# Application of the Segregation Potential Model to Freezing Soil in a Closed System

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## Abstract

**:**

## 1. Introduction

## 2. The Segregation Potential

## 3. Frost Heaving Test

#### 3.1. Material and Method

#### 3.2. Test Results

_{f}(mm) is the frozen depth, t (h) is the freezing time, and a and b are constant parameters. Moreover, the frost heave amount was found to decrease with increasing external pressure owing to the external pressure delay of water migration [14], while the frozen depth was found to increase with increasing external pressure [23] because the unfrozen zone was compressed by the external pressure. In addition, the development of the freezing front can be divided into (I) a fast-freezing stage, during which the soil water is frozen in situ without migration, resulting in a small amount of frost heave and a very slow frost heave curve; (II) a transition stage, during which the water begins to migrate from the unfrozen zone to the frozen zone owing to the decrease of the freezing rate, resulting in an increase in the frost heave amount and the formation of a small amount of ice; and (III) a quasi-steady stage, during which more water migrates from the unfrozen zone to the frozen zone owing to the freezing rate varying very slowly, resulting in the formation of a large amount of ice. This phenomenon was consistent with previous studies [24].

#### 3.3. Calculation of Segregation Potential from the Frost Heaving Test

_{0}coordinates, and the slope is the parameter $\alpha $.

## 4. Determination of the Segregation Potential from Numerical Simulation Results

#### 4.1. Simulation Method

#### 4.1.1. Assumption

- (1)
- Darcy’s law is applicable for water migration during the soil freezing processes.
- (2)
- The soil in the study is isotropic and elastic.

#### 4.1.2. Equilibrium Equations

#### 4.2. Simulation Results

## 5. Approach to Applying the Segregation Potential Model in Closed Systems

#### 5.1. Comparison of the $S{P}_{0}^{\mathrm{M}}$ and $S{P}_{0}^{\mathrm{S}}$

#### 5.2. Applying the Segregation Potential Model in a Closed System

#### 5.3. Further Validation of the Model in a Closed System

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the frost heave model. (

**a**) Open system, (

**b**) Closed system, (

**c**) Temperature distribution in closed system and open system. The solid lines represent the quasi-static state, and the dashed lines represent the dynamic state. The blue lines represent the pore water pressure, the pink line represents the pore water pressure expressed by the Clapeyron Equation, the brown line represents the pore water pressure in the case of terminated water flow, the green lines represent the pore ice pressure, and the red lines represent the soil temperature.

**Figure 2.**(

**a**) Accumulative curve of particle size grading of soil samples, (

**b**) soil freezing characteristic curve of soil sample.

**Figure 4.**Variations in the frost heave displacement (

**a**) and compression amount with freezing time under external pressures (

**b**).

**Figure 5.**Variations of real frost heave and freezing front with freezing time under external pressures.

**Figure 6.**(

**a**) Variations of the average mass ice content with external pressure, (

**b**) variation of in situ frost heave with freezing time under external pressures.

**Figure 7.**(

**a**) Variation of segregation potential under no load with the freezing time, (

**b**) variation of final frost heave with the external pressure.

**Figure 8.**(

**a**) Comparison of simulated and measured frozen depth under no load, (

**b**) profile of water redistribution of the soil samples at different freezing times under no load, (

**c**) variation of the segregation potential with the freezing time.

**Figure 9.**(

**a**) Variation of the $S{P}_{0}^{\mathrm{M}}$ and the $S{P}_{0}^{\mathrm{S}}$ with the freezing time, (

**b**) comparison of initial segregation potential $S{P}_{0}$ calculated by measured data and the simulated data.

**Figure 10.**(

**a**) Segregation potential under external pressure calculated by Equation (7), (

**b**) comparison of calculated and measured frost heave amounts.

**Figure 11.**(

**a**) Variation of frost heave under external load with the freezing time, (

**b**) variation of final frost heave with the external pressure.

$\mathit{L}\left(\mathbf{mm}\right)$ | $\mathit{D}\left(\mathbf{mm}\right)$ | ${\mathit{W}}_{\mathit{L}}$ | ${\mathit{W}}_{\mathit{P}}$ | ${\mathit{W}}_{\mathbf{i}}$ | ${\mathit{\rho}}_{\mathbf{d}}\left(\mathbf{kg}/{\mathbf{m}}^{3}\right)$ | ${\mathit{T}}_{\mathbf{f}}(\mathbb{C})$ | Sand (%) | Silt (%) | $\mathbf{Clay}\text{}(\%)$ |
---|---|---|---|---|---|---|---|---|---|

110 | 100 | 0.351 | 0.228 | 0.351 | 1.43 | −0.18 | 89.8 | 9.92 | 0.28 |

Test Number | External Pressure | Controlled Temperature (°C) | |
---|---|---|---|

Cool End | Warm End | ||

P1-P7 | 0, 13, 51,102, 153, 191, 255 (kPa) | −2 | 1 |

P1-P7 | 0, 13, 51,102, 153, 191, 255 (kPa) | 20 |

$\mathit{P}$(kPa) | RMSE | Average |
---|---|---|

13 | 0.26 | 0.13 |

51 | 0.19 | |

102 | 0.02 | |

153 | 0.04 | |

191 | 0.11 | |

255 | 0.18 |

References | $\mathit{P}$ (kPa) | ${\mathit{T}}_{\mathbf{c}}(\mathbb{C})$ | ${\mathit{T}}_{\mathit{w}}(\mathbb{C})$ | ${\mathit{W}}_{0}(\%)$ | $\mathbf{Grad}\mathit{T}\left(\mathbb{C}/\mathbf{cm}\right)$ |
---|---|---|---|---|---|

[14] | 50 | −1.6 | 1.5 | 20.59 | 0.31 |

100 | −1.6 | 1.5 | 20.52 | 0.31 | |

[29] | 50 | −2.0 | 2.0 | 22.30 | 0.36 |

150 | −2.0 | 2.0 | 22.34 | 0.36 | |

300 | −2.0 | 2.0 | 22.37 | 0.36 | |

500 | −2.0 | 2.0 | 22.16 | 0.36 | |

[30] | 0 | −2.0 | 3.0 | 16.02 | 0.50 |

100 | −2.0 | 3.0 | 16.34 | 0.50 | |

200 | −2.0 | 3.0 | 16.17 | 0.50 |

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**MDPI and ACS Style**

Zhang, X.; Sheng, Y.; Huang, L.; Huang, X.; He, B.
Application of the Segregation Potential Model to Freezing Soil in a Closed System. *Water* **2020**, *12*, 2418.
https://doi.org/10.3390/w12092418

**AMA Style**

Zhang X, Sheng Y, Huang L, Huang X, He B.
Application of the Segregation Potential Model to Freezing Soil in a Closed System. *Water*. 2020; 12(9):2418.
https://doi.org/10.3390/w12092418

**Chicago/Turabian Style**

Zhang, Xiyan, Yu Sheng, Long Huang, Xubin Huang, and Binbin He.
2020. "Application of the Segregation Potential Model to Freezing Soil in a Closed System" *Water* 12, no. 9: 2418.
https://doi.org/10.3390/w12092418