# An Innovative Method to Analyze the Hydraulic Fracture Reopening Pressure of Hot Dry Rock

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Experimental Methodology

#### 2.1. Material Features and Sample Preparation

#### 2.2. Experimental Equipments

#### 2.3. Hydraulic Fracturing Test

- (I)
- Checked and connected the apparatus.
- (II)
- Placed the specimen in a pressurized and heated chamber constructed by loading pads and the heating furnace. In the Z-axis direction, the upper loading pad contains a deflection hole inside to allow fracturing fluid to be pumped into the specimen during loading. For HFS, since the specimen hole opens at one end only, it is enough to seal the gap between the hole and the loading pad with a center-perforated asbestos gasket. For SFS, the specimen hole opens at both ends, so that besides sealing the upper loading pad, a complete asbestos pad is required to seal the gap between the specimen hole and the lower loading pad.
- (III)
- Turned on the heating furnace. Temperature groups T1, T3, and T5 were heated to 100, 300, and 500 °C, respectively. The heating rate was maintained at 1 °C/min to avoid thermal shock damage.
- (IV)
- Held the preset temperature and turned on the pressure pump to load the true triaxial stress. The maximum, minimum, and intermediate principal stresses loaded on the specimen were 30 MPa, 20 MPa, and 25 MPa, respectively. For SFS specimens, to ensure that the fracture extends in the pre-crack direction, the pre-crack should coincide with the preferred fracture direction (PFD) during hydraulic fracturing, which is perpendicular to the minimum principal stress and parallel to the maximum principal stress [35].
- (V)
- Kept the preset temperature and loading pressure, then turned on the advection pump for fracturing. The pump pressure increased to a peak point and then abruptly dropped to a stable value, when fracturing was completed. The injection rate was 10 mL/min during fracturing.
- (VI)
- Took out the specimen after pressure relief and cooling, examined the cracks on the specimen surface, and analyzed the fracturing curve.

#### 2.4. Brazilian Test

## 3. Experimental Results

#### 3.1. Breakdown and Reopening Pressures of Hot Dry Rock in Hydraulic Fracturing

#### 3.2. Fracture Toughness of Hot Dry Rock in Hydraulic Fracturing

_{1}and perpendicular minimum principal stress σ

_{3}, they propose a calculation equation for the reopening pressure P

_{r}at pre-crack extension.

_{IC}represents the pre-crack re-extension stress intensity factor, also defined as fracture toughness; L is the pre-crack length.

#### 3.3. Tensile Strength from Brazilian Test

_{t}in the Brazilian splitting test can be calculated as follows [40]:

_{max}is the peak loading pressure, D is the disc diameter, and t is the disc thickness.

## 4. Calculation Model for Hydraulic Fracturing Reopening Pressure

#### 4.1. Calculation Model

_{h}represents the minimum principal stress, and σ

_{H}represents the maximum principal stress.

_{r}that allows the hydraulic crack to re-extend is [42]:

_{r}can be calculated from Equations (3) and (4) as:

_{y}is the stress in the y-direction.

_{IC}represents the fracture toughness of the type I fracture. Equation (15) reflects that the reopening pressure P

_{r}is related to the pre-crack length, fracture toughness, and stress values rather than just the breakdown pressure and tensile strength as in Equation (5).

#### 4.2. Validation of the Proposed Calculation Model

## 5. Discussion

#### 5.1. Feasibility of the Novel Analysis Method for Hot Dry Rock Reopening Pressure

#### 5.2. Influence Factors of Hot Dry Rock Reopening Pressure

^{2}= 9.999. This demonstrates a good correlation between the reopening pressure and the tensile strength and fracture toughness of the hot dry rock. Overall, the temperature and lithology of hot dry rock not only affect tensile strength and fracture toughness but furthermore result in variations in reopening pressure.

_{IC}

## 6. Conclusions

- The new test and theoretical method for determining the reopening pressure was applicable to hot dry rock with different lithologies and temperatures. Compared to the conventional equation, the reopening pressure calculated by the new equation was closer to the measurement result. Moreover, as the specimen temperature rose, the deviation between the reopening pressure calculated by the conventional equation and that from the tests became larger, while the calculation of the new equation was consistently close to the measurement result.
- The reopening pressure of hot dry rock correlated well with the tensile strength and fracture toughness of the rock tested, showing that the higher the tensile strength and fracture toughness, the larger the reopening pressure during secondary fracturing. In other words, the specimen’s lithology and temperature affect its tensile strength and fracture toughness, and change its reopening pressure with the same regularity.
- Hot dry rock reopening pressure is strictly dependent on breakdown pressure, tensile strength, fracture toughness, geostress, borehole radius, and initial hydraulic crack length, rather than just breakdown pressure and tensile strength as defined by the conventional theoretical model. Therefore, such critical parameters as breakdown pressure, tensile strength, fracture toughness, geostress, borehole radius, and initial hydraulic crack length that affect reopening pressure should be considered in the design of hot dry rock secondary fracturing.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Photos and matrix surfaces of Brazilian disc specimens of the lithologies G1, G2, G3, and S.

**Figure 3.**Micrographs of G1, G2, G3, and S lithological specimens by a single-polarization microscope.

**Figure 4.**X-ray diffraction (XRD) patterns of rock specimen: (

**a**) G1 specimen; (

**b**) G2 specimen; (

**c**) G3 specimen; (

**d**) S specimen.

**Figure 5.**Specimen preparation: (

**a**) HFS specimen for hydraulic fracturing tests; (

**b**) SFS specimen for secondary fracturing tests; (

**c**) BSS specimens for Brazilian splitting tests.

**Figure 6.**Real-time high temperature true-triaxial hydraulic fracturing platform: (

**a**) the heating system; (

**b**) the true triaxial loading system; (

**c**) the water injection and sealing system; (

**d**) the computer monitoring system.

**Figure 7.**Main hydraulic fracturing procedures: (

**a**) HFS or SFS specimen was placed in the fracturing chamber; (

**b**) specimen was heated in the furnace; (

**c**) three directions confining stresses were applied; (

**d**) specimen fractured by water injection.

**Figure 11.**Fracturing curves for HFS and SFS specimens of four lithologies at different temperatures.

**Figure 15.**Variation of measured and calculated reopening pressure with temperature for all lithological samples.

**Figure 16.**Variation of measured and calculated tensile strength with temperature for all lithological samples.

**Table 1.**Breakdown pressure, fracture toughness and tensile strength of specimens at various temperatures and lithologies.

T/°C | Type | P_{b}/MPa | K_{IC}/(MPa × m^{0.5}) | T/MPa | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No.1 | No.1 | No.1 | Ave. | No.1 | No.1 | No.1 | Ave. | No.1 | No.1 | No.1 | Ave. | ||

100 | G1 | 40.1 | 39.6 | 39.7 | 39.8 | 1.49 | 1.55 | 1.60 | 1.55 | 9.4 | 9.2 | 9.1 | 9.2 |

G2 | 38.2 | 38.6 | 37.8 | 38.2 | 1.44 | 1.41 | 1.49 | 1.45 | 7.6 | 8.2 | 7.9 | 7.9 | |

G3 | 36.9 | 37.3 | 36.6 | 36.9 | 1.36 | 1.41 | 1.34 | 1.37 | 7.2 | 6.9 | 7.1 | 7.1 | |

S | 34.7 | 34.3 | 35.2 | 34.7 | 1.05 | 1.07 | 1.00 | 1.04 | 5.6 | 5.8 | 5.4 | 5.6 | |

300 | G1 | 37.9 | 38.0 | 38.5 | 38.1 | 1.19 | 1.29 | 1.20 | 1.23 | 8.3 | 7.9 | 8.3 | 8.2 |

G2 | 36.5 | 36.4 | 36.1 | 36.3 | 1.18 | 1.23 | 1.15 | 1.19 | 6.7 | 6.5 | 6.8 | 6.7 | |

G3 | 35.4 | 35.8 | 35.2 | 35.5 | 1.09 | 1.11 | 1.16 | 1.12 | 5.9 | 5.9 | 5.7 | 5.8 | |

S | 33.6 | 32.9 | 33.3 | 33.3 | 0.79 | 0.85 | 0.90 | 0.85 | 4.8 | 4.8 | 5.0 | 4.9 | |

500 | G1 | 33.7 | 33.2 | 33.1 | 33.3 | 0.86 | 0.92 | 0.87 | 0.88 | 5.4 | 5.6 | 5.2 | 5.4 |

G2 | 32.8 | 32.5 | 32.2 | 32.5 | 0.77 | 0.72 | 0.80 | 0.76 | 4.6 | 4.3 | 4.3 | 4.4 | |

G3 | 31.3 | 31.1 | 31.7 | 31.4 | 0.73 | 0.71 | 0.68 | 0.71 | 3.1 | 3.0 | 2.8 | 3.0 | |

S | 30.2 | 29.9 | 29.4 | 29.8 | 0.55 | 0.65 | 0.49 | 0.56 | 3.5 | 3.2 | 3.3 | 3.3 |

**Table 2.**Reopening pressure measured and calculated by Equations (5) and (18) at various temperatures and lithologies.

T/°C | Type | P_{r}/MPa Equation (5) | P_{r}/MPa Equation (18) | P_{r}/MPa (Measured) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

No.1 | No.1 | No.1 | Avg. | No.1 | No.1 | No.1 | Avg. | No.1 | No.1 | No.1 | Avg. | ||

100 | G1 | 30.7 | 30.4 | 30.6 | 30.6 | 28.8 | 28.9 | 29.5 | 29.1 | 27.8 | 28.3 | 28.7 | 28.3 |

G2 | 30.6 | 30.4 | 29.9 | 30.3 | 28.4 | 28.0 | 28.0 | 28.1 | 27.4 | 27.1 | 27.8 | 27.4 | |

G3 | 29.7 | 30.4 | 29.5 | 29.9 | 26.9 | 28.0 | 26.6 | 27.2 | 26.7 | 27.1 | 26.5 | 26.8 | |

S | 29.1 | 28.5 | 29.8 | 29.1 | 24.2 | 23.8 | 24.6 | 24.2 | 24.0 | 24.2 | 23.6 | 23.9 | |

300 | G1 | 29.6 | 30.1 | 30.2 | 30.0 | 25.7 | 26.9 | 26.4 | 26.3 | 25.2 | 26.1 | 25.3 | 25.5 |

G2 | 29.8 | 29.9 | 29.3 | 29.7 | 25.8 | 26.3 | 25.1 | 25.7 | 25.1 | 25.6 | 24.9 | 25.2 | |

G3 | 29.5 | 29.9 | 29.5 | 29.6 | 24.9 | 25.4 | 25.4 | 25.2 | 24.4 | 24.5 | 25.0 | 24.6 | |

S | 28.8 | 28.1 | 28.3 | 28.4 | 22.2 | 21.9 | 22.4 | 22.2 | 21.8 | 22.3 | 22.7 | 22.3 | |

500 | G1 | 28.3 | 27.6 | 27.9 | 27.9 | 22.1 | 21.9 | 21.8 | 21.9 | 22.4 | 22.9 | 22.5 | 22.6 |

G2 | 28.2 | 28.2 | 27.9 | 28.1 | 21.4 | 21.1 | 21.3 | 21.3 | 21.6 | 21.2 | 21.9 | 21.6 | |

G3 | 28.2 | 28.1 | 28.9 | 28.4 | 21.2 | 20.9 | 21.5 | 21.2 | 21.3 | 21.1 | 20.8 | 21.1 | |

S | 26.7 | 26.7 | 26.1 | 26.5 | 18.4 | 19.1 | 17.4 | 18.3 | 19.7 | 20.6 | 19.2 | 19.8 |

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**MDPI and ACS Style**

Zhuang, D.-D.; Yin, T.-B.; Zhang, Z.-X.; Aladejare, A.; Wu, Y.; Qiao, Y. An Innovative Method to Analyze the Hydraulic Fracture Reopening Pressure of Hot Dry Rock. *Materials* **2023**, *16*, 1118.
https://doi.org/10.3390/ma16031118

**AMA Style**

Zhuang D-D, Yin T-B, Zhang Z-X, Aladejare A, Wu Y, Qiao Y. An Innovative Method to Analyze the Hydraulic Fracture Reopening Pressure of Hot Dry Rock. *Materials*. 2023; 16(3):1118.
https://doi.org/10.3390/ma16031118

**Chicago/Turabian Style**

Zhuang, Deng-Deng, Tu-Bing Yin, Zong-Xian Zhang, Adeyemi Aladejare, You Wu, and Yang Qiao. 2023. "An Innovative Method to Analyze the Hydraulic Fracture Reopening Pressure of Hot Dry Rock" *Materials* 16, no. 3: 1118.
https://doi.org/10.3390/ma16031118