# Simulation of Fracture Morphology during Sequential Fracturing

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Model Assumption

#### 2.2. Discontinuous Displacement Method

_{′x}, f

_{′y}, f

_{′xx}, f

_{′xy}, f

_{′xyy}, f

_{′yyy}, are the derivatives of function f(x,y), respectively; ν is the Poisson ratio; G is the shear modulus, MPa. For a detailed description of rock deformation, please refer to our previous studies [15].

_{f}

_{1}is the injection pressure, Pa. θ is the angle between discrete element and maximum horizontal principal stress, °.

#### 2.3. Deformation of Hydraulic Fracture

_{n}, and tangential stiffness, K

_{s}, which were defined as the ratio of the increment of normal stress and shear stress, respectively, to the related increment of displacement. Zhou [38] tested the stiffness with stress wave propagation. In this paper, we introduce the support stiffness coefficients ${K}_{s}$ and ${K}_{n}$ to simulate the reverse supporting force from proppant on the fracture surface [33,36,39]. Its expression is as follows:

_{i}(i represent different fracturing stage) fracture elements. At the initial stage of fracturing, there is only one fracture, and the fracture surface is affected by both in-situ stress and fluid pressure inside the fracture. Considering that the model size is infinite, the stress equilibrium equation on each fracture element is given by:

_{max}represents the maximum discontinuous displacement of the node when fracture surface opens, m. Introduce the total stress and deformations of element:

#### 2.4. The Initiation and Propagation of Fracture

^{0.5}; E is Young’s modulus, MPa; ν is Poisson’s ratio, and a is the half-length of crack, m.

_{0}) can be obtained by solving the first-order partial derivative for the circumferential stress:

_{IC}is the fracture toughness, MPa∙m

^{1/2}.

#### 2.5. Flowchart

_{n}and D

_{s}of HF until the accuracy requirements are met by Equations (19)–(23) based on different well condition; ④ Check the iterative step, if the iterative step is greater than the preset value, calculate induced stress (Equation (5)) and exit the calculation, otherwise judge whether HF initiation occurs (Equation (26)); ⑤ If initiation occurs, calculate the deflection angle of the element (Equation (25)), increase fracture units and iterative step, and then go to step ②, else go to step ② directly.

#### 2.6. Model Validation

_{f}is the injection pressure, Pa; r, r

_{1}, r

_{2}are respectively the distance between fracture and point Q, m; θ, θ

_{1}, θ

_{2}are the angle between the x-axis and the line distance, °. The specific stress model is shown in Figure 2a:

_{f}) is −3 MPa, the half-length of crack (a) is 1 m, the coordinates of point Q: x = 0.5 m, and y increases from 0 m to 10 m. Calculate the induced stress at point Q under a different vertical distance (coordinate y) from the fracture plane. The normal stress and shear stress obtained by simulation and calculation decrease with the increase of the distance from the fracture plane, and the numerical solution is basically consistent with the analytical solution. In Figure 2b, the solid line represents analytical results, and the dotted line represents the simulation results.

^{1/2}and 0.7765 MPa∙m

^{1/2}. In addition, at the same time, we provided the corresponding simulation results with different numbers of discrete elements. By comparing the simulation results with the analytical results, the relative error of our calculation model is obtained:

_{anal}− K

_{simu})/K

_{anal}

_{anal}is analytical results, MPa∙m

^{1/2}, K

_{simu}is simulation results, MPa∙m

^{1/2}.

_{f}= 1 MPa, θ = 60°.

## 3. Results and Discussion

#### 3.1. Deformation of Hydraulic Fracture and Induced Stress in a Single Well

#### 3.2. Sensitivity Analysis

^{10}, the deformation of HF aperture is obvious. Conversely, when the stiffness is greater than 5 × 10

^{10}, the variation of residual HF aperture is not obvious. The greater the stiffness, the greater the influence of residual deformation on the in-situ stress after the fracture surface is closed.

#### 3.3. Deformation of HF and Distribution of Induced Stress under Zipper Fracturing

## 4. Conclusions

- (1)
- As large quantities of proppant are injected into hydraulic fracture during hydraulic fracturing, fractures will not completely close after the fracturing operation is completed. This residual aperture caused by proppant may produce induced stress and change the distribution of in-situ stress. Induced stress by residual aperture gradually decreases with the increase of vertical distance from the fracture plane and the decrease of residual aperture of fracture.
- (2)
- The residual aperture will also influence the propagation and maximum aperture of subsequent fracture. When the fracture spacing in sequence fracturing is closer, the residual aperture will inhibit the opening degree of the subsequent fracture, which in turn affects the injection volume of the proppant. During sequence fracturing, fractures tend to exclusion and turn away in staged fracturing, on the contrary, which tends to approach and intersect in zipper fracturing.
- (3)
- Subsequent fracturing in turn compresses the previously cracked fracture, resulting in a further reduction in residual aperture, and after the fracture construction is completed, the previously pressurized fracture aperture is rebound. As the number of hydraulic fracture increases, the residual aperture of the previously pressed fracture gradually decreases. However, the fluctuation of fracture aperture mentioned above is small and less than 0.2 mm.
- (4)
- Sensitivity analysis shows that, in staged fracturing, the smaller the fracturing spacing, the more likely subsequent fractures are to be deflected, while in zipper fracturing, the effect of fracture spacing is not obvious. Well spacing can obviously influence the deflection of subsequent fracture in zipper fracture. With the increase of stiffness, the residual aperture of the hydraulic fracture increases, and the subsequent fractures are more likely to be deflected.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Validation of simulation model. (

**a**) mechanical model; (

**b**) comparison of simulation results; “*” represents the numerical solution, solid lines represent analytical solutions.

**Figure 3.**The mechanical model. Where: θ is the angle between fracture and the uniaxial tensile force P.

**Figure 4.**Physical model of sequential fracturing (

**a**) in a single horizontal well; (

**b**) in a double horizontal well, where: d

_{1}represents the perforating depth and d

_{2}represents the fracturing spacing in a well, and d

_{3}represents well spacing. P

_{h}and P

_{H}represent the minimum and maximum horizontal principal stress sequentially. Hydraulic fracture number represents the fracturing sequence.

**Figure 5.**Numerical simulation results of fracture morphology and induced stress of (

**a**) σ

_{xx}produced by residual aperture after completion of fracturing stage 1; (

**b**) σ

_{xx}produced by residual aperture after completion of fracturing stage 2; (

**c**) σ

_{xx}produced by residual aperture after completion of fracturing stage 3; (

**d**) σ

_{xy}produced by residual aperture after completion of fracturing stage 1.

**Figure 6.**Aperture of hydraulic fracture 1. (

**a**) in different fracturing stages; (

**b**) in different shut off stages of the well.

**Figure 7.**(

**a**) Aperture of HF 1 with different stiffness; (

**b**) induced stress near the fracture surface.

**Figure 8.**Deformation results about HF 2. (

**a**) with different stiffness at constant fracture spacing; (

**b**) with different fracture spacing at constant stiffness.

**Figure 9.**Numerical simulation results of fracture morphology and induced stress of (

**a**) σ

_{xx}produced by residual aperture after completion of fracturing; (

**b**) σ

_{xy}produced by residual aperture.

**Figure 10.**Simulation results of (

**a**) the deflection of HF 2 with different well spacing; (

**b**) the deflection of HF 2 with different fracture spacing. Where: the solid line on Well 2 represents HF 2 at different fracture spacing. The colors of HF 1 and HF 2 are corresponding at different spacing.

Element Number | ${K}_{I}/\left(MPa\xb7{m}^{1/2}\right)$ (Analytical Results, 1.3293) | ${K}_{II}/\left(MPa\xb7{m}^{1/2}\right)$ (Analytical Results, 0.7765) | ||
---|---|---|---|---|

The Relative Error | $SimulationResults$ | The Relative Error | $SimulationResults$ | |

r | 5.0% | 1.2622 | 6.1% | 0.7288 |

4 | 2.1% | 1.3016 | 3.2% | 0.7515 |

6 | 1.1% | 1.3151 | 2.1% | 0.7593 |

10 | 0.2% | 1.3261 | 1.4% | 0.7657 |

Young’s modulus (MPa) | 19,830 | Minimum horizontal stress (MPa) | −20 |

Poisson’s ratio | 0.261 | Fracture toughness (MPa·m^{1/2}) | 2.5 |

Maximum horizontal stress (MPa) | −24 | Injection pressure (MPa) | −20 |

Injection angle (°) | 90 | Fracturing cluster number | 3 |

Fracture spacing(m) | 40 | Perforating depth (m) | 1 |

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**MDPI and ACS Style**

Zheng, P.; Gu, T.; Liu, E.; Zhao, M.; Zhou, D.
Simulation of Fracture Morphology during Sequential Fracturing. *Processes* **2022**, *10*, 937.
https://doi.org/10.3390/pr10050937

**AMA Style**

Zheng P, Gu T, Liu E, Zhao M, Zhou D.
Simulation of Fracture Morphology during Sequential Fracturing. *Processes*. 2022; 10(5):937.
https://doi.org/10.3390/pr10050937

**Chicago/Turabian Style**

Zheng, Peng, Tuan Gu, Erhu Liu, Ming Zhao, and Desheng Zhou.
2022. "Simulation of Fracture Morphology during Sequential Fracturing" *Processes* 10, no. 5: 937.
https://doi.org/10.3390/pr10050937