# Numerical Simulations of Radial Well Assisted Deflagration Fracturing Based on the Smoothed Particle Hydrodynamics Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Deflagration Fracturing Theory

#### 2.1. Principle of the SPH Method

#### 2.1.1. Kernel Approximation

#### 2.1.2. Particle Approximation

_{j}is the mass of the SPH particle, ρ

_{j}is the density of the SPH particle, and N is the number of particles in the smooth length.

#### 2.2. JWL State Equation

_{1}and R

_{2}is GPa, A, B, and ω are dimensionless, and they are all constants.

#### 2.3. Criterion for Fracture Propagation in Cohesive Unit

_{n}and t

_{s}are the normal stress, MPa; t

_{t}is the shear stress, MPa; δ is the nominal strain; δ

_{n}and δ

_{s}are the normal strain; δ

_{t}is the shear strain; K is the stiffness matrix; K

_{nn}is the directional stiffness; and K

_{ss}and K

_{tt}are the shear stiffness.

_{max}is the maximum principal stress on the rock, MPa, and P

_{t}is the rock tensile strength, MPa.

_{c}is the cohesion energy; G

_{n}is the normal fracture energy; G

_{s}is the fracture energy in the first shear direction; G

_{t}is the fracture energy in the second shear direction; and η is the material parameter.

## 3. Establishment of the Numerical Model

#### 3.1. Model Characterization

#### 3.1.1. Assumptions

#### 3.1.2. Model Settings

#### 3.2. Model Establishment

_{B}= 0.03 m and L

_{B}= 18 m and is meshed into 4680 C3D8R hexahedrons, as shown in Figure 1. These meshes are converted into particles according to time steps. After 10

^{−8}s, the computation starts, and one mesh is converted into one particle. A total of 4680 particles are generated.

_{x}= 0, U

_{y}= 0 and U

_{z}= 0.

_{rad}= 0.03 m and the length of H

_{rad}= 18 m and is in the center of the reservoir. The radial well has closed ends. The standard TNT explosive is placed in the middle of the radial well, and the center of the charge column is ignited.

#### 3.3. Model Verification

## 4. Analysis of Results

#### 4.1. Effect of the Deflagration Position

^{2}, and the fracture extends in a cross-shape from the deflagration point (green area). Shear failure leads to the fracture area of 58.37 m

^{2}(red area) around the tensile fracture. Comparatively, the reservoir stimulation near the deflagration point is dominated by shear failure. As the distance from the deflagration point increases, the shear failure decreases, the total fracture area decreases, and the reservoir stimulation efficiency decreases, as shown in Table 4. In the sections that are 5 m and 6 m away from the deflagration point, the shear fracture areas of 17.26 m

^{2}and 13.47 m

^{2}and the tensile fracture areas of 50.13 m

^{2}and 47.75 m

^{2}are generated, respectively. The reservoir stimulation areas are reduced by 32.58% and 38.75%, respectively, compared with those at the deflagration point.

#### 4.2. Effect of the Radial Well Azimuth

^{2}to 99.95 m

^{2}, with an increment of 66.33%. As the radial well azimuth changes, the stress distribution on the borehole wall and deflagration stress field change accordingly, which affects the fracture propagation pattern. The larger radial well azimuth indicates lower horizontal stress on the section across the wellbore, and the fractures are prone to initiation and propagation, resulting in a larger stimulation area. Moreover, with the increase in the radial azimuth, the shear fracture area increases gradually, as shown in Figure 7. The tensile fracture area reaches a maximum of 61.15 m

^{2}when α = 75° and decreases to 41.58 m

^{2}when α = 90°. With the increase in the radial well azimuth (α > 75°), the stress in the rock changes, and the fracture mechanism changes from tensile failure to shear failure, as shown in Table 5. This provides a reference for understanding the fracture mechanism of deflagration fracturing.

#### 4.3. Effect of the Horizontal Stress Difference

^{2}to 71.33 m

^{2}, with a reduction of 28.63%. The large stress difference had a significant negative effect on the fracture initiation and propagation, as shown in Table 6. However, in deflagration fracturing, there is still a good capacity for fracture initiation in the rock. With the increase in the horizontal stress difference, the shear failure area decreased rapidly, and the tensile failure area increased to a certain extent. A larger stress difference is conductive to tensile failure.

## 5. Discussion

## 6. Conclusions

- In deflagration fracturing, the deflagration position, the radial well azimuth, and the horizontal principal stress difference have significant effects on the fracture formation area. The closer the distance from the deflagration position is, the larger the radial well azimuth and the smaller the horizontal stress difference are, leading to a larger fracture area, which is conducive to reservoir stimulation. As the distance from the deflagration point increases from 5 m to 11 m, the fracture area decreases by 33.5% and 85.1%, respectively. With the increase in the distance from the deflagration point, the reservoir stimulation efficiency decreases exponentially. When the radial well azimuth increases from 0° to 90°, the fracture area at the deflagration point increases by 66.33%. The larger azimuth is favorable for reservoir stimulation. As the horizontal principal stress difference increases from 2 MPa to 8 MPa, the fracture area at the deflagration point decreases by 28.63%. A higher stress difference has a significant negative effect on fracture initiation and propagation. However, in deflagration fracturing, there is still a good capacity for fracture initiation in the rock.
- In deflagration fracturing, both shear slip fractures and tensile fractures occur. The formation and conversion of shear slip fractures and tensile fractures are related to the deflagration position, radial well azimuth, horizontal principal stress difference, etc. Shear failure is sensitive to the deflagration distance, the radial well azimuth, and the horizontal principal stress. With the increase in the distance from the deflagration point, decrease in the radial well azimuth, and increase in the horizontal principal stress, the shear failure area decreases by 93.39%, 78.31%, and 71.23%, respectively. The tensile failure area is closely related to the deflagration distance, radial well azimuth, and the horizontal principal stress.
- The process of deflagration fracturing can be characterized based on the JWL constitutive equation, the SPH method, and infinite element numerical simulation. The error of the numerical simulation is less than 10%, and the problems of the stress wave rebound and poor calculation accuracy are mitigated.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Transmission of explosive deflagration within the radial borehole: (

**a**) initial state of explosive deflagration; (

**b**) transmission of explosive deflagration.

**Figure 6.**Horizontal fracture morphology at the deflagration point (green indicates failure, and the larger green area indicates the smaller fracture area).

**Figure 8.**Fracture morphology at the deflagration point under various horizontal stress differences.

Parameters | Values | Parameters | Values |
---|---|---|---|

Rock density | 2500 kg/m^{3} | Radial well diameter | 0.03 m |

Horizontal maximum principal stress | 30 MPa | Horizontal minimum principal stress | 28 MPa |

Vertical stress | 35 MPa | Rock tensile strength | 3 MPa |

Poisson’s Ratio | 0.25 | Elastic modulus | 12.9 GPa |

Parameters | Constants | Deflagration Rate | Explosive Density | Initial Specific Internal Energy | ||||
---|---|---|---|---|---|---|---|---|

Unit | A/GPa | B/GPa | R_{1} | R_{2} | ω | D/m·s^{−1} | ρ/kg·m^{−3} | e/GPa |

Values | 371.2 | 3.231 | 4.15 | 0.95 | 0.3 | 6930 | 1600 | 7.0 |

Explosion Center/m | Over-Pressure Peak/MPa | Error/% | |
---|---|---|---|

Measured | Calculated | ||

3.5 | 0.257 | 0.263 | 2.3% |

5.0 | 0.124 | 0.132 | 6.5% |

7.0 | 0.099 | 0.106 | 7.1% |

9.5 | 0.051 | 0.056 | 9.8% |

Positions | Fracture Area at the Left Deflagration Point/m^{2} | Fracture Area at the Middle Deflagration Point/m^{2} | Fracture Area at the Right Deflagration Point/m^{2} | ||||||
---|---|---|---|---|---|---|---|---|---|

Shear Failure Area | Tensile Failure Area | Fracture Area | Shear Failure Area | Tensile Failure Area | Fracture Area | Shear Failure Area | Tensile Failure Area | Fracture Area | |

Left | 57.26 | 40.13 | 97.39 | 16.88 | 47.91 | 64.79 | 4.23 | 10.34 | 14.57 |

Mid | 17.26 | 50.13 | 67.39 | 58.37 | 41.58 | 99.95 | 13.47 | 47.75 | 61.22 |

Right | 3.288 | 13.51 | 16.79 | 13.15 | 45.09 | 58.24 | 56.54 | 40.22 | 96.76 |

Radial Well Azimuths | Fracture Area in the Middle of the Radial Well/m^{2} | ||
---|---|---|---|

Shear Failure Area | Tensile Failure Area | Fracture Area | |

0° | 12.64 | 47.45 | 60.09 |

15° | 17.17 | 50.94 | 68.11 |

45° | 27.43 | 57.59 | 85.02 |

75° | 35.68 | 61.15 | 96.83 |

90° | 58.37 | 41.58 | 99.95 |

Stress Difference/MPa | Fracture Area in the Middle of the Radial Well/m^{2} | ||
---|---|---|---|

Shear Failure Area | Tensile Failure Area | Fracture Area | |

2 | 58.37 | 41.58 | 99.95 |

5 | 22.98 | 51.60 | 74.58 |

8 | 16.8 | 54.53 | 71.33 |

**Table 7.**Quantitative characterization of the fracture morphology on the X-Y plane under variable stress differences.

Stress Difference/MPa | Fracture Area in the Middle of the Radial Well/m^{2} | ||
---|---|---|---|

Shear Failure Area | Tensile Failure Area | Fracture Area | |

2 | 0 | 20.00 | 20.00 |

5 | 1.92 | 18.08 | 20.00 |

8 | 11.25 | 8.75 | 20.00 |

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**MDPI and ACS Style**

Gong, D.; Chen, J.; Wang, W.; Qu, G.; Zhu, J.; Wang, X.; Zhang, H.
Numerical Simulations of Radial Well Assisted Deflagration Fracturing Based on the Smoothed Particle Hydrodynamics Method. *Processes* **2022**, *10*, 2535.
https://doi.org/10.3390/pr10122535

**AMA Style**

Gong D, Chen J, Wang W, Qu G, Zhu J, Wang X, Zhang H.
Numerical Simulations of Radial Well Assisted Deflagration Fracturing Based on the Smoothed Particle Hydrodynamics Method. *Processes*. 2022; 10(12):2535.
https://doi.org/10.3390/pr10122535

**Chicago/Turabian Style**

Gong, Diguang, Junbin Chen, Weibo Wang, Guanzheng Qu, Jianhong Zhu, Xiaoming Wang, and Haoyu Zhang.
2022. "Numerical Simulations of Radial Well Assisted Deflagration Fracturing Based on the Smoothed Particle Hydrodynamics Method" *Processes* 10, no. 12: 2535.
https://doi.org/10.3390/pr10122535