretical model based on the shell to describe the stress state of the inner sleeve. The model also accurately reflects the partial stress and strain law of the inner sleeve of the downhole packer to a certain extent. This study provides theoretical support for the design optimization of the inner sleeve of a pipeline packer, and also provides some guidance for the study of the stress state of its inner sleeve.

1. Introduction

Pipeline packers, as critical equipment widely used in oil and natural gas pipeline systems, are primarily designed to isolate specific pipeline segments for maintenance and operational purposes. The inner sleeve, serving as the core load-bearing component of a pipeline packer, directly determines the performance and service life through its mechanical state during operation. To ensure the stable and reliable operation of the inner sleeve under complex working conditions, it is essential to conduct an in-depth analysis and research on its mechanical behavior. This includes comprehensive considerations of the sleeve material’s mechanical properties, structural design and manufacturing processes. Rational stress analysis and design optimization can enhance the packer’s sealing performance and durability while reducing maintenance frequency and replacement costs, thereby improving economic efficiency.

The current domestic research on downhole packer structures predominantly focuses on components such as sealing rings and central tubes. For instance, Li Xingbiao et al. investigated the sealing performance of packer inner sleeves through experimental analyses, examining the influence of embedded spring rings on sealing capacity [1]. Xu Ruiqi explored the erosion mechanisms of particles in the central tube of a liquid–solid two-phase flow packer, identifying patterns of erosion-induced failures in specific packer configurations [2]. The existing studies on the stress analysis of inner-sleeve structures primarily rely on numerical simulations to obtain stress and strain distributions under specified external loads, with no analytical solutions reported. Geometric analysis of the inner sleeve reveals that its cylindrical shell structure is subjected to uniformly distributed external loads across the keyway region of its outer surface during service. Consequently, the mechanical behavior of the inner sleeve can be analyzed using shell mechanics theories. Significant progress has been made in related mechanical models: Wissler [3] analyzed the geometric characteristics of annular shells and derived fundamental mechanical principles; Flügge [4] established governing differential equations for cylindrical shell displacements under the assumption that shell thickness is negligible compared to curvature radius and other dimensions, thereby ignoring through-thickness stresses—a foundational theory for thin-shell structural analysis; and Parkus [5] formulated equations for cylindrical bodies with arbitrary cross-sections. Additionally, Donnell [6] and Dishinger [7] contributed simplified formulations for cylindrical shell mechanics to enhance practical applicability. These studies collectively provide theoretical references for deriving the geometric and physical equations governing the inner-sleeve structure in this work. Considering the simplified cylindrical shell geometry of the inner sleeve and computational efficiency, this study adopts Flügge’s governing differential equations for cylindrical shells [8] as the theoretical basis to derive tailored geometric and physical equations for the inner-sleeve structure.

In summary, currently, there is no analytical solution for the inner sleeve of pipeline packers, which also leads to the fact that engineers do not have accurate reference values when designing the inner sleeve, and they often can only judge by the rule of thumb, which also causes great inconvenience to the design work. Therefore, in this paper, to address this problem, the theoretical analysis is carried out firstly, on the basis of Flügge’s control differential equations for cylindrical shells [9]; the geometrical equations and physical equations applicable to the inner-sleeve structure are further derived; and the analytical formulas for the stress–strain relationship under the state of compression of the inner sleeve are obtained and verified. At the same time, MATLAB was used to prepare the relevant solution program, and the analytical solutions, such as the stress–strain-displacement of the inner sleeve at the point, can be calculated by setting the material properties [10] and inputting the pressure value at the coordinate point. Subsequently, a three-dimensional model of the inner sleeve and the block was established on three-dimensional simulation software, and the numerical relationship [10] between the stress–strain values of the inner sleeve in the pressurized state was calculated by means of a numerical simulation. Finally, the accuracy of the analytical solution is verified by numerical simulation, and the results show that the root mean square errors between the analytical solution and the numerical solution calculated by the method of this paper are 0.083, 0.074, and 0.086, which indicates that the formulas derived in this paper are able to more accurately describe the stress–strain relationship under the working condition of the inner sleeve [11]. The research of this paper provides a reference basis for the yield strength of the inner sleeve of the packer when it is seated, and also provides important theoretical support for the design and optimization of the inner sleeve of the packer. Meanwhile, this paper simplifies the inner sleeve of the pipeline packer into a cylindrical shell model in the process of the research, which also provides a new method for the optimization and design of other equipment components with similar in the field of petroleum equipment [12].

2. Theoretical Analysis

2.1. Theoretical Model of Pipeline Packers

Taking a commonly used downhole sleeve as an example, its three-dimensional model is illustrated in Figure 1. Unlike standard cylindrical shell structures, the inner sleeve is distinguished by six circumferentially symmetric kidney-shaped keyways distributed across its outer surface. In practical engineering applications, these keyways are engaged with corresponding stoppers and subjected to pressures perpendicular to their surfaces [13]. To simplify the analysis, the keyway structures were removed to form a standard cylindrical shell, enabling the application of Flügge’s governing differential equations for cylindrical shells to derive its stress–strain relationships [14]. The dimensional parameters of the inner sleeve before and after simplification are summarized in

Table 1. Structural dimensions of the inner sleeve.

Configuration	Inner Diameter (mm)	Outer Diameter (mm)	Length (mm)	Wall Thickness (mm)	Keyway Dimensions (mm)
Before simplification	220	253	1500	16.5	Oblong hole Φ18 × 80
After simplification	220	253	1500	16.5	N/A

. Research manuscripts reporting large datasets that are deposited in a publicly available database should specify where the data have been deposited and provide the relevant accession numbers.

The inner-sleeve structure exhibits symmetry, and the external loads acting on it during operation are uniformly distributed across the six keyways on its outer surface. This symmetry permits further simplification in subsequent numerical simulations [1]. Taking the inner sleeve axis as the rotational axis and the plane containing the axis and the long-axis center point of a keyway as the reference plane, a 1/6 sector encompassing one complete keyway structure was extracted by rotating 30° bilaterally from the reference plane. The two symmetric cross-sections relative to the reference plane formed a 60° angle and were treated as fixed ends during the analysis, while both ends of the inner sleeve served as simply supported boundaries. In addition, because the inner sleeve is in the conventional seated sealing case, its stress level is low, the material is in the elastic stage and, at this time, does not take into account the plastic strain and creep effects, you can obtain a more accurate result, and can meet the engineering design and analysis of the accuracy requirements.

2.2. Derivation of Analytical Stress–Strain Formulas for the Inner Sleeve

This paper mainly researches the problem of the yield strength of the inner sleeve of a pipe packer without an analytical solution, and the final purpose is to provide a reference for the yield strength of the inner sleeve of the packer when it is seated, due to the key groove structure of the inner sleeve belonging to a large structural mutation on the surface of the inner sleeve, which will greatly increase the complexity of the computation and the error of the simulation results, taking into consideration the accuracy of the solution as well as the actual demand, so the inner sleeve of the packer is simplified to be a cylindrical shell. Then, any point on the inner sleeve meets the control differential equation derived by Flügge, which is in the form of a set of fourth-order linear partial differential equations:

$$\left.\begin{array}{l}({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}}+{\displaystyle \frac{1-\mu }{2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}})u+{\displaystyle \frac{1+\mu }{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}v}{\mathsf{\partial }\alpha \mathsf{\partial }\beta }}+{\displaystyle \frac{\mu }{R}}{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }\alpha }}=0\\ {\displaystyle \frac{1+\mu }{2}}{\displaystyle \frac{{\mathsf{\partial }}^{2}u}{\mathsf{\partial }\alpha \mathsf{\partial }\beta }}+({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}}+{\displaystyle \frac{1-\mu }{2}}{\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}})v+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }\beta }}=0\\ {\displaystyle \frac{\mu }{R}}{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }\alpha }}+{\displaystyle \frac{1}{R}}{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }\alpha }}+{\displaystyle \frac{w}{R^2}}+{\displaystyle \frac{{\delta }^2}{12}}{\nabla }^{4}w={\displaystyle \frac{1-{\mu }^2}{E\delta }}{q}_3\end{array}\right\}$$

(1)

In the equations, u, v and w represent the displacement functions in the three curvilinear coordinate directions; α, β and γ denote the corresponding curvilinear R corresponding to the radius from the origin to the mid-surface of the shell. The terms $q_i(i=1,2,3)$ correspond to the load distributions in the three coordinate directions. Specifically, q₁ = q₂ = 0, q₃ = q₃(α, β, γ).

A curvilinear coordinate system was established on the simplified shell structure, with the shell domain defined as $\mathrm{A}\in \{0\le \alpha \le a,0\le \beta \le b\}$, and its boundary expressed as $\mathsf{\partial }A=\{\alpha =0,a\cup \beta =0,b\}$, as illustrated in Figure 2.

To simplify the solution process of Equation (1), a displacement auxiliary function F(α, β, γ) was introduced. The displacement functions γ = 0 on the shell surface can be expressed as:

$$\left.\begin{array}{l}u={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\alpha }}({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}}-\mu {\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}})F\\ \nu =-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\beta }}({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}}+(2+\mu ){\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}})F\\ w=R{\nabla }^{4}F\end{array}\right\}$$

(2)

The first two equations in the system satisfy Equation (2), while the third equation holds provided that the following condition is met:

$${\nabla }^{8}F+{\displaystyle \frac{E\delta }{R^{2}D}}{\displaystyle \frac{{\mathsf{\partial }}^{4}F}{\mathsf{\partial }{\alpha }^4}}={\displaystyle \frac{q_3}{RD}}$$

(3)

Thus, by introducing the displacement auxiliary function, the solution of the three equations in Equation (1) was reduced to solving Equation (3). Substituting Equation (2) into the stress-displacement equations of the cylindrical shell yields:

$$\left.\begin{array}{c}{F}_{T_1}=E\delta {\displaystyle \frac{{\mathsf{\partial }}^{4}F}{\mathsf{\partial }{\alpha }^2\mathsf{\partial }{\beta }^2}}\hfill \\ F_{T_2}=E\delta {\displaystyle \frac{{\mathsf{\partial }}^{4}F}{\mathsf{\partial }{\alpha }^4}}\hfill \\ F_{T_{12}}=-E\delta {\displaystyle \frac{{\mathsf{\partial }}^{4}F}{\mathsf{\partial }{\alpha }^3\mathsf{\partial }\beta }}\hfill \\ M_1=-RD({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}}+\mu {\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}}){\nabla }^{4}F\hfill \\ M_2=-RD({\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\beta }^2}}+\mu {\displaystyle \frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{\alpha }^2}}){\nabla }^{4}F\hfill \\ F_{S_1}=-RD{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\alpha }}{\nabla }^{6}F\hfill \\ F_{S_2}=-RD{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\beta }}{\nabla }^{6}F\hfill \end{array}\right\}$$

(4)

In the equations, D represents the bending rigidity of the shell, M denotes moments acting on the shell in different directions and F corresponds to internal forces.

For the boundary conditions in Figure 2, the fixed edges are defined at $\beta =0, \beta =\mathrm{b}$ and $\alpha =0, \alpha =\mathrm{b}$, while the simply supported edges are located at α = 0, α = 0 and α = a, α = a. The latter boundary conditions can be rewritten as:

$$\begin{array}{l}{(F_{T_1},v,w,M_1)}_{\alpha =0}=0\\ {(F_{T_1},v,w,M_1)}_{\alpha =a}=0\end{array}$$

(5)

By applying Levy’s method, the displacement function F(α, β)F(α, β) under the two-dimensional conditions was expressed as a single trigonometric series:

$$F(\alpha ,\beta )={\displaystyle \sum _{m=1}^{\infty }{\Psi }_m(\beta )\sin \frac{m\pi \alpha }{a}}$$

(6)

The radial load on the shell was expanded as a Fourier sine series:

$$q_3={\displaystyle \frac{2}{a}}{\displaystyle \sum _{m=1}^{\infty }[{\displaystyle {\int }_0^a{q}_3(\alpha ,\beta )\sin \frac{m\pi \alpha }{a}d\alpha }]\sin \frac{m\pi \alpha }{a}}$$

(7)

During the service of the pipeline packer, the circumferentially arrayed keyways on the outer surface of the inner sleeve engage with corresponding stoppers, resulting in external loads that are theoretically localized to the keyway regions. This loading characteristic can be modeled using step functions; however, their introduction significantly complicates the solution process. To facilitate analytical treatment, this study represents the external load on the inner sleeve as a two-dimensional Gaussian distribution, expressed as:

$$q_3(\alpha ,\beta )={\displaystyle \frac{1}{(2\pi ){({\sigma }_1^2{\sigma }_2^2)}^{\frac{1}{2}}}}\exp \left\{-{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\alpha -\frac{a}{2}}{{\sigma }_1}}\right)}^2+{\left({\displaystyle \frac{\beta -\frac{b}{2}}{{\sigma }_2}}\right)}^2\right]\right\}$$

(8)

In the equation, ${\sigma }_1$ and ${\sigma }_2$ represent the covariance parameters of the distribution.

Substituting Equation (6) into Equation (3), the coefficient equations are governed by the following ordinary differential equation (ODE):

$${\nabla }^{8}F+{\displaystyle \frac{E\delta }{R^{2}D}}{\displaystyle \frac{{\mathsf{\partial }}^{4}F}{\mathsf{\partial }{\alpha }^4}}={\displaystyle \frac{q_3}{RD}}\left[{\left({\displaystyle \frac{d^2}{d{\beta }^2}}-{\lambda }_m^2\right)}^4+{\displaystyle \frac{E\delta }{R^{2}D}}{\lambda }_m^4\right]{\Psi }_m(\beta )={\displaystyle \frac{2}{RDa}}{\displaystyle {\int }_0^a{q}_3(\alpha ,\beta )\sin \lambda m\alpha d\alpha }$$

(9)

In the equation, ${\lambda }_m=m\pi /a$. The nonhomogeneous solution comprises a homogeneous general solution and a particular solution. The characteristic equation governing the homogeneous general solution is expressed as:

$${({\gamma }_m^2-{\lambda }_m^2)}^4+{\displaystyle \frac{E\delta }{R^{2}D}}{\lambda }_m^4=0$$

(10)

Let ${\Psi }_m^{*}(\beta )$ be defined as a particular solution satisfying this ordinary differential equation. The general solution of the equation is then expressed as:

$$\begin{array}{l}{\Psi }_m(\beta )={\Psi }_m^{*}(\beta )+C_{1m}\cosh a_m\beta \sin b_m\beta \\ +C_{2m}\cosh a_m\beta \sin b_m\beta +C_{3m}\cosh a_m\beta \sin b_m\beta \\ +C_{4m}\cosh a_m\beta \sin b_m\beta +C_{5m}\cosh a_m\beta \sin b_m\beta \\ +C_{6m}\cosh a_m\beta \sin b_m\beta +C_{7m}\cosh a_m\beta \sin b_m\beta \\ +C_{8m}\cosh a_m\beta \sin b_m\beta \end{array}$$

(11)

The eight constant terms C_1m to C_8m in the general solution can be determined by the boundary conditions of the fixed supports. The boundary conditions for the fixed supports are defined as:

$$\left.\begin{array}{c}{(u,v,w,{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }\beta }})}_{\beta =0}=0\\ {(u,v,w,{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }\beta }})}_{\beta =a}=0\end{array}\right\}$$

(12)

By substituting Equation (11) into the displacement function F(α, β) and applying the aforementioned boundary conditions as solution constraints, a system of eight equations governing the constant terms C_1m to C_8m was derived. Ultimately, all constant terms were determined through the numerical resolution of this system.

To derive analytical solutions for the stresses and strains in the shell, the mid-surface displacements u, v and w in Equation (2) were expressed using the displacement function F(α, β). These expressions were subsequently substituted into the stress-displacement equations to determine the internal force distributions. Consequently, the relationships between the stress, strain and displacements for the shell are formulated as:

$$\left.\begin{array}{c}{\sigma }_1={\displaystyle \frac{F_{T_1}}{\delta }}+{\displaystyle \frac{12M_1}{{\delta }^3}}\gamma \hfill \\ {\sigma }_2={\displaystyle \frac{F_{T_2}}{\delta }}+{\displaystyle \frac{12M_2}{{\delta }^3}}\gamma \hfill \\ {\tau }_{12}={\tau }_{21}={\displaystyle \frac{F_{T_{12}}}{\delta }}+{\displaystyle \frac{12M_{12}}{{\delta }^3}}\gamma \hfill \\ {\tau }_{13}={\displaystyle \frac{6F_{S_1}}{{\delta }^3}}({\displaystyle \frac{{\delta }^2}{4}}-{\gamma }^2)\hfill \\ {\tau }_{23}={\displaystyle \frac{6F_{S_2}}{{\delta }^3}}({\displaystyle \frac{{\delta }^2}{4}}-{\gamma }^2)\hfill \end{array}\right\}$$

(13)

$$\left.\begin{array}{ccc}{\epsilon }_1={\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }\alpha }}\hfill & {\epsilon }_2={\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }\beta }}+{\displaystyle \frac{w}{R}}\hfill & {\epsilon }_{12}={\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }\beta }}+{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }\alpha }}\hfill \\ {\chi }_1=-{\displaystyle \frac{{\mathsf{\partial }}^{2}w}{\mathsf{\partial }{\alpha }^2}}\hfill & {\chi }_2=-{\displaystyle \frac{{\mathsf{\partial }}^{2}w}{\mathsf{\partial }{\beta }^2}}\hfill & {\chi }_{12}=-{\displaystyle \frac{{\mathsf{\partial }}^{2}w}{\mathsf{\partial }\alpha \mathsf{\partial }\beta }}\hfill \\ e_1={\epsilon }_1+{\chi }_1\gamma \hfill & e_2={\epsilon }_2+{\chi }_2\gamma \hfill & e_{12}={\epsilon }_{12}+{\chi }_{12}\gamma \hfill \end{array}\right\}$$

(14)

In the equations, the general notation for all second-order tensors is expressed as $X_{ij}(i,j=1,2,3)$, where the face index $i$ denotes the surface normal to the i-axis, and the direction index $j$ represents the direction parallel to the j-axis. According to the reciprocity theorem of shear stresses and the isotropic material assumption, the tensor components remain invariant under the permutation of $i$ and $j$.

For the shell structure, $e_1,e_2$ and $e_3$ correspond to the in-plane strain components tangential to the curved surface, while ${\chi }_1,{\chi }_2$ and ${\chi }_3$ denote the twist curvatures, which characterize the variation in principal curvatures at points on the mid-surface along the two curvilinear coordinate axes.

2.3. Development of Stress–Strain Computational Program for Inner-Sleeve Structure

Based on the aforementioned derivations, a computational program was developed in MATLAB. This program requires only the input of the inner sleeve’s key structural dimensions, material parameters and the pressure applied by the stoppers to calculate the analytical solutions for radial displacements at specified positions, as illustrated in Figure 3. To further obtain stress and strain values at these positions, the computed displacements need only be substituted into Equation (14).

Furthermore, the literature review confirms that the inner sleeve of pipeline packers [2] is predominantly subjected to radial loads from stopper clamping during operation, with other loads being negligible [15]. Consequently, this study focuses on radial loads as the primary observational parameter. Notably, the straight path traversing the load center on the inner sleeve’s pressurized surface represents the region of maximum loading, yielding the most representative data (see Figure 3). To validate the theoretical derivations and program accuracy, radial displacement, stress and strain distribution curves along this path were extracted and compared with the numerical simulation results in the subsequent sections.

By inputting the parameters from Table 1 into the program and defining the pressure applied by the stoppers as 194.23 MPa (based on practical engineering loads), the normal displacement variations along the specified path of the inner sleeve were calculated, as shown in Figure 4. These displacement variations [16] were then substituted into Equation (14) to derive the theoretical stress and strain values at each node along the path, with the results presented in Figure 5. As mentioned in the previous section, the keyway structure on the surface of the inner sleeve belongs to a large structural mutation [17], and the data in this region have little reference significance, and will greatly increase the complexity of the calculation and the error of the results, so the data in this interval are defined as the invalid data interval, and the data in this region are also defined as invalidated in the other studies later on.

Figure 4 illustrates the nodal radial displacement distribution along the α-direction of the inner sleeve. Due to the structural discontinuity at the keyway edges on the outer surface of the inner sleeve—a feature absent in standard cylindrical shells—the analytical solutions exhibit significant deviations in this region, which is designated as an invalid data interval. The radial displacement distribution along the path shows a decaying trend from the loading center toward both ends. The peak radial displacement of 0.44035 mm occurs at the keyway edge of the inner sleeve. A similar pattern is observed for stress and strain distributions along the path, with their peak values also located at the same position: 316.33 MPa (stress) and 0.00213 (strain).

Subsequently, a numerical model of the inner sleeve with identical specifications was established in finite element analysis ABAQUS 2024 software. The displacement, stress and strain distributions under identical external loading conditions were analyzed [18]. Data along the same critical path were extracted to validate the accuracy of the theoretical solution formulas proposed in this study.

3. Numerical Simulation Analysis of Mechanical Behavior in Packer Inner Sleeve

3.1. Geometric Model and Material Properties of the Inner Sleeve

ABAQUS 2024 software is widely used in many engineering simulation fields today. With its excellent performance and wide applicability, it shows an irreplaceable, important position in static simulation and analysis, and the simulation of the pipeline packer’s [17] inner-sleeve sitting sealing process investigated in this paper belongs to one kind of static simulation. So, this study adopted this software for the simulation. Firstly, according to the data collected in the previous paper, modeling was carried out on three-dimensional software. To improve computational efficiency, a one-sixth sector of the inner sleeve was selected for modeling. The geometric models of the inner sleeve and stoppers are shown in Figure 6, with their material properties [19] summarized in

Table 2. Material parameters of components.

Component	Material Type	Yield Strength (MPa)	Tensile Strength (MPa)	Poisson’s Ratio	Young’s Modulus (GPa)
Inner sleeve	2A12 aluminum alloy	361	477	0.3	67
Stoppers	35CrMo steel	758	980	0.3	206

.

3.2. Simulation Parameter Configuration

In practical operations, the upper and lower end faces of the packer inner sleeve [19] are connected to male and female joints. These joints, along with the retaining rings, exhibit elastoplastic deformation under internal pressure. Therefore, their stress–strain relationships must be defined in the numerical simulations [20]. The test male joints, test female joints and retaining ring materials follow a bilinear elastic–plastic constitutive model. Based on the literature review, their tangent moduli were determined as 21 GPa, 21 GPa and 20.9 GPa. During actual loading, both the stoppers and inner sleeve undergo plastic deformation, necessitating the specification of their plastic material properties. The tangent moduli for 35CrMo steel and 2A12 aluminum alloy were identified as 20.6 GPa and 0.67 GPa, respectively, through material property databases. A bilinear elastic–plastic model was configured for both materials using these parameters.

Under operational conditions, the inner sleeve is subjected to an internal pressure of 15 MPa, resulting in an external load of:

$$F=P\cdot S=15\times \pi {\left({\displaystyle \frac{253}{2}}\right)}^2=753.705\mathrm{kN}$$

(15)

The theoretical force exerted by a single stopper is calculated as 125.6175 kN. Accounting for practical factors, such as liquid column weight and dynamic pressure effects, this study defined the applied pressure from a single stopper on the inner sleeve as 150 kN under the operational internal pressure of 15 MPa.

The assembly relationship between the block and the inner sleeve is shown in Figure 7, and a generalized static analysis step was selected to turn on the geometric nonlinearity and set the appropriate initial incremental step and minimum incremental step [17]. According to the actual working conditions in the interaction, the inner sleeve is a fixed constraint, and at the same time the block and the inner sleeve of the contact settings.

A pressure of 150 kN was applied to the upper surface of the stoppers, and both sides of the inner sleeve were defined as fully constrained, as illustrated in Figure 8.

3.3. Mesh Independence Analysis

The mesh size has a great influence on the calculation results [21], so the influence of the mesh size of the solution accuracy needs to be analyzed before the formal calculation. The value range of mesh size is 0.2~2 mm, and the step size is 0.2 mm. The displacement change in a point on the outer surface of the model was calculated under the abovementioned mesh size, as shown in Figure 9. It can be seen that the displacement of the node tends to stabilize when the mesh size is less than 0.8 mm, so in order to reduce the unnecessary calculation volume while ensuring the calculation accuracy, the mesh size of the median (0.6 mm) was selected in the subsequent analysis.

4. Results and Discussion

4.1. Analysis of Simulation Results

This study focuses on a downhole packer inner sleeve, where the material strength of the stoppers significantly exceeds that of the inner sleeve. Consequently, the stoppers remain undamaged prior to the plastic failure of the inner sleeve. Therefore, only the stress, strain and displacement distributions of the inner sleeve are analyzed in subsequent discussions. Figure 10 presents the contact pressure contour of the inner sleeve. The maximum contact pressure in the keyway regions reaches 1374 MPa, far exceeding the material’s yield strength, indicating the onset of plastic deformation in the inner sleeve. Figure 11 displays the equivalent stress (Von Mises) contour, revealing that under operational loading, the equivalent stress distribution is symmetrical to the central plane along the axial direction. The highest stress concentration occurs at the arc-shaped ends of the keyways, reaching 443.1 MPa.

Figure 12 and Figure 13 show the plastic strain contour and displacement contour of the inner sleeve, respectively. The maximum plastic strain occurs at the center of the inner wall surface, measuring 4.25 × 10⁻³. Similarly, the peak plastic strain along the radial direction is observed at the same location, reaching 3.73 × 10⁻³. Notably, the maximum nodal displacement appears at the keyway center on the outer surface of the inner sleeve, with a radial displacement magnitude of 0.403 mm.

The radial displacement, stress and strain data along the path shown in Figure 3 were extracted, with the results presented in Figure 14. The “invalid data interval” in the figure corresponds to the keyway region. During the simulation [21], it was observed that this region exhibits significant structural discontinuity due to abrupt thickness variations caused by the keyway geometry, leading to substantial errors in both the analytical and simulated results. Consequently, the data from this interval were excluded as invalid. The distribution patterns of radial displacement, stress and strain along the path align closely with the analytical solutions presented earlier. The subsequent analysis focuses on assessing the goodness [22] of fit between the theoretical and numerical results.

4.2. Comparative Analysis of Analytical and Numerical Solutions

The numerical and analytical solutions of the radial displacement distribution on the inner set of the path are compared to evaluate the accuracy of the analytical formulation proposed in this paper. In this paper, the root mean square error (RMSE) is used as the evaluation index [23]. RMSE is a common statistical index used to measure the difference between the predicted values and the actual observed values, and it is more sensitive to the error results. Although it is affected by the extreme values, we have already defined the nullification of the data at the structural mutation, so the RMSE is chosen as the error evaluation index in this study. The formula of the RMSE is as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}$$

(16)

In the formula, n represents the number of data points, $y_i$ denotes the actual value (i.e., the numerical solution obtained in this study) of the i-th sample and ${\widehat{y}}_i$ corresponds to the predicted value (i.e., the theoretical solution calculated in this work).

A smaller RMSE value indicates a better agreement between the predicted and observed values. Typically, an RMSE below 0.2 signifies satisfactory fitting accuracy [24]. Figure 15 illustrates the goodness of fit between analytical and numerical solutions for radial displacement, stress and strain along the critical path. Using the numerical solutions as the benchmark, the RMSE values for displacement, stress and strain derived [25] from the proposed analytical formulas were calculated as 0.0770, 0.1428 and 0.0861, respectively. All values are below the 0.2 threshold, demonstrating excellent consistency between the analytical and numerical solutions [26].

5. Conclusions and Outlook

To address the lack of analytical solutions [27] for displacement, stress and strain in downhole packer inner sleeves during service, this study simplified the inner sleeve as a cylindrical shell structure in the shell theory [28]. Analytical formulas under compressive loading were derived based on Flügge’s [29] governing differential equations for cylindrical shells [30]. Comparative analysis with numerical simulations yielded the following conclusions:

(1)

According to Figure 16. The root mean square errors (RMSEs) between the analytical and numerical solutions for the three physical quantities are 0.083, 0.074 and 0.086. These results confirm that the proposed analytical formulas can accurately calculate the mechanical behavior of the inner sleeve under specified external loads.

(2)

Numerical simulations revealed that the maximum equivalent stress (Von Mises) in the inner sleeve during service reaches 466.5 MPa, exceeding the yield strength of the sleeve material.

(3)

The maximum radial displacement (0.40345 mm) and peak plastic strain (0.003727) of the inner sleeve both occur in the central region of the keyways, highlighting this area as the critical zone for structural integrity.

Although significant progress has been made in the force analysis of cylindrical shells, there is still room for further research and innovation, and future work will focus on developing simpler and more accurate computational methods for cylindrical shells with heterogeneous surfaces in various fields or utilizing advanced algorithms to make this type of research more efficient.