# Evaluation on the Seal Performance of SMP-Based Packers in Oil Wells

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## Abstract

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## 1. Introduction

_{g}) and compressed or stretched to reduce its thickness. This programmed shape is fixed after cooling well below T

_{g}. The SMP packer then runs in the well with the mandril. Finally, it is activated by a heater to expand and seal the wellbore at the specified position.

## 2. Experimental Characterization of the Polymer

_{g}) of 80 °C as reported by the manufacturer, accommodating the downhole condition. Figure 2 shows the samples that the shorter one (11.85 mm × 2.83 mm × 3.01 mm) is cut for stress relaxation test and the longer one (49.80 mm × 7.28 mm × 2.86 mm) for thermo-mechanical analysis.

_{g}. The chamber first heated the specimen to the pre-set temperature and held it for 10 min. Then, the machine stretched the sample 0.11 mm, applying a nominal strain of approximately 0.1% for 257 s. Owing to the viscoelastic behavior, the tensile stress in the polymer decreased, and the corresponding modulus was recorded during the test. Figure 3 illustrates the relaxation moduli obtained at elevated temperatures. These curves show that the modulus drops considerably above T

_{g}.

_{g}) induces significant thermal expansion. Because it often encounters high temperatures in the subsurface, it is necessary to consider thermal deformation to design an SMP packer.

## 3. Viscoelastic Parameters for SMP

_{ij}represents the stress components with i, j ∈ {1, 2, 3}, and x

_{j}is the coordinate. For linear, isotropic, and elastic materials, the stress is related to the strain with constitutive equations (incorporating the thermal strain):

_{ij}denotes the strain components, ε

_{kk}= ε

_{11}+ ε

_{22}+ ε

_{33}is the volumetric strain, δ

_{ij}is the Kronecker delta, E and ν are Young’s modulus and Poisson’s ratio, respectively, α is the material’s thermal expansion coefficient, and ΔT represents the temperature difference between the current state and the initial state. Equation (2) indicates that the elastic modulus and thermal expansion coefficient are crucial to characterize the deformation and stress of materials. Instead of the simple linear form of Equation (2), the constitutive equation for viscoelastic materials is a hereditary integral [43]:

_{∞}represents the equilibrium modulus of the material at which it has fully relaxed, E

_{i}and τ

_{i}are the elastic modulus and relaxation time of the ith element, respectively, τ

_{i}= η

_{i}/E

_{i}with η

_{i}denoting the viscosity of the ith dashpot, and N is the element number.

_{ref}is the reference temperature, T represents an arbitrary temperature, and a

_{T}is the time–temperature superposition shift factor. Equation (5) makes it possible to predict the viscoelastic behavior of polymers at different temperatures from the master curve. Naturally, a

_{T}depends on the temperature and is given by the WLF equation [45]:

_{1}and C

_{2}are material constants that can be extracted from the stress–relaxation experiment.

_{1}= 7.38 and C

_{2}= 100.24 °C. The obtained master curve at T

_{g}is shown in Figure 6, from which the parameters of the Prony series in Equation (4) are extracted, as shown in Table 1.

## 4. Simulation Modeling for the Setting Process

- Increasing the temperature above T
_{g}significantly reduced the modulus of the SMP packer. Then, a radial compressive load is applied to the packer to extend its axial length and reduce its outer diameter. The compressed packer clings firmly to the mandril, making it easier to run in the wellbore. - Keep the compressive load and decrease the temperature well below T
_{g}. - Slowly release the load, and the SMP packer store the compressed shape.
- The packer was lowered into the well. When it arrives at the setting position, it is heated again above T
_{g}using a downhole heater. Finally, the packer gradually returns to its original shape. Then, it expands to contact the inner wall of the casing or the formation, which produces contact stress to achieve sealing.

#### 4.1. Material Parameters in FEA

_{R}is the ratio of the shear relaxation modulus to the instantaneous modulus, g

_{i}is the Prony coefficient, and τ

_{i}is the relaxation time. This study assumes that the SMP is incompressible, which makes it easy to transform E

_{i}to g

_{i}from Equation (4) as:

_{0}is the instantaneous tensile relaxation modulus, which can be obtained from Equation (4):

#### 4.2. The Geometry of the Model

#### 4.3. Simulation Steps and Boundary Conditions

- The initial step defines the temperature field as 120 °C (>T
_{g}) and constrains the radial displacement on the left side of the packer. - The first viscoelastic step applies a radial displacement (−10 mm in the radial direction) on the right side to compress the packer under the constant temperature defined in the initial step.
- In the second viscoelastic step, the field temperature decreased to 50 °C with compression.
- During the third viscoelastic step, the model keeps the low temperature unchanged and releases the displacement load. Meanwhile, the rigid casing moves radially to form an annular gap between the tubing and casing.
- The fourth viscoelastic step increases the temperature again above T
_{g}to activate shape recovery. - Considering the stress relaxation of polymers, the fifth step holds the temperature for a while until the contact stress between the packer and the casing reaches a steady value.
- Finally, the extra general step applies pressure (20 MPa) to the top edge of the packer. Under subsurface conditions, the packer must withstand the pressure difference between the annular and pore fluids. In an extreme case, only one end of the packer is subjected to wellbore pressure.

## 5. Results and Discussions

#### 5.1. Base Case Simulation for the Seal Performance

_{g}, the radial stress increases and reaches a maximum of 87 MPa. This stress component gradually decreases during cooling and unloading (Steps 2 and 3), although the material modulus increased as the temperature decreased. In the heating stage (Step 4), the SMP packer expands to contact the casing wall owing to shape recovery and thermal expansion. Then, the contact stress is generated at the interface between the packer and casing (the right side of the packer) with high values at both ends and relatively low but uniform values in the middle. The magnitude of the contact stress is in the range of 6.9~8.5 MPa, similar to Tong et al., which validates the numerical simulation [20]. As expected, the contact stress decreased owing to the stress relaxation in Step 5. Finally, the packer expands laterally under a wellbore pressure of 20 MPa at the top, increasing the contact stress to approximately 15 MPa.

#### 5.2. Effects of Seal Length, Interference, Pre-Compression, Temperature, and Wellbore Pressure

_{s}), and the wellbore pressure (p

_{w}). The base case simulation shows that the contact stress occurs only in Steps 4–6. Hence, we repeated the model simulation with the above parameters to obtain the contact and shear stresses at the packer–casing interface in these steps. The simulation changes one parameter and fixes the others in one calculation.

_{w}increased from 10 MPa to 30 MPa, the maximum contact stress increased from 14 MPa to 26 MPa. A tremendous axial pressure would cause a more lateral expansion in the packer, inducing more significant contact stress. Figure 13j has the same trend whereby the maximum shear stress in the middle of the packer rises from 1.46 MPa to 4.12 MPa when the wellbore pressure changes from 10 MPa to 30 MPa. The enhanced shear stress increases the likelihood of slip failure.

#### 5.3. Sensitivity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Setting process of a shape memory polymer packer: (

**a**) the original shape of the shape memory polymer (SMP) packer, (

**b**) after pre-compression, (

**c**) run in the wellbore, (

**d**) recovery to seal the annular by re-heating.

**Figure 2.**Specimens and test machine for the stress relaxation experiments and thermo-mechanical analysis: (

**a**) relaxation test; (

**b**) thermo-mechanical analysis.

**Figure 8.**Displacement boundary at the right side of the packer and temperature field of the model in each analysis step.

**Figure 9.**Radial stress of the SMP packer for each step: (

**a**) Step 1, (

**b**) Step 2, (

**c**) Step 3, (

**d**) Step 4, (

**e**) Step 5, (

**f**) Step 6.

**Figure 10.**Shear stress of the SMP packer for each step: (

**a**) Step 1, (

**b**) Step 2, (

**c**) Step 3, (

**d**) Step 4, (

**e**) Step 5, (

**f**) Step 6.

**Figure 11.**Distribution of the contact and shear stress after shape recovery under various sealing conditions: (

**a**) contact stress with different seal lengths, (

**b**) shear stress with different seal lengths, (

**c**) contact stress with different interferences, (

**d**) shear stress with different interferences, (

**e**) contact stress for different pre-compression, (

**f**) shear stress for different pre-compression, (

**g**) contact stress at different setting temperatures, (

**h**) shear stress at different setting temperatures.

**Figure 12.**The contact and shear stress after relaxation under different sealing parameters: (a) contact stress with different seal lengths, (

**b**) shear stress with different seal lengths, (

**c**) contact stress with different interferences, (

**d**) shear stress with different interferences, (

**e**) contact stress for different pre-compression, (

**f**) shear stress for different pre-compression, (

**g**) contact stress at different setting temperatures, (

**h**) shear stress at different setting temperatures.

**Figure 13.**The contact and shear stress when the wellbore pressure is applied to the packer: (

**a**) contact stress with different seal lengths, (

**b**) shear stress with different seal lengths, (

**c**) contact stress with different interferences, (

**d**) shear stress with different interferences, (

**e**) contact stress for different pre-compression, (

**f**) shear stress for different pre-compression, (

**g**) contact stress at different setting temperatures, (

**h**) shear stress at different setting temperatures, (

**i**) contact stress under various wellbore pressure, (

**j**) shear stress under various wellbore pressure.

**Figure 14.**Sensitivity of the contact and shear stress to various parameters: (

**a**) the average contact stress; (

**b**) the shear stress.

i | τ_{i} (s) | E_{i} (MPa) |
---|---|---|

1 | 0.01 | 490.60 |

2 | 0.39 | 769.50 |

3 | 3.27 | 313.27 |

4 | 26.67 | 323.98 |

5 | 247.24 | 2.18 |

6 | 6.65 × 10^{5} | 35.31 |

Parameters | Parameter Value (Normalized Value) | ||
---|---|---|---|

Seal length (mm) | 30 (0.5) | 60 (1.0) | 90 (1.5) |

Interference (mm) | 4.37 (0.69) | 6.37 (1.0) | 8.37 (1.31) |

Pre-compression (mm) | 8 (0.8) | 10 (1.0) | 12 (1.2) |

Setting temperature (°C) | 110 (0.92) | 120 (1.0) | 130 (1.08) |

Wellbore pressure (MPa) | 10 (0.5) | 20 (1.0) | 30 (1.5) |

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**MDPI and ACS Style**

Chen, N.; Dong, X.; Ma, Y.
Evaluation on the Seal Performance of SMP-Based Packers in Oil Wells. *Polymers* **2022**, *14*, 836.
https://doi.org/10.3390/polym14040836

**AMA Style**

Chen N, Dong X, Ma Y.
Evaluation on the Seal Performance of SMP-Based Packers in Oil Wells. *Polymers*. 2022; 14(4):836.
https://doi.org/10.3390/polym14040836

**Chicago/Turabian Style**

Chen, Naihan, Xuelin Dong, and Yinji Ma.
2022. "Evaluation on the Seal Performance of SMP-Based Packers in Oil Wells" *Polymers* 14, no. 4: 836.
https://doi.org/10.3390/polym14040836