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

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

- Van Everdigen, R.O. Multi-Language Glossary of Permafrost and Related Ground-Ice Terms; International Permafrost Association, University of Calgary: Calgary, AB, Canada, 1998. [Google Scholar]
- Ran, Y.H.; Li, X.; Cheng, G.D.; Zhang, T.J.; Wu, Q.B.; Jin, H.J.; Jin, R. Distribution of permafrost in China: An overview of existing permafrost maps. Permafr. Periglac.
**2012**, 23, 322–333. [Google Scholar] [CrossRef] - Ma, W.; Zhu, Y.L.; Xu, X.Z. The State Key Laboratory of Frozen Soil Engineering: Review and Prospect. J. Glaciol. Geocryol.
**1998**, 20, 72–80. [Google Scholar] - Harlan, R.L. Analysis of coupled heat-fluid transport in partially frozen soil. Water Resour. Res.
**1973**, 9, 1314–1323. [Google Scholar] [CrossRef] [Green Version] - Newman, G.P.; Wilson, G.W. Heat and mass transfer in unsaturated soils during freezing. Can. Geotech. J.
**1997**, 34, 63–70. [Google Scholar] [CrossRef] - Nixon, J.F. The role of convective heat transport in the thawing of frozen soils. Can. Geotech. J.
**1975**, 12, 425–429. [Google Scholar] [CrossRef] - Taylor, G.S.; Luthin, J.N. A model for coupled heat and moisture transfer during soil freezing. Can. Geotech. J.
**1978**, 15, 548–555. [Google Scholar] [CrossRef] - O’Neil, K.; Miller, R.D. Exploration of a rigid ice model of frost heave. Water Resour. Res.
**1985**, 21, 281–296. [Google Scholar] [CrossRef] - Konrad, J.M.; Morgenstern, R.N. The segregation potential of a freezing soil. Can. Geotech. J.
**1981**, 4, 482–491. [Google Scholar] [CrossRef] - Chen, F.X.; Song, Z.P.; Li, N. Study on Moisture Migrating Force Model of Freezing Soil Base on Adsorption-film Moisture Migration Mechanism. J. Water Resour. Archit. Eng.
**2006**, 4, 1044. [Google Scholar] - Dash, J.G.; Rempel, A.W.; Wettlaufer, J.S. The physics of premelted ice and its geophysical consequences. Rev. Mod. Phys.
**2006**, 78, 695–741. [Google Scholar] [CrossRef] [Green Version] - Ming, F.; Li, D. A model of migration potential for moisture migration during soil freezing. Cold Reg. Sci. Technol.
**2016**, 124, 87–94. [Google Scholar] [CrossRef] - Bai, Q.B.; Li, X.; Tian, Y.H. Equations and numerical simulation for coupled water and heat transfer in frozen soil. Chin. J. Geotech. Eng.
**2015**, 37, 131–136. [Google Scholar] - Li, S.; Zhang, M.; Tian, Y.; Pei, W.; Zhong, H. Experimental and numerical investigations on frost damage mechanism of a canal in cold regions. Cold Reg. Sci. Technol.
**2015**, 116, 1–11. [Google Scholar] [CrossRef] - Hu, Z.; Shan, W. Landslide investigations in the northwest section of the lesser Khingan range in China using combined HDR and GPR methods. Bull. Eng. Geol. Environ.
**2016**, 75, 591–603. [Google Scholar] [CrossRef] [Green Version] - Guo, Y.; Shan, W.; Zhang, C.; Hu, Z.; Wang, S.; Gao, J. Monitoring of permafrost degradation along the Bei’an-Heihe Expressway in China. Bull. Eng. Geol. Environ.
**2021**, 80, 1–10. [Google Scholar] [CrossRef] - Luo, H.; Liu, E.L. Hydro-thermo-mechanical Coupling Analysis of Saturated Soils during the Process of Freezing in Uni-direction. J. Disaster Prev. Mitig. Eng.
**2017**, 34, 586–592. [Google Scholar] - An, L.; Ling, X.; Geng, Y.; Li, Q.; Zhang, F. DEM Investigation of Particle-Scale Mechanical Properties of Frozen Soil Based on the Nonlinear Microcontact Model Incorporating Rolling Resistance. Math Probl. Eng.
**2018**, 2018, 2685709. [Google Scholar] [CrossRef] [Green Version] - Yin, N.; Li, S.Y.; Pei, W.S.; Zhang, M.Y.; Dong, Y. Microscopic deformation mechanisms of triaxial test of frozen clay analyzed by discrete element method. J. Glaciol. Geocryol.
**2016**, 38, 178–185. [Google Scholar] - Ding, F.; Song, L.; Yue, F. Study on mechanical properties of cement-improved frozen soil under uniaxial compression based on discrete element method. Processes
**2022**, 10, 324. [Google Scholar] [CrossRef] - Le, T.; Liu, C.; Tang, C.; Zhang, X.; Shi, B. Numerical Simulation of Desiccation Cracking in Clayey Soil Using a Multifield Coupling Discrete-Element Model. J. Geotech. Geoenviron.
**2021**, 148, 04021183. [Google Scholar] [CrossRef] - Sang, H.W.; Zhang, D.; Liu, C. Numerical Simulation on Heat Transfer of Energy PHC Pile based on Discrete Element Method. J. Disaster Prev. Mitig. Eng.
**2019**, 39, 645–650. [Google Scholar] - Tran, K.M.; Bui, H.H.; Nguyen, G.D. DEM modelling of unsaturated seepage flows through porous media. Comput. Part Mech.
**2022**, 9, 135–152. [Google Scholar] [CrossRef] - Miller, R.D. Freezing and heaving of saturated and unsaturated soils. Highw. Res. Rec.
**1972**, 1, 1–11. [Google Scholar] - Konrad, J.M.; Morgenstern, R.N. Effects of applied pressure on freezing soils. Can. Geotech. J.
**1982**, 19, 494–505. [Google Scholar] [CrossRef] - Zhou, Y.; Zhou, G.; Zhou, J.; Wang, J. Ice lens growth process involving coupled moisture and heat transfer during freezing of saturated soil. Chin. J. Geotech. Eng.
**2010**, 32, 578–585. [Google Scholar] - Mageau, D.W.; Morgensdtern, N.R. Observations on moisture migration in frozen soils. Can. Geotech. J.
**1980**, 17, 54–60. [Google Scholar] [CrossRef] - Gili, J.A.; Alonso, E.E. Microstructural deformation mechanisms of unsaturated granular soils. Int. J. Numer. Anal. Methods
**2002**, 26, 433–468. [Google Scholar] [CrossRef] [Green Version] - Cundall, P.A.; Strack, O.D.L. A discrete numerical model for granular assemblies. Géotechnique
**1979**, 29, 47–65. [Google Scholar] [CrossRef] - Groenevelt, P.H.; Kay, B.D. On the interaction of water and heat transport in frozen and unfrozen soils: II. The liquid phase. Soil Sci. Soc. Am. J.
**1974**, 38, 400–404. [Google Scholar] [CrossRef] - Croney, D.; Coleman, J.D.; Bridge, P.M. Suction of Moisture Held in Soil and Other Porous Materials; Her Majesty’s Stationary Office: London, UK, 1952. [Google Scholar]
- Anderson, D.M.; Tice, A.R. The Unfrozen Interfacial Phase in Frozen Soil Water Systems. Springer Berl. Heidelb.
**1973**, 4, 116–119. [Google Scholar] - Fredlund, D.G.; Xing, A.Q. Equations for soil-water characteristic curve. Can. Geotech. J.
**1994**, 31, 521–532. [Google Scholar] [CrossRef] - Zhang, H.W.; Zhou, Q.; Xing, H.L.; Muhlhaus, H. A DEM study on the effective thermal conductivity of granular assemblies. Powder Technol.
**2011**, 205, 172–183. [Google Scholar] [CrossRef] - Vargas, W.L.; McCarthy, J.J. Heat conduction in granular materials. AIChE J.
**2001**, 47, 1052–1059. [Google Scholar] [CrossRef] - Bathchelor, G.K.; O’Brien, R.W. Thermal or electrical conduction through a granular material. Proc. R. Soc. London. A Math. Phys. Sci.
**1977**, 1682, 313–333. [Google Scholar] - Eucken, A. Allgemeine gesetzmäßigkeiten für das wärmeleitvermögen verschiedener stoffarten und aggregatzustände. Forsch. Auf. Dem. Geb. Des. Ing. A
**1940**, 1, 6–20. [Google Scholar] [CrossRef] - Koop, T.; Luo, B.; Tsias, A.; Peter, T. Water activity as the determinant for homogeneous ice nucleation in aqueous solutions. Nature
**2000**, 406, 611–614. [Google Scholar] [CrossRef] - Ji, Y.; Zhou, G.; Zhou, Y.; Hall, M.R.; Zhao, X.; Mo, P. A separate-ice based solution for frost heaving-induced pressure during coupled thermal-hydro-mechanical processes in freezing soils. Cold Reg. Sci. Technol.
**2018**, 147, 22–33. [Google Scholar] [CrossRef] - Wu, D.; Lai, Y.; Zhang, M. Thermo-hydro-salt-mechanical coupled model for saturated porous media based on crystallization kinetics. Cold Reg. Sci. Technol.
**2017**, 133, 94–107. [Google Scholar] [CrossRef]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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