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The simulation of hydraulic fracturing by the conventional ABAQUS cohesive finite element method requires a preset fracture propagation path, which restricts its application to the hydraulic fracturing simulation of a naturally fractured reservoir under full coupling. Based on the further development of a cohesive finite element, a new dual-attribute element of pore fluid/stress element and cohesive element (PFS-Cohesive) method for a rock matrix is put forward to realize the simulation of an artificial fracture propagating along the arbitrary path. The effect of a single spontaneous fracture, two intersected natural fractures, and multiple intersected spontaneous fractures on the expansion of an artificial fracture is analyzed by this method. Numerical simulation results show that the in situ stress, approaching angle between the artificial fracture and natural fracture, and natural fracture cementation strength have a significant influence on the propagation morphology of the fracture. When two intersected natural fractures exist, the second one will inhibit the propagation of artificial fractures along the small angle of the first natural fractures. Under different in situ stress differences, the length as well as aperture of the hydraulic fracture in a rock matrix increases with the development of cementation superiority of natural fractures. And with the increasing of in situ horizontal stress differences, the length of the artificial fracture in a rock matrix decreases, while the aperture increases. The numerical simulation result of the influence of a single natural fracture on the propagation of an artificial fracture is in agreement with that of the experiment, which proves the accuracy of the PFS-Cohesive FEM for simulating hydraulic fracturing in shale gas reservoirs.

As an important unconventional fossil hydrogen energy source, shale gas reservoirs have been widely developed in recent years. Due to its ultralow permeability, large-scale hydraulic fracturing has become a necessary technology for economic exploitation [

Many scholars have studied the propagation pattern of an artificial fracture intersecting with a single natural fracture by physical modeling and numerical modeling [

The previous numerical study of the intersection of an artificial fracture with a natural fracture mainly adopts linear elastic mechanics, but this method ignores the fluid-solid coupling during hydraulic fracturing. The finite element method (FEM) and the extended finite element method (XFEM) and the discontinuous displacement methods based on the boundary element (DDM) and the discrete fracture network model (DFN) are gradually proposed [

In general, the effect of an intersected natural fracture on artificial fracture propagation is not yet clear. And the conventional ABAQUS cohesive finite element simulating hydraulic fracturing requires a preset fracture propagation path, which restricts its application to the hydraulic fracturing simulation. And the cohesive finite element is only used for simulation of an artificial fracture intersecting with a single natural fracture. While intersected natural fractures are widely distributed in shale reservoirs, the effective connection of a multi-intersected natural fracture, as well as the free propagation of artificial fractures in a rock matrix, has become necessary for numerical simulation of large-scale hydraulic fracturing in shale reservoirs. In this paper, a new dual-attribute element of pore fluid/stress element and cohesive element (PFS-Cohesive) method for a rock matrix is presented based on the further development of a cohesive finite element, which can realize artificial fracture propagation along the arbitrary path. The effect of a multi-intersected natural fracture on artificial fracture propagation is analyzed by this new method. The results obtained can be used to explain the expansion mechanism of artificial fractures in shale reservoirs.

Rock stress equilibrium equation, continuity equation for fluid flow in porous media, and fracture flow model are proposed in this paper, and the effects of matrix poroelastic deformation [

The definition of effective stress of a rock matrix in saturated single-phase liquid is proposed as [

The relationship between stress and strain could be expressed as incremental forms as

The rock stress equilibrium equation could be expressed by the principle of virtual work. The virtual work of the rock is equivalent to the virtual work produced by the force (including body force and surface fore) acting on the rock.

Combining formulas (

After some manipulation, formula (

The flow of liquid in porous media conforms to Darcy flow. Darcy’s law of porous medium flow is written as
^{-15} ^{2}), ^{3}).

The mass conservation for fluid flow in a rock is as follows [

The flow of liquid in a fracture includes tangential flow and normal flow, shown in Figure

Fluid flow in a cohesive element.

Normal flow indicates leakage of fracturing fluid to porous medium and is simulated by defining the leakage coefficient of a rock matrix, as shown in Figure

The leakage coefficient is shown as a permeable layer.

ABAQUS provides a variety of fracture initiation criteria based on stress and strain. The secondary stress criteria are adopted in this model. When the sum of the ratio of the stress in a normal direction and tangential direction to the corresponding critical stress reaches 1, the fracture starts to crack. The fracture initiation criteria can be written as [

Damage evolution refers to the energy needs for further damage of a rock after initial crack. This paper introduces the Benzegagh-Kenane criteria as a damage law of artificial fracture growth during the hydraulic fracturing process, which can be represented as [

The cohesive element is embedded in the pore fluid/stress element, and the dual-attribute element of pore fluid/stress element and cohesive element (PFS-Cohesive) method for a rock matrix is presented, shown in Figure

Cohesive finite element equipped with pore fluid/stress finite element (PFS-Cohesive) in a rock matrix.

The dual element of the rock matrix consists of 6 nodes, of which there are two displacement degrees of freedom (u1-u8) in no. 1-4 nodes. The no. 5 and no. 6 nodes have pore pressure freedom (P1, P2) only, and the injected fracturing fluid pressure and flow rate are dispersed in the middle layer of no. 5 and no. 6. The pore fluid/stress element is used to simulate the property of the reservoir rock, such as the permeability of formation, Young’s modulus, and Poisson’s ratio. The cohesive element is used to simulate the occurrence and propagation of fracture, as well as liquid flow in the fracture. The two element types share the same nodes of nos. 1-4.

Three models are established by using the PFS-Cohesive method (shown in Figure

Schematic model diagram.

Single natural fracture

Two intersected natural fractures

Multi-intersected or discrete natural fractures

Basic model parameters.

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

Maximum in situ horizontal stress | 36 MPa | Minimum in situ horizontal stress | 26/31/36 MPa |

Overburden stress | 40 MPa | Initial pore pressure | 24 MPa |

Young’s modulus | 10 GPa | Poisson’s ratio | 0.24 |

Porosity | 0.02 | Permeability | 0.06 mD |

Tensile strength of the rock | 2.5 MPa | Tensile strength of the natural fracture | 2/1.7/1.5 MPa |

Shear strength of the rock | 2 MPa | Shear strength of the natural fracture | 1.5/1.3/1 MPa |

Normal energy release rate of the rock | 28 N/mm | Normal energy release rate of the natural fracture | 26/24/22 N/mm |

Tangential energy release rate of the rock | 28 N/mm | Tangential energy release rate of the natural fracture | 26/24/22 N/mm |

Filtration coefficient | Approaching angle | 30°/45°/60°/75° | |

Injection rate | 5 m^{3}/min·m |
Viscosity | 15 mPa•s |

The approaching angel is defined as the acute angle between the artificial fracture and natural fracture. According to the first model, the effect of a single natural fracture on artificial fracture propagation morphology with approaching angles of 30°, 45°, 60°, and 75° is studied, when the in situ stress level differences are 0 MPa, 5 MPa, and 10 MPa (shown in Figure

Influence of the approaching angle and in situ horizontal stress difference on artificial fracture propagation morphology.

Under the same approaching angle, a single natural fracture has different effects on the expansion of an artificial fracture at different in situ horizontal stress difference levels. When the approaching angle reaches 30° and the in situ stress difference is less than 10 MPa, the artificial fracture extends along the right side of the natural fracture (shown in Figures

Under the same in situ horizontal stress difference, the effects of the single natural fracture on artificial fracture propagation morphology are different with different approaching angles. As the stress difference is 0 MPa and the approaching angles are 30° and 45°, the artificial fracture propagates along the right side of the natural fracture, while the left wing of the natural fracture is only partly opened (shown in Figures

Figure

Additional resistance effect of a small angle.

The influence of two intersected natural fractures on artificial fracture propagation with approaching angles of 30°, 45°, 60°, and 75° is studied, when the in situ horizontal stress differences are 0 MPa, 5 MPa, and 10 MPa. The results are shown in Figures

Approaching angle of 30°.

When the approaching angle is 30°, the effect of the two intersected natural fractures on the propagation of an artificial fracture with different horizontal stresses is shown in Figure

Comparing with Figure

Figure

Approaching angle of 45°.

The effect of two intersected natural cracks on artificial fracture propagation in different in situ horizontal stress differences with the approaching angle of 60° is shown in Figure

Approaching angle of 60°.

The effect of the two intersected natural fractures on artificial fracture propagation under different in situ stresses with the approaching angle of 75° is shown in Figure

Approaching angle of 75°.

According to the results of the second model, the influences of the two intersected natural fractures on artificial fracture propagation with different approaching angles and in situ stress differences are summarized as shown in Table

The influence of two intersected natural fractures on artificial fracture propagation.

Approaching angle (°) | In situ stress difference ( |
The 1st natural fracture | The 2nd natural fracture |
---|---|---|---|

30 | 0 | Partly deflected | Partly deflected |

5 | Partly deflected | Partly deflected | |

10 | Partly deflected | Partly deflected | |

45 | 0 | Deflected | Partly deflected |

5 | Partly deflected | Partly deflected | |

10 | Crossed | ||

60 | 0 | Deflected | Partly deflected |

5 | Partly deflected | Partly deflected | |

10 | Crossed | ||

75 | 0 | Partly deflected | |

5 | |||

10 | Crossed |

The state of being partly deflected indicates that the artificial fracture can only propagate along one wing of the natural fracture. The state of being deflected indicates that the artificial fracture will propagate along the two wings of the natural fracture. The state of being crossed indicates that the artificial fracture will cross the natural fracture directly.

The third model is used to study the propagation of an artificial fracture under the effect of multi-intersected natural fractures at reservoir scales. The in situ horizontal stress differences are 0 MPa, 5 MPa, and 10 MPa, and the other parameters are based on Table

The effects of in situ horizontal stress difference on fracture propagation.

0 MPa

5 MPa

10 MPa

When the in situ pressure level difference is 0 MPa, the artificial fracture almost extends along the direction of the natural fracture, forming a main fracture that is similar with the strike of the natural fracture. At the same time, the artificial fracture is arrested with a part of the natural fractures, forming some branch fractures. As the in situ horizontal stress difference is 5 MPa, the propagation morphology of the artificial fracture is influenced by the in situ horizontal stress difference and natural fracture synchronously. The artificial fracture extends along the natural fracture at a part of intersections, but the main propagation path of the artificial fracture is deflected to the direction of the maximum in situ horizontal stress. As the in situ horizontal stress difference reaches 10 MPa, the artificial fracture extends along the maximum in situ horizontal stress unaffected by the natural fracture.

According to Table

When the in situ stress differences are 0 MPa, 5 MPa, and 10 MPa, the relationship between the aperture and the length of the artificial fracture with three different natural fracture cementation strengths is showed in Figure

The influence of cementation strength of a natural fracture on artificial fracture propagation.

0 MPa

5 MPa

10 MPa

Figure

The influence of cementation strength of a natural fracture on natural fracture propagation.

In order to affirm the accuracy of the PFS-Cohesive method for numerical simulation of fracture intersection, triaxial experiments are carried out with different approaching angles and in situ horizontal stress differences. Figure

The influence of the approaching angle on artificial fracture propagation.

Many other scholars also studied the propagation morphology of an artificial fracture intersecting with a single natural fracture [

Result comparison between numerical simulation and physical simulation.

Figure

Wang et al. [

Result comparison between PFS-Cohesive method and XFEM method.

The dual-attribute element of a pore fluid/stress element and cohesive element (PFS-Cohesive) method for a rock matrix is presented based on the further development of a cohesive finite element. It solves the limitation of the ABAQUS cohesive finite element method in simulating hydraulic fracturing requiring a preset fracture propagation path. This new method can realize the simulation of artificial fracture propagation along any arbitrary path. And the accuracy of this method is verified by experimental results and XFEM method results

The in situ horizontal stress difference, approaching angle, and cementation of the natural fracture have significant effects on the propagation morphology of an artificial fracture. In general, when the artificial fracture intersects with the natural fracture, a smaller in situ stress difference, a greater approaching angle, and a weaker cementation strength of the natural fracture are favorable to form a relatively complex fracture network

When two intersected natural fractures exist, the second natural fractures will inhibit an artificial fracture extending along the direction of a small angle branch of the first natural fracture. After artificial fracture propagating along the direction of the large angle branch of the first natural fracture, it is easier to continue to propagate along the direction of the large angle branch of the second natural fracture, resulting in only one wing of the first natural fracture being cracked

When the in situ horizontal stress difference is 0 MPa, the artificial fracture almost extends along the natural fracture, forming a main fracture consistent with the strike of the natural fracture. The artificial fracture tends to deflect to the maximum in situ horizontal stress direction with the raise of in situ horizontal stress difference. When the in situ horizontal stress difference is 10 MPa, the artificial fracture tends to extend along the direction of the maximum in situ horizontal stress and it is almost unaffected by the natural fracture

The length and aperture of an artificial fracture in a rock matrix increases with the increase of natural fracture cementation strength. And with the rising of in situ horizontal stress difference, the artificial fracture length in a rock matrix decreases, while the aperture of the artificial fracture increases

The data used to support the findings of this study are available from the corresponding author upon request.

We have no competing interests.

The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (Grant no. 51874338) and express their gratitude to the Fundamental Research Funds for the Central Universities (Grant no. 17CX02077), the Applied Basic Research Project of Qingdao Province (Grant no. 17-1-1-20-jch), and the Innovation Funding Program of the China University of Petroleum (East China) (YCX2018010).