# Numerical Simulation on Hydraulic Fracture Height Growth across Layered Elastic–Plastic Shale Oil Reservoirs

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## Abstract

**:**

## 1. Introduction

^{9}tons [1]. The Jiyang Depression is located in the southeast part of the Bohai Bay Basin, and the predicted shale oil reservoir resource there reaches 40.45 × 10

^{8}tons. The commercial hydrocarbon flow has been obtained in over 40 wells, showing promising prospects [2]. The continental shale is generally in the middle diagenetic stage, and the shale is characterized by rapid variation in lithology, high clay content, and strong rock plasticity, which cause high breakdown pressure, a short HF propagation path, low fracturing fluid recovery, and low stable production. It is very difficult to form a complex fracture network in shale oil reservoirs [3,4,5,6].

^{3}/min) increases the HF height by 40–58% and promotes HF height growth [3]. The stimulation technology of fracture crossing layers is feasible in shale oil reservoirs.

## 2. Methods and Theory

#### 2.1. Equation of Mechanical Equilibrium

**σ**is the second-order Cauchy stress tensor acting on the entire rectangle in Figure 1;

**ε**is the second-order strain tensor,

**u**is the local displacement vector; p(s, t) is the fluid pressure on the fracture surface Γ

_{f}; σ

_{H}, and σ

_{h}are the maximum and minimum horizontal principal stresses in the far-field; and

**n**

_{t}and

**n**

_{f}are the unit normal vectors at the outer boundary and the fracture boundary, respectively. ${\Gamma}_{out}={\Gamma}_{L}\cup {\Gamma}_{R}\cup {\Gamma}_{T}\cup {\Gamma}_{B}$ at the outer boundary and Γ

_{f}at the inner boundary. Ω denotes the rectangle domain in Figure 1.

**D**is the elastic coefficient matrix. For anisotropic rock under plane strain states, we have:

_{p}is the pore pressure acting on the rock matrix; α is the Biot elastic coefficient, α ∈ [0,1]; and

**I**is the unit tensor. The value of α depends on the magnitude of the increment of pore pressure in the fracture and the decrease of total stress around the rock [37,38,39,40]. If the former is higher than the latter, α = 0.95; otherwise, α = 1 [37,38].

#### 2.2. Equation of Flow Pressure in Hydraulic Fracture

_{0}as the injection point inside the fracture. According to the lubrication equation in fluid mechanics, the continuity equation of Poiseuille planar flow between two parallel plates is expressed as:

_{0}is the source term (i.e., the injection rate of fracturing fluid within the wellbore); Ω

_{f}is the fracture area; k is the fracture permeability calculated according to the cubic law in Equation (11);

**u**

^{+}and

**u**

^{−}are the displacement vector on fracture face; and s is the abscissa of the HF. Due to the cubic term of the fracture width, Equation (10) is a nonlinear transient equation.

#### 2.3. Criterion for Fracture Crossing Bedding

_{0}is the cohesion force of the NF surface; τ

_{β}and σ

_{βy}are the shear stress and normal stress on the NF surface, respectively; θ = β or β-π; β is the intersection angle between the NF and the HF; σ

_{H}and σ

_{h}are the maximum and minimum horizontal stress in the far-field, respectively; T = T

_{0}− [(σ

_{H}-σ

_{h})/2]; T

_{0}is the rock tensile strength; K is the root of Equation (17) when σ

_{1}= T

_{0}and σ

_{1}is calculated in Equations (18)–(21); K

_{I}is the I-type stress intensity factor at the fracture tip; r and α are the polar coordinates at the fracture tip; and σ

_{x}, σ

_{y}, and τ

_{xy}are the stress components, expressed as follows:

#### 2.4. Constitutive Equation of Elastic and Plastic Deformation

_{1}> σ

_{2}> σ

_{3}are the principal stress, and c and $\varphi $ are the cohesion force and the internal friction angle, respectively.

_{m}is the average stress, $\overline{\sigma}$ is the equivalent stress, and J

_{2}and J

_{3}are the second and third invariants of stress deviance, respectively: s

_{x}= σ

_{x}− σ

_{m}; s

_{y}= σ

_{y}− σ

_{m}; s

_{z}= σ

_{z}− σ

_{m}.

_{t}is the rock tensile strength.

#### 2.5. Model Validation

_{0}is the injection rate; μ

_{f}is the fracturing fluid viscosity; E′ is the plane–strain modulus, and E′ = 2G/(1 − ν); G is the shear modulus of shale oil reservoir, and ν is Poisson’s ratio.

## 3. Finite Element Model of HF Height Growth

_{H}and σ

_{h}of 59 MPa and 56.8 MPa, respectively, No.2 sublayer at the depth of 2858–2855 m with σ

_{H}and σ

_{h}of 59 MPa and 57 MPa, respectively, No.2 shell layer at the depth of 2860–2858 m with σ

_{H}and σ

_{h}of 58 MPa and 55 MPa, respectively, No.3 sublayer at the depth of 2855–28,526 m with σ

_{H}and σ

_{h}of 60 MPa and 58 MPa, respectively, No.3 shell layer at the depth of 2853–2850 m with σ

_{H}and σ

_{h}of 62 MPa and 60 MPa, respectively, No. 4 sublayer at the depth of 2848–2845 m with σ

_{H}and σ

_{h}of 62 MPa and 60 MPa, respectively, No. 4 shell layer at the depth of 2850–2848 m with σ

_{H}and σ

_{h}of 60 MPa and 58 MPa, respectively, and No. 5 layer at the depth of 2845–2840 m with σ

_{H}and σ

_{h}of 55 MPa and 52 MPa, respectively.

## 4. Results and Analysis

_{h}in each small layer, and the stress loaded vertically is the vertical stress 71 MPa in Table 1. The second step is the Soils hydro-mechanical coupling analysis and simulates HF propagation, the fluid pressure in the fractures, and the HF width, which are solved with the implicit finite element discretization method and the adaptive time step. The parameters in Table 1 are used as the base case, and the parameters are changed in different cases. A large number of nodes in this example, strong nonlinearity in hydro-mechanical coupling problems, and the interaction between HFs and bedding required a large amount of calculation. Numerical simulation was carried out in the high-performance Sugon computing cluster in Beijing Institute of Petrochemical Technology by parallel computing mode using 24 cores.

#### 4.1. Effects of Bedding Shear Strength

#### 4.2. Effects of Internal Frictional Angle

#### 4.3. Effects of Cohesion

#### 4.4. Effects of Bedding Bond Strength

#### 4.5. Effects of Elastic Modulus

#### 4.6. Effects of Layered Stress Contrast

#### 4.7. Effects of Injection Rate

^{3}/min, 6 m

^{3}/min, and 12 m

^{3}/min are simulated. The HF height propagation across the shale oil reservoir is simulated, respectively.

^{3}/min is five times that under the injection rate of 3 m

^{3}/min. The result is similar to Figure 18. The HF height is about 25 m, 10 m, and 5 m under the injection rate of 12 m

^{3}/min, 6 m

^{3}/min, and 3 m

^{3}/min, respectively. The injection pressure increases with the increase of the injection rate (Figure 19b). When the injection rate is 12 m

^{3}/min, the breakdown pressure is about 100 MPa, and the injection pressure gradually increases. When the injection rate is 6 m

^{3}/min and 3 m

^{3}/min, the breakdown pressure is about 85 Mpa, and the injection pressure is stabilized at about 85 Mpa. As the injection rate increases, the ratio of tensile failure increases (Figure 19c). When the injection rate is 6 m

^{3}/min and 3 m

^{3}/min, the ratio of tensile failure is about 40%, showing a mixture of tensile-shear failure. The maximum HF width increases with the injection rate (Figure 19d). Under the high injection rate, the maximum HF width fluctuates initially and then increases. When the injection rate is 6 m

^{3}/min and 3 m

^{3}/min, the maximum HF width has little difference.

#### 4.8. Effects of Fracture Energy

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Numerical solution and analytical solution of hydraulic fracturing: (

**a**) fracture length and (

**b**) fracture width.

**Figure 5.**Cloud picture of displacement under different bedding shear strengths: (

**a**) shear strength of 12 MPa and time t = 55 s; (

**b**) shear strength of 12 Mpa and time t = 100 s; (

**c**) shear strength of 16 Mpa and time t = 55 s; (

**d**) shear strength of 16 Mpa and time t = 100 s;

**e**shear strength of 20 Mpa and time t = 55 s (base case); and (

**f**) shear strength of 20 Mpa and time t = 100 s (base case).

**Figure 6.**HF width profile and maximum HF width under different bedding shear strengths: (

**a**) 12 MPa; (

**b**) 16 MPa; (

**c**) 20 MPa; and (

**d**) maximum HF width.

**Figure 7.**Fracture parameters under different bedding shear strengths: (

**a**) injection pressure and (

**b**) ratio of tensile failure.

**Figure 8.**Cloud picture of displacement under different internal frictional angles of shale oil reservoir: (

**a**) internal frictional angle of 8° and time t = 55 s; (

**b**) internal frictional angle of 8° and time t = 100 s; (

**c**) internal frictional angle of 17° and time t = 55 s; (

**d**) internal frictional angle of 17° and time t = 100 s.

**Figure 9.**Fracture parameter under different internal frictional angles: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 10.**Cloud picture of displacement field under different cohesion values: (

**a**) cohesion of 8 MPa and t = 55 s; (

**b**) cohesion of 8 Mpa and t = 100 s; (

**c**) cohesion of 12 Mpa and t = 55 s; and (

**d**) cohesion of 12 Mpa and t = 100 s.

**Figure 11.**Fracture parameter under different cohesion levels: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 12.**Cloud picture of displacement field under different bond strengths: (

**a**) bond strength of 0.6 MPa (low electric resistivity) and time t = 55 s; (

**b**) bond strength of 0.6 Mpa (low electric resistivity) and time t = 100 s; (

**c**) bond strength of 3 Mpa (medium electric resistivity) and time t = 55 s; and (

**d**) bond strength of 3 Mpa (medium electric resistivity) and time t = 100 s.

**Figure 13.**Fracture parameter under different bond strengths: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 14.**Cloud picture of displacement field under different elastic moduli: (

**a**) elastic modulus of 35 GPa and t = 55 s; (

**b**) elastic modulus of 35Gpa and t = 100s; (

**c**) elastic modulus of 45 Gpa and t = 55 s; and (

**d**) elastic modulus of 45 Gpa and t = 100 s.

**Figure 15.**Fracture parameter under elastic modulus: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 16.**Cloud picture of displacement field under different stress contrast levels: (

**a**) stress contrast of 6 MPa, time t = 55 s; (

**b**) stress contrast of 6 Mpa, time t = 100 s; (

**c**) stress contrast of 8 Mpa, time t = 55 s; and (

**d**) stress contrast of 8 Mpa, time t = 100 s.

**Figure 17.**Fracture parameter under different stress contrast levels: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 18.**Cloud picture of displacement field under different injection rates: (

**a**) injection rate of 3 m

^{3}/min and t = 55 s; (

**b**) injection rate of 3 m

^{3}/min and t = 100 s; (

**c**) injection rate of 6 m

^{3}/min and t = 55 s; and (

**d**) injection rate of 6 m

^{3}/min and t = 100 s.

**Figure 19.**Fracture parameters under injection rate: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

**Figure 20.**Cloud picture of displacement field under different fracture energies: (

**a**) fracture energy of 3000 Pa·m, time t = 55 s; (

**b**) fracture energy of 3000 Pa·m, time t = 100 s; (

**c**) fracture energy of 4000 Pa·m, time t = 55 s; and (

**d**) fracture energy of 4000 Pa·m, time t = 100 s.

**Figure 21.**Fracture parameters under different fracture energy levels: (

**a**) HF width; (

**b**) pressure; (

**c**) ratio of tensile failure; and (

**d**) maximum HF width.

Parameters | Units | Value |
---|---|---|

Porosity | Decimal | 0.05 |

Initial pore pressure | MPa | 49 |

Poisson’s ratio | Decimal | 0.25 |

Elastic modulus | GPa | 25 |

Cohesion | MPa | 17.45 |

Internal friction angle | ° | 34.23 |

Permeability | mD | 0.01 |

Rock tensile strength | MPa | 3 |

Rock shear strength | MPa | 10 |

Bedding tensile strength | MPa | 6 |

Bedding shear strength | MPa | 20 |

Rock fracture energy | Pa·m | 600/600/2000 |

Bedding fracture energy | Pa·m | 1200/1200/4000 |

Leakoff coefficient | m/(Pa·s) | 1 × 10^{−14} |

Liquid viscosity | mPa·s | 10 |

Injection rate | m^{3}/min | 12 |

Vertical stress | Vertical stress/MPa | 71 |

Dilation angle | Dilation angle/° | 0 |

Shell limestone elastic modulus | GPa | 35 |

Shell limestone Poisson’s Ratio | Decimal | 0.15 |

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**MDPI and ACS Style**

Zeng, H.; Jin, Y.; Wang, D.; Yu, B.; Zhang, W.
Numerical Simulation on Hydraulic Fracture Height Growth across Layered Elastic–Plastic Shale Oil Reservoirs. *Processes* **2022**, *10*, 1453.
https://doi.org/10.3390/pr10081453

**AMA Style**

Zeng H, Jin Y, Wang D, Yu B, Zhang W.
Numerical Simulation on Hydraulic Fracture Height Growth across Layered Elastic–Plastic Shale Oil Reservoirs. *Processes*. 2022; 10(8):1453.
https://doi.org/10.3390/pr10081453

**Chicago/Turabian Style**

Zeng, Hao, Yan Jin, Daobing Wang, Bo Yu, and Wei Zhang.
2022. "Numerical Simulation on Hydraulic Fracture Height Growth across Layered Elastic–Plastic Shale Oil Reservoirs" *Processes* 10, no. 8: 1453.
https://doi.org/10.3390/pr10081453