Correlation analysis of mechanical properties of dilated packer cylinder

2.1. Working principle

The working principle of the packer can be briefly described as follows: Under the action of axial load, the rubber cylinder constantly deforms and expands to seal the annulus between tubing and casing. In the process of deformation and expansion, a certain contact stress is formed between the rubber cylinder and the casing wall to ensure its sealing, to achieve the purpose of isolating the production layer, isolating the well fluid pressure and preventing the pressure and fluid interaction between different production layers.

2.2. The geometric model

Packer rubber cylinder, center pipe, casing, rubber cylinder seat and other parts are rotating body parts, and the load is axisymmetric distribution. In order to reduce calculation amount and save calculation time, a two-dimensional axisymmetric model is established for calculation, the model is shown in Fig 1. 35CrMo steel is used for casing, center pipe, rubber cylinder seat and other metal materials. See Table 1 for geometric dimensions and material parameters.

Fig. 1Schematic diagram of calculation model

2.3. Finite element model and cylinder constitutive model and parameters

Based on the grid division standard and grid control method, the calculation accuracy and calculation cost are considered comprehensively. The overall grid type is quadrilateral dominated grid, and the rubber cylinder is the main analysis element and has the characteristics of large deformation, so the rubber cylinder grid is appropriately refined. The mesh of other metal components is thicker than that of the cylinder. The total number of grids is 1090 and the number of points is 3724. Meshing is shown in Fig 2.

Fig. 2Meshing diagram

There are many studies on the constitutive model of rubber. Phenomenological theory assumes that the material has isotropy and incompressibility, and uses strain energy density function $W$ to describe the constitutive model:

where: $I_1$, $I_2$, $I_3$ represent the Cauchy-Green first, second, and third deformation tensor invariants, respectively.

Mooney-Rivlin model is a commonly used constitutive model, which can be applied to the mechanical behavior of most rubber materials, and its strain energy density function is:

where: $C_{10}$ and $C_{01}$ represent the material parameters of the Rivlin model.

When the material is incompressible, the Rivlin coefficient is related to the elastic modulus $E$ of the rubber as follows:

where: the ratio of $C_{01}$ to $C_{10}$ is a constant, ranging from 0.25 to 0.5.

The elastic modulus of the rubber cylinder is 16.95 MPa, and the ratio of $C_{01}$ to $C_{10}$ is 0.5. According to Eq. (3), it can be calculated that $C_{10}=$1.8743 MPa, $C_{01}=$0.93715 MPa.

2.4. Boundary conditions and loads

According to the working conditions, the boundary conditions are as follows :(1) The upper and lower ends of fixed casing and central pipe are fixed; (2) Fix the upper end of the rubber cylinder seat and the lower end of the rubber cylinder seat; (3) Binding constraints are imposed on the rubber cylinder seat and the central pipe; (4) Binding constraints are imposed on the upper and lower ends of the rubber cylinder with the upper and lower rubber cylinder seat respectively; (5) The rubber cylinder is in contact with the central pipe, the surface of the casing and the rubber cylinder seat, and the friction factor is 0.3. The normal contact is set as hard contact; (6) Pressure load is applied inside the rubber cylinder.

3.1. Stress analysis

The stress field distribution of the packer assembly under 20.0 MPa internal pressure was calculated, as shown in Fig. 3. It can be seen from the figure that the stress of metal component is significantly higher than that of rubber cylinder, and the stress level of upper and lower rubber cylinder is the highest, and the maximum Mises stress is 54.86 MPa. Generally, the yield strength of 35CrMo material is 835 MPa, and the safety factor is 15.22. Since this force is primarily static, achieving this factor of safety is generally considered safe.

Fig. 3Mises stress field distribution of packers

Fig. 4Mises stress field distribution of rubber cylinder

Fig. 4 shows the Mises stress field distribution of the rubber cylinder at the internal pressure of 20 MPa. It can be seen that the stress level at both ends of the middle segment of the rubber cylinder is the highest, up to 43.465 MPa. The stress distribution is more uniform in the middle part of the rubber cylinder, and the stress gradient is larger in the smaller areas at both ends of the rubber cylinder.

3.2. Strain analysis

When the internal pressure is 20 MPa and the packer is in the working state, the area with the largest deformation of the rubber cylinder appears at the upper and lower ends of the rubber cylinder. Fig 5 clearly reflects the position of the maximum strain with the locally enlarged shear strain distribution cloud map, and marked the position of the maximum strain element with the unit number, namely the No. 1073 unit at the upper end and the No. 2519 unit at the lower end. The characteristic of easy deformation of the rubber cylinder shoulder is verified by finite element analysis, which provides a theoretical basis for the design of the packer rubber cylinder structure.

Fig. 5Locally enlarged shear strain distribution cloud map

a)

b)

Fig. 6Variation of shear strain with internal pressure

a) Variation of shear strain with internal pressure of unit 1073

b) Variation of shear strain with internal pressure of unit 2519

Fig. 6(a) shows that the shear strain of unit 1073 increases monotonically with the increase of internal pressure, and its change curve shows obvious nonlinear characteristics on the whole. When the internal pressure is small, the shear strain increases rapidly with the increase of the internal pressure. When the internal pressure exceeds 5.0 MPa, the shear strain increases with the increase of internal pressure is obviously slowed down, but the shear strain increases linearly with the increase of internal pressure. When the internal pressure reaches 20 MPa, the engineering shear strain reaches 0.87545.

Fig. 6(b) shows the variation of engineering shear strain with internal pressure in unit 2519, which is similar to that in unit 1073. When the internal pressure reaches 20 MPa, the engineering shear strain of unit 2519 reaches 0.87602. It can be seen that the shear deformation degree of unit 1073 is slightly smaller than that of unit 2519.

Fig 7 shows the forces between the rubber cylinder and the casing under different pressures. When the pressure reaches 0.2 MPa, the force between the rubber cylinder and the casing exceeds 0.3 MPa, then the rubber cylinder starts to unseal, which can play an initial effect of sealing the annulus. However, this situation is only when the contact between the two parts is less, when the internal pressure reaches nearly 20 MPa. The contact part between them basically covers the whole middle section of the rubber cylinder, and the situation can be well sealed to achieve the setting effect.

Fig. 8 and 9 show that, different from shear strain, strain energy density increases significantly with the increase of internal pressure, which is nearly linear in general. When the internal pressure reaches 20.0 MPa, the strain energy densities of unit 1073 and unit 2519 reach 3.81 J/cm³ and 2.83 J/cm³, respectively, which are very high in terms of strain energy densities.

Fig. 7Contact stress diagram on rubber cylinder under different pressure

Fig. 8Variation of strain energy density of unit 1073 with internal pressure

Fig. 9Variation of strain energy density of unit 2519 with internal pressure