Modeling the Hydraulic Fracturing Processes in Shale Formations Using a Meshless Method

Abstract

Complex bedding properties and in situ stress conditions of shale formation lead to complex hydraulic fracturing morphologies. However, due to the limitations of traditional numerical methods, the simulation of hydraulic fracturing in shale formation still needs further development. Based on this, the liquid–solid interaction modes and the SPH governing equations considering liquid–solid interaction force have been introduced. The smoothing kernel function in the traditional SPH method is improved by introducing the fracture mark ξ, which can realize the simulation of rock hydraulic fracturing processes. The stress boundary of the SPH method is applied by stress mapping of “stress particles”, and the feasibility and correctness of the method are verified by two numerical examples. Then, the simulation of hydraulic fracturing processes of bedding shale formations are carried out. With the increase of horizontal stress ratio, the total number of damaged particles decreases, but the initiation and extension pressure increase gradually. The initiation stress of small bedding dip angles (θ < 45°) is larger than that of big bedding dip angles (θ > 45°). The hydraulic fracture propagation range at low horizontal stress ratio is wider and the fracture is along the direction of maximum principal stress, while the hydraulic fracture propagation range at high horizontal stress ratio is limited to the perforation. The hydraulic fracture will propagate through the bedding with small dip angles. However, when the bedding dip angle is larger, the hydraulic fracture will propagate along the bedding direction.

1. Introduction

Hydraulic fracturing is a key technology to increase shale gas production. Shale formation is a kind of ultra-low permeability reservoir, which makes it difficult to meet the requirements of commercial exploitation of natural gas [1,2,3,4]. Therefore, the reservoir needs to be reformed by hydraulic fracturing technology, which can form complex fracture networks in the shale formation [5,6]. Fracture networks are the key indexes during the hydraulic fracturing processes, which can increase formation connectivity, provide a flow path for subsurface fluids, improve overall formation connectivity, increase formation permeability, and allow gas stored in the formation to flow through the fracture network to the wellbore, thus realizing commercial shale gas developments. However, unlike traditional reservoirs, these unconventional reservoirs usually have typical bedding properties due to special diagenesis, as shown in Figure 1 [7]. Therefore, understanding the mechanisms of bedding shale reservoirs under hydraulic fracturing will undoubtedly provide important guidance for shale gas exploitation and production.

Previous studies have made great progress in the formation of hydraulic fracture networks in bedding shale reservoir, mainly concentrating on experimental studies, theoretical studies, and numerical simulations. Experimental studies are the most direct means to obtain the bedding shale formation mechanisms under water-stress coupling, for example, Warpinski et al. [8] conducted field investigations on hydraulic fracturing and found that the fracture modes can be divided into three modes: through (crossover), turn, and obstruction. Janszen et al. [9] applied multi-stage fracturing technology developed in the United States to Dutch Posedonia shale, and carried out pilot tests and optimized hydraulic fracturing parameters. Wang et al. [10] completed 12 stages of hydraulic fracturing in a deep shale gas reservoir in the Dingshan exploration area and discussed the hydraulic fracturing technology applied in deep shale. Sun et al. [11] studied the influence of bedding strength, bedding direction and in situ stress state on hydraulic fracture morphology through laboratory tests. However, laboratory tests have high standards for specimen preparation, which require rock samples to be as complete as possible. Therefore, the representativeness of rock samples obtained remains to be discussed. Theoretical research can quantitatively express the mechanical properties and fracture characteristics of bedding shale under hydraulic fracturing by summarizing and refining mechanical laws in experimental studies. For example, Liu et al. [12] established a full 3D fluid–solid coupling model considering the effect of bedding properties, and studied the vertical extension mechanisms of hydraulic fractures under the opening and shear slip of weak bedding planes. Wang et al. [13] established a 3D elastic-plastic soil model of porous media based on the Mohr–Coulomb plasticity theory and investigated the laws of hydraulic fracturing in shale reservoirs. However, the theoretical model needs to simplify the actual working conditions, and can only analyze extremely abstracted model geometries and boundary conditions. Therefore, it is difficult to directly apply theoretical models into engineering practices.

Numerical simulation overcomes the problems of scale, repeatability, and cost of experimental research; therefore, it is regarded as the “third method” of scientific research. Finite Element Method (FEM) is one of the earliest numerical methods to investigate the formation mechanisms of hydraulic fracturing networks in shale formation. However, FEM needs to locally refine the grids around the fracture tips under hydraulic fracturing processes, and the meshes need to be re-divided at each timestep of fracture evolutions, which requires a large amount of calculation resources. For those numerical models with complex fracture networks, the finite element mesh is easily distorted, resulting in reduced calculation accuracy or even calculation failure [14,15]. In order to overcome these problems, the extended finite element method (XFEM) [16,17] and the cohesive element method [18,19] have subsequently been developed. However, XFEM has some limitations in dealing with fracture crossing, which needs further optimization [20]. The cohesive element method [21] prefabricated zero-thickness element two element surfaces as potential fracture evolution paths, which can well simulate the fracture-crossing behaviors. However, fracture propagation paths are set in advance; therefore, the fracture propagation is not as flexible as XFEM. Discrete Element Method (DEM) is a kind of mesh-less numerical method [22,23], which separates the calculation domain into particles with different particle sizes, and can get rid of the constraints of traditional finite element meshes so as to freely simulate the formations of shale hydraulic fracturing networks. However, DEM has many microscopic parameters with no actual physical meanings, and complex parameter calibrations are required before numerical simulation, which is difficult to apply to engineering practice. Recently, scholars have proposed a series of new numerical methods to simulate the hydraulic fracturing processes, such as numerical manifold method (NMM) [24,25], Discontinuous Deformation Analysis method (DDA) [26,27], PeriDynamics (PD) [28,29], Phase field method (PF) [30,31], and so on. These methods all have great potential in dealing with the formations of hydraulic fracturing networks in shale formation, but also have their own disadvantages. For example: The fracture tips of NMM and DDA still need to fall on the nodes of grids or elements. The Poisson’s ratio of the bond-based PD method is constant, which is not consistent with the actual situation. The determinations of PF model parameters still need to be further discussed. Smooth particle hydrodynamics (SPH) is a pure Lagrangian meshless method, which has its unique advantages in dealing with large deformation and discontinuous problems [32]. SPH overcomes the problems of treatment problems of discontinuous properties existing in FEM. Meanwhile, compared with DEM, the governing equations are all derived from partial differential equations with clear physical meanings. Therefore, there is no need to carry out complex parameter calibrations before numerical simulation. However, there are few applications of SPH method for simulation of shale hydraulic fracturing processes, and there are only some applications in rock fracture mechanics, such as the GPD method proposed by Zhou’s groups [33,34,35,36,37,38,39,40].

Based on the shortcomings of previous studies, the traditional SPH method has been improved to realize the simulation of hydraulic fracturing processes of shale formations. Two numerical examples are carried out showing that the improved SPH method can be well applied to the simulation of hydraulic fracturing, and the correctness is verified by comparing with previous experimental results. Finally, the hydraulic fracturing processes of bedding shale formation under different in situ stress conditions are simulated, and two typical forms of interaction modes between hydraulic fractures and bedding properties are discussed. The research results can provide some references for the optimization of hydraulic fracturing technology in shale gas production, and also provide some references for the applications of the SPH method in shale hydraulic fracturing modelling.

2. Numerical Treatments of Hydraulic Fracturing in SPH

2.1. Liquid—Solid Particle Interaction Modes

There exists liquid–solid contact surface between liquid particles (orange particles in Figure 2) and solid particles (blue particles in Figure 2), where liquid particle i exerts force (water pressure) on solid particle j. The water pressure is oriented along the liquid–solid particle line toward solid particle i and has a magnitude of σ_p. The normal vector between liquid particle i and solid particle j is defined as follows: For any contact pair between liquid particle and solid particle, its normal unit vector can be expressed as [33]:

$$n_{ij}=(x_{ij},y_{ij})/\left|r_{ij}\right|$$

(1)

where n_ij is the normal unit vector between liquid and solid particles, x_ij = x_j − x_i is the difference between the horizontal coordinate of liquid particles and solid particles, y_ij = y_j − y_i is the difference between the ordinate coordinate of liquid particles and solid particles, and $r_{ij}=\sqrt{x_{ij}^2+y_{ij}^2}$ is the distance between liquid particles and solid particles. Then, the following variables can be defined [33]:

$$n_{x,i}=x_{ij}/\left|r_{ij}\right|$$

(2)

$$n_{y,i}=y_{ij}/\left|r_{ij}\right|$$

(3)

Assuming that each liquid particle has the properties of water pressure p_w, the force σ_pij liquid particles on solid particles can be expressed as [33]:

$$\left\{\begin{array}{c}\sigma p_{xx}=p_w\cdot n_{x,i}^2\\ \sigma p_{yy}=p_w\cdot n_{y,i}^2\\ \sigma p_{xy}=p_w\cdot n_{x,i}\cdot n_{y,i}\end{array}\right.$$

(4)

Therefore, according to the liquid–solid interaction relationships, the traditional SPH governing equations can be re-written into the liquid–solid coupling governing equations [33]:

$$\left\{\begin{array}{c}\frac{d{\rho }_i}{dt}={\displaystyle \sum _{j=1}^N{m}_j{v}_{ij}^{\beta }\frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}}\\ \frac{d{v}_i^{\alpha }}{dt}={\displaystyle \sum _{j\in \mathrm{S}}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{{\sigma }_j^{\alpha \beta }}{{\rho }_j^2}+T_{ij})\frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}+{\displaystyle \sum _{j\in W}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{\sigma p^{\alpha \beta }}{{\rho }_j^2}+T_{ij})\frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}+}}\\ \frac{d{e}_i}{dt}=\frac{1}{2}({\displaystyle \sum _{j\in \mathrm{S}}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{{\sigma }_j^{\alpha \beta }}{{\rho }_j^2}+T_{ij})v_{ij}^{\beta }\frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}+{\displaystyle \sum _{j\in W}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{\sigma p^{\alpha \beta }}{{\rho }_j^2}+T_{ij})v_{ij}^{\beta }\frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}})}\\ \frac{d{x}_i^{\alpha }}{dt}=v_i^{\alpha }\end{array}\right.$$

(5)

where j ∈ S are all the solid particles in the liquid–solid contact domain and j ∈ W are all the liquid particles in the liquid–solid contact domain.

2.2. Fracture Treatments under Liquid–Solid Interactions

2.2.1. Failure Criterion

The Mohr–Coulomb criterion with tensile failure is used in this section, which can be expressed as [41]:

$${\sigma }_f={\sigma }_t$$

(6)

$${\tau }_f=c+{\sigma }_f\tan \phi$$

(7)

where σ_f and τ_f are the tensile and shear stress on the failure surface, respectively, σ_t is the tensile strength of rock masses, c is the cohesion of rock masses, and φ is the internal friction angle of rock masses.

2.2.2. Fracture Treatments of Solid Particles

As can be seen from the liquid–solid couple governing Equation (5), its derivative form of the kernel function ∂W_ij,β/∂x_i^β controls the key parameter transfers between different particles such as density, velocity, and energy. In other words, the derivative kernel function ∂W_ij,β/∂x_i^β acts as the “bridge” during parameter transfers. Therefore, controlling the connection and connection type of the “bridge” can realize the “activation” and “failure” of particles, as well as the “conversion” between solid particles and liquid particles. Based on these principals, the fracture mark ξ is introduced in this section. Meanwhile, the improved kernel function considering the particle damage D is defined, whose relationship with the traditional SPH kernel function W can be expressed as [33]:

$$\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}={\xi }_i\cdot \frac{\partial W_{ij,\beta }}{\partial x_i^{\beta }}$$

(8)

When the particle is damaged (the stress of the particle satisfies the failure criterion of Equation (6) or Equation (7)), the fracture mark ξ is set to be 0. When the particle is not damaged (the stress of the particle does not satisfy the failure criterion of Equation (6) or Equation (7)), the fracture mark ξ is set to be 1. The fracture processes of particles can be shown in Figure 3. Therefore, by improving the liquid–solid coupling governing Equation (5), we can obtain the new governing equations considering the particle damage [33]:

$$\left\{\begin{array}{c}\frac{d{\rho }_i}{dt}={\displaystyle \sum _{j=1}^N{m}_j{v}_{ij}^{\beta }\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}}\\ \frac{d{v}_i^{\alpha }}{dt}={\displaystyle \sum _{j\in \mathrm{S}}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{{\sigma }_j^{\alpha \beta }}{{\rho }_j^2}+T_{ij})\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}+{\displaystyle \sum _{j\in W}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{\sigma p^{\alpha \beta }}{{\rho }_j^2}+T_{ij})\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}+}}\\ \frac{d{e}_i}{dt}=\frac{1}{2}({\displaystyle \sum _{j\in \mathrm{S}}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{{\sigma }_j^{\alpha \beta }}{{\rho }_j^2}+T_{ij})v_{ij}^{\beta }\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}+{\displaystyle \sum _{j\in W}^N{m}_j(\frac{{\sigma }_i^{\alpha \beta }}{{\rho }_i^2}+\frac{\sigma p^{\alpha \beta }}{{\rho }_j^2}+T_{ij})v_{ij}^{\beta }\frac{\partial D_{ij,\beta }}{\partial x_i^{\beta }}})}\\ \frac{d{x}_i^{\alpha }}{dt}=v_i^{\alpha }\end{array}\right.$$

(9)

2.3. Application of Stress Boundaries in SPH

The application of stress boundary adopts the method of stress mapping, and more than five layers of “stress particles” are laid outside the solid particles. The “stress particles” should have the following characteristics:

(1) Like solid particles, “stress particles” participate in the internal force calculation of SPH. That is, the density, velocity, energy, and position of particles are updated by Equation (9);

(2) During each calculation step, stress is re-assigned to the “stress particle”, that is, although the “stress particles” participate in the parameter updates of the solid particles, their stress variations still follow the preset of stress boundary requirements;

(3) A layer of “type I virtual particles” with velocity v_inf equal to 0 are set outside the “stress particles”.

Therefore, the distributions of stress particles can be shown in Figure 4.

2.4. Solutions of SPH Method

Similar to traditional numerical software, the architecture of the SPH program mainly includes three modules: (1) input module; (2) computing module, and (3) post-processing module. The input module is shown in Figure 5. The SPH model is composed of fissure generations and regular particle generations. The fissure generations can be carried out by CAD, CATIA, and PROE. The parameter initialization comprises the determination of particle coordinates and particle properties, and the stress particle generation comprises stress particle coordinates and stress particle properties, as illustrated in Section 2.3.

The calculation module comprises 10 parts: (1) Time integral; (2) Particle pairing; (3) Density Pairing; (4) Damage Update Calculation; (5) Artificial Viscosity Calculation; (6) Internal Force Calculation; (7) External Force Calculation; (8) Artificial Heat Calculation; (9) Velocity Correction Calculation; (10) Principal Stress Calculation and Damage Determination. The relationships between the 10 parts can be illustrated in Figure 6.

The post-processing module can be illustrated as follows:

3. Validation of Improved SPH Method

In order to verify the correctness of SPH, two typical numerical examples are selected: (1) Numerical simulation of hydraulic fracturing processes with one vertical fissure without lateral pressure and (2) Numerical simulation of directional hydraulic fracturing with confining pressure. The rationality is verified by comparing the numerical results with previous experimental results.

3.1. One Vertical Fissure without Lateral Pressure

Numerical example 1 comes from reference [42], as shown in Figure 7. The specimen size is 70.7 mm × 70.7 mm, and a 15 mm length vertical fissure is prefabricated in the center of the sample. The whole model is divided into 200 × 200 = 40,000 particles. The loading rate of internal water pressure is 50 Pa/calculation step. The model parameters are set as follows: Particle density is ρ = 2600 kg/m³, Elastic modulus E = 17 GPa, Poisson’s ratio μ = 0.14, Cohesion is set to be 5.95 Mpa, internal friction angle is 40°, and the tensile strength is 2 MPa.

Figure 8 shows the progressive failure processes of the model under the action of internal water pressure. It can be seen that with the increase of internal water pressure, the hydraulic fracture propagates along the prefabricated fissure surface and finally runs through the model. The fracture propagation paths in experimental results are slightly curved due to the fact that concrete is heterogeneous. The comparisons between the experimental results and the numerical results are shown in Figure 9. Previous experimental results also showed these characteristics [42], which verifies the correctness of the improved SPH method.

3.2. Directional Hydraulic Fracturing with Confining Pressure

Numerical example 2 comes from reference [43], as shown in Figure 10. The model size is 300 mm × 300 mm. A circular hole with a diameter of 25 mm is set in the center of the model; meanwhile, directional prefabricated fissures with a length of 10 mm are set on the two sides of the circular hole, whose dip angles are 75°. The horizontal stress is σ_H = 6 MPa, and the vertical stress is σ_h = 5 MPa. The loading rate of internal water pressure is 50 Pa/calculation step. The whole model is divided into 300 × 300 = 90,000 particles. The model parameters are set as follows: Particle density is ρ = 2600 kg/m³, Elastic modulus E = 17 GPa, Poisson’s ratio μ = 0.14, Cohesion is set to be 5.95 Mpa, internal friction angle is 40°, and the tensile strength is 3.2 MPa.

Figure 11 shows the progressive fracture propagation processes of the model under the combined effect of confining pressure and internal water pressure. It can be seen that the fracture initiates from the directional prefabricated fissure tips and propagates along the direction of σ_h = 5 MPa. The comparisons between the experimental results and the numerical results are shown in Figure 12. Previous experimental results also showed these characteristics [43], which verifies the correctness of the improved SPH method.

4. Numerical Simulation of Bedding Shale Hydraulic Fracturing

4.1. Numerical Model and Calculation Parameters

In order to explore the formation rules of a shale hydraulic fracture network under different bedding dip angles and confining pressures, the SPH models containing bedding and hydraulic perforation are established, as shown in Figure 13. The number of layers is defined according to the scale of actual conditions. The model size is 1 m × 1 m. The bedding width is 0.02 m and the bedding distance is 0.08 m. The bedding dip angle θ is defined as the angle between the bedding direction and the horizontal direction. The vertical stress is defined as σ_h, and the horizontal stress is defined as σ_H. The whole model is divided into 200 × 200 = 40,000 particles. The model parameters are set as follows: Particle density is ρ = 2600 kg/m³, Elastic modulus E = 17 GPa, Poisson’s ratio μ = 0.14, Cohesion is set to be 5.95 Mpa, internal friction angle is 40°, and the tensile strength is 5.2 MPa.

The calculation scheme is set as follows: The vertical stress σ_h = 10 MPa is kept unchanged, and the horizontal stress σ_H is set to be 2 MPa, 4 MPa, 6 MPa, 8 Mpa, and 10 MPa, respectively. The ratio of horizontal stress σ_H to vertical stress σ_h is called the horizontal stress ratio. Meanwhile, the bedding dip angle θ is set to be 15°, 30°, 45°, 60°, 75°, respectively. The detailed information is shown in

Table 1. Calculation schemes.

Bedding Dip Angle θ/°	Horizontal Stress σH/MPa	Vertical Stress σh/MPa
15	2, 4, 6, 8, 10	10
30	2, 4, 6, 8, 10	10
45	2, 4, 6, 8, 10	10
60	2, 4, 6, 8, 10	10
75	2, 4, 6, 8, 10	10

. The load application scheme is as follows: (1) Step 0~2000: in situ stress balance period; (2) Step 2000~7000: internal water pressure loading period. The loading rate is set to be 0.01 MPa/calculation step; (3) Step 7000~10,000: keeping internal water pressure p_w = 50 MPa unchanged.

4.2. Effects of Bedding Dip Angles on Particle Failure Counts under Hydraulic Fracturing

The variations of particle failure numbers with calculation steps under hydraulic fracturing are shown in Figure 14. It can be seen that, with the increase of water pressure in perforation, the number of failure particles increases accordingly. Generally, with the increase of horizontal stress σ_H (the increase of horizontal stress ratio), the total counts of failure particles gradually decrease, which indicates that high in situ stress makes the fracturing pressure of bedding shale formation greater. Meanwhile, the increase of confining pressure also make the difference between bedding angle θ = 15°~75° decrease, indicating that high in situ stress level leads to isotropic hydraulic fracture.

In order to quantitatively explore characteristic pressure changes of hydraulic fracturing under different conditions, three typical forms of stress, i.e., (1) initiation stress, (2) extension stress, and (3) buckling stress, are introduced in this section [44]. According to the analysis of numerical simulation results, the initiation stress of hydraulic fracturing is defined as the internal water pressure corresponding to the particle failure number of 50. The extension stress is defined as the internal water pressure corresponding to the particle failure number of 150. The extension stress is defined as the internal water pressure corresponding to the particle failure number of 500.

Figure 15a shows the variations of initiation stress with horizontal stress ratio under different calculation schemes. It can be seen that the initiation stress of the bedding shale model increases with the increase of horizontal stress ratio, varying from 19 MPa to 26 MPa. The expression of the fitting curve is y = a + bx + cx², where a = 18.8432 ± 1.21916, b = 11.96143 ± 4.64542, and c = −4.89286 ± 3.79804. The R² is 0.95965. Under the same horizontal stress ratio, the initiation stress of small bedding dip angles (θ < 45°) is larger than that of big bedding dip angles (θ > 45°). Figure 15b shows the variations of extension stress with horizontal stress ratio under different calculation schemes. The expression of the fitting curve is y = a + bx, where a = 25.8134 ± 0.22906 and b = 6.661 ± 0.34532. The R² is 0.992. It can be seen that extension stress basically increases linearly with the horizontal stress ratio, varying from 25 MPa to 35 MPa. When the horizontal stress ratio is 0.2 and the bedding dip angle θ > 30° or the horizontal stress ratio is 0.4 and the bedding dip angle θ > 45°, the buckling stress will be greater than 40 MPa.

4.3. Effects of Bedding Dip Angles on Formations of Hydraulic Fracturing Networks

Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20 show the morphology of hydraulic fracturing networks under different horizontal stress ratios and different bedding dip angles. It can be seen that the horizontal stress ratio has a great impact on hydraulic fracturing network morphologies. For the circumstance of low horizontal stress ratios, 2~4 main fractures propagate to a certain extent, whose propagation directions are along the direction of maximum principal stress (σ_h = 10 MPa). However, when the horizontal stress ratio is higher, there are multiple propagation directions and the extension length is relatively short. Meanwhile, when the horizontal stress ratio is relatively lower, the hydraulic fracturing length is longer and the formation range of hydraulic fracturing networks is wider. While in the case of high horizontal stress ratio, the hydraulic fractures are confined around the local perforation area and the fracture formation range is smaller.

The bedding dip angle is also another important factor affecting hydraulic fracture morphologies. For the case with small bedding dip angles (θ = 15°), the hydraulic fracture propagates through the bedding. While for the case with larger bedding dip angles (θ = 75°), the hydraulic fracture propagates along the bedding. For the case with bedding dip angle θ ranging from 30° to 60°, the hydraulic fracture morphology is affected by the horizontal stress ratio: When the horizontal stress ratio is smaller, the hydraulic fracture mainly propagates along the bedding. However, when the horizontal stress ratio is larger, the hydraulic fracture will propagate through the bedding.

5. Conclusions

(1)

The interaction modes between liquid particles and solid particles have been introduced and the traditional kernel function in the SPH method has been improved by introducing the fracture mark ξ, which can realize the hydraulic fracturing simulations of particles.

(2)

The application of the stress boundary of SPH is realized by stress mapping. The feasibility of the improved numerical method is validated by two numerical examples, and the correctness of the method is verified by comparisons with previous experimental results.

(3)

The hydraulic fracturing mesh-less numerical model containing bedding properties has been established, and the numerical simulations are carried out on the progressive fracture propagations under different horizontal stress ratios. With the increase of horizontal stress ratio, the total number of damaged particles decreases, but the initiation and extension pressure increase gradually. The initiation stress of small bedding dip angles (θ < 45°) is larger than that of big bedding dip angles (θ > 45°).

(4)

The morphology of hydraulic fracturing networks is greatly affected by bedding dip angles and the horizontal stress ratio. The range of hydraulic fracture propagation at low horizontal stress ratio is wider and the fracture propagation is along the direction of maximum principal stress, while the range of hydraulic fracture propagation at a high horizontal stress ratio is limited to perforation. The hydraulic fracture will propagate through the bedding with small dip angles. However, when the bedding dip angle is larger, the hydraulic fracture will propagate along the bedding direction.