Numerical simulation of frost heaving damage of earth-rock dam berms in cold regions with thermo-hydro-mechanical coupling
冻土地区温湿机械耦合对土石坝坝体冻胀破坏的数值模拟

2.1. Basic assumptions 2.1. 基本假设

When freezing occurs in an earth-rock dam, the microstructure and dynamics of the interaction region involving the dam fill, gravel, water, and ice become more complex. In order to facilitate calculations, the present numerical model performs calculations according to the following assumptions:
当土石坝发生冻结时，涉及坝体填充物、砾石、水和冰的相互作用区域的微观结构和动力学变得更加复杂。为了便于计算，当前的数值模型根据以下假设进行计算：

(1) The soil inside the earth-rock dam model is homogeneous, continuous, and isotropic.
（1）地球岩坝模型内的土壤是均质、连续和各向同性的。

(2) The hydraulic and thermal conductivity coefficients in frozen soil and thawing soil are constant.
（2）结冻土和融土的水力和热导系数是恒定的。

(3) The migration of internal vapor water is not considered, but the water migration of liquid water is considered.
（3）内部水蒸气的迁移不予考虑，但液态水的迁移予以考虑。

(4) Soil particles, gravel bedding, and solid ice are incompressible, and only the volume change caused by frost heaving of water in the soil is considered.
(4) 土粒、碎石垫层和固体冰是不可压缩的，只考虑土壤中水分冻胀引起的体积变化。

(5) The coefficient of permeability is the same for all places in the soil and gravel bedding.
（5）土壤和碎石垫层中的渗透系数在所有地方都相同。

(6) Heat transfer is only considered as heat conduction, and ignores the effect of air convection.
(6) 热传递仅考虑热传导，忽略了空气对流的影响。

(7) The effect of pore water pressure and the external load of the slope protection structure are ignored.
(7) 忽略了孔隙水压力和斜坡防护结构的外部载荷的影响。

2.2. Differential equation for unfrozen water migration
2.2. 不结冰水分迁移的微分方程

According to Richard's equation and taking into account the blocking effect of pore ice on unfrozen water migration, the differential equation for unfrozen water migration is (Lu and Likos, 2012):(1)$\frac{\partial {\theta }_u}{\partial t}+\frac{{\rho }_I}{{\rho }_w}\frac{\partial {\theta }_1}{\partial t}=\nabla \left[D\left(\theta \right)\nabla {\theta }_u+k_g\left({\theta }_u\right)\right]$where ${\theta }_u$ is the volumetric water content of unfrozen water in frozen soil, and k_g is the coefficient of permeability of the soil in the direction of gravitational acceleration.
根据理查德的方程，并考虑到孔隙冰对未冻结水迁移的阻挡效应，未冻结水迁移的微分方程为（Lu 和 Likos, 2012）： (1)$\frac{\partial {\theta }_u}{\partial t}+\frac{{\rho }_I}{{\rho }_w}\frac{\partial {\theta }_1}{\partial t}=\nabla \left[D\left(\theta \right)\nabla {\theta }_u+k_g\left({\theta }_u\right)\right]$ 其中， ${\theta }_u$ 表示冻结土壤中未冻结水的体积含水量，k _g 是土壤在重力加速度方向的渗透系数。

The diffusivity of water in permafrost is calculated as follows (Taylor and Luthin, 1978):(2)$D\left({\theta }_u\right)=\frac{k\left({\theta }_u\right)}{c\left({\theta }_u\right)}·I$where $k\left({\theta }_u\right)$ is the permeability of the soil (m/s), and $c\left({\theta }_u\right)$ is the specific water capacity (1/m). In addition, I is the impedance factor that represents the blocking effect of pore ice on the migration of unfrozen water, and is calculated by the following formula:(3)$I={10}^{-{\theta }_I}$
冻土中水的扩散系数计算如下（Taylor 和 Luthin, 1978）： (2)$D\left({\theta }_u\right)=\frac{k\left({\theta }_u\right)}{c\left({\theta }_u\right)}·I$ 其中， $k\left({\theta }_u\right)$ 是土壤的渗透性（m/s）， $c\left({\theta }_u\right)$ 是特定水容量（1/m）。此外，I 是孔隙冰对未冻结水迁移的阻塞效应的阻抗因子，通过以下公式计算： (3)$I={10}^{-{\theta }_I}$

2.3. Heat transfer control equation
2.3. 热传递控制方程

For the two-dimensional coupled thermo-hydro-mechanical problem, considering the latent heat of phase change as the heat source, the heat transfer control equation can be expressed by Fourier's law as (Tao, 2006):(4)$\mathit{\rho C}\left(\theta \right)\frac{\partial T}{\partial t}=\lambda \left(\theta \right){\nabla }^{2}T+L·{\rho }_{\mathrm{I}}\frac{\partial {\theta }_{\mathrm{I}}}{\partial t}$where $\nabla$is the differential operator which is $\left[\partial /\partial x\partial /\partial y\right]$for a two-dimensional problem; T is the transient temperature of the soil (°C); t is the time (s); $\theta$ is the volumetric water content; ${\theta }_I$ is the pore ice volume content; x and y are horizontal and depth coordinates (m), respectively; $\rho$ and ${\rho }_I$ are the density of soil and ice (kg/m³), respectively; L is the latent heat of the phase transition, which takes the value of 334.5 kJ/kg; $\lambda \left(\theta \right)$ is the coefficient of thermal conductivity (W/m • °C); and $C\left(\theta \right)$ is the volumetric heat capacity (J/kg • °C).
对于二维耦合热-水-力学问题，将相变的潜热作为热源，热传递控制方程可以由傅里叶定律表示为（Tao, 2006）： (4)$\mathit{\rho C}\left(\theta \right)\frac{\partial T}{\partial t}=\lambda \left(\theta \right){\nabla }^{2}T+L·{\rho }_{\mathrm{I}}\frac{\partial {\theta }_{\mathrm{I}}}{\partial t}$ 其中， $\nabla$ 是微分算子，对于二维问题为 $\left[\partial /\partial x\partial /\partial y\right]$ ；T 是土壤的瞬时温度（℃）；t 是时间（s）； $\theta$ 是体积含水量； ${\theta }_I$ 是孔隙冰体积含量；x 和 y 分别是水平和深度坐标（m）； $\rho$ 和 ${\rho }_I$ 分别是土壤和冰的密度（kg/m ³ ）；L 是相变的潜热，其值为 334.5 kJ/kg； $\lambda \left(\theta \right)$ 是热导率的系数（W/m • °C）；和 $C\left(\theta \right)$ 是体积热容量（J/kg • °C）。

2.4. Calculation equation of soil deformation
2.4. 土壤变形的计算公式

During the freezing process of the soil, assuming that the soil particles and gravel bedding are incompressible and isotropic, the expansion deformation of the soil is calculated as follows (Xu et al., 2001):(5)$\epsilon =0.09{\theta }_0-{\theta }_{\mathrm{u}}+1.09\mathit{\Delta \theta }$where $\epsilon$ is the soil expansion deformation, ${\theta }_0$ is the initial water content, ${\theta }_u$ is the unfrozen water content, and $\mathrm{\Delta }\theta$ is the migration water content.
在土壤冻结过程中，假设土壤颗粒和卵石垫层不可压缩且各向同性，土壤的膨胀变形计算如下（徐等，2001）： (5)$\epsilon =0.09{\theta }_0-{\theta }_{\mathrm{u}}+1.09\mathit{\Delta \theta }$ 其中， $\epsilon$ 表示土壤膨胀变形， ${\theta }_0$ 表示初始含水量， ${\theta }_u$ 表示未冻结含水量， $\mathrm{\Delta }\theta$ 表示迁移含水量。

2.5. Phase change dynamic equilibrium relationship
2.5. 相变动态平衡关系

Xu et al. (2001) obtained an empirical relationship for the volume content of unfrozen water in frozen soil through an extensive set of experiments. It can be expressed as follows:(6)$\frac{{\omega }_0}{{\omega }_{\mathrm{u}}}={\left(\frac{T}{T_{\mathrm{f}}}\right)}^B,T<T_{\mathrm{f}}$where $T_f$ is the temperature of the soil in freezing (°C), ${\omega }_0$ is the initial water content of the soil (%), ${\omega }_u$ is the water content of unfrozen water at negative temperature (%), and B is a constant depending on the soil type and salt content.
徐等人（2001 年）通过一系列广泛实验，得到了冻结土壤中未冻结水分体积含量的实证关系。其表达式如下： (6)$\frac{{\omega }_0}{{\omega }_{\mathrm{u}}}={\left(\frac{T}{T_{\mathrm{f}}}\right)}^B,T<T_{\mathrm{f}}$ 其中， $T_f$ 是土壤冻结时的温度（°C）， ${\omega }_0$ 是土壤的初始含水量（%）， ${\omega }_u$ 是负温度下未冻结水分的含水量（%），B 是依赖于土壤类型和盐分含量的常数。

In this study, the linkage equation is established based on the concept of “solid-liquid ratio” proposed in the literature (Bai et al., 2015), which can be expressed as:(7)$B_I=\frac{{\theta }_I}{{\theta }_u}=\left\{\begin{array}{c}1.1{\left(\frac{T}{T_f}\right)}^B-1.1\left(\mathrm{T}<{\mathrm{T}}_f\right)\\ 0\left(\mathrm{T}⩾{\mathrm{T}}_f\right)\end{array}\right.$where $B_I$ is the solid-liquid ratio. Also, the factor 1.1 is the density ratio of water and ice.
在本研究中，基于文献（ Bai 等，2015）提出的“固液比”概念，建立了联系方程，可以表示为： (7)$B_I=\frac{{\theta }_I}{{\theta }_u}=\left\{\begin{array}{c}1.1{\left(\frac{T}{T_f}\right)}^B-1.1\left(\mathrm{T}<{\mathrm{T}}_f\right)\\ 0\left(\mathrm{T}⩾{\mathrm{T}}_f\right)\end{array}\right.$ 其中， $B_I$ 表示固液比。此外，因子 1.1 是水和冰的密度比。

Thus, a coupled equation for pore ice, unfrozen water, and temperature can be obtained as:(8)${\theta }_I=B_I\left(T\right)·{\theta }_{\mathrm{u}}$
因此，可以得到孔隙冰、未冻结水和温度的耦合方程为： (8)${\theta }_I=B_I\left(T\right)·{\theta }_{\mathrm{u}}$

3.1. Model 3.1. 模型

According to the typical profile of an earth-rock dam in the cold region, Comsol Multiphysics finite element software is used to establish the model. The model and arrangement of measurement points are shown in Fig. 2. In general, the Comsol Multiphysics has the property of convergence, and the convergence can be achieved by dividing the domain into smaller elements, regardless of the shape of the element. Therefore, the shape of the elements only affects the convergence rate, and if the number of elements is chosen appropriately, it does not have a significant effect on the accuracy of the calculation results. Here, the discretization of the domain into elements or grids is based on the coupling characteristics of the problem, where a total of 2697 elements are generated, with 44 fixed-point elements and 631 boundary elements. A schematic of the numerical model is shown in Fig. 3.
根据寒冷地区土石坝的典型剖面，使用 Comsol Multiphysics 有限元软件建立模型。模型及测量点的布置如图 2 所示。通常，Comsol Multiphysics 具有收敛性，无论元素的形状如何，通过将域分割成更小的元素，可以实现收敛。因此，元素的形状仅影响收敛率，如果适当选择元素的数量，它对计算结果的准确性影响不大。在这里，基于问题的耦合特性对域进行离散化，生成了总共 2697 个元素，其中 44 个固定点元素和 631 个边界元素。数值模型的示意图如图 3 所示。

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Fig. 2. Computation model.
图 2. 计算模型。

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Fig. 3. Mesh division of numerical model.
图 3. 数值模型的网格划分。

3.2. Boundary conditions 3.2. 边界条件

The upstream dam slope ab is the reservoir level boundary, cd is the downstream water level boundary, and efgh is the impervious boundary.
上游坝坡 ab 是水库水位边界，cd 是下游水位边界，而 efgh 是不透水边界。

3.3. Parameter selection 3.3. 参数选择

4.1. Temperature field analysis
4.1. 温度场分析

Fig. 4 shows the cloud map of the temperature field distribution of the earth-rock dam in the period from November 15, 2021 to February 15, 2022. As can be seen from the figure, during the temperature decrease stage (i.e., from November 15 to January 10), the temperature decreases from 3 °C to −16 °C, the temperature of the surface layer of the dam slope (0–2 m) decreases from 3 °C to −10 °C, and the temperature of the surface layer of the dam slope reaches a minimum value of −10 °C in January 9. But, during the temperature rise stage (i.e., from January 10 to February 15), the temperature increases from −17 °C to 3 °C, and the temperature of the surface layer of the dam slope (0–2 m) gradually increases and reaches a maximum value of −1 °C on February 15.
图 4 显示了从 2021 年 11 月 15 日至 2022 年 2 月 15 日期间地球-岩石坝温度场分布的云图。从图中可以看出，在温度下降阶段（即从 11 月 15 日至 1 月 10 日），温度从 3°C 下降至-16°C，坝坡表层（0-2m）的温度从 3°C 下降至-10°C，坝坡表层的温度在 1 月 9 日达到最低值-10°C。但在温度上升阶段（即从 1 月 10 日至 2 月 15 日），温度从-17°C 上升至 3°C，坝坡表层（0-2m）的温度逐渐上升，在 2 月 15 日达到最高值-1°C。

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Fig. 4. Cloud maps of temperature field distribution on (a) November 15th, (b) January 10th and (c) February 15th.
图 4. 温度场分布的云图，分别为（a）11 月 15 日，（b）1 月 10 日和（c）2 月 15 日。

The maximum frost depth of the earth-rock dam was about 2 m, indicating that the temperature variation in the surface layer of the dam slope (0–2 m) was large, and the temperature variation inside the dam body decreased with increasing distance from the dam slope.
土石坝的最大冻土深度约为 2 米，这表明大坝斜坡表层（0-2 米）的温度变化较大，而大坝主体内部的温度变化随距斜坡距离的增加而减小。

In order to study the temperature change of the surface layer of the dam slope with the air temperature, one temperature measurement point was arranged on the upper level of the dam and three temperature measurement points were arranged on the upstream slope of the dam. The temperature change curve of different measurement points with time is shown in Fig. 5. From the figure, it can be seen that the temperature of measuring points 1, 2, and 3 shows a downward trend and then an upward trend with the change of air temperature. On November 15, the temperatures at measurement points 1, 2 and 3 were all 3 °C, and on January 10, they were − 16, −10 and − 8 °C, showing a decrease of 19, 13 and 11 °C, respectively. In summary, the degree of influence of air temperature on the temperature measurement points from large to small is as follows: measurement point 1 > measurement point 2 > measurement point 3 > measurement point 4.
为了研究大坝边坡表层温度随大气温度的变化，我们在大坝的上部设置了 1 个温度测量点，在大坝的上游斜坡设置了 3 个温度测量点。不同测量点随时间的变化温度曲线如图 5 所示。从图中可以看出，测量点 1、2、3 的温度随大气温度的变化呈现出先下降后上升的趋势。11 月 15 日，测量点 1、2、3 的温度均为 3°C，而 1 月 10 日，它们的温度分别为-16°C、-10°C 和-8°C，分别下降了 19、13 和 11°C。综上所述，大气温度对温度测量点的影响程度从大到小依次为：测量点 1 > 测量点 2 > 测量点 3 > 测量点 4。

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Fig. 5. Temperature time series graph for different measuring points.
图 5. 不同测量点的温度时间序列图。

Since the measuring point 1 was relatively close to the crest surface of the dam, it was more affected by the air temperature and therefore its temperature changes were greater. Measurement points 2 and 3 were located beneath the gravel bed and concrete slab at a considerable distance from the dam slope. With slow temperature transfer and minor impact on the soil temperature, the temperature amplitude at measuring points 2 and 3 was consequently small. Measuring point 4 was located below the water level of the upstream dam slope, and its temperature was basically unchanged due to the small change in water temperature.
由于测量点 1 相对较接近大坝的顶部表面，因此它更受气温影响，因此其温度变化更大。测量点 2 和 3 位于砾石床和混凝土板下，与大坝斜坡相距甚远。由于温度传递缓慢且对土壤温度影响较小，因此测量点 2 和 3 的温度振幅相应较小。测量点 4 位于上游大坝斜坡的水位以下，由于水温变化较小，其温度基本未变。

In summary, the effect of freezing on the influence range of the dam slope was about 2 m, the temperature influence range of the dam slope was mainly related to the freezing depth, and the temperature change of the shallow 0–2 m range of the dam slope was affected by the outside air temperature. Also, the internal temperature of the dam body exhibited a relatively small amplitude compared to the shallow layers of the dam slope and displayed a certain degree of hysteresis. The frost damage to an earth-rock dam berm is different from frost damage to the slopes of highways, railways, water conveyance channels, etc. This distinction primarily arises from the unique characteristics of earth-rock dam structures, where the upper part of the dam consists of concrete slabs, gravel bed, and geomembrane. Additionally, another distinguishing factor is that the berms of the dam are exposed to high water pressure.
摘要，冷冻对大坝边坡影响范围的影响约为 2 米，大坝边坡的温度影响范围主要与冷冻深度有关，大坝边坡浅层 0-2 米范围的温度变化受到外界空气温度的影响。此外，大坝主体的内部温度波动幅度相对较小，与大坝浅层相比，显示出一定程度的滞后性。土石坝的坝肩冻害与公路、铁路、输水渠道等的冻害不同，这种差异主要源于土石坝结构的独特性，其中大坝的上部由混凝土板、碎石层和防渗膜组成。另外，另一个区别是大坝的坝肩承受着较高的水压。

4.2. Moisture field analysis
4.2. 湿度场分析

Fig. 6 shows the cloud map of moisture field distribution of the earth-rock dam in the period from November 15, 2021 to February 15, 2022. As can be seen from the figure, during the temperature decrease stage (i.e., from November 15 to January 10), the moisture in the surface layer of the dam slope (0–2 m) continues to freeze, while the unfrozen water content gradually decreases. Due to the high initial water content in the reservoir level change area, the unfrozen water content decreases significantly. During the temperature rise stage (i.e., from January 10 to February 15), the temperature of the surface layer of the dam slope is still below 0 °C, the ice layer is not yet thawed, and the change in unfrozen water content is relatively small.
图 6 显示了从 2021 年 11 月 15 日至 2022 年 2 月 15 日地球岩坝水分场分布的云图。从图中可以看出，在温度下降阶段（即从 11 月 15 日至 1 月 10 日），坝坡表层（0-2 米）的水分持续冻结，而未冻结的水分含量逐渐减少。由于水库水位变化区域的初始含水量较高，未冻结的水分含量显著减少。在温度上升阶段（即从 1 月 10 日至 2 月 15 日），坝坡表层的温度仍低于 0°C，冰层尚未融化，未冻结水分含量的变化相对较小。

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Fig. 6. Cloud maps of moisture field distribution on (a) November 15th, (b) January 10th and (c) February 15th.
图 6. 湿度分布的云图，分别在(a)11 月 15 日，(b)1 月 10 日和(c)2 月 15 日。

From November 15 to February 15, the unfrozen water content in the surface layer of the dam slope (0–2 m) changed more and the unfrozen water content inside the dam body did not change much, indicating that temperature is an important factor affecting the change of water content.
从 11 月 15 日至 2 月 15 日，大坝斜坡表层（0-2 米）的未冻结水分含量发生了更多变化，而大坝主体内部的未冻结水分含量变化不大，这表明温度是影响水分含量变化的重要因素。

The unfrozen water content of the four typical measurement points over time is shown in Fig. 7. It can be seen that the unfrozen water content of measuring points 1, 2 and 3 shows a trend of decreasing and then increasing. On December 1, the unfrozen water content of the above measurement points was 1.1, 9.1 and 13.1%. Also, the unfrozen water content at this time reached the lowest value at point 1 and decreased by 2.9%. Furthermore, the unfrozen water content of measurement points 2 and 3 dropped by 8.0% and 4.9%. On January 16, the unfrozen water content of measurement points 2 and 3 reached the lowest values of 11.0% and 7.5%. With the increase in air temperature, they increased to 16% and 10% on February 15, while the unfrozen water content of measuring point 1 increased to 3.8%. Measuring point 4 was located below the reservoir water level, and experienced slight changes in water temperature and did not freeze. Throughout the simulation process, the unfrozen water content of measuring point 4 remained constant at 28%.
图 7 显示了四个典型测量点随时间的未冻结水分含量。可以看出，测量点 1、2 和 3 的未冻结水分含量呈现出先减后增的趋势。12 月 1 日，上述测量点的未冻结水分含量分别为 1.1%、9.1% 和 13.1%。此时，测量点 1 的未冻结水分含量达到最低值，减少了 2.9%。此外，测量点 2 和 3 的未冻结水分含量分别下降了 8.0% 和 4.9%。1 月 16 日，测量点 2 和 3 的未冻结水分含量降至最低，分别为 11.0% 和 7.5%。随着气温的升高，到 2 月 15 日，测量点 2 和 3 的未冻结水分含量分别增加至 16% 和 10%，而测量点 1 的未冻结水分含量增加至 3.8%。测量点 4 位于水库水位以下，经历了轻微的水温变化，未冻结。在整个模拟过程中，测量点 4 的未冻结水分含量保持不变，始终为 28%。

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Fig. 7. Cloud map of unfrozen water content distribution.
图 7. 未冻结水分分布的云图。

In summary, the degree of influence of air temperature on the unfrozen water content from large to small is in the order: measurement point 2 > measurement point 3 > measurement point 1 > measurement point 4. Due to the proximity of measuring points 2 and 3 to the infiltration zone of the reservoir water level in the dam berm surface layer, the initial water content of measuring points 2 and 3 was relatively high. When the negative temperature was transferred from the surface layer of the dam slope to the measurement points 2 and 3, the pore water in measurement points 2 and 3 underwent phase transition to ice. As a result, the unfrozen water content of measuring points 2 and 3 rapidly decreased, and this decrease was more significant compared to the measurement point 1. Measuring point 1 was near the top of the dam, with initially had a lower water content. Temperature had a relatively small effect on the unfrozen water content at measuring point 1.
总结而言，空气温度对未冻结水分含量的影响程度从大到小的顺序为：测量点 2 > 测量点 3 > 测量点 1 > 测量点 4。由于测量点 2 和 3 靠近大坝坝肩表面层蓄水位的渗透区域，测量点 2 和 3 的初始水分含量相对较高。当负温从大坝斜坡的表层传递到测量点 2 和 3 时，测量点 2 和 3 中的孔隙水经历了相变成为冰。因此，测量点 2 和 3 的未冻结水分含量迅速减少，这种减少比测量点 1 更为显著。测量点 1 靠近大坝顶部，初始水分含量较低。温度对测量点 1 的未冻结水分含量的影响相对较小。

The infiltration of reservoir water in the surface layer of the dam berm makes the shallow fill layer of the dam slope reach saturation, and the negative temperature makes the shallow pore water phase of the dam slope turn into ice, which is macroscopically manifested in the decrease of unfrozen water content. Also, the cooling intensity during the freezing process is small, and the freezing front moves downward at a slow pace. The phase change of pore water in the shallow soil-rock mixture at the dam slope, the migration of moisture from the dam fill to the dam slope, and the movement of the freezing front to the interior of the dam body are the main factors of the freezing damage of the berm.
大坝坝肩表层的水库水渗透使得大坝斜坡的浅填层达到饱和状态，负温使得大坝斜坡的浅层孔隙水相转化为冰，宏观表现为未冻结水含量的减少。此外，冻结过程中的冷却强度较小，冻结前沿移动速度缓慢。大坝斜坡浅层土石混合物中的孔隙水相变、大坝填料向大坝斜坡的水分迁移以及冻结前沿向大坝主体内部移动是大坝坝肩冻结损害的主要因素。

4.3. Displacement field analysis
4.3. 移动场分析

Fig. 8, Fig. 9 show the cloud map of vertical and horizontal displacement field distribution of the earth-rock dam in the period from November 15, 2021 to February 15, 2022. As can be seen from the figures, during the temperature decrease stage (i.e., from November 15 to January 10), obvious frost deformation occurs in the reservoir level change area of the upstream dam slope of the earth-rock dam, with a maximum amount of 30 cm. During the temperature rise stage (i.e., from January 10 to February 15), the frost deformation area of the upstream dam slope of the earth-rock dam in the reservoir water level fluctuation zone remains almost unchanged, and the amount of frost deformation continues to increase, with a maximum amount of 36 cm.
图 8、图 9 显示了从 2021 年 11 月 15 日至 2022 年 2 月 15 日期间，土石坝在该时间段内的垂直和水平位移场分布的云图。从图中可以看出，在温度下降阶段（即从 11 月 15 日至 1 月 10 日），土石坝上游坝坡的水位变化区域出现了明显的冻胀变形，最大变形量为 30 厘米。在温度上升阶段（即从 1 月 10 日至 2 月 15 日），土石坝上游坝坡在水库水位波动区域的冻胀变形区域几乎保持不变，冻胀变形量继续增加，最大变形量为 36 厘米。

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Fig. 8. Vertical displacement cloud maps on (a) November 15th, (b) January 10th and (c) February 15th.
图 8. (a)11 月 15 日，(b)1 月 10 日和(c)2 月 15 日的垂直位移云图。

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Fig. 9. Horizontal displacement cloud maps on (a) November 15th, (b) January 10th and (c) February 15th.
图 9. (a)11 月 15 日，(b)1 月 10 日和(c)2 月 15 日的水平位移云图。

Fig. 10 shows the variation of vertical displacements against time. It can be seen that with the change of temperature, the vertical displacements of measuring points 1, 2 and 3 increase and stabilize in February. The temperature at the measuring point 4 is 5 °C and no freezing of water occurs at this location. Consequently, no freezing expansion deformation takes place at this point (measuring point 4). It can also be seen that the degree of influence of air temperature on the freezing and expansion deformation from large to small is in the order: measurement point 3 > measurement point 2 > measurement point 1 > measurement point 4.
图 10 显示了垂直位移随时间的变化。可以看出，随着温度的变化，测量点 1、2 和 3 的垂直位移增加并在二月份稳定。测量点 4 的温度为 5°C，此处没有发生水的冻结，因此在这一点（测量点 4）没有发生冻结膨胀变形。也可以看出，从大到小影响空气温度对冻结和膨胀变形的程度顺序为：测量点 3 > 测量点 2 > 测量点 1 > 测量点 4。

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Fig. 10. Vertical displacement time series graph for different measuring points.
图 10. 不同测量点的垂直位移时间序列图。

When the temperature decreases, the vertical displacement of measuring point 1 increases slowly, and the displacement increases from 0 cm to 6 cm on February 15. The vertical displacements of measuring points 2 and 3 show two stages of increase and stabilization, and the incremental displacements are 33 cm and 36 cm respectively on February 15. At measurement point 1, the low initially water content results in a relatively small volume increase produced by water freezing into ice. Therefore, the freezing expansion amount at measuring point 1 is relatively low. Measuring point 2 has a relatively high initial water content and is significantly influenced by temperature. Additionally, under the influence of water migration, water continuously accumulates from the interior of the dam towards the shallow layer of the dam slope. Hence, the freezing expansion deformation at measurement point 2 is significant.
当温度下降时，测量点 1 的垂直位移缓慢增加，从 2 月 15 日的 0cm 增加到 6cm。测量点 2 和 3 的垂直位移显示了增加和稳定两个阶段，2 月 15 日的增量位移分别为 33cm 和 36cm。在测量点 1 处，初始含水量较低，导致由水结冰产生的体积增加相对较小。因此，测量点 1 的结冰膨胀量相对较低。测量点 2 的初始含水量相对较高，受温度影响较大。此外，在水迁移的影响下，水从大坝内部持续向大坝斜坡的浅层积累。因此，测量点 2 的结冰膨胀变形显著。

At the surface of the concrete slabs in the reservoir level change area of the upstream dam slope of the earth-rock dam, the measurement points were marked. The elevations of measurement points 1, 2, 3, and 4 were 1839.9 m, 1833.8 m, 1834.4 m, and 1830.3 m, respectively. Subsequently, the displacement measurements were conducted using a total station according to the monitoring technical specifications of earth-rock dams. The key monitoring period was from November 15, 2021 to February 15, 2022, and measurements of the elevation at measurement points were conducted every 7 days. The monitoring results of vertical displacements at the measurement points are shown in Table 2. The comparative graph of numerical simulation results and measured data is shown in Fig. 11. It can be seen from Table 2 and Fig. 11 that the numerical simulation results of the displacement field for the earth-rock dam are generally consistent with the measured data. The main freezing zone is located within an elevation range of approximately 2 m from the dam ice surface, with a frost heave amount ranging from 20 to 30 cm.
在上游土石坝的水库水位变化区域的混凝土板表面，标记了测量点。测量点 1、2、3 和 4 的高程分别为 1839.9 米、1833.8 米、1834.4 米和 1830.3 米。随后，根据土石坝监测技术规范，使用全站仪进行了位移测量。关键监测期为 2021 年 11 月 15 日至 2022 年 2 月 15 日，每 7 天对测量点的高程进行一次测量。测量点的垂直位移监测结果见表 2。数值模拟结果与测量数据的比较图见图 11。从表 2 和图 11 可以看出，土石坝位移场的数值模拟结果与测量数据总体上是一致的。主要冻结区位于大坝冰面约 2 米的高度范围内，冻胀量在 20 到 30 厘米之间。

Table 2. Calculated and measured vertical displacement for dam slope measuring points at different times.
表 2. 不同时点大坝边坡监测点的计算和测量垂直位移。

Data 数据	Calculate value at measuring point 2 (cm) 计算测量点 2（厘米）的值	Measured value at measuring point 2 (cm) 测量点 2 的测量值（cm）	Calculate value at measuring point 3 (cm) 计算测量点 3（厘米）的值	Measured value at measuring point 3 (cm) 测量点 3 的测量值（cm）
November 22nd 11 月 22 日	3.6	2.8	3.8	2.6
November 29th 11 月 29 日	6.5	5.3	7.4	6.3
December 6th 12 月 6 日	9.2	8.4	11.3	10.2
December 13th 12 月 13 日	12.3	11.2	14.9	14.1
December 20th 12 月 20 日	16.4	14.8	20.6	19.1
December 27th 12 月 27 日	21.1	20.3	26.1	25.3
January 3rd 一月三日	24.1	23.2	29.5	27.9
January 10th 1 月 10 日	28.2	27.3	33.3	31.2
January 17th 1 月 17 日	30.9	29.1	35.2	33.7
January 24th 1 月 24 日	32.5	31.2	36.1	34.9
February 1st 二月一日	33.2	31.9	36.5	35.5
February 8th 2 月 8 日	33.7	32.4	36.7	35.6
February 15th	33.8	32.5	36.8	35.7

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Fig. 11. Comparative graph of numerical simulation results and measured data.

The main cause of the above phenomenon is that in the main freezing zone, the infiltration of reservoir water into the surface layer of the dam berm results in a higher initial water content and a greater freezing depth in the fill of the shallow layer of the dam slope, and the occurrence of frost heave of the gravel bedding and the shallow layer of the dam body is more. During the temperature decrease stage, the negative temperature is continuously transferred from the concrete slab to the gravel bedding and dam body fill. When the temperature drops below zero, the in-situ water in the shallow layer of the dam slope freezes. Simultaneously, the internal water in the dam body overcomes the resistance and continuously migrates towards the freezing front, intensifying frost heave. This results in frost heaving in the soil-rock mixture of the dam body shallow fill and gravel bedding, causing a gradual increase in the displacement of the dam berm. After the freezing of the reservoir surface, the ice pushing and ice pulling contribute to further increase the displacement of the dam berm, with a maximum value of 36 cm.
以上现象的主要原因是，在主冷冻区，水库水渗透到坝肩表层导致该层的初始含水量增加，填筑物的浅层冻土深度加大，坝体浅层的冻胀现象更为明显。在温度下降阶段，负温持续从混凝土板传递到砾石垫层和坝体填筑物。当温度降至零度以下时，坝肩浅层的原位水开始冻结。同时，坝体内部水分克服阻力，持续向冻结前沿迁移，加剧了冻胀现象。这导致坝体浅层填筑物和砾石垫层的土石混合物发生冻胀，引起坝肩逐渐位移增加。水库表面冻结后，冰推和冰拉作用进一步增加了坝肩的位移，最大值达到 36 厘米。

The main cause of frost heaving damage of the dam berm is the combined effect of frost heaving of the soil-rock mixture of the gravel bedding and dam slope fill, along with the force exerted by the ice pushing. Frost heaving results in the uplift and dislocation of the dam berm, as well as an increase in the porosity of the shallow layer fill of the dam slope and gravel bedding. This results in the loss of protective capacity for the berm. Under the continuous scouring of factors such as wind and waves, it eventually leads to the collapse and sliding of the berm.
大坝坝肩冻胀破坏的主要原因是砂垫层和坝坡填料中土石混合物的冻胀效应，以及冰推力的作用。冻胀导致坝肩的抬升和位移，以及坝坡浅层填料和砂垫层的孔隙度增加，从而降低了坝肩的防护能力。在风浪等持续冲刷因素的作用下，最终导致坝肩的坍塌和滑移。