# Numerical Simulation of Heat Transfer of Porous Rock Layers in Cold Sandy Regions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}, the cooling performance of the PRL could offset the adverse impact of climate warming and raise the permafrost table in the first 20 years. Moreover, the closed PRL can be more effective in permafrost regions with colder MAATs. For cold sandy permafrost zones, sand-control measures should be taken to maintain the long-term cooling performance of the PRL. This study is of great significance in guiding porous rock embankment design and road maintenance along the Qinghai–Tibetan Railway.

## 1. Introduction

## 2. Physical and Numerical Model

#### 2.1. Governing Equations

#### 2.1.1. Porous Media Zone

_{x}and v

_{y}are the velocity components of air (m·s

^{−1}) in the directions x and y.

_{p}is the effective average particle size of the medium, φ is the porosity of the medium, and α is a parameter related to the shape characteristics of the medium.

_{0}and T

_{0}are reference values for the density and temperature.

_{m}± ΔT. The C* and λ* are given as [28]:

_{f}and C

_{u}are the volumetric heat capacity of frozen and unfrozen states, respectively; λ

_{f}and λ

_{u}are the thermal conductivity of frozen and unfrozen states, respectively; L is the latent heat per unit volume.

#### 2.1.2. Soil Layer Zone

#### 2.2. Physical Model and Parameters

^{−6}m

^{2}and 840.32 m

^{−1}. The geometric model is shown in Figure 2. Parts I to IV were the aeolian sand-filling PRL, PRL, gravelly soil, and strongly weathered mudstone, respectively.

_{s}, λ

_{w}, λ

_{i}, and λ

_{a}are the thermal conductivity of solid particles (aeolian sand and rock), water, ice, and air, respectively; A and φ are empirical parameters; x

_{s}, x

_{w}, x

_{i}, and x

_{a}denote the volume fraction of solid particles, water, ice, and air, respectively, obtained from Equation (15).

_{d}, ρ

_{s}, and ρ

_{w}are the density of dry sands, solid particles, and water.

#### 2.3. Temperature Boundary Condition

_{a}is the MAAT (°C).

^{2}.

#### 2.4. Modeling Sequence and Calculation Cases

## 3. Results

#### 3.1. Modeling Validation

#### 3.2. Natural Convection Characteristics in the Closed PRL

^{2}[16].

#### 3.2.1. Impact of Sand Filling on the Convection Characteristics of the PRL

_{ac}) when Ra

_{c}= 4π

^{2}. When Ra > 4π

^{2}, the Ra number gradually increases with the increase in the ΔT, and strong natural convection occurs inside the PRL, which enhances the convective heat transfer capacity of the bottom and top. Figure 4b presents the variation in the Ra number with time for different sand-filling cases. Without considering climate warming, the amplitude of the Ra number remained essentially constant with the increase in the operation time, but gradually decreased with the increase in the sand-filling thickness in PRLs. This is because the increase in the sand-filling thickness resulted in a decrease in the effective height of the porous media layer, forcing the ΔT of the PRL to decrease gradually, thereby increasing the difficulty of natural convection. In operation to the 10th year, the maximum values of the Ra number corresponding to the five cases were 123.38, 98.56, 68.52, 43.32, and 11.54, respectively. It can be seen that the Ra number gradually decreases with the increase in the thickness of the sand filling. Under the condition of the sand-filling thickness of 80 cm, only weak natural convection occurred within the closed PRL. When the sand-filling thickness exceeded 80 cm, the predicted Ra number was less than 4π

^{2}, and convection would not occur under this case.

_{ac}and duration of natural convection were also different under each sand-filling case. Table 4 summarizes the ΔT

_{ac}, natural convection period, and maximum Ra number of the four cases under different periods. The simulation results showed that the natural convection in the closed PRL was the strongest in cold seasons, which was consistent with the field monitoring results. Furthermore, the occurrence and end of natural convection were gradually delayed and advanced with the deepening of the sand thickness. In the sand-free case, the predicted duration of natural convection was usually approximately 120 days. When the sand-filling thickness increased to 80 cm, the natural convection duration decreased by approximately 75%, which was approximately 30 days on average.

^{−4}–2.63 × 10

^{−3}m·s

^{−1}, 1.97 × 10

^{−4}–1.78 × 10

^{−3}m·s

^{−1}, 8.47 × 10

^{−5}–7.56 × 10

^{−4}m·s

^{−1}, 9.44 × 10

^{−6}–2.91 × 10

^{−4}m·s

^{−1}, and 8.15 × 10

^{−6}–8.93 × 10

^{−5}m·s

^{−1}, respectively. In the warm season, the variation range of flow velocity within the PRL was 6.14 × 10

^{−6}–1.73 × 10

^{−4}m·s

^{−1}, 7.11 × 10

^{−6}–1.76 × 10

^{−4}m·s

^{−1}, 3.54 × 10

^{−6}–1.43 × 10

^{−4}m·s

^{−1}, 4.85 × 10

^{−7}–8.32 × 10

^{−5}m·s

^{−1}, and 5.65 × 10

^{−7}–4.03 × 10

^{−5}m·s

^{−1}, respectively. The simulation results showed that the air convection velocity within the PRL was larger in the cold season and smaller in the warm season, and thus the convective intensity was greater in the cold season than in the warm season. Meanwhile, driven by an unstable air density gradient attributed to the temperature difference between the top and bottom, the cooler air in the warm season was retained at the bottom of the PRL, which favored the lower underlying soil temperature. In addition, different sand-filling thicknesses influence the thermal state of the underlying permafrost by affecting the strength of air convection within the PRL. With the deepening of the sand-filling thicknesses, the air variation within the PRL decreased significantly, leading to a weakening of the natural convection strength and a reduction in the cooling performance.

#### 3.2.2. Comprehensive Impact of Climate Warming and Sand Filling on the Convection Characteristics of the Closed PRL

_{c}. The prediction results indicated that the cooling effect of the closed PRL disappeared after the 30th and 40th years under scenarios with MAATs of −3.5 °C and −4.5 °C, respectively. In contrast, for a MAAT = −5.5 °C, although natural convection could still occur throughout the simulation period, the duration of natural convection per year was reduced from 27 days in the 1st year to 11 days in the 50th year. Furthermore, for the case of 120 cm sand-filling thickness under three MAATs, the ΔT was far less than the ΔT

_{ac}that allowed natural convection to occur, and no natural convection occurred during the 50-year simulation period.

#### 3.3. Permafrost Thermal Regime Beneath the PRL

#### 3.3.1. Variation in the Permafrost Table

^{−1}, 0.037 m·a

^{−1}, and 0.035 m·a

^{−1}, respectively. Under the case of 80 cm sand filling, the reduction rates of the artificial permafrost table were 0.042 m·a

^{−1}, 0.026 m·a

^{−1}, and 0.025 m·a

^{−1}, respectively. In addition, the permafrost table of the sand-free case was lesser than that of the sand-filling cases, indicating that sand filling has a negative impact on the cooling performance of the PRL. Overall, with the decrease in MAAT, the impact of aeolian sand accumulation on the permafrost table in the lower part of the PRL was weakened, the cooling performance of the PRL became significant, and the duration in which the artificial permafrost table was higher than the natural permafrost table was prolonged. Therefore, closed PRLs are more suitable for cooling reinforcement measures of the subgrade in permafrost regions with low temperatures to maximize the cooling performance of natural convective heat transfer.

#### 3.3.2. Variation in Heat Flux of the Shallow Soil Layer Beneath the PRL

_{1.5}and T

_{2.0}are the soil temperatures at the depths of 1.5 m and 2.0 m. The mean annual heat budget for the subgrade was obtained by integrating the heat flux over time. The thermal budget of the soil layer under different operating cases is shown in Figure 9.

#### 3.3.3. Thermal Changes in the Deep Soil

^{−1}. In the first 10 years, the warming rates of MAGT for the two cases were only 0.013 °C·a

^{−1}and 0.018 °C·a

^{−1}. During subsequent simulation periods, the MAGT at 15 m exhibited a more pronounced warming trend. From the 20th to 50th year, in the sand-free case, the corresponding temperature rise amplitudes of the three MAATs were 0.174 °C, 0.899 °C, and 1.298 °C, and with warming rates of 0.006 °C·a

^{−1}, 0.03 °C·a

^{−1}, and 0.043 °C·a

^{−1}, respectively. In the case of the sand-filling of 80 cm, the three MAAT scenarios had temperature rise amplitudes of 0.166 °C, 0.844 °C, and 1.306 °C, with warming rates of 0.005 °C·a

^{−1}, 0.028 °C·a

^{−1}, and 0.044 °C·a

^{−1}, respectively. Overall, the changes in deep ground temperature under the PRL were mainly affected by climate warming and MAAT, while the aeolian sand had an insignificant effect on the deep ground temperature.

## 4. Discussion

^{−1}·°C

^{−1}[8,32]. The dry sand layer can play the role of heat insulation, hinder the heat transfer between the atmosphere and the subgrade, and promote the warming of the underlying permafrost. On the contrary, the high thermal conductivity of wet sand is conducive to more heat extraction in winter. This shows that the thermophysical properties of aeolian sand play an essential role in the thermal state of permafrost beneath the PRL. Therefore, it is necessary to establish a surface energy balance model and hydro-thermal coupling model to study the serviceability of a PRL with serious sand damage in future research.

## 5. Conclusions

- (1)
- The accuracy of the numerical model was verified using field tests. Natural convection within the closed PRL occurred only in cold seasons, and the convection strength was related to the effective convection height of the rock layer. As the thickness of sand filling increased, the T
_{ac}allowing natural convection to occur increased, and the Ra number decreased, which caused the weakening of the duration and intensity of natural convection. - (2)
- Under a warming scenario of 0.052 °C·a
^{−1}, the cooling performance of a PRL can offset the adverse impacts of climate warming and raise the permafrost table during the first 20 years of operation. However, the cooling performance of the PRL diminishes with the increase in the operation year, and the underlying permafrost continues to degrade over the next several decades. - (3)
- A closed PRL is more suitable for cooling measures of the subgrade in permafrost regions with colder MAATs. In the context of climate change and sand damage, the cooling effect of a PRL on the permafrost can no longer meet the long-term requirements.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Geometric model and strata used for the numerical modeling: I: Aeolian sand-filling PRL, II: PRL, III: Gravelly soil, IV: Strongly weathered mudstone.

**Figure 3.**Model calibration using field test results from June 2013 to June 2015: (

**a**) Temperature variation at depths of 0.5 m and 1.5 m in the PRL, (

**b**) Ra numbers.

**Figure 4.**The time-varying ΔT and Ra number of the closed PRL under the different sand-filling cases: (

**a**) Temperature difference between the bottom and top of the PRL, (

**b**) Ra number.

**Figure 5.**Ra numbers of different sand-filling cases under three MAAT scenarios: (

**a**) MAAT = −3.5 °C, (

**b**) MAAT = −4.5 °C, (

**c**) MAAT = −5.5 °C.

**Figure 6.**The permafrost table changes over time under MAAT = −3.5 °C scenarios: (

**a**) sand-free case, (

**b**) 80 cm thick sand-filling case.

**Figure 7.**The permafrost table changes over time under MAAT = −4.5 °C scenarios: (

**a**) sand-free case, (

**b**) 80 cm thick sand-filling case.

**Figure 8.**The permafrost table changes over time under MAAT = −5.5 °C scenarios: (

**a**) sand-free case, (

**b**) 80 cm thick sand-filling case.

**Figure 9.**Heat budget of the soil layer between depths of 1.5 and 2.0 m beneath the PRL: (

**a**) MAAT = −3.5 °C, (

**b**) MAAT = −4.5 °C, (

**c**) MAAT = −5.5 °C. Note: when q > 0, it means heat absorption from the shallow layer; conversely, q < 0 means heat release from the deep layer of permafrost to the shallow layer.

**Figure 10.**Variations in MAGT at a depth of 15 m beneath the natural ground surface over a period of 50 years under three MAAT (−3.5 °C, −4.5 °C, −5.5 °C) scenarios.

Lithology | Thermal Conductivity (W·m ^{−1}·°C^{−1}) | Volumetric Heat Capacity (J·kg ^{−1}·°C^{−1}) | Latent Heat (J·m ^{−3}) | ||
---|---|---|---|---|---|

Frozen | Unfrozen | Frozen | Unfrozen | ||

Porous rock layer | 0.442 | 0.442 | 1.016 × 10^{6} | 1.016 × 10^{6} | 0 |

Sand-filled porous rock layer | 1.188 | 1.188 | 1.446 × 10^{6} | 1.446 × 10^{6} | 0 |

Gravel soil | 2.720 | 1.870 | 1.864 × 10^{6} | 2.401 × 10^{6} | 2.338 × 10^{7} |

Weathered mudstone | 1.844 | 1.474 | 2.122 × 10^{6} | 2.413 × 10^{6} | 3.811 × 10^{7} |

Thermal insulation board | 0.029 | 0.029 | 2.406 × 10^{5} | 2.406 × 10^{5} | 0 |

**Table 2.**Physical parameters of the air [34].

Physical Domain | c_{p} (J·kg^{−1}·°C^{−1}) | λ_{a} (W·m^{−1}·°C^{−1}) | ρ (kg·m^{−3}) | μ (kg·m^{−1}·s^{−1}) |
---|---|---|---|---|

Air | 1.004 × 10^{3} | 0.02 | 0.641 | 1.75 × 10^{−5} |

_{p}is the specific heat of air.

Case | MAAT (°C) | Climate Change | Sand-Filling Thickness (cm) |
---|---|---|---|

Case 1 | −3.5 | Warm | 0, 20, 50, 80, 120 |

Case 2 | −4.5 | Warm | 0, 20, 50, 80, 120 |

Case 3 | −5.5 | Warm | 0, 20, 50, 80, 120 |

Case 4 | −3.5 | Unwarm | 0, 20, 50, 80, 120 |

**Table 4.**Critical temperature difference, duration, and maximum Ra number of the closed PRL in different periods.

Time | Cases | 0 cm | 20 cm | 50 cm | 80 cm |
---|---|---|---|---|---|

1a | ΔT_{ac} (°C) | 3.50 | 4.49 | 5.50 | 7.92 |

Natural convection period | 11/1−2/28 | 11/7−2/25 | 11/6−2/16 | 12/13−1/13 | |

Maximum Ra number | 124.59 | 99.33 | 68.85 | 43.56 | |

30a | ΔT_{ac} (°C) | 3.47 | 4.47 | 5.48 | 7.90 |

Natural convection period | 11/1−2/28 | 11/7−2/22 | 11/16−2/13 | 12/13−1/13 | |

Maximum Ra number | 123.37 | 98.55 | 68.52 | 43.33 | |

50a | ΔT_{ac} (°C) | 3.92 | 4.44 | 5.58 | 7.98 |

Natural convection period | 11/7−2/30 | 11/11−2/26 | 11/21−2/17 | 11/22−2/27 | |

Maximum Ra number | 121.99 | 98.75 | 68.18 | 44.69 |

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## Share and Cite

**MDPI and ACS Style**

Qiu, K.; Huang, Y.; Han, F.; Yang, Q.; Yu, W.; Cheng, L.; Cao, H.
Numerical Simulation of Heat Transfer of Porous Rock Layers in Cold Sandy Regions. *Atmosphere* **2023**, *14*, 1812.
https://doi.org/10.3390/atmos14121812

**AMA Style**

Qiu K, Huang Y, Han F, Yang Q, Yu W, Cheng L, Cao H.
Numerical Simulation of Heat Transfer of Porous Rock Layers in Cold Sandy Regions. *Atmosphere*. 2023; 14(12):1812.
https://doi.org/10.3390/atmos14121812

**Chicago/Turabian Style**

Qiu, Kaichi, Yong Huang, Fenglei Han, Qiuju Yang, Wenbing Yu, Lu Cheng, and Hang Cao.
2023. "Numerical Simulation of Heat Transfer of Porous Rock Layers in Cold Sandy Regions" *Atmosphere* 14, no. 12: 1812.
https://doi.org/10.3390/atmos14121812