# Hydrological–Thermal Coupling Simulation of Silty Clay during Unidirectional Freezing Based on the Discrete Element Method

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Ideas and Assumptions

## 3. Theories and Methods

#### 3.1. Particle Contact Model

#### 3.2. Hydrological–Thermal Coupling Model

#### 3.2.1. Heat Transfer Model

#### 3.2.2. Water Migration Model

## 4. Materials and Tests

#### 4.1. Indoor Tests Material

#### 4.1.1. Dry the Soil

#### 4.1.2. Configure the Soil

#### 4.1.3. Subsubsection

#### 4.2. Tests Program

#### 4.3. Tests Method and Procedures

#### 4.4. Modeling Process

#### 4.4.1. Build Initial Stacking Model

#### 4.4.2. Cut Model

#### 4.4.3. Set Boundary Conditions

#### 4.5. Hydrological–Thermal Coupling Process

## 5. Results and Discussion

#### 5.1. Temperature Change and Distribution

#### 5.2. Water Migration

## 6. Conclusions

- The discrete element hydrological–thermal coupling model established by introducing thermal conduction, water migration equation, and the relationship between temperature and unfrozen water content is logically tight and has no harsh assumptions. The model can be further developed and customized to meet specific requirements and applications;
- Through indoor tests and discrete element numerical simulation, this paper found that when silty clay undergoes unidirectional freezing, the unfrozen water in the soil will migrate to the freezing front, and the larger the temperature gradient, the smaller the amount of migration. This article can correctly describe the thermal conduction and moisture migration process of silty clay under unidirectional freezing conditions;
- The simulation results obtained by the discrete element hydrological–thermal coupling model established in this paper can describe the hydrological–thermal parameters of soil samples. The parameters such as water content and temperature obtained by numerical simulation can be accurate to each element. After the stress field coupling is added to the subsequent research, the factors such as cracks and consolidation caused by frost heave can be taken into account.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) The arrangement of particles and water distribution in soil; (

**b**) equivalent DEM model; and (

**c**) a schematic diagram illustrating water migration between particles.

**Figure 3.**Unidirectional freezing test system. (i) Cold liquid circulator. (ii, xv) Liquid inlet. (iii, xiii) Liquid outlet. (iv) Thermostat. (v) Top plate. (vi) Temperature sensors. (vii) Bottom plate. (viii) Benchtop. (ix) Insulation material. (x) Silty clay samples. (xi) Data acquisition system. (xii) Computer.

**Figure 4.**Optical and thermographic photographs. (

**a**) Optical photograph. (

**b**) Thermographic photograph of the top surface. (

**c**) Thermal imaging photo of the front side.

**Figure 6.**Temperature distribution of samples after freezing at different cold end temperatures: (

**a**) the cold end temperature is −5 °C; (

**b**) the cold end temperature is −7 °C; (

**c**) the cold end temperature is −10 °C.

**Figure 9.**Distribution of water content along the height of the sample at different cold end temperatures: (

**a**) cold end temperature of −5 °C; (

**b**) cold end temperature of −7 °C; (

**c**) cold end temperature of −10 °C.

Particle Size Range | 1~2 (mm) | 0.5~1 (mm) | 0.25~0.5 (mm) | 0.15~0.25 (mm) | 0.075~0.15 (mm) | <0.075 (mm) |
---|---|---|---|---|---|---|

Percentage | 2.5% | 12.6% | 9.7% | 16.0% | 21.1% | 34.1% |

Test Number | Initial Mass Water Content | Top Plate’s Temperature (°C) | Bottom Plate’s Temperature (°C) |
---|---|---|---|

1 | 20% | −5 | 1 |

2 | 20% | −7 | 1 |

3 | 20% | −10 | 1 |

Parameter | Value | Unit | Parameter | Value | Unit |
---|---|---|---|---|---|

$a$ | $32.957$ | $\backslash $ | ${a}_{w}$ | $0.9985$ | $\backslash $ |

$m$ | $1.825$ | $\backslash $ | ${v}_{2}$ | $0.3$ | $\backslash $ |

$n$ | $0.236$ | $\backslash $ | ${n}_{wi}$ | 1 | $\backslash $ |

${\theta}_{s}$ | $0.512$ | $\backslash $ | $\alpha $ | $0.235$ | $\backslash $ |

${K}_{s}$ | ${10}^{-9}$ | $\mathrm{m}/\mathrm{s}$ | $\beta $ | $-0.127$ | $\backslash $ |

$K$ | $1.32$ | $\mathrm{W}/\left(\mathrm{m}\xb7\mathrm{K}\right)$ | ${L}_{f}$ | $334$ | $\mathrm{kJ}/\mathrm{kg}$ |

${k}_{2}$ | $0.025$ | $\mathrm{W}/\left(\mathrm{m}\xb7\mathrm{K}\right)$ | $K{\left({T}^{*}\right)}_{f}$ | $0.0051$ [40] | ${\mathrm{s}}^{-1}$ |

${C}_{s}$ | $1800$ | $\mathrm{J}/\left(\mathrm{kg}\xb7\xb0\mathrm{C}\right)$ | ${C}_{w}$ | $4184$ | $\mathrm{J}/\mathrm{kg}\xb7\xb0\mathrm{C}$ |

${C}_{ice}$ | $2100$ | $\mathrm{J}/\left(\mathrm{kg}\xb7\xb0\mathrm{C}\right)$ | $\u2206t$ | $2$ | $\mathrm{s}$ |

Cold end Temperature (°C) | Maximum Water Content (%) | Locations of the Maximum Water Content (m) |
---|---|---|

−5 | 25.2 | 0.073 |

−7 | 24.3 | 0.054 |

−10 | 22.7 | 0.036 |

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**MDPI and ACS Style**

Shan, W.; Qu, S.; Guo, Y. Hydrological–Thermal Coupling Simulation of Silty Clay during Unidirectional Freezing Based on the Discrete Element Method. *Water* **2023**, *15*, 1338.
https://doi.org/10.3390/w15071338

**AMA Style**

Shan W, Qu S, Guo Y. Hydrological–Thermal Coupling Simulation of Silty Clay during Unidirectional Freezing Based on the Discrete Element Method. *Water*. 2023; 15(7):1338.
https://doi.org/10.3390/w15071338

**Chicago/Turabian Style**

Shan, Wei, Shiyao Qu, and Ying Guo. 2023. "Hydrological–Thermal Coupling Simulation of Silty Clay during Unidirectional Freezing Based on the Discrete Element Method" *Water* 15, no. 7: 1338.
https://doi.org/10.3390/w15071338