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# Using Cohesive Zone Model to Simulate the Hydraulic Fracture Interaction with Natural Fracture in Poro-Viscoelastic Formation

First, we establish the model and determine the included angle through the establishment of the model also determined here; 60° is taken as an example (Figure 4). The model is then given basic properties such as Young’s modulus and Poisson’s ratio. After determining the nature of the rock to be simulated, assembly integration was performed. Different densities of mash are needed because it can facilitate convergence of calculation results. We define the initial conditions by step, keeping the ground stress balanced, then inject the fluid, set the time and boundary conditions, and start crack propagation.

Figure 4. The mesh of hydraulic fracture interacts with natural fracture at 60°.

## Results and Discussion

Hydraulic fracture geometry is largely dependent on the different parameters and the different parameter sizes have different impacts on hydraulic fracture geometries. The variable includes in situ stress, interaction angle, tensile strength, fracturing fluid injection rate, and fracturing fluid injection fluid viscosity. During every parameter analysis, we only change one parameter; nevertheless, the other parameters remain unchanged according to the basic model in Table 2.

Table 2. Input Parameters.

### Interaction Angle

The interaction angle has enough effects on the hydraulic fracture propagation. By changing the angle, the influence of different angles on the interaction between HF (Hydraulic Fracture) and NF (Natural Fracture) can be obtained. In this paper, change of NFs is compared at 30 degrees, 45 degrees, 60 degrees, and 75 degrees, respectively, under the same other conditions. To verify the results of fracture intersection, the comparison between modeling results and the indoor experiment are shown in Figure 5 [7]. From the picture, the different legend means the result of HF intersection of NF.

It can be seen that numerical simulation results and the lab experimental results are in good agreement. This demonstrates that the CZM could simulate HF intersecting with NF. As is shown in Figure 5, the smaller the interaction angle, the more fractures produced by hydraulic fracturing tend to divert into the NFs and lead to an opening in the NFs. Moreover, for smaller angles of intersection, HF is more likely to divert and propagate along the obtuse angle branch of NF. With high approach angle, the HF is more likely to cross the NF.

Figure 5. The model verification with experimental results.

### In Situ Stress

The in situ stress difference is an essential parameter in hydraulic fracturing design. The horizontal stresses are the result of the poro-elastic deformation of the rocks plus externally applied tectonic forces. The maximum horizontal stress and the minimum horizontal stress always control the HF initiation and propagation.

In this simulation, the maximal horizontal stress is 8 MPa, and we then change the minimal horizontal stress in 8 MPa, 5 MPa, and 2 MPa, respectively in the three cases, as well as keeping the other parameters consistent with the basic model to observe the relationship between HF and NF. It can be seen in Figure 6 that the greater the stress difference, the more easily HFs tend to cross through NFs. The smaller the stress difference, the more conducive to the expansion of NFs [15,16].

Figure 6. Crack geometry under different in situ stress.

### Crack Tensile Strength

Rock tensile strength is a measure of the force required to achieve its point of failure. The tensile strength is a characteristic property of the rock and the crack tensile strength obviously less than the shale. In the simulation, the shale tensile strength is taken as 6 MPa, while the crack tensile strength is 1 MPa, 3 MPa, and 5 MPa, respectively, in the three cases. The different tensile strength NF means different strength of NF effect on HF propagation. The simulation results in Figure 7 show the HF conquers the shale tensile strength, the HF initiation, and propagation.

Therefore, the tensile stress difference ∆St = rock tensile strength—crack tensile strength. When the shale tensile strength is larger than the crack tensile strength, it is easy to generate the bi-wing fracture. When the shale tensile strength is moderately larger than the crack tensile strength, it is easy to generate the single-wing fracture. When the shale tensile strength is almost equal to the crack tensile strength, HF crosses NF [23].

Figure 7. Crack geometry in difference tensile strength.

#### Fracturing Fluid Injection Rate

Injection rates have a great impact on fracture geometry, as shown in Figure 8, and they influence fracture patterns. In this simulation, the fracturing fluid injection rate is taken as 3 m3/min, 6 m3/min, and 12 m3/min. The crack geometries of the three examples are described in Figure 8. When the injection rate is high, HF is more likely to cross NFs, and at the same time fill the entire crack. This increases not only the fracture length but also the fracture width when increasing the fracturing fluid injection rate.

Figure 8. Crack geometry in different fracturing fluid injection rate.

## Fracturing Fluid Injection Viscosity

When injecting hydraulic fluid into the shale, the fracturing fluid penetrates the rock gap and crack, generating fracturing fluid loss. The fracturing fluid loss is mainly decided by fracturing fluid viscosity. In this simulation, the fracturing fluid viscosity is taken as 4 mPa∙s, 8 mPa∙s, and 16 mPa∙s. The simulation results in Figure 9 show the higher fracturing fluid viscosity increase fluid flow resistance and promote the establishment of a cake on the hydraulic rupture surface, thereby reducing fluid loss [24]. When the viscosity is high, HF is more likely to cross NFs. Injecting in a low viscosity is helpful for forming a complex network structure [25].

Figure 9. Crack geometry in different fracturing fluid injection viscosity.

## Conclusions

In this paper, we developed a fully coupled poro-viscoelastic fracture propagation model to investigate the characteristics of hydraulic fracturing in shale formation. The numerical simulation adopted is the finite-element method and detailed implementation steps are presented. Simulations of poro-viscoelastic fracture propagation confirm the generally accepted notion that creep behavior has a detrimental effect on hydraulic fracturing efficiency. From the analysis, it is found that fractures propagating in poro-viscoelastic formation tend to be wider yet shorter than that in poro-elastic formation, which is mainly due to viscous energy dissipation.

By using a cohesive finite-element model, different factors that effect HF interacting with NF have been investigated and clearly simulated. On the one hand, the interaction between HF and NF can promote the expansion of flow paths; on the other hand, it may lead to an extra leak-off. Using TSL, the tip of the fracture process is described in detail. From five different aspects, the interaction between HF and NF has been analyzed. As the angle of interaction continues to decrease (from 75°), HF changes from passing through NF to diverting to NF; reducing the difference between the maximum horizontal stress and the minimum horizontal stress to reduce the differential stress can also achieve a similar effect; injecting in a suitable rate or viscosity is helpful to make full use of NFs and improve the fracturing effect.

### Author Contributions

Y.S. wrote the paper and designed the model, Z.C. conceive the problem, H.Y. editing paper suggestion and D.W. analyzed the data, Y.Z. provide paper editing suggestion.

#### Funding

The authors would like to give their sincere gratitude to the National Science Foundation of China (Grant No. 51804033), China Postdoctoral Science and Foundation (Grant No. 2018M641254), Beijing Postdoctoral Research Foundation (Grant No. 2018-ZZ-045).

#### Conflicts of Interest

The authors declare no conflict of interest.

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#### Contact:

1. Yu Suo: [email protected], Tel.: +610-449-821-012
2. [email protected] (Z.C.)
3. [email protected] (H.Y.)
4. [email protected]
5. [email protected]