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The formation of the fracture network in shale hydraulic fracturing is the key to the successful development of shale gas. In order to analyze the mechanism of hydraulic fracturing fracture propagation in cemented fractured formations, a numerical simulation about fracture behavior in cemented joints was conducted based firstly on the block discrete element. And the critical pressure of three fracture propagation modes under the intersection of hydraulic fracturing fracture and closed natural fracture is derived, and the parameter analysis is carried out by univariate analysis and the response surface method (RSM). The results show that at a low intersecting angle, hydraulic fractures will turn and move forward at the same time, forming intersecting fractures. At medium angles, the cracks only turn. At high angles, the crack will expand directly forward without turning. In conclusion, low-angle intersecting fractures are more likely to form complex fracture networks, followed by medium-angle intersecting fractures, and high-angle intersecting fractures have more difficulty in forming fracture networks. The research results have important theoretical guiding significance for the hydraulic fracturing design.

The porosity and permeability of shale are extremely low. To achieve the effective development of shale gas, fractures are required to provide flow channels for gas. Therefore, hydraulic fracturing is applied to increase fractures in shale. Natural weakness planes contribute to the formation of a complex hydraulic fracture network. Fortunately, many shale outcrops, cores, and image logs show that there are weakness planes including joints, faults, bed-parallel fractures, early compacted fractures, and fractures associated with concretions in shale reservoir [

Weakness planes in rocks.

Cemented fractures of outcrops

Natural fractures network of outcrops

Natural fractures of cores (depth 3635 m)

Cemented fractures of cores (depth 1531 m)

The experimental study primarily focuses on investigating the behaviors of a hydraulic fracture intersecting with a natural fracture. Hydraulic fracturing tests at a laboratory scale have been carried out at different approximation angles and differential stresses, which were run for a condition where the hydraulic fracture opened existing fractures to determine if more constructive crossing interaction could be obtained. For hydraulic fractures, when the angle of approach angle was 30 degrees, existing fractures tend to open and prevent the induced fracture from crossing. At an angle of 60 and 90, hydraulic fractures tend to cross existing fractures when the differential stress was high enough. Based on the results, an elastic solution has been used as a basis for a hydraulic/natural fracture interaction criterion for fracture opening [

Additional numerical analyses have also been performed to develop propagation criteria for determining propagation direction when a hydraulic fracture encounters natural fractures. For example, the discrete element method [

Previous studies mainly focus on the influence of damaged natural fractures on hydraulic fractures. The jointed surface of the weakness planes is not cracked due to cementation, but its strength is weaker. When the fracturing fluid reaches the weakness plane, the initiation and propagation of induced fractures are likely along the weakness plane. Therefore, the interaction between the weakness plane and the hydraulic fracture has an important influence on fracture propagation.

The purpose of this paper is to comprehensively consider the expansion path and morphology of hydraulic fractures intersecting with natural closed fractures under the shear and tensile action of fracture surfaces. The expansion patterns of hydraulic fractures after they meet with intersecting closed fractures are of great significance for the formation of complex fracture networks after fracturing. In this paper, a numerical simulation about fracture behavior in cemented joints was conducted based firstly on block discrete element. Then, the failure modes of hydraulic fractures in fractured formation are discussed, and the failure conditions under different failure modes are given. Based on the different failure conditions of the intersection, the fracture morphology with two boundaries after fracture failure at different intersecting angles is discussed. Besides, the influence of parameters on fracture morphology was analyzed by univariate analysis and response surface method (RSM).

The block discrete element method is a three-dimensional numerical modeling for advanced geotechnical analysis. The block discrete element method simulates the response of discontinuous media (such as jointed rock or masonry bricks) that is subject to either static or dynamic loading. The discontinuous material is represented as an assemblage of discrete blocks. Models may contain a mix of rigid or deformable blocks. Deformable blocks are defined by a continuum mesh of finite-difference zones, with each zone behaving according to a prescribed linear or nonlinear stress-strain law. The relative motion of the discontinuities is also governed by linear or nonlinear force-displacement relations for movement in both the normal and shear directions. Joint models and properties can be assigned separately to individual, or sets of, discontinuities.

The block discrete element method is based on the distinct element method (DEM) for discontinuum modeling. It differs from particle-based methods in its ability to represent a zero-initial porosity condition, as well as interlocked irregular block shapes that provide resistance to block rotation (moments) after contact breakage [

The thickness of the joint is far less than the scale of its plane, so its deformation characteristics are described by the stress-displacement relationship. The constitutive relation of the structural plane mainly studies the relationship between the stress of the structural plane and its normal deformation and tangential deformation. There are two stresses on the structure surface: normal stress

The Goodman element is adopted in this calculation; i.e.,

The Mohr-Coulomb equation for shear strength of the structure plane is

The contact friction joint model and the coulomb slip model are used in this calculation. It is assumed that the normal stress increment

It is assumed that the fracture surface is impermeable and fluid flows only within the fracture surface. The flow of fluid in the crack satisfies the cube law. Using the modified cube law,

In the hydromechanics coupling of fractures, the influence of mechanical deformation on fracture permeability is mainly manifested as the change of fracture aperture. In the elastic stage, the crack aperture is expressed as an equation related to the effective stress.

In the plastic stage, considering the expansion caused by a joint slip, the crack opening can be expressed as

The deformation of blocks is nonnegligible. The block is divided into tetrahedral units. The vertices of a tetrahedral element are called grid difference points. The motion equation is established at each node as follows:

The node force

For every time step, strain and rotation are related with displacement. Their forms are as follows:

The incremental method is applied in calculation. It is not limited to small strain problems. Therefore, the constitution model of deformation blocks takes incremental form. The equation is as follows:

Based on the block discrete element theory, the propagation behavior of hydraulic fractures in rock mass with cemented joints is simulated by the following model. The model including two cutters and one vertical joint that can simulate the behavior of fracture crossing and fracture containment is shown in Figure

Model for numerical simulation: (a) 3D model of rock with joints; (b) a side view of joints; (c) 3D model with elements.

The values of parameters.

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

Elasticity modulus | 20 GPa | Fluid viscosity | 0.0015 cp |

Poisson’s ratio | 0.25 | Fluid density | 1000 kg/m^{3} |

Density of rock | 2600 kg/m^{3} |
Depth | 3000 m |

Joint friction | 20° | Joint tension | 0 MPa |

Joint cohesion | 0 MPa | Initial vertical stress |

In this simulation, the

The time-dependent injection pressure was recorded. The result is shown in Figure

The history of injection pressure with time.

One of the main purposes of numerical simulation is to investigate the behavior of a fluid-driven crack in the rock with cemented joints. In other words, the shape of open fractures after fracking is one of the most important goals. The shape is described by the aperture of fractures in this study. The results are plotted in Figure

The aperture of joints: the spatial distribution of fractures with different angles between the vertical joint and two cutters: (a) 30-30, (b) 45-45, and (c) 90-90; the aperture of vertical joints with different angles between the vertical joint and two cutters: (d) 30-30, (e) 45-45, and (f) 90-90.

Another noteworthy feature different from the 2D model simulation is that fracture height is restrained by two cutters. In the 2D model, the fluid-driven crack must extend forward in their own form, penetrating or captured, when approaching two joints. However, in the 3D model, it is not obligatory for the fluid-driven crack extending forward as in the 2D. Figure

There are many weak surface structures, such as bedding and fractures, in shale due to the action of geological tectonic stress. It is more likely to be damaged in fracturing for the reason that the strength of the weakness plane is low. The fracture will be preferentially expanded along the weakness plane. Therefore, the intersection of the hydraulic fracture and the weakness plane is the key problem of fracture propagation in the formation with the weakness plane.

The intersection model of the hydraulic fracture and the natural fracture is shown in Figure

The intersection model of the hydraulic fracture and the natural fracture.

The failure modes of NF1 and NF2 under fluid action are different with different angles of fracture and principal stress. In general, the failure form of NF1 may be a tensile failure or a shear failure. After tensile failure, cracks will open directly. However, cracks do not necessarily open after shear failure. It also requires that the fluid pressure can overcome the normal stress on the fracture surface and prop up the fracture. The failure mode of NF2 without shear stress on the fracture surface can only be tensile failure. The failure forms are summarized in Table

The criteria for different failure forms of fractures.

Fracture | Failure forms | Criteria |
---|---|---|

NF1 | Shear failure and opening | |

Tensile failure | ||

NF2 | Tensile failure |

Once the critical conditions are determined, the failure forms of fracture propagation can be determined after obtaining the failure conditions of different forms of fractures. Therefore, the critical pressure of different fracture failure forms will be calculated in the following part of this section.

According to Warpinski and Teufel, the critical fluid pressure in the crack during shear failure of NF1 can be calculated.

The strength of the weakness plane disappears after cracking. When the fluid pressure inside the crack is greater than the normal stress on the fracture surface, the fracture surface opens. Therefore, the fluid pressure in the crack after shear failure is

Tensile failure occurs when the fluid pressure in NF1 is greater than the sum of normal stress and tensile strength of the fracture surface. At this point, the critical pressure of fluid in the joint is

Tensile failure occurs when the fluid pressure of fracture NF2 is greater than the sum of normal stress on the surface and tensile strength of the fracture. At this time, the critical fluid pressure of tensile failure for fracture NF2 is

The pressure conditions for crack opening can be determined by the above formulas. When the natural fracture NF1 opens, the hydraulic fracture is captured by fracture NF1. When fracture NF2 opens, the hydraulic fracture passes through fracture NF1. The critical pressure of fracture failure at different angles of NF1 and NF2 is calculated to determine the fracture expansion form at the intersection point.

The critical pressure of different failure forms of cracks can be calculated by the formulas (

The values of parameters.

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

Inherent shear strength, |
2 MPa | Coefficient of friction, |
0.2 |

Maximum principle stress, |
20 MPa | Tensile strength of cemented joint, |
2 MPa |

Minimum principle stress, |
12 MPa |

The critical pressure under different failure conditions.

Obviously, for NF1, the critical pressure of shear failure decreases first and then increases with the increase of the intersecting angle. The critical pressure of tensile failure and opening after shear failure increase with the increase of the intersecting angle. However, the critical pressure of the NF2 tensile failure is independent of the intersection angle.

At a small angle (0° and 10.67°), as the pressure in the crack rises, fluid pressure first reached _{open}, and no fractures are failures now. Then, the pressure continued to rise to

To sum up, with the change of the intersecting angle, the expansion morphology of intersecting cracks can be divided into three categories. At low angles (0-10.67), NF1 and NF2 open at the same time, forming a cross-state. At the middle angle (10.67-37.7), shear cracking occurred in NF1 and no cracking occurred in NF2. The fracture morphology shows turning. At high angles (37.7-90), NF2 cracks and NF1 does not open. The fracture shape is a single fracture extending forward. In summary, the intersection angle can be divided into three parts according to the fracture propagation pattern, namely, low angle, medium angle, and high angle. The boundary between the low angle and the middle angle is defined as the

From Figure

From (

The inherent shear strength influences the shear behavior of cracks. The higher the inherent shear strength is, the greater the shear strength of cracks is. According to the propagation criterion of intersecting cemented cracks, the inherent shear strength only affects

The two angle boundaries vary with

The effect of the inherent shear strength of the interface.

The influence of the friction coefficient on each boundary is shown in Figure

The effect of the friction coefficient.

As mentioned in Section

The effect of the differential stress.

Figure

The tensile strength of cracks affects their tensile cracking. The greater the tensile strength, the higher the critical pressure for tensile cracking. The influence of the fracture tensile strength on each critical pressure is shown in Figure

The effect of the tensile strength of fractures.

As mentioned in Section

The selection and combination of independent variables are called the experimental design. The central composite design (CCD) is a common method for experimental design. The design tables of two experiments are shown in Tables

Design of experiment and response value for the L-M boundary.

No. | Factor 1 | Factor 2 | Factor 3 | Response 1 |
---|---|---|---|---|

R1 | ||||

1 | 8 | 2.4 | 0.3 | 14.62 |

2 | 8 | 2 | 0.5 | 7.79 |

3 | 8 | 2 | 0.3 | 10.91 |

4 | 6.4 | 2 | 0.3 | 14.13 |

5 | 8 | 1.6 | 0.3 | 7.55 |

6 | 8.95137 | 2.23784 | 0.418921 | 9.86 |

7 | 8 | 2 | 0.3 | 10.91 |

8 | 8 | 2 | 0.3 | 10.91 |

9 | 8 | 2 | 0.1 | 13.74 |

10 | 7.04863 | 2.23784 | 0.181079 | 17.16 |

11 | 8 | 2 | 0.3 | 10.91 |

12 | 7.04863 | 1.76216 | 0.181079 | 12.22 |

13 | 9.6 | 2 | 0.3 | 8.92 |

14 | 7.04863 | 2.23784 | 0.418921 | 13.06 |

15 | 8.95137 | 2.23784 | 0.181079 | 12.97 |

16 | 8.95137 | 1.76216 | 0.418921 | 6.26 |

17 | 8.95137 | 1.76216 | 0.181079 | 9.41 |

18 | 7.04863 | 1.76216 | 0.418921 | 8.10 |

19 | 8 | 2 | 0.3 | 10.91 |

20 | 8 | 2 | 0.3 | 10.91 |

Design of experiment and response value for the M-H boundary.

No. | Factor 1 | Factor 2 | Response 1 |
---|---|---|---|

R1 | |||

1 | 6.86863 | 2.28284 | 35 |

2 | 8 | 2 | 30 |

3 | 8 | 2 | 30 |

4 | 9.6 | 2 | 27 |

5 | 8 | 2 | 30 |

6 | 8 | 1.6 | 27 |

7 | 8 | 2 | 30 |

8 | 8 | 2.4 | 33 |

9 | 9.13137 | 2.28284 | 30 |

10 | 9.13137 | 1.71716 | 26 |

11 | 6.86863 | 1.71716 | 30 |

12 | 6.4 | 2 | 34 |

13 | 8 | 2 | 30 |

According to the previous analysis, for the L-M boundary, the influence of the tensile strength of joints is negligible. Therefore, in the analysis of the L-M boundary, three independent variables,

Analysis of variance table.

Source | Sum of squares | Mean square | |||
---|---|---|---|---|---|

Model | 139.16 | 15.46 | 10080.89 | <0.0001 | Significant |

31.69 | 31.69 | 20658.06 | <0.0001 | ||

61.37 | 61.37 | 40010.96 | <0.0001 | ||

43.90 | 43.90 | 28624.17 | <0.0001 | ||

0.94 | 0.94 | 611.84 | <0.0001 | ||

0.48 | 0.48 | 313.07 | <0.0001 | ||

4.5 |
4.5 |
0.29 | 0.5999 | ||

0.67 | 0.67 | 437.30 | <0.0001 | ||

0.052 | 0.052 | 34.02 | 0.0002 | ||

0.040 | 0.040 | 26.35 | 0.0004 |

The model

The factor effect graph shows the linear effect of changing the level of a single factor. It is constructed by predicting the responses for the low (-1) and high (+1) levels of a factor. As shown in Figure

Factor effect graph of variances for the L-M boundary.

From the univariate analysis, it is known that the M-H boundary is not affected by

Analysis of variance table.

Source | Sum of squares | Mean square | |||
---|---|---|---|---|---|

Model | 91.47 | 18.29 | 35479.45 | <0.0001 | Significant |

45.98 | 45.98 | 89173.59 | <0.0001 | ||

44.67 | 44.67 | 86638.72 | <0.0001 | ||

0.20 | 0.20 | 392.73 | <0.0001 | ||

0.55 | 0.55 | 1067.20 | <0.0001 | ||

0.024 | 0.024 | 46.57 | 0.0002 |

The model

The factor effect graph of variances for the M-H boundary is shown in Figure

Factor effect graph of variances for the M-H boundary.

In order to analyze the mechanism of hydraulic fracturing fracture propagation in formations with cemented joints, the critical pressure of three fracture propagation modes under the intersection of hydraulic fracturing fracture and closed natural fracture is derived, and the parameter analysis is carried out. The main conclusions are as follows:

The intersection angle can be divided into three parts according to the fracture propagation pattern, namely, low angle, medium angle, and high angle. Three types of behavior for a fracture are a cross-state at a low angle, turning at a middle angle, and extending forward at a high angle. The first two conditions are conducive to the formation of complex fracture networks

Univariate analysis was performed on each variable by the control variable method. The result shows that the increase of

The relations between the two boundaries and different independent variables are obtained, and their regression formulas are obtained by fitting by RSM

The data and code for numerical simulation and boundary calculation used to support the findings of this study are available from the corresponding author upon request.

The authors declared that they have no conflicts of interest to this work.

We would also like to thank Liuke Huang for his help in the calculation of the block discrete element method. The authors gratefully acknowledge the financial support given by the China National Science and Technology Major Project (Grant No. 2017ZX05037001).