A semi-analytical solution for frost heave prediction of clay soil

Hui Bing ,Ying Zhang,GuoYu Li

State Key Laboratory of Frozen Soil Engineering,Cold and Arid Regions Environmental and Engineering Research Institute,Chinese Academy of Sciences,Lanzhou,Gansu 730000,China

1 Introduction

The effects of harsh climate found in regions such as China,and North-East Asia on civil engineering and important linear engineering cannot be neglected by designers and contractors.For example,on the Qinghai-Tibet Plateau,with freezing indices ranging from 95 to 2,300 degree-days (Jianget al.,2008),frost action mainly develops in frost-susceptible soils,leading to ice lens formation,surface heave,and eventual structural distress.In regions with persistent old climates such as Siberian Russia,freezing indices of 4,000 to 6,000 degree-days are found with frost penetration to depths greater than 2.5 m (Feldmanet al.,1988).Thus,frost heave is an essential problem of construction in cold regions and artificial ground freezing engineering (Tong and Guan,1985;Chenet al.,2000).

During the latter part of the 20th century,numerous studies were performed to understand the mechanism of frost heave.Frost heave is a coupled process of heat and mass flow.When saturated soil is subjected to freezing,ice nucleates in the water-filled pores near the frost front.Some unfrozen water remains as an adsorbed water film on the surface of soil particles,even at temperatures below 0 °C.Through these films,water from unfrozen soil migrates to the frozen zone under the action of a temperature-induced suction gradient.Thus,total frost heave is composed of two components,one is due to the freezing of in-situ pore water and the other is due to the freezing of migrated water.The thickness of the unfrozen water films and permeability decrease as the temperature of the frozen soil decreases.When the permeability is low enough to essentially impede the flow of migratory water a new ice lens starts to form.The zone between the segregation front where the ice lens is growing and the warmest isotherm at which ice exists in the soil pores is known as frozen fringe (Miller,1972).The migration of water depends on suction and permeability of the frozen fringe,which in turn depends on the unfrozen water content.The precise measurement techniques of these parameters within the frozen fringe are not available.Several models of this complex process of frost heave are available in the literature (e.g.,Miller,1978;Gilpin,1980;Konrad and Morgenstein,1980;Shen and Ladanyi,1987;Nixon,1991;Michaloswski,1993).Among them,the rigid-ice model (Miller,1978) is the most comprehensive model but requires complex numerical techniques to solve the governing equations.On the other hand,the model based on the segregation potential(Konrad and Morgenstein,1980) is relatively simple and has been used for many practical engineering applications.

The implementation of SP models into numerical solutions (finite element or finite difference) is available in the literature.Konrad and Morgenstein(1984) developed a numerical algorithm using a finite difference technique to predict frost heave under a chilled gas pipeline.However,detailed discussion of those studies is not given here as the present study is based on the SP model.According to the SP model,precise calculation of the temperature gradient near the frozen fringe is of utmost importance for frost heave prediction.There is a sharp change in temperature gradient at the frozen-unfrozen interface depending on the thermal properties of frozen and unfrozen soil.Therefore,special care must be taken for calculating this temperature gradient.However,the calculated temperature gradient oscillates,especially at the early stage of freezing (Carlson and Nixon,1988).In order to avoid such oscillation in finite element analysis,Konrad and Shen (1996) used the element adjacent to the frost front,which is completely frozen,to determine this temperature gradient.However,this assumption is acceptable only if small elements are used.Moreover,the heat release from in-situ and migrated water in the freezing zone could result in some numerical instability,because the stiffness term is very large in the narrow freezing zone compared to its values in unfrozen or completely frozen zones.In summary,although the flexibility of numerical techniques allows the incorporation of more features in frost heave modeling,the analysis results are very sensitive to modeling of the frozen fringe,which requires considerable expertise.Therefore,the development of simplified analytical or semi-analytical solutions is important for many engineering applications and for the verification of complex numerical solution.The main objective of this study is to show the efficiency of a semi-analytical solution based on a one-dimensional SP model for predicting freezing of clay soil.The prediction results to the two tests with different freezing mode agree well with the tested behavior,which indicates the feasibility of the solution.

2 Segregation potential model

The in-situ heave,arising from the expansion of in-situ pore water freezing,can be calculated as

wherenis porosity of the unfrozen soil;Δyis thickness of freezing soil layer in time interval Δt;εis a factor remains in frozen soil.

Based on theSPmodel,the rate of water intake(v) to the freezing soil and segregational heave can be calculated as

whereSPis segregation potential,gradθis temperature gradient in frozen fringe;Δhsis segregational heave in time interval Δt.

The total frost heave increment in time interval Δtis the sum of these two components.

An increase in normal stress at the freezing front reduces the rate of frost heave,andSPdecreases with increase in applied stress as (Penner and Ueda,1977):

wherePeis normal stress acting on the freezing front,which is equal to the applied stress on the specimen in laboratory frost heave tests;bis a soil constant;andSP0is the maximum value of segregation potential that occurs at zero external pressure acting on the specimen.

The transient freezing in infinite soil medium could reasonably represent the freezing of soil using the model,but the complete analysis to the solution is not available in the literature.Considering the phrase change of in-situ pore water as a half-space problem,the present study modifies the solution incorporating the freezing of migrated water,and shows the efficiency of the prediction of frost heave.

3 Semi-analytical solution of soil frost heave

Assuming the freezing soil as a half-space medium under the transient heat flow change for phrase change of water freezing,the following two partial different equations can be used to represent the temperature distribution in frozen and unfrozen zones for transient heat transfer (Figure 1).

Figure 1 Temperature profile used in semi-analytical solution

For the frozen zone (0

For unfrozen zone (y>y0)

whereTis temperature;yis depth;y0is the depth of the frost front;tis time;αis thermal diffusivity(=K/cv);Kis thermal conductivity;cvis volumetric heat capacity.The subscriptsfandurefer to the frozen and unfrozen state,respectively.

The following initial and boundary conditions are used.The initial conditions

The fixed boundary conditions

whereTp,TGandθare initial ground temperature,ground temperature at infinity,and freezing temperature,respectively.

If the frost front advances a distance dy0in time dt,the amount of heat released by in-situ water is:

whereLis the latent heat of fusion of pore water;ρdis dry density of soil;Ais area;andwis initial water content.

Considering gradθ≈[dTf/dy]y0,the water intake rate to the frozen fringe can be calculated using equation(2).The amount of heat released by migrated water in time dtis

The continuity of heat flow at 0 °C isotherm is the amount of heat flow through the frozen soil is the sum of the heat coming from the unfrozen soil plus the heat liberated by in-situ (equation(11)) and migrated (equation(12)) water.After some algebraic calculation this can be written as:

where

The solutions for equations(6)and(7)can be defined in the form of error function as

when 0

wheny>y0,

whereB1,D1,B2,andD2are constants.

Equations(15)and(16)must satisfy the boundary condition given in equation(10)for all values oft,which implies that the depth of the frost front should be related to time as

The value ofmremains constant throughout the whole freezing process if the input parameters in equation(13)are constant.However,in the process of frost heave the segregation potential (SP) decreases with increase in normal pressure on the freezing front.Therefore,the value ofmis updated with the progress of the freezing front.

Using the boundary conditions defined in equation(17),the constantsB1,D1,B2,andD2are calculated in equations(15)and(16).Substituting those values in equations(15)and(16),the temperature distribution in frozen and unfrozen zones can be represented respectively.

when 0

wheny>y0,

Ts,the segregation temperature of the soil,is different from freezing point which is generally 0 °C when the salt content is small in soil.Tscan be obtained from experiment test.

Differentiating equations(18)and(19)with respect toyand equation(17)with respect tot,and then substituting them into equation(13)the following function of one unknownmis obtained.

The formulation described above requires step-by-step computation.The algorithm used to solve the above equations using MATLAB is as follows:

1) For a given location of the frost front calculate the normal pressure acting on frost front and then determine the segregation potential using equation(5).

2) Calculateusing equation(14),and find the value ofmfrom equation(20).

3) Calculate the depth of the frost front (y0) using equation(17).

4) Differentiating equation(16)calculate the temperature gradient at the frost front

5) Substituting the value of gradθinto equation(2),and then using equations(1),(3)and(4)calculate the heave displacement.

wheret1andt2are start and end of the time increment,respectively.

The computation is then shifted to the next time step and the process is repeated until the assigned time is completed.

4 Frost heave calculation using the semi-analytical solution

The progress of frost heave of two different freezing modes to clay soil is simulated in this study.Table 1 shows the parameters used in the analyses and figure 2 shows the predicted and tested frost heave displacement of the clay soil.As shown in this figure,predicted results agree well with the experimental results to the different freezing mode.In other words,the simple semi-analytical solution can successfully predict the observed frost heave for the soil with two different freezing modes,which indicates the feasibility of the semi-analytical solution.

Table 1 Computation parameters

Figure 2 Relation of frost heave amounts with time (calculation and experimental results)

5 Conclusion

Frost heave is an important issue in engineering designs,which is the main problem in frozen ground and has serious effects on engineering construction and maintenance.Currently,several models are available for calculating frost heave,among them the Konrad and Morgenstern segregation potential model is suitable for engineering purposes because of its simplicity.However,the numerical methods are often cumbersome and expensive for using numerical techniques.Therefore,a simplified semi-analytical solution could be useful for many engineering applications.Also,the predicted results agree well with experimental results to the different freezing mode,indicated the feasibility of the semi-analytical solution.

This project is supported by the National Natural Science Foundation of China (No.41371090,No.41023003,No.40901039),and the Project from the State Key Laboratory of Frozen Soil Engineering of China (SKLFSE-ZT-08).All support is gratefully acknowledged.

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