_{2}-Driven Hydraulic Fracturing Trajectories across a Preexisting Fracture

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Defining the trajectory of hydraulic fractures crossing bedding planes and other fractures is a significant issue in determining the effectiveness of the stimulation. In this work, a damage evolution law is used to describe the initiation and propagation of the fracture. The model couples rock deformation and gas seepage using the finite element method and is validated against classical theoretical analysis. The simulation results define four basic intersection scenarios between the fluid-driven and preexisting fractures: (a) inserting—the hydraulic fracture inserts into a bedding plane and continues to propagate along it; (b) L-shaped crossing—the hydraulic fracture approaches the fracture/bedding plane then branches into the plane without crossing it; (c) T-shaped crossing—the hydraulic fracture approaches the fracture/bedding plane, branches into it, and crosses through it; (d) direct crossing—the hydraulic fracture crosses one or more bedding planes without branching into them. The intersection scenario changes from (a) → (b) → (c) → (d) in specimens with horizontal bedding planes when the stress ratio

The production of environmentally friendly unconventional gas has increased rapidly in recent years including shale gas, gas from tight sandstones, and coal bed methane [^{3} [_{2} is 4-20 times of that of methane [_{2} and the cosequestration of CO_{2}.

Shale is a sedimentary rock containing bedding planes [

A variety of studies focus on the behavior of hydraulic fractures intersecting preexisting bedding planes, including theoretical analyses, laboratory experiments, and numerical simulations. Multiple crossing criteria have been proposed using different methods, such as linear elastic fracture mechanics [

Laboratory hydraulic fracturing experiments conducted on rock containing bedding planes investigate the controls on intersection behavior from different perspectives. Hydraulic fracturing experiments in prefractured shale indicate that hydraulic fractures tend to cross a preexisting fracture only under high differential stresses and at high angles of approach [

Numerical simulation is an effective way to study the intersection relation between hydraulic fractures and bedding planes. Such approaches have explored the physics of fracture-inhomogeneity interactions, indicating that hydraulic fracture branching and diversion are the result of inhomogeneity [

In this work, a coupled hydraulic-mechanical model is proposed where a damage evolution law is employed. It is then solved by FEM using COMSOL and MATLAB and utilized to simulate the fracturing processes. The intersection scenarios between hydraulic fractures and bedding planes under several conditions are numerically researched.

We develop a damage-based hydraulic-mechanical model that follows the evolution of damage around a borehole with a propagating fluid-driven fracture. The model couples fluid and mechanical deformations to define the geometry of the resulting fluid-driven fracture.

According to the elastic theory, the constitutive equation considering the influence of pore pressure can be written as

Combining the modified constitutive equation, the geometric equation, and the equilibrium equation, the modified Navier-type equation can be written as

The governing equation for CO_{2} flow based on mass balance can be defined as
_{2} mass per volume of rock, _{2} density, _{2}, _{2}, and

According to Darcy’s law, the seepage velocity of CO_{2} can be defined as
_{2}.

The shale rock is assumed saturated with the injected CO_{2}; therefore, CO_{2} mass per volume of rock can be written as _{2} will transfer from the gaseous to the supercritical state when the pressure reaches 7.43 MPa (the temperature is kept at 350 K). As shown in Figure _{2} change dramatically when the phase change occurs. Base on the above, the first item of equation (_{2} which can be calculated from Figure

The evolution of density and viscosity of CO_{2} versus pressure at 350 K.

Substituting equations (_{2} continuity equation is shown as

A damage evolution law based on representative elemental volume (REV) is used in this study to describe the initiation and propagation of hydraulic fracture in numerical samples with bedding planes. The evolution of stress-strain of REV under uniaxial tension or compression is shown in Figure

The damage constitutive criterion of REVs under uniaxial stress conditions.

For a damaged REV, the elastic modulus decreases but the permeability increases correspondingly as the damage variable increases. The evolution of elastic modulus and permeability with damage variable

Shale is a kind of sedimentary rock which is heterogeneous. Previous studies indicated that the heterogeneity of rock plays an important role on the propagation of microcracks [

We use a finite element method to solve the proposed numerical model and obtain numerical results, and we compare the numerical results with two classical analytical results to verify the effect of the proposed model.

The numerical model is established in the previous section. Due to the complex coupled relationship between solid mechanics field and fluid seepage field, it is difficult to obtain an analytical solution. Therefore, the finite element method (FEM) is adopted to solve these coupled equations. The primary procedures are summarized as shown in Figure

After setting up the model geometry, the geometry is discretized into a series of REVs. Then, the initial mechanical parameters are defined at the REV scale and the boundary conditions are applied correspondingly

Numerical calculation is conducted by COMSOL Multiphysics at the initial load step. After calculation, the stress and strain of REVs are obtained for the following analysis

According to criterions (

The damage variable of damaged REVs can be calculated according to equation (

Numerical simulation is conducted with the updated parameters, and the simulation results are compared with results of the former iteration step. If the damage area expands, steps (c)–(e) are repeated; otherwise, step (f) is applied

The boundary conditions are updated in the next load increment

Primary solution procedures for the numerical model.

In this part, the proposed model for simulating the propagation of the hydraulic fracture is validated. There are two classical theoretical solutions for forecasting the breakdown pressure in terms of far-field stresses, tensile strength, and initial pore pressure. One is proposed by Hubbert and Willis [

In this part, the tensile strength of the shale rock is 6 MPa, the initial pore pressure of the rock is 1 MPa, the initial permeability of the rock is 10^{-18} m^{2}, the Biot coefficient is 0.1, and Poisson’s ratio of the rock is 0.225. The model geometry for verification is shown in Figure

The numerical geometry for verification.

The breakdown pressures under different tectonic stress coefficients obtained by H-W solution, H-F solution, and numerical simulation.

Shale rock naturally contains bedding planes with different directions owing to geological deposition and folding. These bedding planes in different directions play a significant role in the propagation of the hydraulic fracture. Thus, numerical samples with bedding planes in different directions are adopted in this work. As shown in Figure ^{3}/s. The parameters used in the simulations can be found in Table

The numerical geometry for fracturing experiments.

The parameters used in simulations.

Parameters | Rock matrix | Bedding plane |
---|---|---|

Average elastic modulus of REVs (GPa) | 36 | 18 |

Poisson’s ratio | 0.225 | 0.25 |

Average tensile strength of REVs (MPa) | 6.2 | 3.1 |

Average compressive strength of REVs (MPa) | 62 | 31 |

Initiate porosity | 0.01 | 0.015 |

Initiate permeability (m^{2}) | 10^{-18} | |

Internal friction angle (rad) | 0.368 | 0.368 |

Initiate pore pressure (MPa) | 1 | 1 |

Heterogeneity coefficient | 6 | 6 |

The physical processes for CO_{2}-driven hydraulic fracture trajectories across a bedding plane are simulated. Besides, the evolution of intersection scenarios between hydraulic fractures and bedding planes under several conditions (stress ratio, bedding plane angle, and bedding plane stiffness) is obtained and discussed.

A series of numerical simulation tests were conducted to study the impact of stress ratio on the propagation of hydraulic fractures. Eight stress ratios were used in this study which are

In Figure

The fracture initiation pressure of specimen versus stress ratio

Figure

The distribution of hydraulic fractures in horizontal bedding plane samples under different stress ratios. (a)

The horizontal radius and vertical radius of hydraulic fractures under different stress ratios.

Understanding the complexity of the fracture network is of vital importance for hydraulic fracturing design. The fracture network near the gas reservoirs is formed through propagation and combination of basic intersection scenarios. Thus, the basic intersection types between the hydraulic fracture and bedding plane are summarized in this subsection.

Based on the simulation results above, four types of intersection scenarios between hydraulic fractures and bedding planes are shown in Figure

Four intersection scenarios between hydraulic fractures (HF) and bedding planes (BP): (a) inserting, (b) L-shape crossing, (c) T-shape crossing, and (d) direct crossing.

Numerical simulations were performed to study the effect of the bedding plane angle on the propagation of hydraulic fractures. Six kinds of specimens with different bedding plane angles (

The distribution of hydraulic fractures in specimens with different bedding plane angles is shown in Figure

The distribution of hydraulic fractures in specimens with different bedding plane angles: (a)

Three numerical tests were performed to research the behavior of hydraulic fractures propagating in specimens with different levels of stiffness of bedding planes. The stiffness (elastic modulus

Figure ^{2} at time ^{2} and 0.00243 m^{2}, respectively, whilst the highest acoustic emission decreases to 550 and 364, respectively. The results indicate that hydraulic fractures are formed more easily, and a relatively more complex intersection scenario is obtained with a lower stiffness of the bedding plane.

The distribution of hydraulic fractures in specimens with different levels of stiffness of bedding planes: (a)

The development of seepage area and acoustic emission with different levels of stiffness of bedding planes: (a)

Understanding the mechanism of the intersection scenario between hydraulic fractures and bedding planes is of vital importance for creating a complex fracture network and improving the recovery of unconventional resources. In this work, a coupled hydraulic-mechanical model is developed where a damage evolution law is used to describe the initiation and propagation of hydraulic fractures. This model is then used to conduct a series of numerical simulations to investigate the propagation of hydraulic fractures in specimens with bedding planes. The following conclusions can be obtained:

Stress ratio has a vital impact on the intersection scenario between a hydraulic fracture and a bedding plane. Four types of intersection scenarios are summarized based on the study: (a) inserting—the hydraulic fracture inserts into a bedding plane and continues to propagate along it; (b) L-shaped crossing—the hydraulic fracture approaches the fracture/bedding plane then branches into the plane without crossing it; (c) T-shaped crossing—the hydraulic fracture approaches the fracture/bedding plane, branches into it, and crosses through it; (d) direct crossing—the hydraulic fracture crosses one or more bedding planes without branching into them. The simulation results indicate that intersection types vary from (a) → (b) → (c) → (d) in specimens with horizontal bedding planes when the stress ratio

The bedding plane angle can also greatly affect the propagation of hydraulic fractures. The results show that the intersection type changes from (d) → (c) → (a) with the increase of the bedding plane angle

We also investigate the influence of bedding plane stiffness on the propagation of hydraulic fractures. The results indicate that the intersection type changes from the T-shaped crossing to the direct crossing with the increase of the bedding plane stiffness. When the stiffness ratio

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

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (51804339).

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