# Numerical Analysis of the Influence of Foundation Replacement Materials on the Hydrothermal Variation and Deformation Process of Highway Subgrades in Permafrost Regions

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## Abstract

**:**

## 1. Introduction

^{6}km

^{2}to 16 × 10

^{6}km

^{2}[1,2], and approximately 11% of permafrost disappeared permanently, leading to more subgrade thawing and settlement deformation [3,4,5]. The survey results of subgrade diseases in permafrost regions such as Siberia, Alaska and the Qinghai–Tibet Plateau show that the settlement caused by the thawing of shallow frozen soil on the ground surface has become the main form of subgrade damage, and the amount of deformation has increased with the increasing thawing depth of frozen soil [6,7,8]. The sixth IPCC climate change assessment report points out [9] that the mean annual air temperature (MAAT) between 2011 and 2020 was 1.09 °C higher than that from 1850–1900, making it the fastest warming decade in human history. In 2019, the highest air temperature in Siberia reached 38 °C, and the degradation of permafrost is accelerating at an unprecedented rate. As a result of climate and environmental change, permafrost is extremely sensitive to temperature change. On the one hand, the strength of the foundation soil decreases after permafrost thaws, and the uneven distribution of water in the subgrade will lead to uneven settlement deformation. Once the deformation exceeds the material limit, longitudinal cracks will occur [10,11,12]. On the other hand, the freezing and thawing action in the active layer is essentially a dynamic process of mutual coupling of water, heat and deformation. Even if there is no external load, the hydrothermal variation in frozen soil can cause obvious stress and deformation. The combined actions of these two aspects may eventually reduce subgrade stability [13,14,15,16]. Considering the background on the accelerated degradation of permafrost [9,17], it is of great practical significance to explore the interaction mechanism between water and heat and the deformation of the permafrost subgrade and establish a road structure system in cold regions to adapt to permafrost degradation.

## 2. Study Area

## 3. Model Description

#### 3.1. Geometric Model

#### 3.2. Mathematical Model

_{w}is the volumetric liquid water content, expressed as ${\theta}_{w}=a{\left|T\right|}^{b}$; θ

_{v}and θ

_{i}are the volumetric contents of vapor and ice, respectively; ρ

_{w}, ρ

_{v}and ρ

_{i}are the densities of liquid water, vapor and ice, respectively; h is the height of the water head; K

_{wh}and K

_{wT}are the isothermal and nonisothermal hydraulic conductivities caused by the matrix potential and temperature gradient, respectively; K

_{vh}and K

_{vT}are the isothermal and nonisothermal vapor conductivities, respectively.

_{s}is a saturated permeability coefficient.

_{wT}, the vapor permeability coefficient K

_{vT}caused by the temperature gradient, and the vapor permeability coefficient K

_{vh}caused by the matrix potential gradient are listed as follows:

_{wT}is the empirical parameter for evaluating the influence of temperature on the soil–water characteristic curve and γ is the surface tension at 25 °C. M and g are the molar mass of water and the acceleration of gravity, respectively. R is the gas constant, H

_{r}is the relative humidity, η is the water vapor diffusion enhancement factor, and D is water vapor diffusion in the soil.

_{v}is the latent heat of evaporation of water. The expressions of other hydrothermal–related parameters are shown in Table 1.

_{T}] is the elasticity matrix related to temperature and water content, E

_{T}is the elastic modulus and v

_{T}is Poisson’s ratio. {Δε}, {Δε

_{vp}} and {Δε

_{v}} are the total strain increment vector, the viscoplastic strain increment vector, and the frost heave strain increment vector, respectively.

_{T}is the viscosity parameter, Q is the plastic potential function, and Φ(F) is a scalar function, which can be written as [49]

_{0}is the yield stress. Assuming that the plasticity criterion applies to frozen soil, according to the D–P yield criterion, Q is equivalent to F.

_{0}is the initial volumetric water content, Δθ

_{L}is the volumetric content of migrated water, and δ is defined as follows:

_{v}} can be expressed as follows [49]:

#### 3.3. Parameters of the Soil Layers

_{1}–a

_{4}, b

_{1}–b

_{4}, c

_{1}–c

_{5}, d

_{1}–d

_{3}and e

_{1}–e

_{3}are all test fitting parameters.

#### 3.4. Boundary Conditions

_{0}is the initial phase angle, determined by the completion time of the subgrade. ΔT is the climate warming rate, taken as 0.052 °C/a [17]. The mean annual temperature T

_{0}and annual temperature amplitude A for each boundary are shown in Table 4. The side ABCDE is a symmetrical boundary, the temperature boundary FGHI is adiabatic, and the geothermal flux at the bottom boundary EF is 0.03 W/m

^{2}[43]. The water boundary IJK is permeable, AK and FGHI are impermeable [50,60,61], and the bottom boundary EF has a 15.20% liquid water supply. The horizontal displacement for the lateral FGHI and the vertical displacement of the bottom boundary EF are restrained. The top surface of the subgrade, the side slope of the subgrade and the natural ground surface (boundaries AK, JK and IJ) are all free boundaries, and the displacement is not restricted.

#### 3.5. Initial Conditions and Model Verification

## 4. Results

#### 4.1. Verification of the Temperature Field after Subgrade Completion

#### 4.2. Laboratory Test Verification

#### 4.3. Analysis of the Variation in Thermal Conditions in Different Subgrade Structures

#### 4.4. Analysis of the Variation in Liquid Water Content in Different Subgrade Structures

#### 4.5. Deformation Analysis of Different Subgrade Structures

## 5. Discussion

#### 5.1. Comparison of Permafrost Table Changes in Three Subgrade Structures in Case 1 and Case 2

#### 5.2. Comparison of the Volumetric Liquid Water Content of the Three Subgrade Structures in Case 1 and Case 2

#### 5.3. Comparison of the Deformation at the Road Centerline and the Shoulder of the Three Subgrade Structures in Case 1 and Case 2

## 6. Conclusions

- (1)
- The replacement block–stone subgrade structure has a higher permafrost table in warm seasons and a thinner thawed interlayer in cold seasons. Compared with the replacement breccia layer and the replacement gravel layer, the replacement block–stone layer can effectively reduce the total heat entering the deep foundation in warm seasons. Its good thermal regulation performance reduces the impact of seasonal temperature changes on subgrade temperature to improve the permafrost table.
- (2)
- The low thermal conductivity of the block–stone layer reduces the temperature gradient between the block–stone layer and the active layer. In addition, the very small matrix potential of the block–stone layer leads to a very small driving force of water migration. The reduction in the driving force of water migration reduces the water supply from the thawed interlayer to the active layer, which reduces the liquid water content in the active layer and finally, reduces the transverse and vertical deformations of the subgrade.
- (3)
- The replacement block–stone layer can effectively resist the local uneven deformation of the foundation, so the settlement of the replacement block–stone subgrade structure is the smallest. After the thawing of the permafrost in the breccia layer and the gravel layer, the strength of these two replacement materials decreases rapidly, and it is difficult for them to resist the uneven deformation of the foundation. Therefore, the cumulative settlement of the two subgrades is relatively large. In addition, the liquefaction characteristics of the gravel lead to a greater settlement of the replacement gravel subgrade than that of the replacement breccia subgrade.
- (4)
- In cold seasons, when the water in the active layer freezes, two forms of dispersed ice crystals and continuous ice lenses form, which have different retardation effects on water migration. We discussed these effects and corrected the subgrade deformation. The results show that from 2019 to 2039, the maximum cumulative settlement and the maximum transverse deformation of the replacement block–stone subgrade are –0.211 cm and +0.111 cm, respectively. The maximum cumulative settlement and the maximum transverse deformation of the replacement breccia subgrade are –23.467 cm and −1.209 cm, respectively. The maximum cumulative settlement and the maximum transverse deformation of the replacement gravel subgrade are –33.793 cm and –2.207 cm, respectively. The replacement block–stone subgrade structure can not only reduce the cumulative settlement and frost heave but also reduce the transverse deformation and longitudinal cracks to improve the overall stability of the subgrade. In contrast, the vertical and transverse deformation of the replacement breccia subgrade and the replacement gravel subgrade are too large, and even the subgrade fill layer will undergo transverse deformation in the opposite direction, which will cause sliding failure. Therefore, these two subgrade structures cannot be used in permafrost regions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Permafrost distribution in Northeast China and the study area and project location in this paper, with data from Jaroslav O. et al. (2019) [41].

**Figure 2.**Changes in the surface freezing (thawing) index and surface frost number in the study area from 1972 to 2018.

**Figure 3.**Geometric model and calculation unit division of the subgrade structure. A, B, C, and D are node numbers of the geometric model (Unit: m).

**Figure 4.**Site photos: (

**a**) photo of surface soil layer and (

**b**) compaction photo of the block–stone layer.

**Figure 5.**Comparison between the measured and simulated ground temperature values at the 1 m borehole outside the right slope toe from July 2018 to October 2019: (

**a**) 15 July 2018, (

**b**) 15 October 2018, (

**c**) 15 January 2019, (

**d**) 15 March 2019, (

**e**) 15 July 2019, and (

**f**) 15 October 2019.

**Figure 6.**Comparison between the measured and simulated ground temperature values at the 1 m borehole outside the right slope toe from January 2020 to October 2021: (

**a**) 15 January 2020, (

**b**) 15 July 2020, (

**c**) 15 October 2020, (

**d**) 15 January 2021, (

**e**) 15 July 2021, and (

**f**) 15 October 2021.

**Figure 7.**Comparison of the measured and simulated values of hydrothermal variations and deformation processes in one–sided freezing tests: (

**a**) comparison of the total volumetric water contents at different moments, (

**b**) comparison of the temperatures at different moments, and (

**c**) comparison of the variations in the frost heave.

**Figure 8.**Temperature contour maps on October 15 of the three subgrade structures in different years: (

**a1**–

**a4**) replacement block–stone subgrade structure, (

**b1**–

**b4**) replacement breccia subgrade structure, and (

**c1**–

**c4**) replacement gravel subgrade structure. The 0 °C isotherm represents the location of the permafrost table. (Unit: °C).

**Figure 9.**Temperature contour maps on March 15 of the three subgrade structures in different years: (

**a1**–

**a4**) replacement block–stone subgrade structure, (

**b1**–

**b4**) replacement breccia subgrade structure, and (

**c1**–

**c4**) replacement gravel subgrade structure. The 0 °C isotherm represents the thawed interlayer range. (Unit: °C).

**Figure 10.**Vertical distribution of the volumetric liquid water content under the road centerlines of the three subgrade structures in different years: (

**a1**–

**a4**) in cold seasons and (

**b1**–

**b4**) in warm seasons.

**Figure 11.**Vertical deformation contour maps of three subgrade structures in different years in warm seasons: (

**a1**–

**a4**) replacement block–stone subgrade structure, (

**b1**–

**b4**) the replacement breccia subgrade structure, and (

**c1**–

**c4**) the replacement gravel subgrade structure. (Unit: cm).

**Figure 12.**Transverse deformation contour maps of three subgrade structures in different years in warm seasons: (

**a1**–

**a4**) replacement block–stone subgrade structure, (

**b1**–

**b4**) replacement breccia subgrade structure, and (

**c1**–

**c4**) replacement gravel subgrade structure (Unit: cm).

**Figure 13.**Comparison of permafrost table changes under the road centerlines of the three subgrade structures in Case 1 and Case 2.

**Figure 14.**In Case 2, comparison of the vertical distributions of the volumetric liquid water contents under the road centerlines of the three subgrade structures in cold seasons and warm seasons in different years: (

**a1**–

**a4**) in cold seasons and (

**b1**–

**b4**) in warm seasons.

**Figure 15.**Comparison of the vertical distributions of volumetric liquid water contents under the road centerlines in Case 1 and Case 2 on March 15 of different years for the three subgrade structures: (

**a1**–

**a4**) replacement block–stone subgrade structure, (

**b1**–

**b4**) replacement breccia subgrade structure, and (

**c1**–

**c4**) replacement gravel subgrade structure.

**Figure 16.**Vertical deformation curves at the road centerline and the shoulder of the three subgrade structures in Case 1: (

**a**) at the road centerline and (

**b**) at the shoulder.

**Figure 17.**Vertical deformation curves at the road centerline and the shoulder of the three subgrade structures in Case 2: (

**a**) at the road centerline and (

**b**) at the shoulder.

**Figure 18.**Transverse deformation curves at the road centerline and the shoulder of the three subgrade structures in Case 1: (

**a**) at the road centerline and (

**b**) at the shoulder.

**Figure 19.**Transverse deformation curves at the road centerline and the shoulder of the three subgrade structures in Case 2: (

**a**) at the road centerline and (

**b**) at the shoulder.

**Figure 20.**Actual vertical deformation curves at the centerline and the shoulder of the three subgrade structures: (

**a**) at the road centerline and (

**b**) at the shoulder.

**Figure 21.**Actual transverse deformation curves at the road centerline and the shoulder of the three subgrade structures: (

**a**) at the road centerline and (

**b**) at the shoulder.

Parameters | Expression |
---|---|

Surface tension | $\mathit{\gamma}=75.6-0.1425T-2.38\times {10}^{-4}{T}^{2}$ |

Saturated vapor density | ${\rho}_{\mathrm{v}}=\mathrm{exp}\left(31.37-6014.79/T-7.92\times {10}^{-3}T\right)\times {10}^{-3}/T$ |

Enhancement factor | $\eta =9.5+3{\theta}_{\mathrm{w}}{\theta}_{\mathrm{sat}}{}^{-1}-8.5\left(1/\mathrm{exp}{\left(1+2.6{\left({f}_{\mathrm{c}}\right)}^{-0.5}{\theta}_{\mathrm{w}}{\theta}_{\mathrm{s}}{}^{-1}\right)}^{4}\right)$ |

Relative humidity | ${H}_{\mathrm{r}}=\mathrm{exp}\left(hM\mathrm{g}/RT\right)$ |

Water head | $h={L}_{\mathrm{i}}\left(T-{T}_{\mathrm{f}}\right)/\mathrm{g}{T}_{\mathrm{f}}$ |

Equivalent specific heat of the soil | $C={C}_{\mathrm{s}}{\theta}_{\mathrm{s}}+{C}_{\mathrm{w}}{\theta}_{\mathrm{w}}+{C}_{\mathrm{i}}{\theta}_{\mathrm{i}}+{C}_{\mathrm{v}}{\theta}_{\mathrm{v}}$ |

Latent heat of water evaporation | ${L}_{\mathrm{v}}=2.501\times {10}^{6}-2369.2T$ |

Effective thermal conductivity Volumetric vapor content Vapor diffusion | $\lambda ={\lambda}_{\mathrm{s}}{}^{{\theta}_{\mathrm{s}}}{\lambda}_{\mathrm{w}}{}^{{\theta}_{\mathrm{w}}}{\lambda}_{\mathrm{i}}{}^{{\theta}_{\mathrm{i}}}{\lambda}_{\mathrm{v}}{}^{{\theta}_{\mathrm{v}}}\phantom{\rule{0ex}{0ex}}{\theta}_{\mathrm{v}}={\rho}_{\mathrm{v}}{H}_{\mathrm{r}}\left({\theta}_{\mathrm{sat}}-{\theta}_{\mathrm{w}}\right)/{\rho}_{\mathrm{w}}\phantom{\rule{0ex}{0ex}}D=2.12\times {10}^{-5}{\left(T/273.15\right)}^{2}{\theta}_{\mathrm{v}}^{10/3}/{\theta}_{\mathrm{sat}}^{2}$ |

Unfrozen water flux | ${q}_{\mathrm{w}}=-{K}_{\mathrm{wh}}\left(\mathit{\nabla}h+1\right)-{K}_{\mathrm{wT}}\mathit{\nabla}T$ |

Vapor flux | ${q}_{\mathrm{v}}=-{K}_{\mathrm{vh}}\mathit{\nabla}h-{K}_{\mathrm{vT}}\mathit{\nabla}T$ |

_{s}, λ

_{w}, λ

_{i}and λ

_{v}are the thermal conductivity of soil particles, unfrozen water, ice and vapor, respectively. f

_{c}is the mass fraction of clay in the soil, and L

_{i}is the latent heat of ice–water phase change. θ

_{s}is the volume content of soil particles, θ

_{sat}is the saturated water content of the soil, and T

_{f}is the freezing temperature of soil (273.15 K). C

_{s}and C

_{i}are the specific heat of soil particles and ice, respectively.

Soil Layers | λ_{s} (W/m·K) | C_{s} (J/kg·K) | K_{s} (m/s) | ρ (kg/m^{3}) | θ_{0} (%) | n_{0} | a | b |
---|---|---|---|---|---|---|---|---|

Embankment fill | 2.116 | 1028.6 | 8.522 × 10^{−6} | 1940 | 14.2 | 0.233 | 0.082 | −0.28 |

Peat clay | 1.215 | 1404.7 | 3.043 × 10^{−6} | 1300 | 18.6 | 0.438 | 0.123 | −0.20 |

Breccia | 1.741 | 1226.6 | 7.496 × 10^{−6} | 1710 | 21.2 | 0.481 | 0.133 | −0.25 |

Moderately weathered andesite | 2.022 | 1161.4 | 5.137 × 10^{−6} | 1800 | 15.2 | 0.352 | 0.107 | −0.18 |

Gravel | 2.253 | 981.2 | 4.525 × 10^{−5} | 2080 | 14.7 | 0.313 | 0.085 | −0.29 |

Block–stone layer | 2.642 | 923.0 | 8.404 × 10^{-}^{4} | 2700 | 10.2 | 0.350 | 0.067 | −0.21 |

Soil Layers | a_{1} (MPa) | b_{1} | a_{2} | b_{2} | a_{3} (^{o}) | b_{3} | a_{4} (MPa) | b_{4} | c_{1} (MPa) | d_{1} (MPa) | e_{1} (MPa) | c_{2} | c_{3} (^{o}) | d_{2} (^{o}) | e_{2} (^{o}) | c_{4} | c_{5} (MPa) | d_{3} (MPa) | e_{3} (MPa) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Embankment fill | 48.43 | 27.26 | 0.35 | −0.007 | 15 | 0.75 | 0.014 | 0.032 | −31 | −5.0 | 48.43 | 1.1 | −3 | −2 | 15 | 1.1 | −0.013 | −0.008 | 0.014 |

Peat clay | 4.52 | 2.16 | 0.40 | −0.008 | 12 | 0.67 | 0.068 | 0.108 | −4 | −0.4 | 4.52 | 1.2 | −6 | −4 | 12 | 1.2 | −0.025 | −0.016 | 0.068 |

Breccia | 6.27 | 2.84 | 0.38 | −0.006 | 18 | 0.55 | 0.046 | 0.065 | −6 | −0.6 | 6.27 | 1.2 | −7 | −5 | 18 | 1.2 | −0.031 | −0.028 | 0.046 |

Moderately weathered andesite | 40.11 | 22.41 | 0.25 | −0.004 | 26 | 0.84 | 0.140 | 0.149 | −21 | −4.1 | 40.11 | 1.2 | −4 | −3 | 26 | 1.2 | −0.047 | −0.035 | 0.140 |

Gravel | 5.38 | 2.56 | 0.42 | −0.004 | 8 | 0.08 | 0.001 | 0.046 | −96 | −13.2 | 5.38 | 1.1 | −15 | −8 | 8 | 1.1 | −0.098 | −0.079 | 0.001 |

Block–stone layer | 86 | 24 | 0.30 | −0.001 | 30 | 0.86 | 0 | 0.005 | −16 | −0.2 | 86 | 1.1 | −1 | −0.4 | 30 | 1.1 | −0.001 | −0.001 | 0 |

Variables | T_{0} (°C) | A (°C) |
---|---|---|

Natural ground surface: IJ | 0.29 | 16.24 |

Side slope of the subgrade: JK | 0.52 | 18.11 |

Top surface of the subgrade: AK | 1.13 | 19.62 |

**Table 5.**Actual maximum deformations of the top surface of the three subgrade structures from 2019 to 2039.

Subgrade Structures | Cumulative Settlement at the Road Centerline (cm) | Cumulative Settlement at the Shoulder (cm) | Maximum Transverse Deformation at the Road Centerline (cm) | Maximum Transverse Deformation at the Shoulder (cm) |
---|---|---|---|---|

Replacement block–stone subgrade structure | −0.153 | −0.211 | −0.077 | +0.111 |

Replacement breccia subgrade structure | −23.467 | −15.971 | +0.495 | −1.209 |

Replacement gravel subgrade structure | −33.793 | −22.068 | +0.395 | −2.207 |

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## Share and Cite

**MDPI and ACS Style**

Shan, W.; Ma, M.; Guo, Y.; Zhang, C. Numerical Analysis of the Influence of Foundation Replacement Materials on the Hydrothermal Variation and Deformation Process of Highway Subgrades in Permafrost Regions. *Water* **2022**, *14*, 2642.
https://doi.org/10.3390/w14172642

**AMA Style**

Shan W, Ma M, Guo Y, Zhang C. Numerical Analysis of the Influence of Foundation Replacement Materials on the Hydrothermal Variation and Deformation Process of Highway Subgrades in Permafrost Regions. *Water*. 2022; 14(17):2642.
https://doi.org/10.3390/w14172642

**Chicago/Turabian Style**

Shan, Wei, Min Ma, Ying Guo, and Chengcheng Zhang. 2022. "Numerical Analysis of the Influence of Foundation Replacement Materials on the Hydrothermal Variation and Deformation Process of Highway Subgrades in Permafrost Regions" *Water* 14, no. 17: 2642.
https://doi.org/10.3390/w14172642