# Experimental Study on the Coefficient of Internal Frictional Resistance in the Annular Gap during the Plunger Gas Lift Process

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Procedure and Equipment

^{3}/h and an accuracy of ±1%. Meanwhile, water was supplied by a pump and metered by a turbine flow meter at a range of 0~0.3 m

^{3}/h and accuracy of ±0.3%. High-pressure compressed air and water were thoroughly mixed in the mixer to form a mixed air–water fluid medium, which then entered the test section. At the end of the experiment, water was recovered and air was vented through a gas–liquid separation device attached to the outlet of the test tube section.

## 3. Experimental Results and Analysis

#### 3.1. Plunger Movement Analysis

#### 3.2. Analysis of Frictional Resistance in the Plunger Annular Gap

- Plunger upper surface pressure: the hydrostatic pressure of the liquid and the hydrostatic column pressure on the top of the plunger due to the liquid above the plunger, $\pi {d}^{2}{p}_{1}/4$.
- Plunger lower surface pressure: at the bottom of the plunger, the dynamic column pressure from the annulus and formation is applied, $\pi {d}^{2}{p}_{2}/4$.
- Gravity of the plunger and liquid column: since both the plunger and liquid column have a certain mass, they are subject to a constant force of gravity, $({m}_{p}+{m}_{l})g$.
- Frictional resistance of the liquid column and inner wall of the tubing: when the liquid column moves upward with the plunger, the friction between the liquid and inner wall of the tubing hinders the upward movement of the plunger, ${F}_{f1}$.
- Fluid friction in the annular gap: since gases and liquids move turbulently in the annular gap and move upward with the plunger, the friction in the annular gap also affects the plunger’s movement, ${F}_{f2}$.

^{2}; $d$ is the inner diameter of the tubing, m; ${P}_{1}$ and ${P}_{2}$ are the pressure levels on the upper and lower surfaces of the plunger, respectively, Pa; and $g$ is the acceleration of gravity, m/s

^{2}.

^{3}; and μ is the dynamic viscosity of the fluid, m

^{2}/s, with an experimental water dynamic viscosity of 10

^{−6}m

^{2}/s used for the calculation.

^{4}~2.4 × 10

^{4}, and its annular gap frictional-resistance coefficient was mainly concentrated at 0.066~0.072. The Reynolds number interval at 0.25 MPa was mainly concentrated at 2.4 × 10

^{4}~2.9 × 10

^{4}and its annular gap frictional-resistance coefficient was mainly concentrated at 0.066~0.070. The Reynolds number interval at 0.3 MPa was mainly concentrated in the range of 2.9 × 10

^{4}~3.4 × 10

^{4}and its annular gap frictional-resistance coefficient was mainly concentrated in the range of 0.066~0.069.

^{4}to 2.4 × 10

^{4}; in the interval of 2.4 × 10

^{4}~2.9 × 10

^{4}, the decrease in the annular gap frictional-resistance coefficient gradually slowed down and showed an excessive value; and in the interval of 2.9 × 10

^{4}~3.4 × 10

^{4}, the annular gap tended to flatten out, showing a steady state. The overall trend was a gradual shift from a rapid decline to a constant smoothness.

^{−20}. Therefore, we rejected the 0 hypothesis (the regression coefficient was not significant) and accepted the alternative hypothesis (the regression coefficient was significant).

#### 3.3. Comparative Analysis of Different Frictional-Resistance Coefficient Formulas

^{−6}; Jain’s and Churchill’s formulas had resistance coefficients in the range of 0.05445 to 0.05544, with a decrease of 0.00216; and Chen’s formula had resistance coefficients in the range of 0.054 to 0.05535, with a decrease of 0.00135. Compared to the magnitude of the decrease in these computational models, the experimental data have a range of friction coefficients between 0.0651 and 0.072. These computational models have a relatively smooth variation in the friction coefficient. However, by comparing the absolute errors of the results of these computational models with the experimental data, we obtained the following absolute errors: 25.99%, 20.31%, 19.09%, 18.99%, and 19.91%. These results show that none of the abovementioned five models used for calculating the frictional-resistance coefficient can accurately reflect the plunger annular gap frictional-resistance coefficient, as shown in Figure 8.

## 4. Modeling of Annular Gap Frictional-Resistance Coefficient

^{3}; $v$ is the plunger’s velocity, m/s; ${D}_{eff}$ is the equivalent diameter, m; and $\mu $ is the fluid dynamic viscosity, kg/(m·s).

^{2}), mean absolute percentage error (MAPE), and root mean square error (RMSE), were used for the model comparison. Where the RMSE was used to measure the deviation between the predicted and experimental values, the smaller the error was, the better the predictive ability of the model. These error calculation methods can help us evaluate the performance of each model to determine that model that is more suitable for describing the trend of the resistance coefficient of the annular gap in the plunger lift. R

^{2}, MAPE, and RMSE values are calculated as follows:

## 5. Case Study

^{2}. Over time, the plunger gradually reaches dynamic equilibrium, fluid continues to pour into the tube, and the acceleration tends to zero.

^{2}, and when the annular gap frictional resistance is considered, the acceleration of the plunger increases again at approximately 95 s, with a peak close to 0.2 m/s

^{2}. This is because, at this point, the liquid in the annulus has completely entered the tubing, and the subsequent entry is the gas in the annulus.

## 6. Conclusions

- In this paper, although the experimental results indicate that the turbulent sealing effect inside the grooves can mitigate the impact of liquid fallback, the analysis of the actual data suggests that the velocity difference between the plunger and liquid column is the primary reason for leakage. This velocity difference cannot be ignored, the leakage phenomenon cannot be simply attributed to fluid slippage or self-weight, and the existence of this velocity difference requires a consideration of the effect of frictional resistance in the annular gap.
- Through the analysis of the relationship between the Reynolds number and annular gap frictional-resistance coefficient, we found that the annular gap frictional-resistance coefficient showed a decreasing trend with the increase in the Reynolds number. Under different pressure conditions, the changes in the annular gap frictional-resistance coefficient showed a pattern of a rapid decrease, followed by a gradual slowing down, ultimately converging in a stable mode. These findings contribute to a better understanding of the effect of frictional resistance on the plunger lift process and provide valuable suggestions for process optimization purposes.
- Through an analysis and experimental data comparison of five friction coefficient calculation models, it was observed that these models showed significant deviations when describing the annular gap frictional-resistance coefficients in plunger lift systems, making them unable to accurately reflect real-world conditions. This was because the frictional forces within the annular gap were primarily created by the velocity difference between the plunger and liquid column, leading to liquid leakage. These forces are determined by the combined effects of turbulent flow and liquid film friction. Therefore, it is necessary to establish a new model tailored to the annular gap frictional-resistance coefficient to provide a more accurate description of this flow condition.
- In the vertical wellbore, the model for calculating the annular gap frictional-resistance coefficient showed a negative correlation trend with an increasing Reynolds number, which was more in line with the variation in the annular gap frictional-resistance coefficient during the plunger lift process. Through error calculation methods, such as the coefficient of determination, average absolute percentage error, and root mean square error, the results indicate that the newly proposed model in this paper shows greater accuracy and reliability in predicting the annular gap frictional-resistance coefficient in plunger lift technology.
- A comprehensive analysis of the simulation results reveals that, when considering the annular gap frictional resistance, the plunger’s upward traveling time is close to the actual traveling time, and the plunger’s acceleration shows distinct variations at different stages. Furthermore, through the comparison of the pressure and velocity, it is evident that the annular gap frictional resistance plays a significant role in the plunger’s ascent process.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Plot of plunger annular gap friction coefficient results (The blue dots represent experimental data points and the red line represents the data trend line).

**Figure 7.**Comparison of calculated and experimental values of different frictional-resistance coefficient models.

**Figure 8.**Comparison of absolute errors of different models for calculation of friction coefficients.

**Figure 11.**Comparison between experimental and calculated values of annular gap frictional-resistance coefficients.

**Figure 12.**Comparison of relative errors between calculated and experimental values (The blue dots represent the relative error between the experimental and predicted values).

**Figure 13.**Plot of plunger reach time vs. upward velocity (Points A and B in the figure represent the annulus fluid entering the tubing; points C and D in the figure represent the start of fluid discharge from the wellbore).

Measurement Equipment | Measurement Range | Measurement Precision (%) |
---|---|---|

Gas flow meter | 34.72 m^{3}/min | ±1 |

Liquid flow meter | 0–20 m^{3}/h | ±0.3 |

Differential pressure sensor | −10–100 KPa | 0.025–0.04 |

Pressure transmitter | 0–5 MPa | ±0.5 |

Temperature sensor | 0~90 °C | ±1 |

P_{wf} (KPa) | P_{c} (KPa) | P_{t} (KPa) | Average Velocity | Liquid Column Movement Speed | Plunger Movement Speed | Relative Speed |
---|---|---|---|---|---|---|

219 | 171 | 161 | 4.80 | 4.83 | 5.00 | 0.17 |

214 | 159 | 158 | 5.01 | 5.00 | 5.38 | 0.38 |

205 | 151 | 148 | 4.34 | 4.83 | 5.00 | 0.17 |

202 | 150 | 146 | 5.54 | 4.24 | 5.83 | 1.59 |

202 | 147 | 141 | 4.74 | 4.38 | 5.60 | 1.23 |

202 | 150 | 144 | 4.34 | 3.78 | 5.19 | 1.40 |

211 | 159 | 151 | 5.95 | 4.52 | 5.60 | 1.08 |

202 | 150 | 144 | 5.63 | 4.52 | 5.19 | 0.67 |

252 | 201 | 189 | 5.34 | 4.83 | 5.60 | 0.77 |

255 | 200 | 193 | 5.86 | 5.19 | 6.67 | 1.48 |

264 | 212 | 186 | 5.63 | 5.19 | 6.09 | 0.90 |

270 | 215 | 198 | 5.89 | 5.19 | 6.67 | 1.48 |

270 | 216 | 199 | 6.05 | 5.38 | 6.67 | 1.28 |

252 | 201 | 193 | 4.47 | 6.09 | 8.24 | 2.15 |

257 | 197 | 190 | 6.15 | 4.12 | 5.83 | 1.72 |

261 | 201 | 196 | 5.63 | 5.00 | 6.09 | 1.09 |

255 | 198 | 192 | 5.81 | 5.00 | 6.36 | 1.36 |

317 | 256 | 256 | 5.15 | 6.09 | 7.00 | 0.91 |

308 | 251 | 244 | 7.11 | 5.60 | 7.37 | 1.77 |

302 | 241 | 242 | 6.45 | 5.38 | 6.67 | 1.28 |

313 | 257 | 245 | 7.05 | 6.09 | 7.37 | 1.28 |

305 | 251 | 239 | 6.61 | 5.19 | 7.00 | 1.81 |

313 | 257 | 251 | 6.81 | 6.09 | 7.00 | 0.91 |

305 | 242 | 235 | 5.81 | 5.00 | 6.67 | 1.67 |

307 | 247 | 242 | 4.93 | 5.00 | 6.67 | 1.67 |

310 | 248 | 245 | 5.76 | 5.00 | 7.00 | 2.00 |

Bottomhole Pressure (KPa) | Casing Pressure (KPa) | Tubing Pressure (KPa) | Speed of Movement (m/s) | Annular Gap Friction (N) | Annular Gap Friction Factor |
---|---|---|---|---|---|

202 | 148 | 138 | 5.11 | 4.977153 | 0.06751 |

204 | 156 | 146 | 5.71 | 6.383483 | 0.06914 |

205 | 157 | 144 | 5.86 | 6.473506 | 0.06671 |

204 | 153 | 141 | 5.15 | 5.128638 | 0.06838 |

204 | 160 | 143 | 5.29 | 5.1804 | 0.06556 |

205 | 160 | 146 | 5.11 | 5.307037 | 0.07198 |

204 | 150 | 144 | 5.06 | 4.724593 | 0.06536 |

204 | 150 | 144 | 5.11 | 5.195129 | 0.07047 |

205 | 150 | 140 | 5.29 | 5.46595 | 0.06897 |

Name | Model | Re Adaptation Range | $\mathit{\epsilon}/\mathit{d}$ Range of Adaptation |
---|---|---|---|

Moody | ${f}_{D}=5.5\times {10}^{-3}[1+{(2\times {10}^{4}\epsilon /D+{10}^{6}/\mathrm{Re})}^{1/3}]$ | [4 × 10^{3}, 10^{7}] | [1 × 10^{−5}, 0.05] |

Wood | ${f}_{D}=0.094{(\frac{\epsilon}{D})}^{0.225}+0.53(\frac{\epsilon}{D})+88{(\frac{\epsilon}{D})}^{0.4}{(\mathrm{Re})}^{-1.62}{(\frac{\epsilon}{D})}^{0.134}$ | [4 × 10^{3}, 10^{7}] | [4 × 10^{−5}, 0.05] |

Jain | $\frac{1}{\sqrt{{f}_{D}}}=1.14-2\mathrm{log}(\frac{\epsilon}{D}+\frac{21.25}{{\mathrm{Re}}^{0.9}})$ | [5 × 10^{3}, 10^{7}] | [4 × 10^{−5}, 0.05] |

Churchill | ${f}_{D}=8{[{(\frac{8}{\mathrm{Re}})}^{12}+\frac{1}{{(A+B)}^{3/2}}]}^{1/12}$ $A={{\{2.457\mathrm{ln}[(\frac{7}{\mathrm{Re}})}^{0.9}+0.27\frac{\epsilon}{D}]\}}^{16}$ $B={(\frac{37530}{\mathrm{Re}})}^{16}$ | All scopes | All scopes |

Chen | ${f}_{D}=-2.0\mathrm{log}[\frac{\epsilon /D}{3.7065}-\frac{5.0452}{\mathrm{Re}}\mathrm{log}(\frac{{(\epsilon /D)}^{1.1098}}{2.8257}+\frac{5.8506}{{\mathrm{Re}}^{0.8981}})]$ | [4 × 10^{3}, 4 × 10^{8}] | [5 × 10^{−7}, 0.05] |

**Table 5.**Calculation error of the frictional-resistance coefficients for different plunger annular gaps.

Model Name | R^{2} | MAPE | RMSE |
---|---|---|---|

Moody | 0.051 | 25.99% | 1.77 |

Wood | 0.088 | 20.31% | 1.39 |

Jain | 0.089 | 19.09% | 1.31 |

Churchill | 0.089 | 18.99% | 1.29 |

Chen | 0.085 | 19.91% | 1.36 |

Article | 0.74 | 2.20% | 0.39 |

Mean Experimental Value | 0.06809 | Mean Predicted Value | 0.06810 |

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**MDPI and ACS Style**

Shi, H.; Chen, Z.; Liao, R.; Liu, J.; Li, J.; Jin, S.
Experimental Study on the Coefficient of Internal Frictional Resistance in the Annular Gap during the Plunger Gas Lift Process. *Processes* **2023**, *11*, 3246.
https://doi.org/10.3390/pr11113246

**AMA Style**

Shi H, Chen Z, Liao R, Liu J, Li J, Jin S.
Experimental Study on the Coefficient of Internal Frictional Resistance in the Annular Gap during the Plunger Gas Lift Process. *Processes*. 2023; 11(11):3246.
https://doi.org/10.3390/pr11113246

**Chicago/Turabian Style**

Shi, Haowen, Zhong Chen, Ruiquan Liao, Jie Liu, Junliang Li, and Shan Jin.
2023. "Experimental Study on the Coefficient of Internal Frictional Resistance in the Annular Gap during the Plunger Gas Lift Process" *Processes* 11, no. 11: 3246.
https://doi.org/10.3390/pr11113246