# Advanced Grouting Model and Influencing Factors Analysis of Tunnels with High Stress and Broken Surrounding Rock

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multi-Field Coupling Grouting Mechanism

#### 2.1.1. Influence Characteristics of Grouting Materials’ Physical Property Change on Permeability Coefficient

- Viscosity-Time-Molarity 3D Surface Equation for Grouting Materials

^{3}, 1530 kg/m

^{3}, and 1630 kg/m

^{3}, respectively; the molarities of tricalcium silicate were calculated to be 1628 mol/m

^{3}, 1864 mol/m

^{3}, and 2234 mol/m

^{3}, respectively (calculated according to the proportion of tricalcium silicate is 50%). An NDJ-5T rotational viscosimeter was adopted to test the time-varying data of viscosity, and the linear interpolation algorithm [36] is adopted to drive the 3D surface functions of grouting material viscosity versus time and molarity:

^{3}, ranging within 0–1628 mol/m

^{3}, 0–1864 mol/m

^{3}, and 0–2234 mol/m

^{3}, respectively. As shown in Figure 1, 3D surface functions are plotted for the three cement grouting materials with distinct water–cement ratios.

- 2.
- Variation of density with molarity during the migration of cement slurry

^{3}, 1864 mol/m

^{3}, and 2234 mol/m

^{3}, and the densities are 1485 kg/m

^{3}, 1530 kg/m

^{3}, and 1630 kg/m

^{3}, respectively; the molarity of water is 0 mol/m

^{3}and the density is 1000 kg/m

^{3}. Assume that the molarity of the slurry is proportional to the density; thus, the density of the three slurries can be expressed by the following equation.

#### 2.1.2. Variation of Voids’ Seepage Parameters with Infiltration Pressure

#### 2.1.3. Flow and Mass Transfer Characteristics Control Equations

- Fluid Flow Control Equation

^{2}, dependent only on the solid’s skeleton structure; $\mu $ is the fluid’s viscosity in Pa∙s; $p$ is fluid pressure in Pa; $\rho $ is the fluid’s density in kg/m

^{3}; $\nabla D$ is the unit vector in the direction of gravity; and $D$ is the vertical coordinate.

- 2.
- Basic Equation for Solute Migration

^{th}component in mol/m

^{3}; ${D}_{i}$ is the molecular diffusion coefficient in m

^{2}/s; and ${R}_{i}$ is the reaction rate of the i

^{th}component in mol/(m

^{3}∙s). Since cement hydration reaction rate is extremely low during the quiescent stage, the effect of cement hydration reaction on molarity of components is ignored; $\mathbf{u}$ is Darcy seepage flow velocity in m/s.

#### 2.2. Engineering Background and Numerical Simulation

#### 2.2.1. Engineering Background

#### 2.2.2. Numerical Model

#### 2.2.3. Model Parameter Values

## 3. Results and Discussions

#### 3.1. Basic Law of Grouting Migration for Tunnels with High-Stress Broken Surrounding Rocks

^{3}) of grouting materials can be taken as an indicator of permeation radius of grouting materials; 10%–95% of the initial molarity (162.8–1546.6 mol/m

^{3}) of grouting materials is defined as the transition region; and 95%–100% of the initial molarity as the raw slurry region.

#### 3.2. Influence Characteristics of Water–Cement Ratio on Grouting Migration Law

#### 3.3. Influence Characteristics of Grouting Pressure on Grouting Migration Law

#### 3.4. Influence Characteristics of Initial Permeability on Grouting Migration Law

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The three-dimension curved surface of linear interpolation of cement paste viscosity with respect to time and concentration for three water–cement ratios; (

**a**) w:c = 1.0; (

**b**) w:c = 0.8; (

**c**) w:c = 0.6.

**Figure 2.**Project overview of downstream segment TBMb of Xianglushan Tunnel.The red refers to TBM section; the green refers to drilling and blasting section; the blue refers to Jifucun Aqueduct.

**Figure 3.**Illustrations of the numerical model. Panel (

**a**) displays the 3D section view of the model; (

**b**) displays the 2D abbreviated drawing of the model.

**Figure 4.**Basic law of grouting migration (initial permeability 5D, grouting pressure 2 MPa, and water–cement ratio 1.0). Panel (

**a**) displays the molarity distribution characteristics of grouting materials at different times; (

**b**) displays the viscosity distribution characteristics of grouting materials at different times; (

**c**) displays the permeability coefficient distribution characteristics of grouting materials at different times; (

**d**) displays the grouting radius and range of transition region of grouting materials at different times; (

**e**) displays the molarity contour surface of grouting material migration for 40 min.

**Figure 5.**Law from the indoor simulation test by Du, X. [42]. Panel (

**a**) displays the variation of retention rate of cement particles with migration distance at distinct water–cement ratios; (

**b**) displays the variation of permeability with migration distance at distinct water–cement ratios.

**Figure 6.**Comparison between numerically simulated values and theoretically calculated values of grouting radius at distinct water–cement ratios; (

**a**) w:c = 1.0; (

**b**) w:c = 0.8; (

**c**) w:c = 0.6.

**Figure 7.**Comparison between numerically simulated values and theoretically calculated values of grouting radius at distinct grouting pressures; (

**a**) 2MPa; (

**b**) 1MPa; (

**c**) 0.5MPa.

**Figure 8.**Comparison between numerically simulated values and theoretically calculated values of grouting radius at distinct initial permeabilities; (

**a**) 5D; (

**b**) 0.5D; (

**c**) 0.05D.

Composition Name | Tricalcium Silicate | Dicalcium Silicate | Tricalcium Aluminate | Tetracalcium Aluminoferrite | Gypsum |
---|---|---|---|---|---|

Content proportion | 40–60% | 15–37% | 7–15% | 10–18% | few |

Order | Parameter Name | Parameter Symbol | Unit | Value or Expression |
---|---|---|---|---|

1 | Elasticity Modulus of Rocky Soil | E | (Pa) | 1 × 10^{8} |

2 | Poisson Ratio of Rocky Soil | $\upsilon $ | (1) | 0.34 |

3 | Density of Rocky Soil | $\gamma $ | (kg/m^{3}) | 2200 |

4 | Cohesion of Rocky Soil | C | (Pa) | 0.05 × 10^{6} |

5 | Internal Friction Angle of Rocky Soil | $\phi $ | (°) | 28.8 |

6 | Initial Porosity of Rocky Soil | ${{\displaystyle n}}_{0}$ | (1) | 14% |

7 | Initial Permeability of Rocky Soil | ${k}_{0}$ | (m^{2}) | 5 × 10^{−12}, 5 × 10^{−13}, 5 × 10^{−14} |

8 | Dynamic Porosity of Rocky Soil | $n$ | (1) | Formula (7) |

9 | Dynamic Permeability of Rocky Soil | $k$ | (m^{2}) | Formula (8) |

10 | Fluid Viscosity | $\mu $ | (Pa·s) | Formulae (1), (2), (3) |

11 | Fluid Density | $\rho $ | (kg/m^{3}) | Formulae (4), (5), (6) |

12 | Molecular Diffusion Coefficient [40] | ${D}_{F,c}$ | (m^{2}/s) | 1.2 × 10^{−11} |

13 | Mechanical Dispersion Coefficient [41] | ${D}_{\mathrm{D}}$ | (m^{2}/h) | 8.4 × 10^{−4} |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Z.; Pan, D.; Zhang, S.; Yin, Z.; Zhou, J.
Advanced Grouting Model and Influencing Factors Analysis of Tunnels with High Stress and Broken Surrounding Rock. *Water* **2022**, *14*, 661.
https://doi.org/10.3390/w14040661

**AMA Style**

Jiang Z, Pan D, Zhang S, Yin Z, Zhou J.
Advanced Grouting Model and Influencing Factors Analysis of Tunnels with High Stress and Broken Surrounding Rock. *Water*. 2022; 14(4):661.
https://doi.org/10.3390/w14040661

**Chicago/Turabian Style**

Jiang, Zhixiong, Dongjiang Pan, Shuhao Zhang, Zhiqiang Yin, and Jianjun Zhou.
2022. "Advanced Grouting Model and Influencing Factors Analysis of Tunnels with High Stress and Broken Surrounding Rock" *Water* 14, no. 4: 661.
https://doi.org/10.3390/w14040661