# The Impact of Oriented Perforations on Fracture Propagation and Complexity in Hydraulic Fracturing

^{1}

^{2}

^{*}

## Abstract

**:**

^{−3}Pa·s to 1 Pa·s. While for the gas fracturing, the critical perforation angle remains 62° to 63°. This study is of great significance in further understanding the near-wellbore impacts on hydraulic fracture propagation and complexity.

## 1. Introduction

## 2. Conceptual Model

_{H}and minimum horizontal stress σ

_{h}. The preexisting perforation length L is assumed to be very small relative to the wellbore radius r—therefore the impact of preexisting perforation length on hydraulic fracture path can be neglected. The perforation is defined by its azimuth θ with respect to the orientation of maximum horizontal stress σ

_{H}(x-axis). If the perforation tunnel is oriented in the direction of the maximum principal stress ($\theta =0$), the hydraulic fractures would propagate directly in this (x-axis) direction. When the tunnels are oblique to this direction ($\theta \ne 0$) the hydraulic fractures must ultimately reorient themselves parallel to the direction of the maximum horizontal stress when they exit the stress shadows of the near-wellbore region and of the perforation. Beyond such stress-shadows, the hydraulic fractures tend to propagate in the plane of least resistance perpendicular to the minimum horizontal stress.

## 3. Formulation of the Conceptual Model

#### 3.1. Governing Equations for Mechanical Response

_{i}(m) and fluid pressure p (Pa), can be expressed as:

#### 3.2. Governing Equations for Fluid Flow

#### 3.2.1. Slightly Compressible Fluids

_{f}(Pa

^{−1}), as:

_{f}is constant over a prescribed range of pressure, after integration, Equation (4) can be written as:

^{−3}) at the reference pressure p

_{0}(Pa). According to Taylor series expansion, this may be expressed as:

_{0}. Similarly, this may be approximated as:

^{2}) and ${\mu}_{f}$ is the fluid viscosity (Pa·s).

^{−3}), $q$ is the Darcy velocity vector and ${Q}_{s}$ is the source or sink of the compressible fluid defined in terms of mass rate (kg·m

^{−3}·s

^{−1}).

#### 3.2.2. Compressible Fluids

_{g}is both much larger than for a slightly compressible fluid (liquid) and variable with pressure. In such a case, the pressure-volume-temperature (PVT) relation for a non-ideal or real gas can be written as:

^{−1}), R is the universal gas constant (J·mol

^{−1}·K

^{−1}), and T is the temperature (K). Assuming that the gas compressibility and viscosity are constant, then the governing relation is defined as:

#### 3.3. Governing Equations Accommodating Rock Heterogeneity and Damage Evolution

_{0}is related to the average of the element parameter, and m is the shape parameter of the Weibull distribution function. m is defined as the degree of rock homogeneity and called the homogeneity index. The variations of $f\left(u\right)$ with respect to m are shown in Figure 2. Obviously, a higher m value represents a more homogeneous rock.

_{t}

_{0}and f

_{c}

_{0}are uniaxial tensile and compressive strength (Pa), respectively, ${\sigma}_{1}$ and ${\sigma}_{3}$ are first and third principal stresses (Pa), respectively, θ is the internal frictional angle (°), and F

_{1}and F

_{2}are two damage threshold functions (Pa).

_{0}are the Young’s modulus (Pa) of the damaged and the undamaged element, respectively. In the present study, the element, as well as its damage, is assumed isotropic, so the E, E

_{0}and D parameters are all scalars. Under any stress and initial conditions, the tensile stress criterion is applied preferentially. In other words, the maximum tensile stress criterion is first applied to judge whether the elements are first damaged in tension or not. Only elements that survive this test will be checked for damage in shear using the Mohr–Coulomb criterion.

_{1}< 0 and F

_{2}< 0 the applied stress is insufficient to satisfy the maximum tensile stress criterion and the Mohr–Coulomb failure criterion, respectively. F

_{1}= 0 and dF

_{1}> 0 implies rock damage in the tensile mode when the stress state satisfies the maximum tensile stress criterion and the rock is still under load. F

_{2}= 0 and dF

_{2}> 0 implies rock damage in the shear mode when the stress state satisfies the Mohr–Coulomb failure criterion and the rock remains loaded.

_{1}> 0 or dF

_{2}> 0) and remain unchanged during unloading (dF

_{1}< 0 or dF

_{2}< 0). In this respect, the damage, defined by Equation (20), reduces the elastic modulus E and the shear modulus G of the rock, via to Equations (20) and (21).

## 4. Model Validation against Experimental Observations

#### 4.1. Comparisons of Breakdown Pressure and Fracture Geometry

#### 4.2. Geometric Model and Boundary Conditions

_{H}applied on the top boundary and σ

_{h}applied on the right boundary with rollers applied along both the left side and the base. There are no-flux conditions on all boundaries except for the borehole wall into which a constant fluid injection rate is applied. The fluid injection rate in the experiments is 2.1 × 10

^{−9}m

^{3}·s

^{−1}. The numerical mechanical properties for the simulation are listed in Table 2.

#### 4.3. Effects of Preexisting Perforation Orientation

## 5. Analysis of Near-Wellbore Hydraulic Fracture Complexity

#### 5.1. Effects of Horizontal Differential Stress

_{h}is held constant at 20 MPa. Stress ratio has a significant effect on both hydraulic facture propagation and reorientation, as shown in Figure 9, where different stress ratios correspond to different fracture morphologies. When the stress ratio is equal to unity (hydrostatic) the hydraulic fractures initiate and propagate along the direction of the oriented perforation. As the stress ratio λ is proportional to the difference between σ

_{H}and σ

_{h}, the larger this difference, the larger the propensity for the hydraulic fractures to reorient themselves to the maximum horizontal stress direction [35]. A larger stress ratio results in both a smaller curvature during reorientation and occurs at a shorter distance for the fractures to reorient their direction to align with the direction of the maximum horizontal stress. Moreover, based on the results presented in Figure 10 and Table 3, it appears that the stress ratio also has a significant influence on the initiation pressure and breakdown pressure during hydraulic fracturing. It is apparent that there are clear reductions both in the initiation pressure and the breakdown pressure with increasing stress ratio.

#### 5.2. Effects of Initial Pore Pressure

_{h}(20 MPa) and σ

_{H}(32 MPa), the initial pore pressure is varied from 2 MPa to 16 MPa. The purpose of these numerical simulations is to examine the effects of initial pore pressure on initiation pressure and breakdown pressure. An increase of fluid pore pressure can decrease static friction and thereby facilitate fracture propagation on favorably oriented planes in a deviatoric stress field. As shown in Figure 12 and Figure 13, the resulting differential initiation pressure (P

_{c}-P

_{0}) and breakdown pressure (P

_{b}-P

_{0}) decrease with an increase in the initial pore pressure. This is consistent with the D–C (Detournay–Cheng) criterion [36]. To incorporate an effective stress law into the D–C criterion, the geomechanical model proposed in this study correctly describes the relationship between breakdown pressure and the far-field stress in hydraulic fracturing [37]. The effects of pore pressure on the initiation pressure are illustrated by the initiation equation based on poroelastic theory [38]. The initiation pressure decreases with an increase in the initial pore pressure [39]. There are two classical approaches to define initiation pressure in terms of far-field stresses [38,40]. These represent behavior for both nonpenetrating injection fluids [40], and for penetrating fluids [38], with initiation pressures given by:

_{H-W}and P

_{H-F}are the initiation pressures related to Hubbert–Willis [40] and Haimson–Fairhurst [38] equations, respectively, T is the tensile strength of the rock, and σ

_{h}and σ

_{H}are the far-field principal stresses. As $0\le \varphi \le \alpha \le 1$ ($\varphi $ is rock porosity) and $0\le \nu \le 0.5$ ($\nu $ is Poisson’ ratio of rock) for rock, then obtain $0\le \alpha \left(1-2\nu \right)/\left(1-\nu \right)\le 1$. Therefore, the initiation pressure for a penetrating fluid is always smaller than (or equal to) that for a nonpenetrating fluid. It can be seen from Figure 12 that the resulting differential initiation pressure (P

_{c}-P

_{0}) during gas fracturing is significantly different from that for liquid fracturing. The obvious effect of gas fracturing is in the reduction of the initiation pressure. This is in good agreement with Equation (22). In addition, as shown in Figure 12 and Figure 13, for water-based fracturing, the resulting P

_{c}-P

_{0}and P

_{b}-P

_{0}decrease linearly with an increase in the initial pore pressure. However, for gas fracturing, the resulting P

_{c}-P

_{0}and P

_{b}-P

_{0}indicate a nonlinear decrease with an increase in initial pore pressure. The influences of gas penetration complicate the mechanism of the fracturing process. The gas penetration not only alters the pore pressure in the reservoir, but also the gas adsorption-induced damage modifies the mechanical properties of the reservoir rock [27,41,42]. According to the Langmuir adsorption isotherm, as shown in Equation (2), with an increase in pore pressure, the sorption capacity and volumetric strain simultaneously increase with the adsorptive strain potentially resulting in additional rock damage. This adsorption-induced damage is fully coupled to the gas fracturing model proposed in this paper. As a consequence, both the initiation pressure and breakdown pressure of gas fracturing show nonlinear decreases with increasing pore pressure.

#### 5.3. Effects of Fracturing Fluids

## 6. Conclusions

^{−}

^{4}Pa·s. Notably, the critical perforation angle, remaining 62° to 63°, and does not change a lot with the varies of gas viscosity. The critical perforation angle is sensitive to the viscosity of liquid and is insensitive to the viscosity of gas.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Distributions of rock properties for different homogeneous indices (Mechanical parameter ${u}_{0}$ is 1).

**Figure 4.**Simulation of uniaxial compression of a sample compared with experimental observations by Chen et al. [11].

**Figure 5.**Comparison of model simulation results with experimental results. (

**a**) Experimental fracture pattern from Chen et al. [11]. (

**b**) Fracture pattern obtained from numerical simulation.

**Figure 6.**Geometry of the numerical model used to investigate near-wellbore hydraulic fracturing processes.

**Figure 7.**Numerical results of near-wellbore fracture patterns for different preexisting perforation angles.

**Figure 9.**Simulation results of fracture tortuosity/complexity of hydraulic fracturing produced at various stress ratios.

Symbol | Experimental Data | Numerical Data | $\mathbf{\Delta}\mathit{E}/\mathit{E}$ | $\mathbf{\Delta}{\mathit{\sigma}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{c}}$ | $\mathbf{\Delta}{\mathit{\sigma}}_{\mathit{t}}/{\mathit{\sigma}}_{\mathit{t}}$ |
---|---|---|---|---|---|

Young’s modulus, E (GPa) | 8.402 | 8.51 | 1.29% | - | - |

Uniaxial compressive strength, σ_{c} (MPa) | 28.34 | 28.57 | - | 0.81% | - |

Tensile strength, σ_{t} (MPa) | 2.59 | 2.56 | - | - | −1.16% |

Symbol | Value | Unit |
---|---|---|

Homogeneity index, m | 10 | - |

Mean value of the elasticity modulus, E | 8.737 | GPa |

Mean value of uniaxial compressive strength, ${\overline{f}}_{c}$ | 45.53 | MPa |

Mean value of uniaxial tensile strength, ${\overline{f}}_{t}$ | 5.69 | MPa |

Poisson ratio, $\nu $ | 0.23 | - |

Initial porosity, ${\varphi}_{0}$ | 1.85 | % |

Initial permeability, ${k}_{0}$ | 1.0 × 10^{−16} | m^{2} |

Initial pore pressure, ${p}_{0}$ | 0.1 | MPa |

Viscosity, $\mu $ | 133 | mPa·s |

Stress Ratio | Perforation Angle (^{o}) | Initiation Pressure (MPa) | Breakdown Pressure (MPa) |
---|---|---|---|

1.0 | 45 | 30.3 | 54.5 |

1.2 | 45 | 29.4 | 53.7 |

1.4 | 45 | 28.3 | 52.3 |

1.6 | 45 | 27.9 | 51.6 |

1.8 | 45 | 26.5 | 49.5 |

2.0 | 45 | 25.2 | 46.8 |

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**MDPI and ACS Style**

Liu, L.; Li, L.; Elsworth, D.; Zhi, S.; Yu, Y.
The Impact of Oriented Perforations on Fracture Propagation and Complexity in Hydraulic Fracturing. *Processes* **2018**, *6*, 213.
https://doi.org/10.3390/pr6110213

**AMA Style**

Liu L, Li L, Elsworth D, Zhi S, Yu Y.
The Impact of Oriented Perforations on Fracture Propagation and Complexity in Hydraulic Fracturing. *Processes*. 2018; 6(11):213.
https://doi.org/10.3390/pr6110213

**Chicago/Turabian Style**

Liu, Liyuan, Lianchong Li, Derek Elsworth, Sheng Zhi, and Yongjun Yu.
2018. "The Impact of Oriented Perforations on Fracture Propagation and Complexity in Hydraulic Fracturing" *Processes* 6, no. 11: 213.
https://doi.org/10.3390/pr6110213