Introduction

The freezing and thawing of soil have long been highlighted as significant contributors to various types of geo-structural failures [1]. From a Multiphysics perspective, the mechanical behavior of soil is closely linked to the changes in temperature and water/ice content, which are driven by environmental factors such as meteorological, hydraulic, and geothermal dynamics. Extensive research has been conducted to investigate the detrimental environmental effects on geo-infrastructure in cold regions, encompassing slope diseases [2,3,4,5], the loss of bearing capacity [6,7,8], damage to embankments [9,10,11], deterioration of pavements [12,13,14,15,16], and buckling and breakage of pipelines [17,18,19,20].

The response of frozen and unfrozen soil to different environmental factors is complex and highly coupled. Numerous of numerical models have been proposed to investigate this intricate process, generally falling into two categories: thermal-hydro (TH) model and thermal-hydro-mechanical (THM) models [21]. While a majority of the TH [22,23,24,25,26,27,28] and THM [5, 29,30,31,32,33,34,35] have been proven to be effective tools in quantifying the spatial and temporal distribution of temperature and moisture within the soil, few have focused on studying the effects of integrated internal and external environmental loads on geo-structures, like the complex interactions between the atmosphere and frozen soil [3].

Climate factors are widely recognized as significant inducements of geo-structure failures. With the rise of global warming, temperature and precipitation abnormality have led to phenomena such as increased active layer thickness, permafrost degradation, and irregular freeze–thaw cycles, which have been widely observed and reported [36, 37]. Since the end of the last century, plenty of research [8, 13, 14, 38,39,40,41,42,43,44,45,46] has been conducted to assess the impact of climatic variations on geo-infrastructure in cold region. However, most of these studies give more weight to temperature and rainfall, while the other meteorological parameters (e.g., wind, solar, humidity, and evaporation) are not thoroughly considered. Moreover, the interactions between geo-infrastructure and environment are not only associated with a variety of meteorological factors, but also with geothermal and hydrological forces (e.g., drainage, runoff, ground water, vegetation transpiration, ground heat flux, and moisture or heat source). Figure 1 provides a detailed illustration of soil interactions with the environment, showing the exchange of energy and moisture between the soil and environmental factors. The transfer of energy is represented by red arrows, while the transfer of moisture is represented by blue arrows.

Fig. 1
figure 1

The sketch of soil interactions with environment

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As shown in Fig. 1, the transport of water in soil is driven by both exogenous soil-atmosphere interactions and endogenous hydrological forces. The soil-atmosphere exchanges are contributed by distinct components on the soil surface, and the net water flux can be estimated using the equation [21]:

$$NF=P-E-R-{T}_{r}-D$$
(1)

where NF is the net water flux in m/s; P is precipitation in m/s; E is the evaporation in m/s; R is the run-off in m/s; \({T}_{r}\) is the vegetation transpiration in m/s; and D is drainage in m/s. The endogenous hydrological forces are exerted on domains or boundaries inside of soil, mainly related to drainage, ground water table (GWT) change, and other moisture sources (e.g., pipe leakage and water injection). Similarly, heat alternation in soil is primarily induced by exogenous atmospheric and endogenous geothermal factors on surface and non-surface domains or boundaries. The heat flux balance on the soil-atmosphere surface can be expressed as [47]:

$$Q={Q}_{s}-{Q}_{l}+{Q}_{c}-{Q}_{e}$$
(2)

where \(Q\) is the net heat flux in W/m2; \({Q}_{s}\) is the shortwave radiation in W/m2; \({Q}_{l}\) is the outgoing longwave radiation in W/m2; \({Q}_{c}\) is the conductive heat flux in W/m2; \({Q}_{e}\) is the evaporation heat flux in W/m2. The geothermal heat refers to the energy exchanged on the non-surface boundaries or domains of the analyzed soil body. The heat source can originate from chemical reactions, microbial activity, water phase change, adjacent structures (such as thermal piles, wastewater pipes, and electrical cables), and the earth at deep levels or on the sides. Notably, the magnitude of heat from the earth is dependent on the depth of the soil. To be specific, the heat exchange is more active at shallow depths where the thermal gradient is relatively large. Whereas, below a certain depth, the high thermal inertia of the soil often keeps the soil temperature nearly constant, and the heat source from the earth is typically neglected [48].

The current study introduces a comprehensive hydro-thermal model that integrates Fourier's law and a modified form of Richards' equation with formulations of environmental dynamics. This model aims to capture the interactions between the soil and its surroundings, considering meteorological, geothermal, and hydrological factors. By incorporating the energy and mass exchanges between the soil-atmosphere and soil-earth interfaces, the proposed model enables the simulation of geo-infrastructure responses under various environmental effects in cold regions. To solve the highly non-linear partial differential equations of the model, a commercial Finite Element Method (FEM) software, COMSOL Multiphysics 5.5, is employed. In order to validate the proposed model, two case studies on pavement structures are conducted, and the obtained results are compared with field data, which ensures the applicability and reliability of the model predictions.

Theoretical background

Governing equations

The energy conservation in porous media is expressed by a modified Fourier’s equation:

$${\mathrm{C}}_{\mathrm{a}}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}=\nabla \left(\uplambda \nabla \mathrm{T}\right)+{L}_{f}{\rho }_{i}\frac{\partial {\theta }_{i}}{\partial t}$$
(3)

where \({\mathrm{C}}_{\mathrm{a}}\) is the volumetric heat capacity in J/(m3 \(\cdot\) K); T is the temperature in K; t is time; \(\uplambda\) is thermal conductivity in W(m \(\cdot\) K); \({\mathrm{L}}_{\mathrm{f}}\) is latent heat in J/kg; \({\theta }_{i}\) is the volumetric ice content; \({\uprho }_{\mathrm{i}}\) is the density of ice in kg/m3. In Eq. (1), the coupling variables \({\mathrm{C}}_{\mathrm{a}}\) and λ depend upon the proportions of phases (e.g., soil, frozen water, air, and unfrozen water) in the soil. \({\mathrm{C}}_{\mathrm{a}}\) is estimated by:

$${C}_{a}={C}_{s}{\theta }_{s}+{C}_{w}{\theta }_{w}+{C}_{i}{\theta }_{i}+{C}_{v}\left(n-{\theta }_{w}-{\theta }_{i}\right)$$
(4)

where θ represents volumetric content; C represents heat capacity in J/(m3 \(\cdot\) K); subscripts s, w, i, and v represent soil mass, unfrozen water, frozen water (or ice), and air phase separately. The \(\uplambda\) is evaluated by geometric mean method which takes into account the effect of soil particles, liquid water ice, and air on thermal transport [49]:

$$\lambda ={\lambda }_{s}^{{\theta }_{s}}{\lambda }_{w}^{{\theta }_{w}}{\lambda }_{i}^{{\theta }_{i}}{\lambda }_{a}^{{\theta }_{a}}$$
(5)

where \(\lambda\) denotes thermal conductivity and subscripts has same meaning as those of heat capacity in Eq. (4).

The fluid transport in the unsaturated porous media is governed by an extended Richards’ equation:

$$\frac{\partial {\theta }_{w}}{\partial t}+\frac{{\rho }_{i}}{{\rho }_{w}}\frac{\partial {\theta }_{i}}{\partial t}=\nabla ({K}_{Lh}\nabla h+{K}_{Lh}i+{K}_{LT}\nabla T)$$
(6)

where \({\uprho }_{\mathrm{w}}\) represents density of water in kg/m3; \({\mathrm{K}}_{\mathrm{Lh}}\) is the hydraulic conductivity associated with pore pressure head in m/s; \({\mathrm{K}}_{\mathrm{LT}}\) is the hydraulic conductivity caused by temperature gradient in m/s; h is the matric suction in m (water pressure unit); i is the unit vector of the gravity direction.

The Van Genuchten’s equations [50] of soil–water characteristics curve (SWCC) is used to describe the relationship between suction and unfrozen water content for either unfrozen and frozen soil [51]. In addition, the hydraulic conductivity in Eq. (6) is evaluated by Van Genuchten’s equations via [22]:

$${S}_{e}=\frac{\theta -{\theta }_{r}}{{\theta }_{s}-{\theta }_{r}}={\left(1+{\left|\alpha h\right|}^{n}\right)}^{-m}$$
(7)
$${K}_{Lh}={K}_{s}{S}_{e}^{l}[1-{\left(1-{S}_{e}^\frac{1}{m}\right)}^{m}]$$
(8)
$${K}_{LT}={K}_{Lh}(\frac{h{G}_{wT}}{{\gamma }_{0}}\frac{d\gamma }{dT})$$
(9)
$${K}_{fLh}={10}^{-\Omega Q}{K}_{Lh}$$
(10)

where \({\mathrm{S}}_{\mathrm{e}}\) denotes the effective saturation; \({\theta }_{s}\) and \({\theta }_{r}\) are the saturated and residual water content in percentage separately; α, m, n, and l are material-specified constants determined by SWCC curve; \({K}_{s}\) is the saturated hydraulic conductivity in m/s; \(\upgamma\) is the surface tension of soil water in \(g{s}^{2}\), which depends on temperature and is evaluated as \(\upgamma =75.6-0.145\mathrm{T}-2.38\times {10}^{-4}{T}^{2}\)(T is in degC); \({K}_{fLh}\) is the hydraulic conductivity associated with pore pressure head of frozen soil in m/s, which considers the impedance impact of ice on moisture migration; \(\Omega\) is impedance factor related to material type; and Q is the ratio of \({\theta }_{i}\) to \({\theta }_{i}+{\theta }_{s}-{\theta }_{r}\). The volumetric ice content in Eq. (6) is evaluated by empirical equations [52, 53]

$$B\left(T\right)=\frac{{\theta }_{i}}{{\theta }_{w}}=\left\{\begin{array}{c}\frac{{\rho }_{w}}{{\rho }_{i}}\left({\left|\frac{T}{{T}_{f}}\right|}^{b}-1\right) \left(T<{T}_{f}\right) \\ 0 \left(T\ge {T}_{f}\right)\end{array}\right.$$
(11)
$${\theta }_{i}=B\left(T\right)*{\theta }_{w}$$
(12)

where B is the volumetric content ratio of ice and unfrozen water content; T is the temperature in \(K\); \({T}_{f}\) is the freezing point in K; b is the empirical coefficients associated with the soil type (0.56 for clay, 0.47 for silt, and 0.61 for sand and gravel).

Environmental factors

The proposed model integrates various environmental effects through boundary condition equations. For the thermal field, effects of solar short-wave radiation, upward longwave radiation, heat convection, ground heat flux are considered by Neumann boundary conditions. Meanwhile, for the hydraulic field, the model incorporates the influence of precipitation, groundwater elevation change, and drainage using either Neumann or Dirichlet boundary conditions.

In the thermal field, the absorption of short waver radiation on the soil surface can be described by [54]:

$${q}_{s}=\left(1-albedo\right)*S$$
(13)

where \({q}_{s}\) is the short-wave absorption of the solar radiation; the albedo is the solar reflectivity; S is the solar radiation in (W/m2). The total long wave radiation (including outgoing radiation and counter-radiation) on the structure surface follows the Stefan-Boltzmann law and is described as [55]:

$${q}_{l}=\epsilon \sigma {T}_{s}^{4}-{\epsilon }_{a}\sigma {T}_{sky}^{4}$$
(14)
$${T}_{sky}={\left(0.754+0.0044{T}_{dp}\right)}^{0.25}*{T}_{amb}$$
(15)
$${T}_{dp}={T}_{amb}-\frac{100-RH}{5}$$
(16)

where total \({q}_{l}\) is the long-wave radiation; \(\epsilon\) is the emission coefficient; \({\epsilon }_{a}\) is the absorption coefficient of pavement; here the assumption is \(\epsilon ={\epsilon }_{a}\) to simplify the analysis; \(\sigma\) is the Stefan–Boltzman constant equals to \(5.68*{10}^{-8} W*{m}^{-2}*{K}^{-4}\); \({T}_{s}\) is the pavement surface temperature in K; \({T}_{sky}\) is the effective ambient temperature above the structure in K; \({T}_{dp}\) is the dewpoint temperature in \(K\) which is the temperature needed to cool and make the air saturated; RH is relative humidity in percentage. \({T}_{amb}\) is the ambient air temperature in K. In addition to by Eq. (15), the effect of ambient air temperature is also reflected by convective heat flux on structure surface through Newton’s law of cooling [56]:

$$n\cdot \left(\lambda \nabla T\right)={h}_{c}({T}_{amb}-T)$$
(17)
$${h}_{c}=\left\{\begin{array}{c}5.6+4*{v}_{wind}\;for\;{v}_{wind}\le 5m/s\\ 7.2+4*{v}_{wind}^{0.78}\;for\;{v}_{wind}>5m/s\end{array}\right.$$
(18)

where n is the normal unit vector of the boundary \(; {\lambda }_{c}\) is the thermal conductivity in W/(m \(\cdot\) K), T is the temperature at boundaries in K; and \({\mathrm{h}}_{\mathrm{c}}\) is the convection heat transfer parameter in W/(m2 \(\cdot\) K). As shown in Eq. (17) and 18, not only the ambient air temperature but also the wind speed together with soil surface temperature worked concurrently to determine the convective heat exchange. The geothermal effect by surrounding earth is calculated by:

$${q}_{g}={\lambda }_{botom}\nabla {T}_{bottom}$$
(19)

where \({\lambda }_{botom}\) and \(\nabla {T}_{bottom}\) are the thermal conductivity in W/(m \(\cdot\) K) and temperature gradient in K/m of the material on the non-surface boundaries of the analyzed soil domain. The heat source other than geothermal earth heat is not incorporated into the current model.

In the hydraulic field, the extended Horton empirical equation [57] is used to evaluate the infiltration rate by:

$$I=\left\{\begin{array}{c}RI\;for\;RI<RC\\ RC\;for\;RI>RC\end{array}\right.$$
(20)
$$RC={i}_{f}+({i}_{0}-{i}_{f})\mathrm{exp}[-\left(\frac{RI}{{i}_{f}}\right){\left(\frac{{K}_{s}}{{i}_{f}}\right)}^{0.5}t ]$$
(21)

where I is infiltration rate in mm/s; the RI is the rainfall intensity in mm/s; RC is the infiltration capacity in mm/s; \({i}_{0}\) is initial infiltration capacity;\({i}_{f}\) is the final infiltration capacity; \({K}_{s}\) is the saturated hydraulic conductivity in m/s. \({i}_{0}\) can be associated with in-situ suction and \({i}_{f}\) are usually assumed to be equal or smaller than the \({K}_{s}\). The GWT is normally located at the interface where the positive and negative pore water pressure is separated with zero water pressure on. The position of GWT can indicate water pressure along vertical direction of the soil. When the GWT is higher than the bottom boundary, the positive saturated pore water pressure is exerted on it. Whereas, when the GWT is lower than the bottom boundary, the negative suction pressure under static condition (moisture movement balance between gravity and suction gradient) is applied on it. In the proposed model, either Dirichlet boundary condition or Neumann boundary condition can be added on the lower boundary condition to reflect the groundwater effect.

Verication by case study

The coupled model with integrated environmental force expressions shows considerable nonlinearity, posing a challenge for solution. To address this, the software COMSOL Multiphysics 5.5 with an advanced solver is employed in this study to numerically solve the model equations. Two separate scenarios of pavement in cold regions are analyzed. The first case is an hourly-based 1-D analysis spanning one month during a non-freezing season, which aims to validate the performance of the model under non-freezing conditions and demonstrates its sensitivity to environmental inputs of relatively small time scale. The second case entails a 2-D analysis conducted on a daily basis, extending over more than half a year. The aim is to test the applicability of the model in simulating the response of geo-infrastructures to environmental loads of cold regions. The parameters utilized in the simulation are found from literature [21, 58] or through calibration, as detailed in Tables 1, 2 and 3. Since the pavement surface material is asphalt concrete, a material with very low permeability, only thermal field calculations are conducted for this layer, while moisture transfer is not considered during the simulation. However, for the layers beneath the asphalt (base and subgrade layer), both thermal and hydraulic processes are evaluated.

Table 1 The common simulation parameters for the two cases
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Table 2 The simulation used parameters for Case 1
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Table 3 The parameters used for simulation Case 2
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The COMSOL software incorporates robust and versatile solvers capable of handling strong nonlinearities inherent in governing equations. Given this capability, COMSOL was chosen to address the model under study, characterized by significant nonlinearity. While COMSOL provides a fixed time step option, it also offers an adaptive time-stepping mechanism. This feature dynamically adjusts the time step size based on a predefined tolerance, enhancing stability, accuracy, computational efficiency, and convergence, especially in complex nonlinear simulations. To enhance model convergence, the adaptive time step was employed for both cases. Additionally, specific solver settings within COMSOL were adjusted. The absolute tolerance was set using the global method and scaled to a factor of 1e-5. Within the fully coupled solver, the non-linear approach was designated as “Automatic (Newton),” with maximum number of more than four iterations. Mesh sizes for the two cases were determined by mesh refinement studies. The built-in mesh setting labeled “Extra fine (pre-defined)” was found to strike a balance between computational efficiency and error stabilization, and was thus applied to both cases.

Case 1: hourly -based non-freezing season simulation

Case 1 analysis is conducted using data from the LTPP Sect. 50–1002. This site is situated in Vermont (44°07′10.6"N 73°10′45.8"W). The pavement layer information is summarized in Table 4. Parameters utilized for each layer in the simulation are presented in Table 2.

Table 4 The pavement layer information for Case 1
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At this site, hourly-monitored meteorological, subsurface temperature, and subsurface moisture content data are collected from the LTPP database. The data are employed for the 1-D case analysis with an hourly time scale. The simulation starts from 05/02/2001 6:00am and ends at 05/31/2001 6:00 am. Initial conditions for the simulation are based on MRC Thermistor and TDR measurements of temperature and moisture content from the simulation start date. The effects of environmental factors, such as air temperature, precipitation, solar short-wave radiation, wind speed, relative humidity, geothermal temperature gradient, and ground water table elevation, are considered through corresponding boundary conditions. Figure 2 shows the temporal fluctuations in air temperature, precipitation, solar short-wave radiation, wind speed, relative humidity, and geothermal temperature gradient over the simulation period. On the lower point boundary of the model at 2m depth, as shown in Fig. 2 (c), the temperature gradient is calculated based on the site-measured temperature above and below this depth and is then used to estimate the time-series geothermal flux following Eq. (19). To simulate the soil-atmosphere energy exchange process, the heat flux induced by solar short-wave radiation, long wave radiation, and air convection are applied to the top point boundaries of the 1-D model. Geothermal heat flux is added to the bottom point boundary to simulate geothermal energy migration. The GWT is around 1.1m below the pavement surface and remained constant during the simulation. The precipitation-induced water flux and ground water-related Dirichlet boundary (constant positive water pressure) are added to the top and bottom point boundaries, respectively, to simulate water transfer driven by environmental forces.

Fig. 2
figure 2

Environmental factors variation with time from 05/02/2001 to 05/31/2001: a temperature and rainfall; b Solar shortwave radiation and wind speed; c relative humidity and geothermal related temperature gradient

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It’s interesting to note that the meteorological data are likely correlated with each other. For example, the amplitude of temperature fluctuation decreases when precipitation is present, as illustrated in Fig. 2 (a). Furthermore, on rainy days, the wind speed tends to have low value, as shown in Fig. 2 (b), and the daily relative humidity fluctuation range also exhibits a reduction. Shortwave radiation, ambient temperature, and relative humidity display daily periodic variation trends. The peaks of radiation, temperature, and relative humidity usually occurred in noon, afternoon, and after midnight, respectively, with corresponding troughs transpiring approximately 12 h later. The increase or decrease in radiation crest usually follows a consistent, albeit time-lagged, increase or decrease of the temperature crest. After accounting for the time lags, a positive correlation between radiation and temperature becomes evident. These observations implies that there is greater energy influx into the soil from atmosphere during daytime compared to nighttime.

The model estimated results are presented and compared with the site measured data in Figs. 3 and 4. Overall, the simulated temperature aligns well with the site data. In Fig. 3, the temperature fluctuated periodically in both asphalt concrete layer (near surface) and the base layer (in the middle of the layer). It is believed that these periodic temperature trends stem from the cyclic heat exchange of radiation (both shortwave ingoing and longwave outgoing) and convection. The timing of the simulated temperature peak and troughs matches perfectly with the site data in in Fig. 3, indicating the sufficient model sensitivity and adaptability to respond to rapidly changing energy dynamics. The LTPP database does not provide hourly-monitored temperatures below 0.8m but does offer daily average temperature profiles at greater depths. To assess the performance of the model in deeper soil, the simulated temperature profile on day 15(a mid-point of the simulation) and day 28 (near the end of the simulation) are plotted and compared with the LTPP site data in Fig. 4. As shown in Fig. 4, the temperature profile predictions generally align with those of the actual site, demonstrating the capability of the model to predict temperatures at various depths after considering complex environmental inputs.

Fig. 3
figure 3

The comparison between simulation and site measurement: a temperature at 0.03 m depth in asphalt layer; b temperature at 0.68 m depth in base layer

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Fig. 4
figure 4

The simulated temperature profiles on day 15 (a middle time of the simulation) and on day 28 (a nearly end time of the simulation) vs. the site monitored temperature profiles

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The hydraulic field comparison results are displayed in Fig. 5. Coherent water content variation trends are observed between the predictions and site measurements for the two examined depths. As shown in Fig. 5, the moisture content remains nearly constant before day 10. The surge of water content after day 10 is very likely induced by rainfall as shown in Fig. 2(a).The water drains following the day 10 surge, resulting in declining water content before day 20. After the day 20, the water content fluctuated due to a series of precipitation in the following days. The timing of the simulated moisture content peaks basically aligns with the site data, demonstrating the model's ability to effectively capture the water flux disturbances caused by rainfall. Due to the lack of detailed soil data, the simulation assumed homogeneous base layer with consistent parameters. The slightly underestimated water content in Fig. 5 (a) after day 10 may be attributed to uneven hydraulic properties at the actual site. Another possible reason for the underestimated water content after day 10 could be that the model does not incorporate vapor condensation and evaporation processes.

Fig. 5
figure 5

The comparison between simulation and site measurement: a volumetric water content at 0.37 m depth in shallow base layer; b volumetric water content at 0.66 m depth in deeper base layer

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Case 2: daily based seasonal simulation

The analysis for Case 2 employed data from the LTPP Sect. 46–0804 in South Dakota (45°55′40.7"N 100°24′31.7"W). In Case 2, daily monitored data are used for a 2-D and daily-time-scale analysis. The pavement geometry is shown in Fig. 6 and consists of a three-layer system with a 1:3 slope ratio. The detailed layer information is summered in Table 5. It is assumed the pavement system is symmetric, and only the right-hand side of the pavement is modeled.

Fig. 6
figure 6

The geometry of the three-layer pavement system for case 2

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Table 5 The pavement layer information for case 2
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An analysis spanning 194 days is conducted, starting from 09/01/2020 and ending on 03/15/ 2021. Initial conditions for the simulation are determined using average daily temperature and moisture content data measured by MRC Thermistor and TDR on the simulation start date. Similar as Case 1, boundary conditions for distinct environmental factors are incorporated in Case 2, from which the effects of air temperature, precipitation, solar short-wave radiation, wind speed, relative humidity, geothermal temperature gradient, and groundwater table elevation are evaluated. Figure 7 displays the variation of these environmental factors over time. To simulate the energy exchange between pavement surface and atmosphere, flux by short-wave radiation, long wave radiation, air convection, and precipitation, are applied to the upper three boundaries (boundary 1 to 3 as shown in Fig. 6). In Fig. 7 (c), the temperature gradient on lower boundary is evaluated using the site-measured temperature near the 2m depth, by which the geothermal flux variation with time is calculated. Geothermal effects are then simulated by setting Neumann boundary with this flux on boundary 7. Given the relatively low rainfall intensity and relatively high soil saturated permeability, it is assumed the rainfall intensity is always lower than the infiltration capacity, so the precipitation induced water flux equals to the rainfall intensity. The precipitation boundary is applied on boundary 5, 6 and 3 in Fig. 6. The GWT data for the site are unavailable, so the GWT is assumed to be located 1.25m below the pavement surface at the beginning of the simulation. The GWT level changes to 2m on day 75 and to 2.5m on day 200, and the effect of GWT is simulated using a Dirichlet boundary with linearly changing pressure head applied to boundary 7. The right boundary 4 in Fig. 6 is assumed to be thermal insulated and hydraulic insulated.

Fig. 7
figure 7

Environmental factor variations with time from 09–01-2000 to 03–15-2021: a temperature and rainfall; b solar shortwave radiation and wind speed; c relative humidity and geothermal related temperature gradient

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The water content, temperature, and frost depth variation with depth together with time are calculated by the coupling model. The simulation results are compared with site-monitored daily ground temperature and daily moisture of the Sect. 46–0804. Figure 8 compares the measured and simulated temperature variations at different depth (with burried MRC Thermistor) along the pavement axis of symmetry (The left side boundary in Fig. 6). The selected MRC Thermistor burry depth are 0.08 m in asphalt layer, 0.49 m in base layer, and 1.1 m in subgrade layer below pavement surface. Similarly, the measured and simulated unfrozen moisture content variations at different TDR depth (0.33 m in base layer, and 1.1 m in subgrade layer) are compare in Fig. 9. The comparison between simulated and field measured frost depth with time is shown in Fig. 10. As seen from Figs. 8, 9 and 10, the simulation results well match the site measured values at the selected depth, temporally and spatially. This validates the performance of the model to capture the thermal as well as hydraulic filed response of soil to environmental factors during the non-freezing state, freezing state, and the transition process from non-freezing to freezing state.

Fig. 8
figure 8

Simulated and measured temperatures versus time: a at 0.08 m in asphalt layer; b at 0.49 m in base layer; c at 1.1 m in subgrade layer (along pavement axis of symmetry)

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Fig. 9
figure 9

Simulated and measured unfrozen volumetric moisture content versus time at 0.33 m in base layer and 1.1 m in subgrade layer (along pavement axis of symmetry)

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Fig. 10
figure 10

Frost depth and volumetric ice content (along pavement axis of symmetry) versus time

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According to the observations in Fig. 9, both the simulated and site-monitored water content at a depth of 0.33 m exhibited fluctuations prior to day 70, which are likely induced by precipitation, as all water content crests occurred sooner after rainfall (see Fig. 7). After day 70, the water content sharply declined. Since TDR can only measure unfrozen water content, not the total water content, such abrupt decrement is explained as the conversion of unfrozen water to ice under low temperature. Conversely, at a depth of 1.1 m, the water content slightly decreased from day 1 until approximately day 137, which is likely the results of the dropped GWT. After day 137, the water content exhibited a more pronounced decrease, which is also illustrated as the results of soil freezing.

Figure 10 depicts the temporal variations in volumetric ice content (VIC) at depths of 0.33 and 1.1 m, along with a comparison between the site-measured frost depth (FD) and the simulated FD. The comparison reveals a generally consistent but a slightly underestimated FD. Moreover, a comparison of the results presented in Figs. 9 and 10 indicates that the timing of the sudden decrease in unfrozen water content at depths of 0.33 and 1.1 m coincides with the timing of FD penetrating to these depths and the initiation of the increase in VIC at these depths. These observations proves that the primary cause of the noticeable decline in unfrozen water content, as presented in Fig. 9, is the freezing of the soil. The results presented in Fig. 10 effectively showcase the ability of the model to estimate the timing of ice formation and evaluate the temporal evolution of FD under the environmental dynamics of cold region.

Conclusions

In this study, a comprehensive coupling model was developed to assess the response of geo-infrastructure to different environmental dynamics. Two case analyses were conducted to evaluate the model's performance under different conditions, including geometry dimensions (1D and 2D), input time scale (hourly and daily), simulation durations (1 month and more than half a year), and seasons (non-freezing and transition from non-freezing to freezing). In Case 1, the model demonstrated its commendable sensitivity and adaptability in handling rapidly changing heat inputs. The predicted temperatures at different depths aligned well with the observed data, and the variation trend of water content was generally captured. In Case 2, the model performance was evaluated during the non-freezing, freezing, and transition stages. The model effectively simulated the thermal and hydraulic reactions of the pavement structure to environmental factors under different stages. The two case analyses highlight the potential of the model to provide valuable insights into evaluating the effects of environmental factors on geo-infrastructures in cold regions. The model shows promise in providing recommendations for design adaptations in response to climate change, predicting environmental geohazards, and addressing engineering concerns with geo-infrastructure in cold regions.