Frost heave and freezing processes of saturated rock with an open crack under different freezing conditions

Abstract

Frost heave experiments on saturated sandstone and tuff with an open crack are conducted under uniform and unidirectional freezing conditions. Frost heave of crack in sandstone with high permeability is more significant under uniform freezing condition than that under unidirectional freezing condition. However, frost heave of crack in tuff with low permeability is more significant under unidirectional freezing condition. To illustrate the reasons for this phenomenon, a numerical model on the freezing processes of saturated rock with an open crack considering the latent heat of pore water and water in crack is proposed and confirmed to be reliable. Numerical results show that a frozen shell that blocks the migration of water in crack to rock develops first in the outer part of the rock before the freezing of water in crack under uniform freezing condition. However, the migration path of water in crack to the unfrozen rock under freezing front exists under unidirectional freezing condition. The freezing process and permeability of rock together determine the migration of water in crack and lead to the different frost heave modes of crack for various permeable rocks under different freezing conditions. The frost heave modes of crack in rock with low or high permeability are similar under uniform freezing condition because water migration is blocked by a frozen shell and is irrelevant to rock permeability. For high permeability rock, the frost heave of crack will be weakened due to water migration under unidirectional freezing condition; however, the frost heave of crack would be more significant for low permeability rock because water migration is blocked under unidirectional freezing condition. Therefore, the freezing condition and rock permeability determine the frost heave of rock with crack together, and this should be concerned in cold regions engineering applications.

Keywords

frost heave / rock with crack / freezing process / freezing condition / frost heave mode

Introduction

The frost heave of rock mass is a common problem that exists in geotechnical engineering applications, such as tunnels [¹–³] or slopes [⁴] in cold regions and the storage of liquefied natural gas in underground rock chambers [⁵]. For example, the frost heave of the rock mass is considered a cause for the landslide disaster at the Toyohama Tunnel in Japan [⁶].

Numerous studies on the frost heave of rock have been conducted. If a saturated rock with a porosity n freezes in a closed system without water supply, then the potential volumetric freezing strain is 0.09n given that pore water is completely frozen and the elastic modulus of rock is zero [⁷]. However, the existing unfrozen pore water, the redistribution of pore water during the freezing process, and the constraint of rock skeleton to the expansion of pore ice lead to the phenomenon that the substantial frost heaving strain is much smaller. For example, Huang et al. [⁸] collected the experimental results of several researchers and proposed that the volumetric freezing strain of saturated rock is approximately 2.17%n. Huang et al. [⁹] proposed a model based on saturated poroelasticity theory to estimate unfrozen water content by measuring the freezing strain of saturated rock. Lv et al. [¹⁰] proposed a coupled thermal–mechanical model of frost heave for saturated rock in a closed system, in which a constraint coefficient is used to consider the susceptibility of the internal rock grain structure to the expansion of pore ice.

Akagawa and Fukuda [¹¹] conducted frost heave experiments on welded tuff and verified that segregation freezing occurs in the rock under temperature gradient with sufficient water supply. Moreover, the tensile strength of rock influences the initiation temperature of ice lens [¹²,¹³]. Besides, Laura [¹⁴] performed freezing experiments on saturated sandstone cores and suggested that ice segregation also occurs under isothermal conditions. Neaupane et al. [¹⁵] established a THM model to estimate the deformation behavior of the freezing and thawing processes of rock considering the phase change (PC) of pore water. However, the model is only applicable to elastic limit. Later, Duca et al. [¹⁶] developed a THM model that incorporates the thermodynamics of ice-water mixtures to reproduce ice segregation within the rock under temperature gradient and water supply conditions.

However, joints and cracks are ubiquitous in the rock mass [¹⁷–²⁰], and much attentions were paid on the crack propagation [²¹–²⁵] to reveal the significant influence of the crack on the mechanical properties of rock mass. Hence, the influence of joints and cracks on the frost heave of the rock mass should be considered in engineering applications. Walder and Hallet [²⁶] presented a model to predict crack–growth rates based on fracture mechanics and ice segregation theory, and the model delineates the role of material parameters (elastic modulus and crack size), environmental conditions (temperature and temperature gradient), and time in frost damage to the rock. Tharp [²⁷] estimated the geometric conditions of rectangular crack and tapered crack in rocks for crack propagation. Huang et al. [²⁸] developed a model for predicting frost heaving pressure in the elliptical fractures of rock and confirmed that the flux of water, aspect ratios of fracture, and elastic modulus of rock significantly influence the frost heaving pressure.

In experiments, Davidson and Nye [²⁹] measured the ice pressure induced by the expansion of ice in a slot of perspex under unidirectional freezing condition; however, perspex is non-permeable, which may induce different results from real rocks. Matsuoka [³⁰] measured the opening of the artificial crack in granite blocks during the freezing process of water in crack under unidirectional freezing condition. Arosio et al. [³¹] measured the ice pressure on artificial crack wall in limestone under uniform freezing condition. Bost and Pouya [³²] conducted freeze–thaw experiments on three types of notched limestone specimens under uniform freezing condition and established a model of stress evolution along the depth of an open crack. Previous frost heaving experiments of rock with crack were conducted under uniform or unidirectional freezing condition. However, minimal attention is paid on the issue that the different freezing conditions may induce significant differences in the freezing process and frost heave, since freezing direction plays a key role in determining the frost heave of a crack [³³]. Moreover, the abovementioned experiments were performed on low permeability rocks, whereas the permeability of rocks may influence the water migration during the freezing process. Thus, the influence of the permeability of rocks on the frost heave of crack should be concerned.

Therefore, in this study, frost heave experiments on low and high permeability rocks with an open crack are conducted under uniform and unidirectional freezing conditions. Then, a numerical model on the freezing processes of saturated rock with an open crack considering the latent heat of pore water and water in crack is proposed. Comparisons between the numerical and experimental results demonstrate the reliability of the model. Finally, the influence of the freezing process and the permeability of rock on the frost heave modes of crack under uniform and unidirectional freezing conditions are analyzed to assist engineering applications.

Experimental materials and methods

Experiments are conducted on sandstone and tuff specimens, and the physical and mechanical parameters of sandstone and tuff are listed in Table 1. The tuff is dense with very low porosity and permeability. However, the porosity of the sandstone is large, and its permeability is far higher than that of the tuff. The specimens are cut into cuboid with the dimension of 100 mm × 80 mm × 80 mm (length × width × height). A 40 mm deep, 80 mm long, and 4 mm wide crack is formed in the middle of the specimen by using a saw and a file (Fig. 1). Both ends of the crack are sealed with silicone rubber to prevent the escape of water from the crack. The flexibility of the silicone rubber allows the deformation of the crack without constraint from both ends. Moreover, coat the surfaces of the specimens with epoxy resin to prevent the water loss during the experiments.

Two pairs of waterproof strain gauges are attached across the crack on the front and back faces of a specimen to measure the freezing strain of the crack (Fig. 1). In each pair, the two strain gauges are 10 and 30 mm away from the top surface, respectively. Moreover, compensating strain gauges are attached to the intact block at the same horizontal line as active strain gauges, as shown in Fig. 1, and the deformation of the rock is eliminated by incorporating the active and compensating strain gauges into Wheatstone bridges. Thus, strains only due to the deformation of the crack are measured. Two HK-PT1000 thermistor temperature sensors are set in the crack with 20 mm (Point A) and 36 mm (Point B) away from the top surface, as shown in Fig. 1, to measure temperature variations during the experiments. Point A is 20 mm away from the front surface, and Point B is 20 mm away from the back surface. Data of strains and temperature are collected every 2 min by a data logger.

The specimens are saturated through the evacuation method before experiments. The crack is filled with water, and the water level is approximately 2 mm below the top surface because a higher water level may lead to the overflow of water during PC. The specimens are covered with polyvinyl chloride membrane to prevent water evaporation during experiments.

The uniform freezing experiments are performed in a low temperature test chamber, and the temperature maintains constant at –10°C during the experiments. The unidirectional freezing experiments are carried out in a freezing test chamber with top and bottom plates, as shown in Fig. 2. Controlled liquids circulate through the cold and warm plates separately to maintain the temperature of the plates at –10°C and+ 1°C, respectively. Moreover, the side surfaces of the specimens are thermally insulated by covering with a polyurethane layer. Thus, the specimens freeze from top to bottom surface. Moreover, the specimens freeze and expand freely without lateral constraint during the experiments.

Experimental results of frost heave and freezing processes

Frost heave of the open crack

The frost heaving strains of the crack in saturated sandstone and tuff specimens are shown in Figs. 3 and 4, correspondingly. The temperature of Point A is also shown in Figs. 3 and 4. {\epsilon }_{\mathrm{1}} and {\epsilon }_{\mathrm{2}} represent the strains measured by the strain gauges placed at 10 and 30 mm deep in the crack, respectively; 1# and 2# correspond to strain gauge pairs 1# and 2# in Fig. 1, respectively.

Figure 3(a) demonstrates four stages in the variation curves of {\epsilon }_{\mathrm{1}} and {\epsilon }_{\mathrm{2}} under uniform freezing condition for saturated sandstone with an open crack. In the first stage, the strains vary in a small range because temperature variations at the locations of active strain gauges slightly lag behind those at the locations of compensating strain gauges. In the second stage, the observed strains become negative, which do not indicate that the crack contracts. In fact, experiments conducted on the sandstone with no crack demonstrate that frost heave will occur when it is frozen. Thus, the frost heave of saturated sandstone occurs first when the temperature at locations of compensating strain gauges drops to below 0°C; however, at the time, temperature at locations around the crack is higher, and no or less frost heave occurs. Therefore, the strains of active gauges are less than the strains of compensating gauges; thus, the observed strains become negative. In the third stage, the observed strains increase and become positive drastically with the temperature of water in crack dropping to below 0°C, which denotes that the frost heaving strain of crack is significantly greater than that of sandstone. Finally, the strains become stable when the temperature of Point A drops to below –2°C. Moreover, {\epsilon }_{\mathrm{1}} is much greater than {\epsilon }_{\mathrm{2}}.

Compared Fig. 3(b) with Fig. 3(a), variations of {\epsilon }_{\mathrm{1}} and {\epsilon }_{\mathrm{2}} under unidirectional freezing condition are significantly different from those under uniform freezing condition. As the fractured rock freezes, {\epsilon }_{\mathrm{1}} and {\epsilon }_{\mathrm{2}} decrease to negative. This indicates that the strain at the location of active gauge is less than that at location of compensating gauges. Hence, the frost heave of crack is less than that of saturated sandstone under unidirectional freezing condition.

As shown in Fig. 4, the curves of {\epsilon }_{\mathrm{1}} and {\epsilon }_{\mathrm{2}} under uniform and unidirectional freezing conditions for the saturated tuff with an open crack are similar in the variation process, and three stages are observed. In the first stage, the strains vary in a small range because the temperature variations at the locations of the active strain gauges lag behind those at the locations of the compensating strain gauges. In the second stage, the observed strains increase and become positive drastically with the temperature of water in crack dropping to below 0°C. This phenomenon is different from that of sandstone. The frost heave experiments conducted on the saturated tuff with no crack demonstrate that no frost heave occurs and only thermal contraction displays when it is frozen due to its low porosity and large elastic modulus. Hence, only the frost heave of the crack occurs in the saturated tuff with an open crack.

Compared Fig. 3 with Fig. 4, the frost heaving strain of crack is significantly greater than that of saturated sandstone under uniform freezing condition, while the frost heave of crack is less than that of saturated sandstone under unidirectional freezing condition. However, the frost heaving strain of crack under unidirectional freezing condition is greater than that under uniform freezing condition for saturated tuff with an open crack. Significant differences in frost heaving characteristics exist between saturated sandstone and tuff with an open crack under uniform and unidirectional freezing conditions. To illustrate the reasons that induce the differences, the freezing processes of saturated sandstone and tuff with an open crack are investigated in the following sections.

Freezing process of water in crack

The measured temperature variations of saturated sandstone and tuff with an open crack under uniform and unidirectional freezing directions are shown in Figs. 5 and 6, respectively. Four stages are observed in the freezing processes of water in crack under the two conditions: linear cooling down (LCD) stage, PC stage, gradual cooling down (GCD) stage, and steady stage.

The temperature variation processes of Points A and B are completely synchronized and the temperature of Points A and B are equal for sandstone and tuff under uniform freezing condition, as shown in Figs. 5(a) and 6(a). Taking temperature variation of sandstone as an example, the LCD stage sustains approximately 80 min, and the temperature decreases rapidly in this stage. The PC stage sustains approximately 130 min, and the temperature of water in crack remains nearly constant due to the release of latent heat. In the GCD stage, the temperature of water and ice in crack starts to decrease again because the released latent heat is insufficient to balance the heat loss of specimen due to heat transfer. The temperature decreases much slower in the GCD stage than that in the LCD stage because the difference between the temperature of specimen surfaces and the ambient temperature is smaller, thereby reducing the convection heat transfer rate. Moreover, the PC stage sustains much longer in sandstone than that in tuff because the pore water in saturated sandstone releases considerable latent heat, whereas the porosity of the tuff is only 0.61%, and little latent heat is released by pore water.

The variation processes are unsynchronized under unidirectional freezing condition, and the variation process of Point B is always lagging behind, which is different from the uniform freezing experiments, as shown in Figs. 5(b) and 6(b), although the temperature variation processes of Points A and B are similar, Moreover, the PC stage starts earlier and sustains a shorter time in Point A than that in Point B for sandstone and tuff.

Figures 5 and 6 present that significant differences exist in the freezing processes of water in crack under uniform and unidirectional freezing conditions. First, the PC stage sustains a much longer time under uniform freezing condition than under unidirectional freezing condition. More importantly, Point A in the middle of the crack freezes simultaneously with Point B at the bottom of the crack under uniform freezing condition. However, Point A freezes earlier than point B under unidirectional freezing condition. The difference in the freezing sequence of water in crack under uniform and unidirectional freezing conditions may be one of the reasons that induces the different frost heave characteristics of crack in rock. However, only the freezing processes of several points can be monitored and demonstrating the freezing process of the whole rock specimen through experiments is difficult due to the limitations of the measuring point arrangement. Therefore, numerical simulations on the freezing processes of saturated rock with an open crack are conducted in the following section to demonstrate the freezing processes of saturated rock and water in crack.

Numerical model of the freezing process for rocks with an open crack

To demonstrate the freezing processes of saturated rock and water in crack and further illustrate the development of the different frost heave characteristics under different freezing conditions, numerical simulations on the freezing processes of saturated rock with an open crack are conducted.

Governing equations of the freezing process

To simplify the governing equations of the freezing process for rocks with an open crack, the following assumptions are introduced.

1) The thermal conductivity of rock block, pore water, and water in crack is isotropic.

2) The heat transfer between saturated rock block and water in crack occurs by conduction only, and the convection is disregarded.

The freezing process of rock with an open crack consists of two parts, namely, the freezing of saturated rock block and the freezing of water in crack. The heat transfer governing equation of saturated rock block considering the latent heat of pore water can be expressed as [¹⁰]:

(1)C_{\mathrm{eq}}{\displaystyle \frac{\partial T}{\partial t}}-\nabla ({\lambda }_{\mathrm{ef}}\nabla T)=\mathrm{0},

where T and t are the temperature and time, respectively; C_{\mathrm{eq}} is the equivalent volume thermal capacity of saturated rock considering the latent heat of pore water; and {\lambda }_{\mathrm{ef}} is the effective thermal conductivity for mixtures of rock grains, unfrozen water, and ice.

C_{\mathrm{eq}} can be defined as:

(2)C_{\mathrm{eq}}=\{\begin{array}{l}C+L{\rho }_{\mathrm{w}}{\displaystyle \frac{\partial {\theta }_{\mathrm{w}}}{\partial T}}T\le T_{\mathrm{0}}\\ CT>T_{\mathrm{0}}\end{array},

where L is the latent heat of water per unit mass, and the value is 334 kJ/kg; {\rho }_{\mathrm{w}} is the density of water; {\theta }_{\mathrm{w}} is the volumetric unfrozen water content; and T_{\mathrm{0}} is the freezing point of pore water and is considered to be 0°C in the present study. Moreover, C is the volumetric thermal capacity of rock composite and takes the following form [³⁴]:

(3)C=(\mathrm{1}-n)C_{\mathrm{s}}+{\theta }_{\mathrm{w}}{C}_{\mathrm{w}}+{\theta }_{\mathrm{i}}{C}_{\mathrm{i}},

where C_{\mathrm{s}}, C_{\mathrm{w}}, and C_{\mathrm{i}} correspond to the volumetric thermal capacities of rock grains, water, and ice; n is the porosity of rock; and {\theta }_{\mathrm{i}} is the volumetric ice content and can be calculated by {\theta }_{\mathrm{i}}=n-{\theta }_{\mathrm{w}}.

A logarithmic law is adopted to calculate the effective thermal conductivity [³⁴]:

(4){\lambda }_{\mathrm{ef}}={\lambda }_{\mathrm{s}}^{\mathrm{1}-n}{\lambda }_{\mathrm{w}}^{{\theta }_w}{\lambda }_{\mathrm{i}}^{{\theta }_i},

where {\lambda }_{\mathrm{s}}, {\lambda }_{\mathrm{w}}, and {\lambda }_{\mathrm{i}} are the thermal conductivity of rock grains, water, and ice, respectively.

{\theta }_{\mathrm{w}} can be defined as:

(5){\theta }_{\mathrm{w}}={\displaystyle \frac{(\mathrm{1}-n){\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}}\omega ,

where {\rho }_{\mathrm{s}} is the density of rock grains, and \omega is the unfrozen water content.

Equation (6) is employed to describe the presence of unfrozen water content \omega in the freezing rock [³⁵]:

(6)\omega ={\omega }^{\ast }+({\omega }_{\mathrm{0}}-{\omega }^{\ast })e^{a(T-T_{\mathrm{0}})},

where {\omega }^{\ast } is the residual unfrozen water content at a certain low reference temperature, {\omega }_{\mathrm{0}} is the minimum unfrozen water content at freezing point T_{\mathrm{0}}, and a is a material parameter that describes the decay rate of unfrozen water.

Furthermore, the heat transfer governing equation of water in crack considering the latent heat of water during PC can also be expressed as Eq. (1). But, here for water in crack, take the value of n = 1 when calculating C_{\mathrm{eq}} and {\lambda }_{\mathrm{ef}} with Eqs. (2)-(4). Assuming that the PC temperature range of water in crack is 0°C to –2°C and unfrozen water content varies linearly with temperature, the volumetric unfrozen water content {\theta }_{\mathrm{w}} of water in crack can be expressed as:

(7)

Numerical model and boundary conditions

The governing equations of the freezing process are solved using COMSOL Multiphysics. The numerical results show that the freezing process of rock and water in crack are similar for saturated sandstone and tuff under the same freezing condition. Hence, the numerical models and results of sandstone are demonstrated in the present study as an example to avoid the unnecessary repetition of similar content.

The 3D numerical model under uniform freezing condition is shown in Fig. 7(a). The water in crack builds as solid elements, and the water level is approximately 2 mm below the top surface. The boundary conditions of the external surfaces of the model are convective heat transfer boundaries, and can be expressed as:

(8)-n\cdot (-{\lambda }_{\mathrm{ef}}\nabla T)=h(T_{\mathrm{ext}}-T),

where n is the outward unit normal vector on the boundaries, T_{ext} is the temperature of the external environment (experiment temperature), h is the convective heat transfer coefficient between the surfaces of the specimen and the air, and the value is 9.5 W·m^-2·°C^-1.

The 3D numerical model under unidirectional freezing condition is shown in Fig. 7(b). To facilitate the arrangements of the boundaries, 10 mm thick solid elements are built at the top and bottom of the specimen to simulate the cold and warm plates. 10 mm is the thickness of the walls of the plates. Controlled liquids circulate through the cold and warm plates, hence the boundary conditions between the liquids and the walls of the plates are convective heat transfer boundaries. Therefore, the boundaries at the top and bottom surfaces of the model can be expressed as Eq. (8). However, here the thermal conductivity in Eq. (8) is the conductivity of the plates; the convective heat transfer coefficients h₁ and h₂ at the top and bottom surfaces of the model are 250 and 100 W·m^-2·°C⁻¹, respectively; T_{ext} is the temperature of the cold or warm plate. In addition, the side surfaces of the specimens and the plates are thermally insulated in the experiments, hence the side surfaces of the model are thermally insulated boundaries.

Furthermore, continuous conditions should be satisfied at the interfaces between crack walls and the water in crack:

(9)\{\begin{array}{l}{T}_{\mathrm{1}}=T_{\mathrm{2}}\\ {\left(\lambda {\displaystyle \frac{\partial T}{\partial \overrightarrow{n}}}\right)}_{\mathrm{1}}={\left(\lambda {\displaystyle \frac{\partial T}{\partial \overrightarrow{n}}}\right)}_{\mathrm{2}}\end{array}.

The required parameters of the different materials are listed in Table 2. The initial temperature is 20.1°C. Moreover, the sandstone specimens in this study is the same as those used in the study of Lv et al. [¹⁰], hence the values of the following parameters can be acquired by referring to the literature: {\omega }^{\ast }=\mathrm{0.0062}, {\omega }_{\mathrm{0}}=\mathrm{0.0884}, and a = 0.5.

To ensure the accuracy of the numerical simulation, the maximum element size of water in crack is 0.002 m. The fully coupled scheme is used in calculation and the time step is taken by solver freely.

Validation of the numerical model

The numerical and experimental results of the temperature variation curves of Points A and B under uniform freezing condition are shown in Fig. 8, and the results under unidirectional freezing condition are shown in Fig. 9. The numerical results of three cases, considering no latent heat, considering the latent heat of water in crack only, and considering the latent heat of pore water and water in crack are shown in Figs. 8 and 9 for comparison. The numerical results considering no latent heat significantly deviate from the experimental results and cannot reflect the PC stage. The PC stage of the numerical results considering the latent heat of water in crack only sustains a much shorter time than that of the experimental results. Hence, the numerical method considering the latent heat of water in crack only cannot accurately simulate the freezing process of the rock mass. Therefore, the PC of pore water and water in crack would significantly influence the freezing process of the rock mass, and disregarding the latent heat of PC will induce significant errors.

Furthermore, Figs. 8 and 9 present that the numerical results considering the latent heat of pore water and water in crack under uniform and unidirectional freezing conditions are consistent with the experimental results. Hence, the numerical results considering the latent heat of pore water and water in crack can describe the freezing process of rock mass accurately and can be used in the comparative analysis of the freezing process of saturated rock with an open crack under uniform and unidirectional freezing conditions.

Freezing process and frost heave mode of rock with an open crack

Freezing process and frost heave mode under uniform freezing condition

The numerical results of temperature variations for points in the crack with a depth of 2, 10, 20, 30, and 40 mm in section y= 0.02 m are shown in Fig. 10. Only the data from 50 to 300 min are presented to highlight the PC process. Figure 11 shows the temperature distribution of section y= 0.02 m at 100, 150, and 200 min under uniform freezing condition. The water in crack with a depth of 2 mm starts to freeze at approximately 100 min; at the time, the temperature of water in crack with a depth of 10 to 40 mm remains positive and is nearly the same. When the water in crack with a depth of 10 mm starts to freeze at approximately 150 min, the temperature of the water in crack with a depth of 2 mm has been –0.7°C already. PC occurs simultaneously for water in crack with a depth of 20 to 40 mm at approximately 200 min; meanwhile, the temperature of the water in crack with a depth of 2 and 10 mm has been –2.0°C and –0.9°C, respectively. Therefore, during the freezing process of saturated rock with an open crack under uniform freezing condition, water in crack with a depth of 2 to 10 mm freezes first, and thus ice wedge forms first in the upper part of the crack, thereby confining the upward flow of water due to frost expansion in the lower part of the crack. Thereafter, the water in crack with a depth of 20 to 40 mm freezes and expands simultaneously. Moreover, Fig. 10 demonstrates that the PC stage of water in the upper part of the crack sustains a shorter time than that of water in the lower part of the crack.

The freezing front is a curved surface that is developed from outside to inside under uniform freezing condition, as shown in Fig. 11, and a frozen shell develops first in the outer part of rock before the freezing of water in crack. The permeability coefficient is typically far smaller in the frozen saturated rock than in the unfrozen rock, and can be assumed as approximately zero [³⁶]. Hence, the seepage and migration path of water in crack to the saturated rock is blocked by the frozen shell that forms before the freezing and expansion of water in crack.

Therefore, the freezing process determines the migration of water in crack under uniform freezing condition, and thus influences the frost heave mode. Figure 12 shows the frost heave mode of saturated rock with an open crack under uniform freezing condition. The frozen shell develops earlier than the freezing of water in crack, and water migration through the frozen rock can be ignored regardless of the low or high permeability of rocks. In addition, ice wedge forms first in the upper part of the crack, and the enclosed space is generated before the freezing of water in the middle and lower parts of the crack. Hence, the water in the middle and lower parts of the crack freezes simultaneously and expands in a closed space, and thus induces the frost heave of the open crack. Moreover, the frost heave modes of the open crack in rocks with low or high permeability are similar under uniform freezing condition because water migration is blocked by a frozen shell and is irrelevant to rock permeability. This result is consistent with the experimental results that the frost heave variations of the crack in sandstone and tuff are similar under uniform freezing condition, as shown in Figs. 3(a) and 4(a), although the permeability is far higher in the sandstone than in the tuff.

Furthermore, the special freezing mode under uniform freezing condition cannot reflect the influence of the permeability of rock on water migration and the frost heave of water in crack. Hence, the uniform freezing experiments are inapplicable to studies, such as migration of water in crack during freezing process, the influence of permeability on the freeze–thaw damage or the frost heave of rock with cracks.

Freezing process and frost heave mode under unidirectional freezing condition

The numerical results of the temperature variations for the points in the crack with a depth of 2, 10, 20, 30, and 40 mm in section y= 0.02 m are shown in Fig. 13. Only the data from 0 to 250 min are shown to highlight the PC process. Figure 14 shows the temperature distribution of section y= 0.02 m at 25, 50, and 100 min under unidirectional freezing condition. Figures 13 and 14 illustrate that temperature of water in crack increases with depth. At approximately 25 min, the water in crack with a depth of 2 mm starts to freeze, while the temperature of the water in crack with a depth of 40 mm is about 10.5°C. When the water in crack with a depth of 10 mm starts to freeze at 50 min, the temperature of the water in crack with a depth of 2 mm has been –5.5°C. The PC occurs for water in crack with a depth of 30 to 40 mm at approximately 100 min; at the time, the temperature of water in crack with depth of 20 mm has been –1.8°C. Thus, a 2.5 mm deep ice wedge has developed, which would confine the upward flow of water in the lower part of the crack. Thereafter, the PC occurs for water in crack with a depth of 40 mm at 150 min. Moreover, Fig. 13 indicates that the shallower the water in crack is, the shorter time the PC stage sustains, and the earlier the stable stage reaches.

The freezing front is a trumpet-shaped surface developed from top to bottom under unidirectional freezing condition, as shown in Fig. 14. Moreover, the freezing front of the water in crack lags behind that of saturated rock, and the hysteresis increases gradually as time elapses. The freezing front of the water in crack lag behind that of saturated rock approximately 3, 5, and 11 mm at 25, 50, and 100 min, respectively. During the developing process of freezing front, the seepage and migration path of the water in crack to the unfrozen saturated rock under the freezing front exists all along.

Therefore, the freezing process and rock permeability determine the migration of water in crack together, and thus lead to the different frost heave modes of crack for different permeability rocks under unidirectional freezing conditions. Figure 15 shows the frost heave modes of saturated rocks with an open crack under unidirectional freezing condition. Ice wedge also forms first in the upper part of the crack under unidirectional freezing condition and confines the upward flow of the water due to frost expansion in the lower part of the crack. Although the freezing front develops quicker in rock than that of water in crack, the rock at both sides and bottom of the crack under freezing front is unfrozen. During the freezing process of the water in crack, the frost expansion of water under freezing front induces water pressure. For high permeability rocks, the seepage migration of water in crack into the unfrozen rock will occur driven by the water pressure. Thus, the frost heave of the crack for high permeability rocks will be weakened due to water migration, as shown in Fig. 15. This outcome is the reason why the frost heave of the crack in the sandstone with high permeability is less than the frost heave of the saturated sandstone under unidirectional freezing condition, as shown in Fig. 3(b). However, for low permeability rocks, water migration is blocked. Hence, the frost heave of crack for low permeability rock would be more significant under unidirectional freezing condition. This finding is consistent with the experimental results that the frost heave of the crack in the tuff with low permeability is significant under unidirectional freezing condition, as shown in Fig. 4(b), although the elastic modulus of tuff is much higher than that of ice.

In addition, ice wedge forms first in the upper part of the crack under uniform and unidirectional freezing conditions. However, the temperature of ice wedge in the upper part of the crack under uniform freezing condition is higher than that under unidirectional freezing condition, which induces that the strength and stability of the ice wedge under uniform freezing condition are much lower than that under unidirectional freezing condition [³⁷]. Hence, the constraint of the ice wedge on the frost expansion of water in the lower part of the crack under uniform freezing condition is gentler than that under unidirectional freezing condition, and ice extrusion is more significant under uniform freezing experiments. Therefore, the frost heave of the crack under uniform freezing condition is less than that under unidirectional condition for low permeability rocks. This finding is consistent with the experimental results that the frost heave of the crack in the tuff under unidirectional freezing condition is greater than that under uniform freezing condition, as shown in Fig. 4.

Conclusions

Freezing experiments are conducted on saturated sandstone and tuff with an open crack under uniform and unidirectional freezing conditions to investigate the frost heave of the crack in various permeability rocks under different freezing conditions. Moreover, numerical simulations on the freezing processes of saturated rock with an open crack are performed to illustrate the influence of the freezing process on the frost heave mode. The following conclusions are obtained:

1) Frost heave of crack in rock with high permeability is more significant under uniform freezing condition than that under unidirectional freezing condition. However, frost heave of crack in rock with low permeability is more significant under unidirectional freezing condition.

2) The numerical model considering the latent heat of pore water and the water in crack can accurately describe the freezing process of rocks with an open crack under uniform and unidirectional freezing conditions.

3) The freezing processes of saturated rock with an open crack are significantly different under uniform and unidirectional freezing conditions. The freezing process and rock permeability determine the migration of water in crack together, and thus lead to the different frost heave modes of the crack for different permeability rocks under uniform and unidirectional freezing conditions.

4) The frost heave modes of the open crack in rocks with low or high permeability are similar under uniform freezing condition because water migration is blocked by a frozen shell and is irrelevant to the permeability of the rock. For high permeability rocks, the frost heave of the crack would be weakened due to water migration under unidirectional freezing condition. However, water migration is blocked for low permeability rock, and thus the frost heave of the crack would be more significant under unidirectional freezing condition. Therefore, the influence of freezing condition and permeability of rock on the frost heave of rock mass should be concerned in engineering applications.

5) The work only focuses on the frost heave of rock with crack under different freezing conditions, and attentions should be paid on the pressure induced on the wall of the crack and the crack propagation during the freeze of fractured rock in the next step work.

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