# Confining Stress Response to Hydraulic Fracturing Volumetric Opening on the Representative Volume Element (RVE) Scale

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## Abstract

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## 1. Introduction

## 2. The Model of Confining Stress Response to the HVO

#### 2.1. Staged Expressions of the HVO

#### 2.2. Breakdown Criterion and Evolving Laws of State Variables

#### 2.3. Model of Confining Stress Response on RVE

## 3. Incorporating the Regimes of Hydraulic Fracturing Propagation

## 4. Examples

#### 4.1. The Process for Obtaining the Parameters of the Confining Stress Response Model

#### 4.1.1. Parameter Measurement

#### 4.1.2. Parameter Conversion

#### 4.1.3. Computation Time

#### 4.1.4. Curve Fitting

#### 4.2. Physical Meanings of the Curve Fitting

#### 4.3. Results and Validation

#### 4.3.1. Hydraulic Volumetric Openings

#### 4.3.2. Confining Stress Response

#### 4.3.3. Back Stress Effects

#### 4.4. The Mechanisms of Confining Stress Response in the Four Limiting Regimes

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Hydraulic fracturing process in the RVE. The ‘

**A**’ is original compression state; ‘

**B**’ represents the stress neutralization state; ‘

**C**’ is the critical fracture state; ‘

**D**’ is the fracture opening-steady state.

**Figure 2.**Quasi-static evolving path of the skeleton stress. The parameter ${\sigma}_{I}^{\prime}$ signifies the contrast between the effective stresses. ${\sigma}_{coh}$ represents cohesive stress. ${P}_{p}$ denotes pore pressure. $\alpha {P}_{p}$ represents the effective pore pressure, controlled by stretch bulk strain ${\epsilon}_{I}$ and fluid injection time $t$. The OCD curve represents a cohesive traction decomposition model [33]. The blue curve represents evolution law of ${P}_{p}-t$; The red curve represents evolution law of ${\sigma}_{I}^{\prime}-{\epsilon}_{t}$.

**Figure 6.**Evolutions of the effective stress values and fluid pressure in the four limiting regimes.

**Figure 7.**Contrast between the evolutions of the effective stress and damage via principal strain in four limiting regimes.

**Figure 9.**Contrast between evolutions of breakdown damage, confining stress, and fluid pressure via a hydraulic volumetric opening in four limiting regimes.

Fracturing Cases | ${\mathit{\sigma}}_{\mathit{t}}$/MPa | ${\mathit{\sigma}}_{0}$/MPa | ${\mathit{\sigma}}_{\mathit{h}}$/MPa | k_{0}/mD | $\mathit{\varphi}$/% | ${\mathit{\alpha}}_{0}$ | Regimes |
---|---|---|---|---|---|---|---|

1 | 0.36 | 7.3 | 5.8 | 0.01 | 2 | 0.6 | $M$ |

2 | 0.36 | 7.3 | 5.8 | 1000 | 10 | 0.6 | $\tilde{M}$ |

3 | 28.00 | 7.3 | 5.8 | 0.01 | 2 | 0.6 | $K$ |

4 | 28.00 | 7.3 | 5.8 | 200 | 10 | 0.6 | $\tilde{K}$ |

Basic Parameters | Conversion | Parameters for KGD Model |
---|---|---|

${\sigma}_{t}$ | ${K}_{IC}={\sigma}_{t}\sqrt{2\pi {r}_{0}}\approx 0.12{\sigma}_{t}$ | ${K}^{\prime}=4{\left(\frac{2}{\pi}\right)}^{1/2}{K}_{Ic}$ |

$E,v$ | — | ${E}^{\prime}=\frac{E}{1-{\nu}^{2}}$;${v}^{\prime}=12v$ |

${k}_{0}$,${\sigma}_{0}$,$\alpha $,$\varphi $,$\mu $,${C}_{pp},{C}_{f}$ | $k={k}_{0}{e}^{-({\sigma}_{0}-\alpha {P}_{p})}$ $c=\frac{k}{\varphi \mu ({C}_{pp}+{C}_{f})}$ ${C}_{l}\approx \frac{k{\sigma}_{0}}{\mu \sqrt{\pi c}}$ | ${C}^{\prime}=2{C}_{l}$ |

${Q}_{m},h,N$ | — | ${Q}_{0}={Q}_{m}/Nh$ |

Fracturing Cases | ${\mathit{E}}^{\prime}$$/\mathbf{G}\mathbf{p}{\mathbf{a}}^{-1}$ | ${\mathit{K}}^{\prime}$$/\mathbf{M}\mathbf{P}\mathbf{a}\cdot {\mathbf{m}}^{1/2}$ | ${\mathit{\mu}}^{\prime}$$/\mathbf{P}\mathbf{a}\cdot \mathbf{s}$ | ${\mathit{C}}^{\prime}$$/\mathbf{m}\cdot {\mathbf{s}}^{-1/2}$ | ${\mathit{K}}_{\mathit{m}}$ | ${\mathit{C}}_{\mathit{m}}$$\mathbf{or}{\mathit{C}}_{\mathit{k}}$ |
---|---|---|---|---|---|---|

$M$ | 14.25 | 0.13787 | 0.012 | 4.5931 × 10^{−6} | 0.075 | ${C}_{m}$: 0.1 |

$\tilde{M}$ | 14.25 | 0.13787 | 0.012 | 1.9 × 10^{−3} | 0.050 | ${C}_{m}$: 4 |

$K$ | 14.25 | 10.724 | 0.012 | 4.5931 × 10^{−6} | 5.800 | ${C}_{k}$: 0.0542 |

$\tilde{K}$ | 14.25 | 10.724 | 0.012 | 8.5653 × 10^{−4} | 3.880 | ${C}_{k}$: 1.421 |

Fracturing Cases | $\mathit{n}$ | $\mathit{m}$ | ${\mathit{\epsilon}}_{\mathit{b}}/{\mathit{\epsilon}}_{\mathit{t}0}$ | ${\mathit{\epsilon}}_{\mathit{s}}/{\mathit{\epsilon}}_{\mathit{b}}$ | ${\mathit{k}}_{\mathit{b}}/{\mathit{E}}_{0}$ | ${\mathit{\sigma}}_{\mathit{h}\mathit{t}}/{\mathit{\sigma}}_{\mathit{t}}$ | $\mathit{N}$ |
---|---|---|---|---|---|---|---|

$M$ | 1.36 | 2.90 | 1.80 | 1.22 | −1.179 | 0.97 | 40 |

$\tilde{M}$ | 2.31 | 2.80 | 1.10 | 1.61 | 1.940 × 10^{−5} | 0.82 | 10 |

$K$ | 3.65 | 1.00 | 0.38 | 2.00 | 0 | 0.19 | 50 |

$\tilde{K}$ | 2.62 | 1.00 | 0.08 | 2.00 | 0 | 0.04 | 10 |

Fracturing Cases | ${\mathit{V}}_{\mathit{p}}^{\mathit{d}}$ $\mathbf{}{\mathbf{m}}^{3}$ × 10 ^{−4} | ${\mathit{V}}_{\mathit{p}}^{\mathit{e}}$ $\mathbf{}{\mathbf{m}}^{3}$ × 10 ^{−4} | ${\mathit{V}}_{\mathit{p}}^{}$ $\mathbf{}{\mathbf{m}}^{3}$ × 10 ^{−4} | ${\mathit{V}}_{\mathit{p}}^{\mathit{d}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % | ${\mathit{V}}_{\mathit{p}}^{\mathit{e}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % | ${\mathit{V}}_{\mathit{p}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % |
---|---|---|---|---|---|---|

$M$ | 6.11 | 1.18 | 7.89 | 3.10 | 0.91 | 4.01 |

$\tilde{M}$ | 7.29 | 5.50 | 13.00 | 0.73 | 0.55 | 1.28 |

$K$ | 16.00 | 9.34 | 25.00 | 7.87 | 4.74 | 12.61 |

$\tilde{K}$ | 21.00 | 11.00 | 32.00 | 2.09 | 1.14 | 3.23 |

**Table 6.**Total hydraulic volumetric openings for the multi-fracturing system in four limiting regimes.

Fracturing Cases | ${\mathit{V}}_{\mathit{p}}^{\mathit{d}}$$/{\mathbf{m}}^{3}$ × 10 ^{−3} | ${\mathit{V}}_{\mathit{p}}^{\mathit{e}}$$/{\mathbf{m}}^{3}$ × 10 ^{−3} | ${\mathit{V}}_{\mathit{p}}^{}$$/{\mathbf{m}}^{3}$ × 10 ^{−3} | ${\mathit{V}}_{\mathit{p}}^{\mathit{d}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % | ${\mathit{V}}_{\mathit{p}}^{\mathit{e}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % | ${\mathit{V}}_{\mathit{p}}/{\mathit{\varphi}}_{\mathit{i}\mathit{n}\mathit{i}}$ % |
---|---|---|---|---|---|---|

$M$ | 24.40 | 4.714 | 31.56 | 123.92 | 36.24 | 160.17 |

$\tilde{M}$ | 7.29 | 5.499 | 13.00 | 7.30 | 5.50 | 12.80 |

$K$ | 80.00 | 46.60 | 125.00 | 393.39 | 236.94 | 630.34 |

$\tilde{K}$ | 21.00 | 11.00 | 32.00 | 20.88 | 11.39 | 32.28 |

Fracturing Regimes | ${\mathit{P}}_{\mathit{c}\mathbf{max}}/{\mathit{P}}_{\mathit{p}\mathbf{max}}$ | ${\mathit{P}}_{\mathit{c}}/{\mathit{P}}_{\mathit{p}}$ | ${\mathit{P}}_{\mathit{c}\mathbf{max}}/{\mathit{\sigma}}_{\mathit{h}}$ | ${\mathit{P}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathit{h}}$ | ${\mathit{P}}_{\mathit{c}\mathbf{max}}/{\mathit{\sigma}}_{\mathbf{Im}\mathit{a}\mathit{x}}^{\prime}$ | ${\mathit{P}}_{\mathit{c}}/{\mathit{\sigma}}_{\mathbf{I}}^{\prime}$ |
---|---|---|---|---|---|---|

$M$ | −1.0059 | −1.8932 | 2.0500 | 1.8994 | −33.9551 | −1672 |

$\tilde{M}$ | −1.1676 | −1.6918 | 1.9600 | 1.8314 | −38.3442 | −35.8941 |

$K$ | −0.8258 | −0.7355 | 2.7126 | 1.6378 | −2.9573 | −1.7858 |

$\tilde{K}$ | −1.1099 | −1.0349 | 1.7448 | 1.2846 | −8.8151 | −6.4904 |

$\mathit{M}$ | $\tilde{\mathit{M}}$ | $\mathit{K}$ | $\tilde{\mathit{K}}$ | |
---|---|---|---|---|

${\overline{\chi}}_{tip}={\sigma}_{b\mathrm{max}}/{P}_{p\mathrm{max}}$ | 0.52 | 0.57 | 0.52 | 0.47 |

$\overline{\chi}={\sigma}_{b}/{P}_{p}$ | 0.90 | 0.77 | 0.28 | 0.23 |

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## Share and Cite

**MDPI and ACS Style**

Guo, S.; Cao, Y.; Wang, L.; Zhang, X.; Zhang, W.; Lin, H.; Cao, Z.; Meng, B.
Confining Stress Response to Hydraulic Fracturing Volumetric Opening on the Representative Volume Element (RVE) Scale. *Water* **2023**, *15*, 4184.
https://doi.org/10.3390/w15234184

**AMA Style**

Guo S, Cao Y, Wang L, Zhang X, Zhang W, Lin H, Cao Z, Meng B.
Confining Stress Response to Hydraulic Fracturing Volumetric Opening on the Representative Volume Element (RVE) Scale. *Water*. 2023; 15(23):4184.
https://doi.org/10.3390/w15234184

**Chicago/Turabian Style**

Guo, Shuaifang, Yunxing Cao, Li Wang, Xinsheng Zhang, Wenying Zhang, Haixiao Lin, Zhengzheng Cao, and Bingbing Meng.
2023. "Confining Stress Response to Hydraulic Fracturing Volumetric Opening on the Representative Volume Element (RVE) Scale" *Water* 15, no. 23: 4184.
https://doi.org/10.3390/w15234184