# Numerical Simulation of Hydraulic Fracture Propagation Guided by Single Radial Boreholes

^{*}

## Abstract

**:**

_{f})’ was introduced for the first time to effectively quantify the radial borehole guidance. The guidance of nine factors was evaluated through gray correlation analysis. The experimental results were consistent with the numerical simulation results to a certain extent. The study provides theoretical evidence for the artificial control technology of directional propagation of hydraulic fracture promoted by the single radial borehole, and it predicts the guidance effect of a single radial borehole on hydraulic fracture to a certain extent, which is helpful for planning well-completion and fracturing operation parameters in radial borehole-promoted hydraulic fracturing technology.

## 1. Introduction

## 2. Establishment of a Fluid-Solid Coupling Mathematical Model and Its Finite Element Discretization

#### 2.1. Stress Balance Equation

#### 2.2. Continuity Equation

#### 2.3. Boundary Conditions

#### 2.4. Finite Element Discretization Method and Stress-Seepage Coupling Equation

## 3. Numerical Simulation of Propagation of Hydraulic Fracture

#### 3.1. Introduction to the Model

#### 3.1.1. Simulation of Initial Fracture

#### 3.1.2. Level-Set Simulation of Fracture Propagation

_{i}(crack terminal point function) to simulate fracture propagation. Renewal and evolution of φ is actually a progress of simulating the fracture propagation [29]. The nodal value of enrichment function is determined by two level-set functions, as shown in Figure 2.

#### 3.1.3. Criteria for Initiation of Fracture

#### 3.1.4. Damage Evolution Law

_{n}, t

_{s}and t

_{t}are the actual withstand stresses in the three corresponding directions.

#### 3.1.5. Energy Release Rate Criterion

#### 3.1.6. Solution

#### 3.2. Assumptions

- (1)
- Only a strip of hydraulic fracture is generated during fracturing, which is initiated along the azimuth of the radial borehole [34].
- (2)
- The formation rock is isotropic.

#### 3.3. Fundamental Parameters of Model

## 4. Analysis of Simulation Results

#### 4.1. Azimuth of Radial Borehole

_{f})” is introduced to characterize the guidance of the radial borehole on a hydraulic fracture. The guidance factor “G

_{f}” is defined as the ratio of the area surrounding a hydraulic fracture and radial borehole and its round boundary to the whole flat area in a 2D plane in bird’s-eye view, namely G

_{f}= S

_{p}/S in Figure 6.

_{p}is the area surrounding the hydraulic fracture and radial borehole and its round boundary, and S is total area of 2D plane of reservoir model. During numerical simulation, the circular boundary restrains the displacement in three directions. A hydraulic fracture will stop propagating when it extends to the boundary, but there may be some distance between the front end of the hydraulic fracture and the model boundary. Only through establishment of a hydraulic fracture extension line based on the late propagation trend, the area (S

_{p}) surrounded by hydraulic fracture (and extension line), radial borehole and the round boundary can be calculated. Using the lasso tool of “Adobe Photoshop” software, you can select the area with any shape (S

_{p}) that needs to be calculated, and then get the pixels of the selected area, the pixels ratio of two areas “S

_{p}” and “S” is G

_{f}. The use of the pixel calculation method is very accurate and convenient. G

_{f}is between 0 and 0.25, and a lower value indicates a stronger guidance of the radial borehole on the hydraulic fracture.

_{f}is 0, which has the best guidance. When the angle between the hydraulic fracture and radial borehole is 90°, G

_{f}is 0.25, which means no guidance. A large amount of numerical simulation shows that it is valid to use the guidance factor G

_{f}to determine the guidance strength of a radial borehole on a hydraulic fracture. In Model B, the guidance factors corresponding to the radial borehole azimuths of 15°, 30° and 45° are respectively 0.014, 0.026, 0.037, which shows that under the conditions of horizontal in-situ stress differences of 5 MPa, well length of 20 m and borehole diameter of 0.05 m, the radial boreholes with azimuths between 15°and 45° can create a guiding effect. Moreover, the guidance strength of the radial borehole decreases as the azimuth of the radial borehole increases. When the azimuth of a radial borehole increases by 30°, the guidance factor increases 2.6 times. When the azimuth of a radial borehole is 45°, the hydraulic fracture deflects and propagates along the maximum principle stress line after extending for 6.02 m. Thus, a single radial borehole with an azimuth of 45° does not significantly guide a hydraulic fracture to propagate along itself. Therefore, in order to make the research more practical, the follow-up studies take an azimuth angle of 30° as a basic condition.

#### 4.2. Horizontal In-Situ Stress Differences

_{H}= 41 MPa, and the same other parameters. The minimum horizontal main stresses are respectively 39, 36 and 33 MPa, and the simulation results are shown in Figure 7. The guidance factors corresponding to horizontal in-situ stress differences of 2, 5 and 8 MPa are respectively 0.010, 0.026, and 0.036, which shows that larger horizontal in-situ stress difference creates worse guidance strength of the radial borehole on the hydraulic fracture. Under the given parameters, when the horizontal in-situ stress difference is 8 MPa, the hydraulic fracture deflects and propagates along the maximum principle stress after extending for 7.53 m, with the smallest diversion radius and the largest deflection angle of fracture, and the weakest radial borehole guidance. When the horizontal in-situ stress difference is 5 MPa, the hydraulic fracture deflects and propagates along the maximum principle stress after extending for 10.12 m. Moreover, when the horizontal in-situ stress difference is 2 MPa, the hydraulic fracture diverts after extending for 18.50 m and propagates along the maximum main stress direction. The hydraulic fracture presents the largest diversion radius because it is affected by the radial borehole stress, and the deflection angle of the fracture is small and the guidance is obvious. When the horizontal in-situ stress difference increases by 6 MPa, the guidance factor increases 3.6 times. Therefore, in order to make the research more practical, the follow-up studies take the horizontal in-situ stress differences of 3 MPa as a basic condition to ensure an effective radial borehole guidance.

#### 4.3. Radial Borehole Diameter

#### 4.4. Length of Radial Borehole

#### 4.5. Young’s Modulus of Reservoir Rock

#### 4.6. Poisson’s Ratio of Reservoir Rock

#### 4.7. Reservoir Permeability

^{−3}μm

^{2}, 10 × 10

^{−3}μm

^{2}and 100 × 10

^{−3}μm

^{2}, and the same other parameters were established, and the simulation results are shown in Figure 12. When the reservoir permeability is 1 × 10

^{−3}μm

^{2}, the hydraulic fracture basically propagates along the orientation of the radial borehole without deflection, and its guidance factor is 0.004, which is the strongest guidance strength. When the reservoir permeability is 10 × 10

^{−3}μm

^{2}, the hydraulic fracture also propagates along the radial borehole orientation without obvious deflection, and its guidance factor is 0.010, and its distance between hydraulic fracture and radial borehole is larger than that of the model with 1 × 10

^{−3}μm

^{2}reservoir permeability. When the reservoir permeability is 100 × 10

^{−3}μm

^{2}, the hydraulic fracture deflects after extending for 16.72 m, and its guidance factor is 0.017, which represents the weakest guidance strength. When the reservoir permeability increases by 100 times, the guiding factor increases 4.3 times as much. Thus, the increased reservoir permeability weakens the radial borehole guidance. The smaller permeability tends to create a pressured area around the radial borehole and guide the propagation of a hydraulic fracture toward an ideal orientation.

#### 4.8. Fracturing Fluid Viscosity

#### 4.9. Fracturing Fluid Injection Rate

^{3}/min were established to analyze the influence of fracturing fluid injection rate on hydraulic fracture propagation, and the simulation results are shown in Figure 14. When the injection rate fracturing fluid is 1 m

^{3}/min, the hydraulic fracture deflects after extending 13.12 m along the radial borehole orientation, and its guidance factor is 0.023, which is the weakest guidance strength among the four injection rates. When the fracturing fluid injection rate is 3 m

^{3}/min, the hydraulic fracture deflects after extending 15.71 m, and its guidance factor is 0.018. When the fracturing fluid injection rate is 6 m

^{3}/min, the hydraulic fracture deflects after extending for 17.14 m, and its guidance factor is 0.011. When the fracturing fluid injection rate is 9 m

^{3}/min, the hydraulic fracture basically propagates along the radial borehole orientation without deflection, and its guidance factor is 0.002. When the fracturing fluid injection rate increases by 8 m

^{3}/min, the guiding factor decreases by about 91%. Thus, an increased fracturing fluid injection rate strengthens the radial borehole guidance and creates better guidance for hydraulic fracture propagation. This conclusion is a very important basis for guiding field fracturing.

#### 4.10. Gray Correlation Analysis of Guidance Factors

## 5. Experimental Verification

^{3}) [36]. The rock mechanics test results of the artificial cores show an average Young modulus of 14 GPa, average Poisson's ratio of 0.22, and average tensile strength of 2.7 MPa. The vertical borehole is modeled with a steel tube, which has an OD of 28 mm, ID of 23 mm, and length of 270 mm. In order to model an open radial borehole resulting from a radial hydraulic jet, and guarantee unlikely deformation of pre-set radial boreholes while pouring cement mortar, a polytetrafluoroethylene (PTFE) tube is chosen to model the radial borehole after several trials, and a combination of drilling holes and cutting sieves in PTFE tube is adopted to obtain properties close to those of actual radial boreholes, which not only ensures enough strength to prevent deformation and stop cement mortar flowing into the PTFE tube when casting specimens, but also guarantees enough flowing channel. The PTFE tube has a length of 130 mm, and a diameter of 10 mm, is preset in the modeling wellbore.

## 6. Conclusions and Suggestions for Future Work

- (1)
- The influence of in-situ stress could be overcome by scientifically arranging the single radial borehole under certain reservoir conditions, which realizes directional propagation towards target area. Thus, the problem that the hydraulic fracture only propagates along the direction parallel to horizontal maximum in-situ stress, and the available wellbores fail to develop the remaining oil and trap reservoir, and complex multi-fractures tend to generate in the immediate vicinity of wellbore, which makes it hard to realize the deep penetration of fractures, are solved to improve the effectiveness of fracturing operations and recovery efficiency in oil fields.
- (2)
- The concept of ‘guidance factor’ is introduced for the first time to quantify the guidance of a radial borehole on hydraulic fractures. A large amount of simulation shows that the ‘guidance factor’ reflects the guidance of a radial borehole on hydraulic fractures, and larger guidance factor reflects weaker guidance strength.
- (3)
- A smaller radial borehole azimuth, horizontal in-situ stress difference and larger radial borehole diameter and length create stronger guidance strength, and vice versa. When the azimuth of the radial borehole increases from 15°to 45°, the guidance factor increases 2.6 times as much; when the horizontal in-situ stress difference increases from 2 MPa to 8 MPa, the guidance factor increases 3.6 times; when the wellbore diameter increases from 3 cm to 7 cm, the guidance factor decreases 75%; when the well length increases from 10 m to 20 m, the guidance factor decreases 69%.
- (4)
- Both reservoir physical properties and fracturing operation parameters influence the guidance of radial boreholes on hydraulic fractures. The increased Poisson’s ratio and injection rate strengthen the radial borehole guidance, and the increased Young modulus and permeability weaken the radial borehole guidance, both excessive high and low viscosity go against radial borehole guidance of hydraulic fractures, and a fracturing fluid viscosity between 50–100 mPa·s creates the best guidance on propagation of hydraulic fractures.
- (5)
- The gray correlation analysis results show that the influence level (from strong to weak) of the above factors on radial borehole guidance may be listed as follows: horizontal in-situ stress differences > azimuth > borehole diameter > length > fracturing fluid injection rate > Young modulus of rock > reservoir permeability > fracturing fluid viscosity > Poisson’s ratio. The parameters of the radial borehole, physical property parameters of the reservoir and fracturing operation parameters together influence the guidance strength of a radial borehole on hydraulic fractures.
- (6)
- The numerical model is based on real rock parameters from a practical field. It is recommended that the numerical model of different fields and areas should be established based on their rock physical mechanical parameters, and the influence of different factors on guidance be analyzed to obtain radial borehole parameters and fracturing operation parameters applicable to their conditions, which is beneficial to improving the fracturing success rate.
- (7)
- The experimental results show that the guidance of a single radial borehole on hydraulic fracture propagation is limited by the radial borehole azimuth and horizontal in-situ stress difference. A single radial borehole with larger azimuth and larger horizontal in-situ stress difference has poor guidance of the directional propagation of hydraulic fractures. The experimental results are consistent with the numerical simulation results, which shows that the numerical simulation results are reliable to some extent.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic diagram of hydraulic fracture-directed propagation guided by a single radial well.

**Figure 2.**Schematic diagram of fracture surface simulation and level set function on the calculation point.

**Figure 3.**Wireframe view: (

**a**) A model without radial well and (

**b**) B model with the single radial well.

**Figure 5.**Simulation results of hydraulic fracture propagation guided by the single radial well in different azimuths: (

**a**) 15°; (

**b**) 30°; and (

**c**) 45°.

**Figure 7.**Simulation results of hydraulic fracture propagation guided by the single radial well under the different horizontal principal stress differences: (

**a**) σh = 39 MPa; (

**b**) σh = 36 MPa; and (

**c**) σh = 33 MPa.

**Figure 8.**Simulation results of hydraulic fracture propagation guided by a single radial well under radial well different diameters: (

**a**) Φ = 0.03 m; (

**b**) Φ = 0.05 m; and (

**c**) Φ = 0.07 m.

**Figure 9.**Simulation results of hydraulic fracture propagation guided by the single radial well under the different radial well lengths: (

**a**) 10 m; (

**b**) 15 m; and (

**c**) 20 m.

**Figure 10.**Simulation results of hydraulic fracture propagation guided by a single radial well under different Young’s modulus of reservoir rock: (

**a**) 13 GPa; (

**b**) 23 GPa; and (

**c**) 33 GPa.

**Figure 11.**Simulation results of hydraulic fracture propagation guided by a single radial well under different Poisson ratios of reservoir rock: (

**a**) 0.15; (

**b**) 0.2; and (

**c**) 0.25.

**Figure 12.**Simulation results of hydraulic fracture propagation guided by a single radial well under different reservoir permeabilities: (

**a**) 1 × 10

^{−3}μm

^{2}; (

**b**) 10 × 10

^{−3}μm

^{2}; and (

**c**) 100 × 10

^{−3}μm

^{2}.

**Figure 13.**Simulation results of hydraulic fracture propagation guided by a single radial well under different fracturing fluid viscosities: (

**a**) 1 mPa·s; (

**b**) 50 mPa·s; (

**c**) 100 mPa·s; and (

**d**,

**b**) 150 mPa·s.

**Figure 14.**Simulation results of hydraulic fracture propagation guided by the single radial well under the different fracturing fluid injection rates: (

**a**) 1 m

^{3}/min; (

**b**) 3 m

^{3}/min; (

**c**) 6 m

^{3}/min; and (

**d**) 9 m

^{3}/min.

**Figure 15.**Fracture morphology after hydraulic fracturing of sample 1#: (

**a**) the overall picture before opening; (

**b**) the inner picture after opening.

**Figure 16.**Fracture morphology after hydraulic fracturing of sample 2#: (

**a**) the overall picture before opening; (

**b**) the fracture initiates in the heel of radial borehole; (

**c**–

**e**) the multi-branch fractures occur in the core.

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

Reservoir saturation | 1 | Poisson ratio of rock | 0.25 |

Initial pore pressure | 20 MPa | Young’s modulus of rock | 12.9 GPa |

Initial porosity | 0.16 | Reservoir permeability | 60 × 10^{−3} μm^{2} |

Horizontal maximum principal stress (σ_{H}) | 41 MPa | Filtration coefficient | 10^{−10} m s^{−1} |

Horizontal minimal principal stress (σ_{h}) | 36 MPa | Injection rate of fracturing fluid | 3.2 m^{3} min^{−1} |

Overburden stress | 45 MPa | Fracturing fluid viscosity | 50 mPa·s |

Tensile strength of rock | 3.0 MPa | Fracturing fluid density | 9525 kg/m^{3} |

Casing diameter | 139.7 mm | Reservoir model size (diameter) | 40 m |

No. | Parameters | Correlation Coefficient |
---|---|---|

1 | Radial well azimuth | 0.7680 |

2 | Radial well diameter | 0.7537 |

3 | Radial well length | 0.7485 |

4 | Horizontal principal stress difference | 0.7921 |

5 | Young’s modulus of rock | 0.7465 |

6 | Poisson ratio of rock | 0.5312 |

7 | Reservoir permeability | 0.7367 |

8 | Fracturing fluid viscosity | 0.7354 |

9 | Injection rate of fracturing fluid | 0.7476 |

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## Share and Cite

**MDPI and ACS Style**

Guo, T.; Qu, Z.; Gong, F.; Wang, X.
Numerical Simulation of Hydraulic Fracture Propagation Guided by Single Radial Boreholes. *Energies* **2017**, *10*, 1680.
https://doi.org/10.3390/en10101680

**AMA Style**

Guo T, Qu Z, Gong F, Wang X.
Numerical Simulation of Hydraulic Fracture Propagation Guided by Single Radial Boreholes. *Energies*. 2017; 10(10):1680.
https://doi.org/10.3390/en10101680

**Chicago/Turabian Style**

Guo, Tiankui, Zhanqing Qu, Facheng Gong, and Xiaozhi Wang.
2017. "Numerical Simulation of Hydraulic Fracture Propagation Guided by Single Radial Boreholes" *Energies* 10, no. 10: 1680.
https://doi.org/10.3390/en10101680