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According to the theory of plane mechanics involving the interaction of hydraulic and natural fractures, the law of hydraulic fracture propagation under the influence of natural fractures is verified using theoretical analysis and RFPA^{2D}-Flow numerical simulation approaches. The shear and tensile failure mechanisms of rock are simultaneously considered. Furthermore, the effects of the approach angle, principal stress difference, tensile strength and length of the natural fracture, and elastic modulus and Poisson’s ratio of the reservoir on the propagation law of a hydraulic fracture are investigated. The following results are obtained: (1) The numerical results agree with the experimental data, indicating that the RFPA^{2D}-Flow software can be used to examine the hydraulic fracture propagation process under the action of natural fractures. (2) In the case of a low principal stress difference and low approach angle, the hydraulic fracture likely causes shear failure along the tip of the natural fracture. However, under a high stress difference and high approach angle, the hydraulic fracture spreads directly through the natural fracture along the original direction. (3) When natural fractures with a low tensile strength encounter hydraulic fractures, the hydraulic fractures likely deviate and expand along the natural fractures. However, in the case of natural fractures with a high tensile strength, the natural fracture surface is closed, and the hydraulic fracture directly passes through the natural fracture, propagating along the direction of the maximum principal stress. (4) Under the same principal stress difference, a longer natural fracture corresponds to the easier initiation and expansion of a hydraulic fracture from the tip of the natural fracture. However, when the size of the natural fracture is small, the hydraulic fracture tends to propagate directly through the natural fracture. (5) A smaller elastic modulus and larger Poisson’s ratio of the reservoir result in a larger fracture initiation pressure. The presented findings can provide theoretical guidance regarding the hydraulic fracturing of reservoirs with natural fractures.

In recent years, hydraulic fracturing has been widely used in the engineering practices of petroleum, natural gas, shale gas, and coal mining [

Results of numerical simulation and experimental testing have indicated that the horizontal principal stress difference and approach angle corresponding to the hydraulic and natural fractures are the main factors affecting the trends of hydraulic fractures [

The equivalent plane model is one kind of the equivalent models. With the help of the equivalent plane model in this paper, the actual process is transformed and abstracted into an equivalent, simple, and easy mathematical model, which is convenient for theoretical analysis. In this study, an equivalent plane model of hydraulic and natural fractures is established by combining the stress-seepage theory with seepage-damage mechanics. Moreover, the mechanisms of shear fractures and tensile failure are considered. The RFPA^{2D}-Flow software is used to examine the mechanism of hydraulic fracture propagation under the action of natural fractures. Furthermore, the trends of the hydraulic fracture propagation under the action of the approach angle, principal stress difference, tensile strength, and length of the natural fracture are clarified. These findings not only help improve the established theory of hydraulic fracture networks but also provide theoretical support for engineering practices, such as the optimization of the arrangement of field fracturing boreholes, minimization of the interference of natural fractures in the expansion of hydraulic fractures, effective enhancement of the increase in permeability owing to the hydraulic fractures, and the development of hydraulic fractures in a fractured reservoir.

The influence of the natural fractures on hydraulic fracture propagation depends on the position of the natural fractures. When a hydraulic fracture meets a natural fracture located near a borehole, the natural fracture gradually expands as the fluid pressure is higher than the normal stress acting on the surface of the natural fracture. The following 4 situations may occur during the continuous expansion of the hydraulic fracture: (1) When the natural fracture is in the closed state, shear failure occurs at the intersection (Figure

Plane models demonstrating the intersection of hydraulic and natural fractures. (a) Hydraulic fracture propagation when shear fracture occurs in the natural fracture. (b) Hydraulic fracture penetrates the natural fracture in the closed state directly. (c) Hydraulic fracture penetrates the natural fracture. (d) Hydraulic fracture expands from one end of the natural fracture.

According to the relevant theories of hydraulic fracture propagation, the main hydraulic fracture of coal and rock mass expands in the direction perpendicular to the minimum horizontal principal stress after its initiation. When the hydraulic fracture propagates along the direction of the maximum horizontal principal stress, it intersects with a natural fracture. In Figure _{1} and _{3} denote the maximum and minimum horizontal principal stresses, respectively.

Plane model of a hydraulic fracture intersecting a natural fracture.

When a hydraulic fracture intersects a natural fracture, the natural fracture does not inflate if the fluid pressure at the tip of the hydraulic fracture is less than the normal stress

According to the results of Blanton’s research, the shear stress in Equation (

By substituting Equation (

When Equation (

Considering the ground stress field and the orientation of the natural fractures shown in Figure

When the shear stress acting on the natural fracture surface is greater than the shear strength of the natural fracture surface, shear failure occurs in the natural fracture [_{0} is the cohesion of the natural fracture and

By substituting Equations (

The fluid pressure in the natural fracture surface must be lower than the normal stress acting on the natural fracture surface; otherwise, the fracture is opened, that is, the fluid pressure satisfies the following relation:

According to the theory of fracture propagation, the Griffith linear fracture propagation requires the minimum fluid pressure. Assuming that the fracture is a Griffith fracture, the water pressure at the tip of the hydraulic fracture can be expressed as follows:

Hence, the following expression can be obtained:

According to Equation (

In this paper, RFPA^{2D}-Flow software, a seepage-stress coupling analysis system for rock fracture instability, is used to analyze the interaction mechanism between natural fractures and hydraulic fractures based on the damage mechanics theory, in which both tensile and shear failure criteria of the rock are chosen[_{1} in the horizontal direction is 10 MPa, and the minimum principal stress _{3} in the vertical direction is 5 MPa. The initial water pressure applied in the borehole is 0 MPa, and the step increment is 0.5 MPa.

Geometric model.

To ensure that the numerical calculation can more closely simulate the real physical experiment, the actual physical parameters of the experimental sample are adopted in the numerical simulation as much as possible. The relevant parameters utilized in this numerical simulation are presented in Table

Parameters used in the numerical simulation.

Parameter | Value and unit | Parameter | Value and unit |
---|---|---|---|

Tensile strength | 6.5/10.5 MPa | Internal friction angle | 30° |

Elastic modulus | 50 GPa | Residual strength | 0.1% |

Compression tension ratio | 10 | Pore water pressure coefficient | 0.1 |

Permeability coefficient | 0.000864 m/d | Maximum compression strain coefficient | 200 |

Maximum tension strain coefficient | 1.5 | Water increment | 0.5 MPa |

Initial water pressure | 0 MPa | Vertical principle stress | 5 MPa |

Horizontal principle stress | 10 MPa | Approach angle | 45/90° |

Figure

Propagation processes of a hydraulic fracture under the action of a natural fracture. (a) Propagation of a hydraulic fracture along a natural fracture. (b) Propagation of a hydraulic fracture penetrating the natural fracture. (c) Propagation of a hydraulic fracture through and along a natural fracture.

The numerical results indicate that the hydraulic fracture can propagate via three paths under the action of natural fractures. (1) The hydraulic fracture extends completely along the natural fracture. The fractures are generated around the natural fracture, and their expansion is not evident. (2) The hydraulic fracture penetrates the natural fracture and expands along the original direction. The length of the generated hydraulic fracture is considerably larger than that of the natural fracture; however, few short fractures are present around the natural fracture. (3) The hydraulic fracture propagates through and along the natural fracture simultaneously, including cases (1) and (2), resulting in a larger number of fractures and a wider distribution range. These three cases are consistent with the theoretical plane model of the intersection of the hydraulic and natural fractures. In the exploitation of oil and gas resources, the third case leads to the formation of a hydraulic fracture network and can reduce the engineering quantity, thereby improving the exploitation efficiency of the oil and gas resources.

Renshaw and Pollard et al. examined the critical approach angle of a hydraulic fracture passing through a natural fracture under different stress conditions by conducting a physical experiment and obtained the critical curve of the hydraulic fracture passing through the natural fracture (black solid line in Figure

Comparison of numerical simulation and experimental results.

The homogeneity, size, number of cells, borehole aperture, and basic mechanical parameters of the model were kept unchanged, and the single variable method was used to examine the influence of the approach angle, principle stress difference, tension strength and length of the natural fracture, elastic modulus, and Poisson’s ratio on the hydraulic fracture propagation.

When the approach angle _{3}. After the hydraulic fracture intersects the natural fracture, it extends along the natural fracture and propagates from one end of the natural fracture. Subsequently, the fracture deviates and continues expanding in the direction of the maximum horizontal principal stress _{1}. For a smaller angle

Process of hydraulic fracture propagation under different approach angles. (a) Morphology of hydraulic fracture propagation. (b) Acoustic emission signal. (c) Cloud map of pore water pressure.

Considering the minimum main stress _{3} as 5 MPa and the maximum main stress _{1} as 5 MPa, 10 MPa, 15 MPa, 20 MPa, 25 MPa, and 30 MPa, the corresponding horizontal stress difference

Process of hydraulic fracture propagation under different horizontal principle stress differences

When the principal stress difference _{1}. The branch fractures at the tip of the hydraulic fracture are not significant. At this time, the hydraulic fracture is more likely to expand after penetrating the natural fracture.

When the tension strength

Process of hydraulic fracture propagation under different tensile strengths

The hydraulic fracture may extend along the natural fracture. After extending for a certain distance, the branches of the hydraulic fractures tend to deviate, which is beneficial for the hydraulic fracture expansion. In some cases, the hydraulic fracture extends through the natural fracture. Under the same principal stress difference, a smaller natural fracture length

Process of hydraulic fracture propagation under different natural fracture lengths

The elastic modulus ^{2D}-Flow software, both the nonuniformity of the rock strength and elastic modulus are considered. Therefore, the initiation and expansion processes of a hydraulic fracture under the action of a natural fracture, simulated using the RFPA^{2D}-Flow software, are sufficiently realistic.

The relationship between the elastic modulus of the reservoir rock and the opening pressure of the natural fracture is shown in Figure

Opening pressure of the natural fracture under different elastic moduli.

The relationship between the Poisson’s ratio

Opening pressure of the natural fracture under different Poisson’s ratios.

The influence of natural fractures on hydraulic fracture propagation has been always a hot topic. In this paper, only two natural fractures are chosen. In fact, the number of the natural fractures is far more than two natural fractures in the fractured reservoir. In the next research, more natural fractures should be adopted in the numerical model. Of course, the plane model and propagation mechanism of the intersection of hydraulic and natural fractures will become more complex.

RFPA^{2D}-Flow is a good numerical computational software, which can realize the visualization of the hydraulic fracture process. However, the process of the numerical simulation depends on the increment step, not the time. Therefore, much more dependent variable, such as water pressure, length, and width of the hydraulic fracture and the like, which regard time as independent variable, cannot be obtained.

Considering the tensile and shear failure mechanisms of rock rupture, a theoretical model for the hydraulic fracture propagation under the action of a natural fracture is established based on the equivalent plane fracture theory of hydraulic and natural fractures. The morphology of the hydraulic fracture determined using the numerical simulation is in agreement with the existing physical experiment results.

In the case of a large approach angle, the hydraulic fracture directly passes through the natural fracture, and the initiation pressure of the natural fracture gradually increases. When the approach angle is 90°, the initiation pressure of the natural fracture reaches its maximum value. When the approach angle is small, the hydraulic fracture expands along the natural fracture, propagates from one end, and deviates and continues expanding along the direction of the maximum horizontal principal stress.

Under a low principle stress difference, the hydraulic fracture tends to expand along the natural fracture, and the development of the hydraulic fracture is relatively complex. However, with the increase in the principal stress difference, the hydraulic fracture tends to penetrate the natural fracture. This aspect shows that under a low principal stress difference, the natural fracture is more likely to be opened, which facilitates the development of the hydraulic fracture.

When the tensile strength of the natural fracture is large, the hydraulic fracture cannot induce shear and tensile failure, which impedes the opening of the natural fracture and the propagation of the hydraulic fracture. In contrast, when the tensile strength of the natural fracture is small, the weak surface features of the natural fracture are prominent, which facilitates the opening of the natural fracture.

With the increase in the elastic modulus or decrease in the Poisson’s ratio, the critical opening pressure of the natural fracture decreases. Since the reservoir rock with a higher elastic modulus usually has a smaller Poisson’s ratio, the influence of the elastic modulus and Poisson’s ratio on the opening pressure of the natural fracture is consistent.

The data in the manuscript can be available on request through Weiyong Lu, whose email address is

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The authors would like to particularly thank Dalian Mechsoft Co. Ltd for their help in providing the free RFPA^{2D}-Flow software for three months. This work was supported by the Transformation of Scientific and Technological Achievements Programs (TSTAP) of Higher Education Institutions in Shanxi (No. 2020CG050), the Special Project of 2019 Plan for the Introduction of High-Level Scientific and Technological Talents in Development Zone of Lvliang City (development of automatic disassembly platform for hydraulic support pin shaft) (No. 2019L0002), the Science and Technology Project of Lvliang City in 2019 (pressure relief and permeability improvement technology by integrated hydraulic flushing and cutting for low permeability coal seam containing methane) (No. 2019L0008), and the National Natural Science Foundation of China (Nos. 51774111 and 51974105).

_{2}fracturing in unconventional gas reservoir