# A Numerical Study on the Diversion Mechanisms of Fracture Networks in Tight Reservoirs with Frictional Natural Fractures

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

#### 2.1. Governing Equations of Hydraulic Fracturing Problems

#### 2.2. Crack Propagation Criterion

#### 2.3. The Cross Criterion between HF and Frictional NF

#### 2.4. XFEM and Discretization of the Governing Equations of the Hydraulic Fracturing Problem

## 3. Results and Discussion

#### 3.1. Verification of the XFEM Model

#### 3.2. Effect of the Location of Natural Fractures on the Diversion of Fracture Network Propagation

#### 3.3. Effect of Horizontal Stress Differences on the Diversion of Fracture Network Propagation

#### 3.4. Effect of the NF–HF Intersection Angle on the Diversion of Fracture Network Propagation

#### 3.5. Effect of Fluid Viscosity on the Diversion of Fracture Network Propagation

## 4. Conclusions

- (1)
- Fracture diversion propagation will occur near the two tips of the opened NF after an HF is intersecting with an NF. The numerical results show that some key factors such as the NF position, the NF–HF intersection angle, the horizontal stress differences, and the fluid viscosity have a significant impact on the diversion propagation in the upper and lower parts of the opened NF.
- (2)
- For a constant length of NF (7 m), the upper length of the DF decreases by about 2 m with a 2 m increment of the upper length of the NF (${L}_{upper}$), while the length of the DF increases 9.06 m, with the fluid viscosity increased from 1 to 100 mPa.s; (2) the deflection angle in the upper parts increases by 30.8° with the stress difference increased by 5 MPa, while the deflection angle increases by 61.2° with the intersection angle decreased from 75° to 45°.
- (3)
- The longer the upper parts of the original NF are, the more difficult it is for the opened NF to divert away from the upper tip of the NF under the conditions of an isotropic stress state, while the lower parts of the original NF is more easily diverted to the right-hand side than the upper parts of the original NF. The NF–HF intersection angle will have a significant impact on the diversion propagation of the primary HF and the secondary opened NFs.
- (4)
- In general, the distributions of fracture aperture, net pressure, and flow rate reveal asymmetrical characteristics for the secondary hydraulically driven fractures. For the distribution of Von-Mises stress, there is usually a concentrated stress zone area near the turning point of the secondary cracks, which corresponds to the inflection points on the curves of the fracture aperture and net pressure.
- (5)
- The diversion mechanisms of the fracture network are the results of the combined action of all factors. This will provide a new perspective on the mechanisms of fracture network generation. Future work should determine the primary and secondary relations of various factors by means of experiments and numerical calculation.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

XFEM | Extended Finite Element Method |

DEM | Discrete Element Method |

NMM | Numerical Manifold Method |

SRV | Stimulated Reservoir Volume |

HF | Hydraulic Fracture or Hydraulically Driven Fracture or Hydro-Fracture |

NF | Natural Fracture |

DF | Diverted Fracture |

PFP | Preferred Fracture Plane |

ELP | Enhanced Local Pressure |

## Appendix A

Input Parameter | Value |
---|---|

Young’s Modulus, E | 20 GPa |

Poisson’s ratio, $\nu $ | 0.2 |

Fracture toughness, ${K}_{\mathrm{IC}}$ | 0.1 $\mathrm{MPa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{\frac{1}{2}}$ |

The consistency index of fracturing fluid, K | 0.84 $\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{\mathrm{n}}$ |

Injection rate, ${Q}_{0}$ | 0.001 ${\mathrm{m}}^{2}/\mathrm{s}$ |

Viscosity, $\mu $ | 0.1 $\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

Dimensionless fracture toughness, ${K}_{m}$ | 0.313 |

Injection time, t | 30 s |

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**Figure 5.**The crack propagation paths at different lengths of lower and upper parts of the NF. In this figure, the black, blue, and red dotted lines, respectively, denote the original HF, the initial NF, and the diverted fracture (DF).

**Figure 7.**The fracture aperture curves of the DF along the fracture length at different lengths of the lower and upper parts of the NF. The distance in the x-axis is along the direction from the lower parts to the upper parts of the DF.

**Figure 8.**The net pressure curves of the DF along the fracture length at different lengths of the lower and upper parts of the NF. The distance in the x-axis is along the direction from the lower parts to the upper parts of the DF.

**Figure 9.**The flow rate curves in the DF along the fracture length at different lengths of the lower and upper parts of the NF. The distance in the x-axis is along the direction from the lower parts to the upper parts of the DF.

**Figure 10.**The crack propagation paths at different levels of remote horizontal principle stress difference.

**Figure 11.**Von-Mises stress distributions at different levels of remote horizontal principle stress difference.

**Figure 12.**The fracture aperture curves of the DF along the fracture length at different levels of remote horizontal principle stress difference. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 13.**The net pressure curves of the DF along the fracture length at different levels of remote horizontal principle stress difference. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 14.**The flow rate curves of the DF along the fracture length at different levels of remote horizontal principle stress difference. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 16.**Von-Mises stress distributions at different levels of intersection angle between HF and NF.

**Figure 17.**The fracture aperture curves of the DF along the fracture length direction at different levels of intersection angle between HF and NF. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 18.**The net pressure curves of the DF along the fracture length direction at different levels of intersection angle between HF and NF. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 19.**The flow rate curves of the DF along the fracture length direction at different levels of intersection angle between HF and NF. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 22.**The fracture aperture curves of the DF along the fracture length at different levels of viscosity of fracturing fluid. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 23.**The net pressure curves of the DF along the fracture length at different levels of viscosity of fracturing fluid. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

**Figure 24.**The flow rate curves of the DF along the fracture length direction at different levels of viscosity of fracturing fluid. The distance in the x-axis is along the direction from the lower part to the upper part of the DF.

Input Parameter | Value |
---|---|

Young’s Modulus, E | 20 GPa |

Poisson’s ratio, $\nu $ | 0.2 |

Rock density, $\rho $ | 2460 $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Friction coefficient of NF, ${\mu}_{\mathrm{f}}$ | 0.3 |

Cohesion of the NF, ${S}_{0}$ | 0 MPa |

Fracture toughness, ${K}_{\mathrm{IC}}$ | 1.0 $\mathrm{MPa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{\frac{1}{2}}$ |

Tensile strength, ${T}_{0}$ | 1.5 MPa |

Unconfined compression strength, $\mathrm{UCS}$ | 100 MPa |

Apparent viscosity of fracturing fluid, $\mu $ | 0.1 $\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$ |

The consistency index of fracturing fluid, K | 0.84 $\mathrm{Pa}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{\mathrm{n}}$ |

The flow behavior index of fracturing fluid, n | 0.53 |

Dynamic viscosity index, m | 2.0 |

Fluid pump rate, ${Q}_{0}$ | 0.001 ${\mathrm{m}}^{2}/\mathrm{s}$ |

Pore pressure, ${P}_{0}$ | 5 MPa |

Maximum horizontal stress, ${\sigma}_{\mathrm{H}}$ | 5 MPa |

Minimum horizontal stress, ${\sigma}_{\mathrm{h}}$ | 5 MPa |

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## Share and Cite

**MDPI and ACS Style**

Wang, D.; Shi, F.; Yu, B.; Sun, D.; Li, X.; Han, D.; Tan, Y. A Numerical Study on the Diversion Mechanisms of Fracture Networks in Tight Reservoirs with Frictional Natural Fractures. *Energies* **2018**, *11*, 3035.
https://doi.org/10.3390/en11113035

**AMA Style**

Wang D, Shi F, Yu B, Sun D, Li X, Han D, Tan Y. A Numerical Study on the Diversion Mechanisms of Fracture Networks in Tight Reservoirs with Frictional Natural Fractures. *Energies*. 2018; 11(11):3035.
https://doi.org/10.3390/en11113035

**Chicago/Turabian Style**

Wang, Daobing, Fang Shi, Bo Yu, Dongliang Sun, Xiuhui Li, Dongxu Han, and Yanxin Tan. 2018. "A Numerical Study on the Diversion Mechanisms of Fracture Networks in Tight Reservoirs with Frictional Natural Fractures" *Energies* 11, no. 11: 3035.
https://doi.org/10.3390/en11113035