# Numerical Investigation of Hydraulic Fracture Propagation Based on Cohesive Zone Model in Naturally Fractured Formations

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. Fluid Flow within the CZM

#### 2.2. Damage Initiation and Evolution of Cohesive Elements

_{n}, T

_{s}and T

_{t}are the normal and shear stress components predicted by the elastic traction–separation behavior for current separation without damage.

_{m}, that accounts for combining the effects of δ

_{n}, δ

_{s}and δ

_{t}:

_{m}is the relative effective displacement attained over the course of loading. The superscripts max, f and 0 represent the max relative displacement, displacement when fractures are completely damaged, and the displacement at damage initiation, respectively.

#### 2.3. Equilibrium Equation

_{w}. According to the principle of virtual work, the finite element method equilibrium equation can be formulated as

_{w}are the Boit effective stress and pore pressure, respectively; $\delta \epsilon $ and $\delta v$ are the virtual strain rate and virtual velocity, respectively; t and f are the surface displacement per unit area and body force per unit volume, repectively; and I is the unit matrix.

## 3. Results and Discussion

#### 3.1. Interference between a Single Natural Fracture and Hydraulic Fractures

^{3}/s, all the HFs propagate along the NF with two fracture directions. However, as $\nabla \sigma $ increases, the HF can open the NF, but only in the direction of the large intersection angle. When the stress difference is large enough, all the HFs will cross the NF to form a long planar fracture, which demonstrates that stress difference is the key factor when HFs intersect with the NF. The HFs tend to cross the NF at any approaching angle when the stress difference is high. Furthermore, a HF is more likely to open and propagate along the NF when the stress difference is low. The smaller the approaching angle of the NF, the higher the critical difference stress for HF crossing will be. The lower the approaching angle of the NF, the easier it is to open two sides of the NF. The numerical simulation results are consistent with experimental results that have been published [25,28,29], as shown in Figure 7. For better visualization, we connected the critical opening pressure at different approaching angles, which is presented as a red line and acts as a comparison standard of our numerical model. It can be seen that HF is more likely to cross the NF above the red critical line; however, the HF is inclined to open the NF below the critical line.

#### 3.2. Horizontal Stress Difference

^{3}/s, and µ = 1 mPa·s. All models in this section adopted a two-clusters configuration with a spacing of 40 m. The results show that HFs tend to cross the NFs and form a less complex fracture network as the horizontal stress difference increases, which agrees well with the findings of Zou et al. [7]. In the isotropic case of ∇σ = 2 MPa, the HFs that initiate from the two perforation clusters mainly open and propagate along the NFs, resulting in a highly complex fracture network near the horizontal wellbore. However, the geometry of HFs change with the propagation process owing to stress interference. As shown in Figure 8a, two HFs propagate in opposite directions, deviating from each other, which leads to more opportunities for the right HFs to open NFs and for the left HFs to cross the NFs. When the stress difference increases, the complexity of the fracture network reduces, and fractures propagate in a more straight manner within the same injection process [11]. In addition, during a propagation process, HF growth can be suppressed because of back stress induced by other propagating HFs (Figure 8b,c). Under a horizontal stress difference of 8 MPa (Figure 8d), almost all NFs are crossed by HFs instead of deflecting, which causes less complex fracture networks.

#### 3.3. Spacing of Perforation Clusters

^{3}/s, µ = 1 mPa·s, $\nabla \sigma $ = 3 MPa were set for the simulation. It can be seen that different cluster spacing has a significant effect on fracture morphology. As the cluster spacing increases, the influence of the stress shadow among adjacent fractures is alleviated, which results in more opening phenomena among HFs and NFs. Given S = 30 m in Figure 10a, the multiple HFs initiate from the wellbore and then intersect with the NFs. With the fracturing process, the stress shadow becomes more severe and obvious suppression can be observed in one of the NFs. Further, the suppressed fractures propagate, becoming wider and shorter. However, unsuppressed fractures extended rapidly, resulting in a longer and narrower fracture network. When the cluster spacing increases (Figure 10b–d), the HFs propagate quickly from the wellbore and experience less suppression compared with the case of S = 30 m. The interference of each cluster decreases as cluster spacing increases. Meanwhile, the orientation of the crack tip is also alleviated under this condition, especially when S is larger than 60 m. For the purpose of maximizing the fracture network, deeper propagation should be provided to connect more NFs. Therefore, production can be improved by increasing the perforation cluster spacing within a relative range.

#### 3.4. Injection Rate of Fracturing Fluid

^{3}/s, Q = 0.0005 m

^{3}/s and Q = 0.001 m

^{3}/s. It can be seen that the fracturing of two clusters initiated at the wellbore and reactive NFs at an earlier stage of injection. With the fracturing procedure, the fractures divert in opposite directions at lower injection rates, as shown in Figure 12a,b. However, the morphologies of the two-cluster HFs change less when the fluid rate increases from 0.0001 to 0.0005 m

^{3}/s. When the injection rate reaches 0.001 m

^{3}/s, it has a significant influence on the complexity of the fracture network. As shown in Figure 12c, HFs propagate in a straighter manner perpendicular to the horizontal minimum principle stress, which results in more crossing phenomena and a less complex fracture network.

^{3}/s, which resulted in a reduction of flow channel and make the release of gas and oil difficult, as shown in Figure 13. However, a high injection rate can create larger far-field propagation of the fracture network [34], which is mainly due to reduced connection in the near wellbore and deeper propagation of the fracture network. Therefore, various injection rates should be adopted to maximize the complexity of the fracture network. The large fluid pressure within a fracture due to high injection rates can be used to form straighter fracture networks, and deeper propagation at the early stage of fracturing. A low injection rate can open more NFs and creates more diversions in the wider control area in the later period of fracturing. By controlling the injection rates, a large filed of fracture network can be realized.

#### 3.5. Viscosity of the Pumping Fluid

^{3}/s and 40 m, respectively. Figure 14a shows that HFs tend to open more NFs and propagate along the NFs at a low viscosity condition of µ = 1 mPa·s. Meanwhile, the preference for diverting to the opposite direction for each cluster fracture is enhanced with low viscosity fracturing fluid. However, Beguelsdijk et al. [35] found that a fracturing fluid with a high viscosity could induce separate propagation on an optimal fracture surface. For this reason, more opening of NFs may increase the complexity of the fracture network, which will increase the possibility of separate propagation. In addition, more straight fractures will be formed along the direction of maximum horizontal stress, which will result in the formation of a simple fracture network, as shown in Figure 14b,c. The morphology and complexity of fractures will change obviously when the viscosity increases from 1 to 50 mPa·s; however, few transformations can be obtained when the viscosity varies from 50 to 100 mPa·s. Hence, it can be concluded that lower range viscosities have a greater effect on fracture morphology than higher range viscosities.

#### 3.6. Number of Perforation Clusters

^{3}/s, µ = 1 mPa·s, S = 40 m and $\nabla \sigma $ = 3 MPa, the fracture propagation morphologies are comparable in these cases. In this section, three types of perforation configurations (two, three and four clusters) were modeled to investigate the effect of perforation application. The results reveal that the interference of the stress shadow plays a major role in fracture propagation, as it may make the propagation undesirable for production. The resulting mechanical action alters the value and direction of in situ stress, which results in unexpected suppression of the fracture’s propagation. As shown in the two-clusters perforation configuration (Figure 16(a1,a2)), back stress causes the left fracture to be suppressed, exhibiting shorter and wider networks compared with the right fracture. The middle fracture seems to be significantly influenced, forming shorter and wider HFs in Figure 16(b1,b2), compared with the fractures at either side. For the four-cluster configuration (Figure 16(c1,c2)), one fracture can break though and propagate into the deep reservoirs rapidly, while the residuary fractures tend to be suppressed more or less. Further, it can be observed from Figure 16 that the fracturing time will affect the morphology of the HFs to differing degrees. All of the configurations show that there is less influence on the morphology of HFs in the early stage of fracturing (i.e., T = 500 s), which is probably due to less spreading of net pressure in the fracture network.

## 4. Conclusions

- (1)
- The horizontal differential stress dominates main propagation patterns of the hydraulic fracture network. The morphology of a hydraulic fracture may change from being randomly diverted to more straight as the horizontal differential stress increases. The fluid driven fracture nearly crosses NFs to extend simple transverse fractures when the differential stress exceeds 8 MPa at an approaching angle of 60°, according to our FEM models.
- (2)
- The viscosity and injection rate of the fracturing fluid affect the development of the fracture network at different fracturing times. The fracture network area can reach its maximum with various injection rates and viscosities during different fracturing stages. Using a higher injection rate and fluid viscosity appears to raise the net pressure, which will increase the capacity of a HF to cross the NFs near the wellbore and facilitate a deeper propagation into the formations.
- (3)
- Using a short cluster spacing at 30 m will result in unexpected consequences where one or more fractures will be suppressed to form short and simple transverse fractures. Nevertheless, the phenomenon of suppression could weaken when the cluster spacing distance increases from 40 to 60 m.
- (4)
- Cluster spacing should be taken into consideration as the increase of cluster numbers causes enhanced stress interference. Due to tremendous suppression and merging of fractures under multiple stress interference, increasing the cluster number is meaningless with lower cluster spacing.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 6.**Effect of approaching angle and stress difference on hydraulic fracture propagation. (

**a1**–

**a3**) Approaching angle = 45°: stress difference = 2, 8, and 12 MPa, respectively; (

**b1**–

**b3**) approaching angle = 60°: stress difference = 2, 6, and 8 MPa, respectively; (

**c1**–

**c3**) approaching angle = 90°: stress difference = 2, 2.5, and 3 MPa, respectively.

**Figure 7.**Comparison between numerical simulation and experimental results with different angles of approach and horizontal differential stresses.

**Figure 8.**Effect of stress difference on hydraulic fracture propagation. (

**a**) Stress difference = 2 MPa; (

**b**) stress difference = 4 MPa; (

**c**) stress difference = 6 MPa; (

**d**) stress difference = 8 MPa.

**Figure 10.**Effect of cluster spacing on hydraulic fracture propagation. (

**a**) Cluster spacing = 30 m; (

**b**) cluster spacing = 40 m; (

**c**) cluster spacing = 50 m; (

**d**) cluster spacing = 60 m.

**Figure 12.**Effect of fluid rate on hydraulic fracture propagation. (

**a**) Injection rate = 0.0001 m

^{3}/s; (

**b**) injection rate = 0.0005 m

^{3}/s; (

**c**) injection rate = 0.001 m

^{3}/s.

**Figure 14.**Effect of viscosity on hydraulic fracture propagation. (

**a**) Fluid viscosity = 1 mPa∙s; (

**b**) fluid viscosity = 50 mPa∙s; (

**c**) fluid viscosity = 100 mPa∙s.

**Figure 16.**Effect of cluster numbers on hydraulic fracture propagation. (

**a1**,

**a2**) Two clusters fracturing scenario at fracturing time T = 500, 1200 s; (

**b1**,

**b2**) three clusters fracturing scenario at fracturing time T = 500, 1200 s; (

**c1**,

**c2**) four clusters fracturing scenario at fracturing time T = 500, 1200 s.

Categories | Variables | Values |
---|---|---|

Rock properties | Young’s modulus E (GPa) | 17.2 |

Poisson’s ratio v | 0.175 | |

Permeability k (mD) | 1 | |

Porosity (%) | 3.65 | |

Cohesive zone properties | Normal nominal stress | 1.4 (natural fracture) |

t_{n} (MPa) | 6 (hydraulic fracture) | |

First shear nominal stress | 8 (natural fracture) | |

t_{s} (MPa) | 12 (hydraulic fracture) | |

Second shear nominal stress | 8 (natural fracture) | |

t_{t} (MPa) | 12 (hydraulic fracture) | |

Normal fracture energy | 300 (natural fracture) | |

G_{n} (J/m^{2}) | 2000 (hydraulic fracture) | |

First shear fracture energy | 1500 (natural fracture) | |

G_{s} (J/m^{2}) | 3000 (hydraulic fracture) | |

Second shear fracture energy | 1500 (natural fracture) | |

G_{t} (J/m^{2}) | 3000 (hydraulic fracture) | |

Pumping parameters | Fluid viscosity µ (mPa·s) | 1/50/100 |

Injection rate (m^{3}/s) | 0.0001/0.0005/0.001 | |

Initial conditions | Pore pressure (MPa) | 51.2 |

In situ stress | Horizontal stress (MPa) | 51.2–66.2 |

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**MDPI and ACS Style**

Li, J.; Dong, S.; Hua, W.; Li, X.; Pan, X.
Numerical Investigation of Hydraulic Fracture Propagation Based on Cohesive Zone Model in Naturally Fractured Formations. *Processes* **2019**, *7*, 28.
https://doi.org/10.3390/pr7010028

**AMA Style**

Li J, Dong S, Hua W, Li X, Pan X.
Numerical Investigation of Hydraulic Fracture Propagation Based on Cohesive Zone Model in Naturally Fractured Formations. *Processes*. 2019; 7(1):28.
https://doi.org/10.3390/pr7010028

**Chicago/Turabian Style**

Li, Jianxiong, Shiming Dong, Wen Hua, Xiaolong Li, and Xin Pan.
2019. "Numerical Investigation of Hydraulic Fracture Propagation Based on Cohesive Zone Model in Naturally Fractured Formations" *Processes* 7, no. 1: 28.
https://doi.org/10.3390/pr7010028