A model of unfrozen water content in rock during freezing and thawing with experimental validation by nuclear magnetic resonance

Zhouzhou Su,Xinjun Tn,Weizhong Chen,Hiling Ji,Fei Xu,d

a State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan,430071,China

b University of Chinese Academy of Sciences,Beijing,100049,China

c College of Architecture and Civil Engineering,Xi’an University of Science and Technology,Xi’an,710054,China

d Structural Health Monitoring and Control Institute,Shijiazhuang Tiedao University,Shijiazhuang,050043,China

Keywords:Freezing and thawing Unfrozen water content Super-cooling and hysteresis Nuclear magnetic resonance(NMR)Unfrozen water calculation model Red sandstone

A B S T R A C T The unfrozen water content of rock during freezing and thawing has an important influence on its physical and mechanical properties.This study presented a model for calculating the unfrozen water content of rock during freezing and thawing process,considering the influence of unfrozen water film and rock pore structure,which can reflect the hysteresis and super-cooling effects.The pore size distribution curves of red sandstone and its unfrozen water content under different temperatures during the freezing and thawing process were measured using nuclear magnetic resonance(NMR)to validate the proposed model.Comparison between the experimental and calculated results indicated that the theoretical model accurately reflected the water content change law of red sandstone during the freezing and thawing process.Furthermore,the influences of Hamaker constant and surface relaxation parameter on the model results were examined.The results showed that the appropriate magnitude order of Hamaker constant for the red sandstone was 10-19 J to 10-18 J;and when the relaxation parameter of the rock surface was within 25-30 μm/ms,the calculated unfrozen water content using the proposed model was consistent with the experimental value.

1.Introduction

Rock engineering in cold regions experiences serious seasonal freezing and thawing erosion each year.The loss of rock strength caused by frost heaving force during freezing and thawing cycles can cause geological disasters in geotechnical engineering,including debris flow,landslide,high steep slope collapse,and tunnel cracking(Tan et al.,2011;Kang et al.,2014;Huang et al.,2018;Li et al.,2018).For example,with the construction of the Sichuan-Tibet Railway in cold regions,many rockfall disasters occurred due to the degradation of slope rock induced by the freezing and thawing process,posing an extreme threat to the safety of the railway(Huang et al.,2020;Sun et al.,2020;Fan et al.,2021).Long-term freezing and thawing corrosion leads to rock weathering,making particles fall off at the joints,and resulting in the deterioration of shear/tensile strength.When severe frost heaving occurs,steep rock slope collapses(Sandeep and Senetakis,2018;Kasyap and Senetakis,2020).The freezing and thawing damage of rock is mainly caused by the frost heaving force,the magnitude of which is closely related to the unfrozen water content in rock.Therefore,study on the unfrozen water content of rock during the freezing and thawing process is essential.

Currently,there are many methods for measuring unfrozen water contents in soil or rock.The principles and the advantages and disadvantages of various testing methods are summarized in Table 1.Each testing method has its specific application conditions.Among these methods,the nuclear magnetic resonance(NMR)has been widely promoted and applied because it is fast,accurate,and convenient(Smith and Tice,1988;Ishizaki et al.,1996;Watanabe and Mizoguchi,2002).Therefore,the NMR method was selected to measure the unfrozen water content in this study.

With the continuous development of testing technology and theoretical investigation,the correlations between unfrozen water content and measured physical indicators have become increasingly clear.In the past 30 years,many theoretical models have been proposed.Some typical models are summarized in Table 2.Mostmodels are relatively simple,with only two models considering the super-cooling effect(Topp et al.,1980;Kozlowski,2003;Liu et al.,2016a)and no model considering the hysteresis effect.Although these two phenomena have been confirmed in many experimental studies(Koopmans and Miller,1966;Bittelli et al.,2003),the super-cooling and hysteresis effects are physical processes that must be considered for accurately calculating the unfrozen water content.Therefore,it is necessary to propose a new model.

Table 1Testing methods for unfrozen water content.

Table 2Formulae for calculating the unfrozen water content.

In the present study,a model was presented to calculate the unfrozen water content of rock during the freezing and thawing process.This model considered the influence of the unfrozen water film and rock pore structure,which reflected the hysteresis andsuper-cooling effects.A freezing and thawing cycle test of red sandstone under a specific temperature gradient was designed,and the unfrozen water content of red sandstone under different temperatures during the freezing and thawing process was measured using NMR technology(Tice et al.,1981).The model results were compared with the experimental data,and measurements of the Hamaker constant and surface relaxation parameter were performed to determine their influences on the model results.

2.Development of theoretical model

2.1.Basic assumptions

To determine the unfrozen water content in low-temperature rock,the pore structure of the rock and composition of the unfrozen water must be studied firstly.According to previous research,the following assumptions were made in the present study:

(1)The cylindrical pores with different diameters are randomly distributed.Before freezing,water is fully distributed in primary and secondary pores(Jia et al.,2020),as shown in Fig.1a.

(2)In the freezing process,the phase change of water to ice first occurs in pores with a larger radius and then in pores with smaller radius as decreasing temperature due to the interface curvature effect(Fig.1b).Thus,for a certain freezing temperature,a corresponding pore radius value exists,i.e.the critical frozen pore radius.Only when the pore radius is greater than this critical value,water will freeze into ice.Otherwise,the water remains in liquid state,which is called the free unfrozen water(Horiguchi,1985).

(3)There is a nano-scale unfrozen water film between the ice in the frozen pores due to the interface pre-thawing effect(Fig.1c),which is called the non-free unfrozen water(Ishizaki et al.,1996).

Fig.1.Distribution of unfrozen water in rock pores:(a)Water distribution before freezing;(b)Initial freezing;and(c)Unfrozen water distribution after freezing.

Fig.2.Schematic diagram of pore size distribution.

Based on the above assumptions,the unfrozen water inside pores can be divided into free and non-free unfrozen water.To calculate the unfrozen water content,the critical frozen pore radius(rc),the film thickness of unfrozen water and the distribution of the pore size must be obtained.Schematic diagram of pore size distribution is shown in Fig.2,wheref(r)is the pore volume ratio function.The shadow part in the figure denotes the pore with the radius smaller than the critical one,which contains free unfrozen water.

2.2.Critical frozen pore radius

Classical thermodynamic theory can be used to describe the process of water solidification,i.e.the change from liquid to solid.At the balanced phase boundary,the specific Gibbs free energy is the same in the two phases on both sides.From the Gibbs-Duhem equation(Reif,1965),the phase transition equilibrium equation on the ice-water interface can be obtained:

where ρsis the ice density,ρlis the water density,Lis the released latent heat,Tmis the critical freezing temperature,Tis the core temperature of the testing sample,ΔTis the difference between the test temperature and the core temperature during measurement,pmis the hydrostatic pressure,plis the water pressure at the interface,andpsis the ice pressure at the interface.

We ignore pressure difference by settingpl=pm,Eq.(1)can be simplified as

According to capillary theory,when the temperature drops belowTm,the ice will not immediately penetrate the pores of rock.The pressure difference between the curved ice-water interface can be given as follows according to the Young-Laplace equation:

where γiwis the free energy of the water-ice interface.According to Eqs.(2)and(3),we can obtain:

The ice body at this time is regarded as a cylinder and the critical radius of the circular ice cap is

2.3.Film thickness

The thickness of the adsorption film is related to the surface and liquid properties,geometry,and chemical potential.The liquid film adsorbed on the plane only interacts with long-range molecules according to Iwamatsu and Horii(1996)and Doppenschmidt and Butt(2000):

wherehis the film thickness,andAsvlis the Hamaker constant involved in the intervention of liquid-solid-gas interaction.

From Eqs.(2)and(6),Eq.(7)can be obtained:

wherePTis the surface pressure.

The film thickness is expressed as

2.4.Calculation model of unfrozen water content

In pores with the radius less than the critical value,there is free and non-free unfrozen water.Assuming that the free unfrozen water freezes into a cylindrical shape,the volume is

whereV1，randlare the volume,radius,and height of the water cylinder,respectively.

The total volume of all cylinders with the radiusris

whereVris the total volume of all cylinders with the radiusr,andVsis the total volume of the voids.

The number of cylinders with the radiusrisn1,then we have

Subsequently,the free unfrozen water contentW1is calculated as

whereW1is the free unfrozen water content.

Similarly,for non-free unfrozen water,the volume of the unfrozen water film can be written as

whereV2is the volume of the non-free unfrozen water cylinder.The thickness of the unfrozen water film is negligible relative to the length of the cylindrical pore,thus Eq.(13)becomes

The total volume of the water-containing membrane cylinder is

whereV3is the total volume of the water-containing membrane cylinder.

The number of cylinders with the water film thicknesshisn2,then we have

whereVhis the volume of the cylinder containing the unfrozen water film,andf(r+h)is the pore volume ratio function of the cylinder containing the unfrozen water film.The non-free unfrozen water contentW2is

where λ=r／h(Watanabe and Mizoguchi,2002).

Then the unfrozen water content at the time of freezing,Wfis given as follows:

In the same way,in the thawing stage,the absolute value of the thawing temperature is only half of the freezing temperature,and the critical radius is half of the critical radius when freezing(Anderson et al.,2009):

wherer′cis the critical thawed pore radius.

Then,the unfrozen water content at the time of thawing is

2.5.Correction of the proposed model

The unfrozen water content curve during the freezing process exhibited a super-cooling phenomenon in previous studies(Style et al.,2011;Zhou et al.,2015)because when the temperature reaches 0°C,there will be tiny crystals and ordinary crystals in the pore water of rock.The surfaces of the tiny crystals have a high Gibbs free energy and chemical potential,and the tiny crystals will automatically melt.At the freezing point,there is a balance between the formation of ordinary crystals and the melting of the tiny crystals in the water.Then,the crystal nuclei in the water are in a metastable state.As the temperature continues to decrease within a certain range,tiny crystal nuclei are formed,the metastable balance is broken,and more and more ice crystals gather,which eventually manifests as freezing.When the water freezes,it will release latent heat,the local temperature will suddenly rise,and there will be a jump in the temperature-time chart(Fig.3a).The temperature after the jump is called the freezing temperature,which will be lower than the actual super-cooling temperature.The proposed theoretical model(shown as Eq.(19))cannot reflect this physical process and must be corrected.

Fig.3.Correlation images of super-cooling degree during freezing:(a)Freezing temperature over time;and(b)Change in super-cooling degree with pore radius.

The determination of the“super-cooling effect”was based on the fact that the unfrozen water content kept constant in a small range below 0°C and the mutation occurred at a certain temperature.Several factors affected the occurrence of the super-cooling phenomenon:

(1)Freezing rate.The super-cooling of the pore water usually do not occur under rapid freezing.

(2)Environmental disturbance for freezing caused by intermittent freezing and mechanical vibration,where the supercooling phenomenon is not evident.

(3)Moisture content.When the moisture content in rock is low(typically saturation degree＜60%),the super-cooling phenomenon cannot be observed(Jia et al.,2019).

The degree of super-cooling is also related to its pore structure and the solute,which can be characterized by the Gibbs-Thomson equation(Dash et al.,1995;Ishizaki et al.,1996):

where σ is the surface tension of ice water.

Then,the degree of super-cooling is obtained as follows:

For red sandstone with saturated water content,the value of super-cooling is mainly related to the freezing radius.Fig.3b shows the relationship between super-cooling and freezing radius.After correction,the unfrozen water content during the freezing process can be written as

3.Measurement of pore size distribution and unfrozen water content using NMR

3.1.Sample preparation

A red sandstone block was taken from Linyi,Shandong Province,China for sample preparation.The sample was processed to a cylinder with the diameter of 50 mm and the height of 100 mm.The basic data of the sample are shown in Table 3.

Table 3Basic data of the sample.

3.2.Experiment design

The temperature was set at 20°C,10°C,and 5°C to measure the pore size distribution using NMR.The unfrozen water content during freezing and thawing was measured at different temperatures(0°C,-0.5°C,-1°C,-1.5°C,-2°C,-3°C,-4°C,-5°C,-6°C,-7°C,-8°C,-9°C,-10°C,-12°C,-14°C,-16°C,-18°C,and-20°C)with NMR.

3.3.Experiment procedure

The tests were conducted in the following the procedure:

(1)The rock sample was placed in the equipment,and cold bath was used to control its temperature.The rock pore size distribution was measured at temperatures of 5°C-20°C,and the unfrozen water content was measured in both the freezing and thawing processes from-20°C to 0°C.The temperature was adjusted slowly.After each adjustment,the rock temperature was stable.

(2)NMR measurements were performed using the Carr-Purcell-Meiboom-Gill(CPMG)pulse at a field of 11 MHz(0.25 T).The magnetic field inhomogeneity is less than 0.004‰.The optimized acquisition parameters are as follows:spin echoes of 2000,90°pulse duration of 2.8 μs,shortwave of 250 kHz,deadtime of 50 μs,and radiofrequency pulses with a 300 W amplifier.A total of 64 scans and an echo time(TE)of 0.1 ms or 1 ms were used.

(3)The transverse relaxation time(T2)spectrum distributions were obtained for the CPMG sequence at each temperature during the NMR measurements and then processed for T2distribution using the inductive logic programming(ILP)method with the uniform penalty(UPEN)algorithm(Fleury et al.,2013;Birdwell and Washburn,2015)

(4)According to the NMR principle,fluid free relaxation time and diffusion relaxation can be ignored;therefore,the NMR crosswise relaxation rate was given as(Jia et al.,2019):

whereVis the pore volume,Sis the pore surface area,and φ is the surface relaxation parameter.

Eq.(24)can be further transformed into the relationship between T2relaxation time and pore radius:

whereFsis the geometry factor,andFs=3 for the spherical pores andFs=2 for the columnar pores.LetC=Fsφ,Eq.(26)can be rewritten as

whereCis the conversion coefficient.

(5)According to steps(1)-(3),the temperature and NMR peak curves were obtained.The paramagnetic regression line wasobtained by connecting and extending the points in the positive temperature region to the negative temperature region.The unfrozen water content at a certain temperature is equal to the measured signal intensity divided by the signal intensity of the paramagnetic regression line multiplied by the water content.Finally,the unfrozen water content for each temperature under the freezing and thawing conditions was obtained.

Fig.4.Pore size distribution curve.

3.4.Experimental results

3.4.1.Pore size distribution

The pore size distribution is shown in Fig.4.The radium of most pores of red sandstone is less than 10 μm,and the proportion of pore with the radius greater than 100 μm is very small,which can be ignored.Therefore,the maximum pore radius of red sandstone is identified as 100 μm.

3.4.2.Unfrozen water content

The NMR T2spectrum was transformed into an NMR pore throat distribution graph to quantitatively characterize the distribution of different levels of pore radii.Based on the methodology described by Tice et al.(1983),the NMR data were converted into the unfrozen water content(Fig.5).

As shown in Fig.5c,when the temperature was above 0°C,the unfrozen water content was 3.47%.When the temperature decreased to-3.5°C,an inflection point appeared on the unfrozen water content curve in the freezing process,and the unfrozen water content started to decrease.In particular,when the temperature decreased from-3.5°C to 1°C,the unfrozen water content decreased sharply from 3.47% to 2%.When the temperature decreased from-5°C to-20°C,the unfrozen water content decreased slowly.When the temperature dropped to-20°C,the unfrozen water content decreased to 1%.When the temperature increased to 0°C,the unfrozen water content returned to 3.4%.The unfrozen water content curve in the freezing process showed an evident super-cooling section(Style et al.,2011;Zhou et al.,2015),and the unfrozen water content in the thawing process was lower than that during freezing for the same temperature,resulting in a non-overlapping unfrozen water content curve in the freezing and thawing process and a hysteresis zone.There are at least two possible mechanisms for hysteresis in the freezing characteristics of porous media.The first one is the super-cooling effect.When the temperature drops below the freezing point,the water in the pores does not freeze immediately but reaches a super-cooled state.However,there is no similar process during thawing.The other mechanism is the ink-bottle effect of the pore.The movement of the ice-liquid interface in the pores is similar to the movement of the gas-liquid interface;therefore,bottleneck effects will cause hysteretic behavior(Koopmans and Miller,1966;Spaans and Baker,1996;Bittelli et al.,2003).

4.Validation of the theoretical model

The parameters required for model calculation are listed in Table 4(Liu et al.,2016b)and the results are shown in Fig.6.

Table 4Calculation parameters.

As shown in Fig.6a,experimental measurements were consistent with the theoretical results,and the errors were within 5%.The super-cooling degree and hysteresis zone can be calculated from the theoretical curve and observed in the experimental results.From the freezing curve,two important temperature points at temperatures of-3.5°C and-5°C could be found.As the temperature decreased from-3.5°C to-5°C,the unfrozen water content quickly decreased from 3.46%to 2%.At temperatures higher than-3.5°C,the unfrozen water content remained unchanged,whereas it slowly decreased from 2% to 1% as the temperature decreased from-5°C to-20°C.For the thawing curve,it had the similar trend as to the freezing curve;however,the two important points occurred at temperatures of 0°C and-5°C.The unfrozen water content at the same temperature was lower than that during freezing,indicating super-cooling in the freezing process of water in the rock but not in the thawing process.Therefore,the freezing and thawing curves showed a hysteresis zone.When freezing,the super-cooling temperature was determined as-3.5°C.

As shown in Fig.6b,when the temperature dropped from 0°C to-5°C,the free unfrozen water decreased by 1.4%,whereas the non-free unfrozen water increased by only 0.2%.When the temperature was lower than-5°C,the free unfrozen water slowly decreased and reached a stable value of 0.96% at-20°C,and the non-free unfrozen water stabilized at 0.2%.Therefore,the unfrozen water content was mainly influenced by the free unfrozen water content and the effect of the non-free unfrozen water content was negligible.Both free and non-free unfrozen water remained constant after the temperature reached-20°C.

5.Discussion

The unfrozen water content is related to temperature,the Hamaker constant of water in the rock,and the surface relaxation parameter of the rock during the freezing and thawing process.

5.1.Influence of Hamaker constant

Hamaker constant is a key parameter to characterize van der Waals forces,which is commonly used to calculate the surface force and surface energy constant.It affects the distribution of water molecules on the surfaces of rock and ice particles.This influences the thickness of the unfrozen water film,thereby affecting the nonfree unfrozen water content.At present,there are no exact Hamaker constant values for geotechnical materials.As shown by Watanabe and Osada(2017)and Markus and Dani(2003),when water is in contact with the surface of rock and soil,the constantAsvlshould be between-10-20J and-10-19J,whereas Vlahou and Worster(2010)showed that when the rock was in contact with water,Asvl=-10-18J.Thus,it is necessary to discuss the influence of different Hamaker constants on the theoretical results of unfrozen water content.Hamaker constant is related to the dielectric constant.The value range given by previous scholars was appropriately expanded and examined(Liu et al.,2016a).The present study considered the values ofAsvlranging from-10-21J to-10-17J.According to the different values of Hamaker constant for various rocks,the present study used different values ofAsvlto compare the theoretical and real results,as shown in Fig.7.

Fig.5.NMR data converted to unfrozen water content curves:(a)T2 spectrum signal peak;(b)Paramagnetic regression line(a,b,and c are the corresponding points on the three curves at the same temperature);and(c)Unfrozen water content.

Fig.6.(a)Corrected experimental and theoretical unfrozen water contents;and(b)Free and non-free unfrozen water contents during freezing and thawing.

During freezing and thawing,whenAsvl＜10-19J,the non-free unfrozen water increased before reaching a constant value,and the larger the value ofAsvl,the larger the constant value(Fig.7).However,the non-free unfrozenwater content after stabilizationwas only 1/3-1/4 that of the experimental data.WhenAsvl＞10-18J,the non-free unfrozen water content in the freezing and thawing process increased rapidly with the decrease of temperature and reached the extreme value,and then it decreased.Eventually,the theoretical result of the non-free unfrozen water content was 3-4 times higher than the experimental value.When 10-18J＜Asvl＜10-19J,the theoretical result of the non-free unfrozen water in the freezing and thawing process was close to the experimental data.When the temperature decreased from 0°C to-10°C,the non-free unfrozen water content increased,whereas the increase rate was slow.When the temperature decreased to-10°C,the non-free unfrozen water content was 1.9%.When the temperature decreased again,the non-free unfrozen water content remained at approximately 1.9%.In summary,the appropriate Hamaker constant for red sandstone material inthe freezing and melting stages wasinthe range of 10-19J to 10-18J.In this range,the experimental results of the unfrozen water content were consistent with the theoretical ones.

5.2.Influence of surface relaxation parameter

The calculation of the unfrozen water content in rock requires the pore size distribution curve,which can be derived from an NMR T2spectrum.Eqs.(25)-(27)show that the pore size distribution is proportional to the surface relaxation parameter,which varies according to lithology.The surface relaxation parameter needs to be determined through a large number of experiments.As it has not been determined for red sandstone,this study uses empirical values.Through experiments,the surface relaxation parameter of sandstone has been found to range within 10-30 μm/ms(George et al.,1999).This study selected four different values of the surface relaxation parameters within this range,i.e.10 μm/ms,20 μm/ms,25 μm/ms,and 30 μm/ms.The calculated results of the unfrozen water content under different surface relaxation parameters are shown in Fig.8.

Through trial calculations,it was found that the surface relaxation parameter has a great influence on the unfrozen water content,as shown in Fig.8.For different values of surface relaxation parameter,the unfrozen water content decreases with temperature in a consistent trend,indicating that the variation in φ does not affect the freezing and melting processes.The surface relaxation parameter is various for different types of rocks,and reflects the response of rock particles to the nuclear magnetic relaxation time and,ultimately,only affects the data during the calculation process.When φ＜25 μm/ms,the pore throat distribution curve converted from the NMR spectrum curve was below that obtained from the experiment.Therefore,the unfrozen water content obtained by the theoretical calculation was smaller than that obtained in laboratory.Through calculation,it was found that when the surface relaxation parameter was 25-30 μm/ms,the unfrozen water content of the rock was within±10% of the experimental value.Therefore,the theoretical data were consistent with the experimental results.

6.Conclusions

In the present study,the unfrozen water content in rock was theoretically investigated during freezing and thawing.Then,the unfrozen water content of rock was measured at different temperatures by NMR.The test results were compared with the theoretical ones.Finally,the influences of different values of Hamaker constant and surface relaxation parameter on the model were discussed.The following conclusions are obtained:

(1)A model for calculating the unfrozen water content was deduced,considering the unfrozen water film and rock pore structure during the freezing and thawing process,which can reflect the hysteresis and super-cooling effects.

Fig.7.Effect of different Hamaker constants on the non-unfrozen water under(a)Freezing and(b)Thawing conditions.

(2)Super-cooling and hysteresis effects are the physical processes that must be considered to accurately calculate the unfrozen water content.The super-cooling temperature of red sandstone is-3.5°C.

(3)The unfrozen water of rock consists of free and non-free unfrozen water.When the temperature decreases,the free unfrozen water content decreases,and the non-free unfrozen water content increases.When the temperature is lower than-20°C,the free and non-free unfrozen water content remains unchanged.

(4)For red sandstone,the appropriate order of magnitude for Hamaker constant was in the range of 10-19J to 10-18J.

(5)When the surface relaxation parameter was 25-30 μm/ms,the unfrozen water content was consistent with the experimental value.

Fig.8.Free unfrozen water contents during(a)Freezing and(b)Thawing.

The formation and changes of unfrozen water inside rock are very complicated physical processes.The present study only discussed the theoretical calculation of the unfrozen water content of unsaturated rock.However,unsaturated conditions of rock are a more common state in engineering.For unsaturated conditions,the influence of water proportion in pores with different size distributions and freezing shapes on the calculation of unfrozen water content requires further research.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors gratefully acknowledge the support of the Second Tibetan Plateau Scientific Expedition and Research Program(STEP)of China(Grant No.2019QZKK0904),the National Outstanding Youth Science Fund Project of National Natural Science Foundation of China(Grant No.51922104),Youth Innovation Promotion Association CAS and Open Research Fund of State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences(Grant No.Z018014).

List of symbols

ρsIce density

TmCritical freezing temperature

plWater pressure at the interface

psIce pressure at the interface

pmHydrostatic pressure

ψ Resistivity

rcCritical frozen pore radius

AsvlHamaker constant

φ Surface relaxation parameter

hFilm thickness

lCylindrical height

rCylindrical radius

f(r) Pore volume ratio function

rmaxThe maximum radius of the rock sample

VhVolumes of film thickness

σ The surface tension of ice water

r′cCritical thawed pore radius

WfUnfrozen water content at the time of freezing

θ Negative temperature

LLatent heat is released when a unit mass of water freezes

TThe core temperature of the test rock sample

ΔTDifference between test temperature and core temperature

γiwFree energy of the water-ice interface

pTSurface pressure

θuThe ratio of unfrozen water weight to sample dry weight

VsEntire sample’s total volume

n1Number of cylinders with this radiusr

n2Number of cylinders whose water film thickness ish

V1Volumes of each free unfrozen water cylinder

V2Volumes of each non-free unfrozen water cylinder

VrThe total volume of all cylinders with a radius γ

W1Free unfrozen water contents

W2N on-free unfrozen water contents

WuUnfrozen water content

TfSuper-cooling degree

WtUnfrozen water content at the time of melting