# Impact of Wellbore Cross-Sectional Elongation on the Hydraulic Fracturing Breakdown Pressure and Fracture Initiation Direction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Definition, Mixed Criterion

#### 2.1. Elliptical Wellbores

#### 2.2. Breakout Wellbores

#### 2.3. The Numerical Model

#### 2.4. The Mixed Criterion, Procedure for Finding Hydraulic Fracture Breakdown Pressure and Initiation Length

## 3. Results for Elliptical Wells

#### 3.1. The Effect of the Wellbore Shape Parameter on Fracture Initiation Position and Breakdown Pressure

#### 3.2. The Effect of Wellbore Size on Breakdown Pressure Behavior

#### 3.2.1. The Isotropic Stress Field

#### 3.2.2. The Anisotropic Stress Field

## 4. Results for Breakout Wellbores

#### 4.1. The Effect of Breakout Depth on Breakdown Pressure

#### 4.2. The Effect of Breakout Width on Breakdown Pressure

#### 4.3. The Effect of Deviatoric In Situ Stress on Breakdown Pressure

#### 4.4. The Effect of Wellbore Size on Breakdown Pressure

## 5. Discussion

## 6. Conclusions

- For each deviatoric stress, there is a critical shape factor. If the wellbore shape factor is less than the critical shape factor, the breakdown pressure increases with the increase in the shape factor; for this case, the fracture initiation site is aligned with the major principal stress (like circular wellbores) or minor diameter of the ellipse. If the wellbore shape factor is greater than the critical shape factor, the breakdown pressure will decrease as the shape factor increases, and in this situation, the fracture initiation site is aligned with the minor principal stress or major diameter of the wellbore, unlike the circular wellbores.
- According to Equation (6), the critical eccentricity is in control of the deviatoric stress and the rock tensile strength, it increases by increasing the deviatoric stress or decreasing the tensile strength of the rock.
- The mixed model strongly represents the impact of wellbore size on the breakdown pressure behavior. The analyses performed on elliptical wellbores with the same shape factor but different size show that in large wellbores, the tensile fracture mechanism and breakdown pressure are dominated by the rock tensile strength criterion, but in small wellbores, the fracture mechanism is dominated by fracture toughness.

- The main finding of the section of breakout wellbores is that it was observed that for a given in situ stress, the breakdown pressure in the breakout wellbore is always lower than that of the circular wellbore. As shown in the following results, wellbore elongation has a significant effect on both breakdown pressure and failure initiation position.
- Where $0\le D<0.9$, fracture initiation is aligned with the minor in situ stress or breakout tip, and the ratio of breakdown pressure in the breakout wellbore to the breakdown pressure of the circular wellbore is between 0.47 and 1, depending on the deviatoric stress and the width and depth of the breakout zone. In this case, with the rise of the dimensionless deviatoric stress, the breakdown pressure increases.
- Where $0.9\le D\le 2.5$, the initiation of fracture is aligned with the maximum in situ stress, while, the ratio of the breakdown pressure of the breakout wellbore to the breakdown pressure in the circular wellbore is between 1 and 0.04, depending on the deviatoric stress and the width and depth of the breakout zone. In this case, with the rise of the dimensionless deviatoric stress, the breakdown pressure declines.
- As the breakout depth increases, the breakdown pressure declines.
- The effect of wellbore size in breakout wellbores with the same breakout depth and width but different diameters was investigated and it was seen that similar to elliptical wellbores, the breakdown pressure in large-diameter wellbores is dominated by the tensile strength theory, while the breakdown pressure in small wellbores is dominated by the fracture toughness criterion.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Numerical Model Verification

**Figure A1.**Configuration of circular wellbore with two symmetrical cracks along maximum in situ stress.

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**Figure 5.**Illustration of calculating the breakdown pressure: (

**a**) cross-section of a breakout wellbore; (

**b**) wellbore pressure versus crack length.

**Figure 6.**Plot of breakdown pressure against the shape parameter for different values of dimensionless deviatoric stress.

**Figure 8.**Breakdown pressure obtained from the mixed criterion versus the minor axis of the ellipse for different shape parameters $\left(D=0\right)$.

**Figure 9.**Dimensionless breakdown pressure versus the dimensionless size of the wellbore (made dimensionless by relation (8)) $\left(D=0\right)$.

**Figure 11.**Breakdown pressure versus the minor axis of the ellipse for $e=0$ and $e=0.1$ $\left(D=0.6\right)$.

**Figure 12.**Breakdown pressure versus the minor axis of the ellipse for $e=0.2$ and $e=0.3$ $\left(D=0.6\right)$.

**Figure 13.**Dimensionless breakdown pressure versus the minor axis of the ellipse (made dimensionless by relation (10)) $\left(D=0.6\right)$.

**Figure 14.**Dimensionless breakdown pressure versus the minor axis of the ellipse (made dimensionless by relation (11)) $\left(D=0.6\right)$.

**Figure 16.**Dimensionless breakdown pressure versus $\chi $ (made dimensionless by relation (11))$\left(e=0.3\right)$.

**Figure 25.**Cross section of breakout wellbores with the same breakout depth and width but different diameters, a: $a=0.2\text{}\mathrm{m}$, b: $a=0.1\text{}\mathrm{m}$, c: $a=0.05\text{}\mathrm{m}$, d: $a=0.02\text{}\mathrm{m}$, e: $a=0.006\text{}\mathrm{m}$, f: $a=0.001\text{}\mathrm{m}$.

$E$ | $\upsilon $ | $T$ | ${K}_{Ic}$ |

$\mathrm{GPa}$ | --- | $\mathrm{MPa}$ | $\mathrm{MPa}\text{}\sqrt{\mathrm{m}}$ |

59 | 0.25 | 8 | 1.16 |

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**MDPI and ACS Style**

Jolfaei, S.; Lakirouhani, A.
Impact of Wellbore Cross-Sectional Elongation on the Hydraulic Fracturing Breakdown Pressure and Fracture Initiation Direction. *Processes* **2024**, *12*, 848.
https://doi.org/10.3390/pr12050848

**AMA Style**

Jolfaei S, Lakirouhani A.
Impact of Wellbore Cross-Sectional Elongation on the Hydraulic Fracturing Breakdown Pressure and Fracture Initiation Direction. *Processes*. 2024; 12(5):848.
https://doi.org/10.3390/pr12050848

**Chicago/Turabian Style**

Jolfaei, Somaie, and Ali Lakirouhani.
2024. "Impact of Wellbore Cross-Sectional Elongation on the Hydraulic Fracturing Breakdown Pressure and Fracture Initiation Direction" *Processes* 12, no. 5: 848.
https://doi.org/10.3390/pr12050848