# Shale Gas Productivity Prediction Model Considering Time-Dependent Fracture Conductivity

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Time-Dependent Effect of Fracture Conductivity

#### 2.1. Common Formulas and Problems

- (1)
- An inconsistent experiment process:Figure 1 shows the long-term conductivity test results of shale reservoirs [20,21]. Figure 1a shows the results of a conventional long-term conductivity test in which ${F}_{c0}$ was equal to the initial conductivity at the start of the test. The gas was injected throughout the test, and the damage to the conductivity caused by the immersion of the fracturing fluid was not considered. In fact, due to the influence of factors such as fracturing and shut-in, the fracture rate of the proppant changes after being soaked in fracturing fluid, and this change affects the change law of conductivity with time.

- (2)
- Variable experiment parameters:The practicability of Equation (1) is poor. When the test conditions (i.e., field construction parameters) are changed, the test needs to be carried out again to determine the values of $\eta $ and c in Equation (1) under the new test conditions, which undoubtedly increases the time it takes to obtain results.

#### 2.2. Long-Term Conductivity Test Experiment

^{2}= 0.9471, of the regression model indicates that the model fit well and the model could be used for prediction.

#### 2.3. Amendment to the Formula of Conductivity

## 3. Shale Gas Physicomathematical Statements

^{3}; ${\varphi}_{m}$${\varphi}_{f}$ are the matrix and the fracture porosity, respectively; ${q}_{g}$ is the adsorption quality for shale gas, cm

^{3}/s; ${k}_{m}$${k}_{f}$ are the corrected matrix and the fracture permeability, respectively, $k={f}_{1}{f}_{2}{f}_{3}{k}_{0}$, D; ${\mu}_{g}$ is the viscosity of the shale gas in the reservoir, mPa·s; $q$ is the source and sink, cm

^{3}/s; ${q}_{mf}$ is the leakage term between matrix and fractures, cm

^{3}/s.

#### 3.1. Gas Adsorption

^{3}; ${V}_{L}$ is the Langmuir volume, m

^{3}/kg; ${p}_{L}$ is the Langmuir pressure, MPa.

#### 3.2. Gas Slip and Spread

#### 3.3. Gas Non-Darcy Flow

#### 3.4. Time-Dependent Conductivity

## 4. The EDFM Numerical Model

#### 4.1. Numerical Discrete

#### 4.2. Non-Neighbor Connections

#### 4.2.1. Fracture–Matrix NNC

^{2}; ${l}_{xy}$ is the distance from area element to fracture in matrix mesh, m.

#### 4.2.2. Fracture–Fracture NNC

#### 4.2.3. Fracture–Well NNC

#### 4.3. Analysis of Model Calculation Results

#### 4.3.1. Validation

#### 4.3.2. Analysis of Flow and Pressure Changes in Fractures

#### 4.3.3. Effects of Complex Fractures

#### 4.3.4. Effects of Natural Fractures

## 5. Application Case

^{−4}mD, the porosity was 5%, the compressibility was 1.05 × 10

^{−4}MPa

^{−1}, the fracture length was 120 m, the height was 40 m, and the width was 0.003 m. The fracturing parameters of this well were used in Equations (4)–(6) to calculate ${R}_{fg}$ = 72.01%, $\eta $ = 0.4, $c$ = 0.003. The production performance of the well was evaluated using the established productivity prediction model, considering the effect of variable conductivity. The comparison results are shown in Figure 13.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Long-term conductivity test. (

**a**) Long-term conductivity test. (

**b**) Long-term conductivity test (considering liquids).

**Figure 6.**Formation pressure at different production times. (

**a**) Pressure of production, 50d (constant conductivity). (

**b**) Pressure of production, 50d (variable conductivity). (

**c**) Pressure of production, 180d (constant conductivity). (

**d**) Pressure of production, 180d (variable conductivity). (

**e**) Pressure of production, 1y (constant conductivity). (

**f**) Pressure of production, 1y (variable conductivity).

Level | Sand Concentration (kg/m ^{2}) | Fluid Viscosity (mPa·s) | Fluid Retention Time (h) |
---|---|---|---|

−1 | 5.5 | 1 | 10 |

0 | 7 | 3 | 12 |

1 | 8.5 | 5 | 14 |

Number | Sand Concentration (kg/m^{2}) | Fluid Viscosity (mPa·s) | Fluid Retention Time (h) | Conductivity Retention Rate (%) |
---|---|---|---|---|

1 | 7 | 3 | 12 | 43.03 |

2 | 7 | 3 | 12 | 39.96 |

3 | 5.5 | 1 | 12 | 47.61 |

4 | 7 | 3 | 12 | 42.38 |

5 | 8.5 | 3 | 14 | 47.43 |

6 | 7 | 1 | 10 | 56.53 |

7 | 7 | 5 | 10 | 32.89 |

8 | 7 | 1 | 14 | 51.71 |

9 | 8.5 | 1 | 12 | 60.59 |

10 | 8.5 | 3 | 10 | 51.08 |

11 | 5.5 | 5 | 12 | 26.43 |

12 | 7 | 3 | 12 | 41.85 |

13 | 5.5 | 3 | 14 | 32.46 |

14 | 7 | 3 | 12 | 41.88 |

15 | 8.5 | 5 | 12 | 40.34 |

16 | 7 | 5 | 14 | 29.11 |

17 | 5.5 | 3 | 10 | 36.21 |

Parameters | Value | Parameters | Value |
---|---|---|---|

Model size | 200 × 200m | Fracture permeability | 10,000 mD |

Number of grids | 100 × 100 | Fracture porosity | 1 |

Matrix porosity | 0.03 | Fracture length | 100 m |

Matrix permeability | 0.0001 mD | Fracture width | 0.003 m |

Langmuir pressure | 9 MPa | Initial formation pressure | 30 MPa |

Langmuir volume | 0.004 m^{3}/kg | Bottom hole pressure | 15 MPa |

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**MDPI and ACS Style**

Pan, Y.; Xu, Y.; Yang, Z.; Wang, C.; Liao, R.
Shale Gas Productivity Prediction Model Considering Time-Dependent Fracture Conductivity. *Processes* **2022**, *10*, 801.
https://doi.org/10.3390/pr10050801

**AMA Style**

Pan Y, Xu Y, Yang Z, Wang C, Liao R.
Shale Gas Productivity Prediction Model Considering Time-Dependent Fracture Conductivity. *Processes*. 2022; 10(5):801.
https://doi.org/10.3390/pr10050801

**Chicago/Turabian Style**

Pan, Yuan, Yiwen Xu, Ze Yang, Chunli Wang, and Ruiquan Liao.
2022. "Shale Gas Productivity Prediction Model Considering Time-Dependent Fracture Conductivity" *Processes* 10, no. 5: 801.
https://doi.org/10.3390/pr10050801