﻿ A mathematical approach to evaluate maximum frost heave of unsaturated silty clay
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 寒旱区科学  2017, Vol. 9 Issue (5): 438-446  DOI: 10.3724/SP.J.1226.2017.00438 0

### Citation

Geng L, Ling XZ, Tang L, et al. 2017. A mathematical approach to evaluate maximum frost heave of unsaturated silty clay. Sciences in Cold and Arid Regions, 9(5): 438-446. DOI: 10.3724/SP.J.1226.2017.00438.
[复制英文]

### Correspondence to

Liang Tang, Professor of School of Civil Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150090, China. E-mail: hit_tl@163.com; gl_hit@163.com

### Article History

Accepted: June 22, 2017
A mathematical approach to evaluate maximum frost heave of unsaturated silty clay
Lin Geng 1,2,3, XianZhang Ling 2,3, Liang Tang 2,3, Jun Luo 2,3, XiuLi Du 1
1. Beijing Key Lab of Earthquake Engineering and Structural Retrofit, Beijing University of Technology, Beijing 100124, China;
2. School of Civil Engineering, Harbin Institute of Technology, Harbin, Heilongjiang 150090, China;
3. State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou, Gansu 730000, China
Abstract: Maximum frost heave of unsaturated frost-susceptible soils, in conjunction with a high water table, is an important consideration for the design of foundations in seasonally frozen regions. Therefore, it is necessary to evaluate accurately and efficiently the maximum frost heave for a given soil. For this purpose, a series of one-sided freezing experiments was conducted on unsaturated silty clay in an open system. Multistage cooling of sufficient duration was applied to the soil sample's top, while constant above-zero temperatures were maintained at the bottom. Then, a simple methodology for calculating maximum frost heave at a given cooling temperature was derived utilizing information obtained within the limited time allotted for each stage. On this basis, an empirical equation for defining maximum frost heave as a function of cooling temperature and overburden pressure was determined. Overall, this study provides a simple and practical procedure that is applicable to the evaluation of maximum frost heave of unsaturated frost-susceptible soils.
Key words: mathematical approach    maximum frost heave    multistage freezing experiment    unsaturated silty clay

1 Introduction

In regions with seasonally frozen ground or permafrost, frost heave is an uneven upward movement of the ground surface during extended exposure to subfreezing temperatures (Cheng and He, 2001; Xu et al., 2001 ). Experimental observations indicate that when fine-grained soils, usually called frost-susceptible soils, are subjected to subfreezing temperatures, substantial frost heave may be observed in the field or laboratory (Konrad and Morgenstern, 1980; Sheng et al., 2013 ). Frost heave is attributed to the solidification of a portion of the water in soil pores and is associated with water migration, particularly in the presence of adequate water supply (Zhao et al., 2009 ; Zhou and Li, 2012).

In practice, the detrimental effect of frost action in frost-susceptible soils is an inevitable and significant concern in geotechnical engineering (Penner and Ueda, 1977; Peppin et al., 2011 ). In general, frost heave in cold regions is a primary cause of damage that can lead to the failure of various types of infrastructure such as roadways, bridges, pipelines, and foundations. Failures primarily occur when footings are located in frost-susceptible soils above the final frost-penetration line (Everett, 1961; Wang and Hu, 2004; Abzhalimov and Golovko, 2009).

For this reason, numerous efforts have been made over decades to explain the frost heave mechanism of a variety of soils; these efforts have included laboratory tests on small specimens, model experiments, and field investigation (Taber, 1930; Singh and Chaudhary, 1995; Ma and Wang, 2012; Zheng and Kanie, 2015). Recently in China, there has been renewed interest in researching the frost-heave mechanism and its adverse effects on engineering in frost-susceptible soils (Zhou and Zhou, 2012; Lai et al., 2014 ; Ma et al., 2015 ).

In this regard, typical methods to estimate the amount of frost heave are based on assumptions about saturated soils; proposed methods include the capillary model, the rigid-ice model, the segregation-potential model, thermo-mechanical models, and the semi-empirical model (O'Neil and Miller, 1985; Fremond and Mikkola, 1991; Nixon, 1991; Black, 1995; Michalowski and Zhu, 2006). Most of these approaches, requiring the use of many parameters, virtually explain the complicated phenomenon of saturated soils to predict accurately the amount of frost heave. However, these models may inherently show reduced accuracy when applied to the engineering problem of predicting frost-heave amounts of unsaturated soils. In this work, a series of one-sided, multistage freezing experiments involving an unsaturated silty clay column in an open system were conducted. Here, experimental results are presented and analyzed. On this basis, a mathematical procedure is proposed to predict the maximum frost heave with satisfactory accuracy and simplicity. Finally, some conclusions are drawn and future research directions proposed.

2 Test and materials 2.1 Sample preparations

Frost-susceptible silt clay obtained from the city of Harbin in Northeast China was used to prepare all reconstituted samples. A grain-size distribution of this clay is shown in Figure 1; and the clay exhibits the following grain-size characteristics: curvature coefficient, Cc=1.25; soil non-uniformity index, 10.04; and plasticity index, 11.57%. All circular solid samples were prepared with a diameter of 10 cm and height of 15 cm. These soil samples were compacted using slight vibration at an optimal moisture content of approximately 14% to achieve a density of about 1,670 kg/m3. The degree of saturation of the soil samples was 59%.

 Figure 1 Grain size distribution of the silty clay
2.2 Equipment

Figure 2 shows the experiment setup. The apparatus includes four main components: temperature control, water supply, loading, and data acquisition (Figure 3). The surface of the inner container was covered with Vaseline to minimize side friction and adhesion between the sample and the container. Before testing, the sample was placed into a plexiglas container with inner dimensions of 20 cm in height and 10 cm in diameter. To enhance the cooling efficiency, the container circumference was insulated with a 50-mm-thick layer of insulating Styrofoam. All prepared specimens in the container were placed in a temperature chamber to avoid heat loss and kept at a constant environmental temperature of −2 °C.

 Figure 2 Experiment setup

 Figure 3 Schematic diagram of the experiment apparatus

Alcohol, at temperatures ranging from −40 °C to +30 °C with an accuracy of ±0.1 °C, was used as the controlled liquid circulating through the plate ( Figure 3). Throughout the test, the tops of the samples were the cooling end, subjected to subzero temperatures (Tc) through a copper plate; the bottoms of the samples were the warm end, subjected to the above-zero temperature (Tw), 2 °C, through another copper plate.

Additionally, the water-supplement system had a Mariotte flash connected to the bottom plate through a plastic tube. In all experiments, an unpressurized water-supplement process was employed.

2.3 Determination of freezing duration and test procedures

There were six freezing-test cases (i.e., Cases FT0 to FT5) conducted in this study (Table 1). Various overburden-stress conditions were enforced on the top of the soil column. All test specimens began to freeze after standing for 24 hours under corresponding load with no water supplement. A displacement transducer was installed at the plate to continuously monitor the accumulated frost heave at the specimen surface. The side of the container included four holes, accommodating four thermistors for each sample. These thermistors were inserted laterally through the holes, approximately 3 cm into the sample, at 3-cm intervals to measure internal soil temperature (Tsoil); two additional thermistors were placed at the top and bottom of each sample.

Table 1 Experiment cases

Usually, evaluating maximum frost heave requires the maximum possible time duration at a given Tc. To minimize and reasonably determine freezing duration, the top of the sample in Case FT0 was successively cooled to a Tc of −5 °C over a 168-hour period. The frost heave and frost-heave rate with time elapsed in Case FT0 are presented in Figure 4. For ease of presentation, experimental data for frost heave is marked as S(t), where t is the freezing duration. From Figure 4, frost heave, SFT0(t), accumulated with t and approached a maximum value as the t approached 168 hours of freezing. Nevertheless, the frost-heave rate, dS(t)/dt, gradually reduced as t lapsed; and it tended toward zero by the end of the freezing period (i.e., t=168 hours). Using data of SFT0(t), a straight line is obtained by plotting 1/S(t) versus 1/t in Figure 5. Thus, the relationship between S(t) and t through the freezing period could be approximately expressed using a hyperbolic form.

 Figure 4 Variation of frost heave and frost heave rate with time elapsed in Case FT0

 Figure 5 Variation of 1/S(t) versus 1/t (t varies in the range of 0~12 hours)

Further, the correlation between frost heave and freezing time for various ranges of elapsed time is shown in Figures 6a to 6g. From these figures, we see a correlation by using just the data from 0 to 72 hours; thus, the validity of the function and parameters used is confirmed. For this purpose, the 72-hour freezing duration for every stage in the following multistage freezing test cases was determined.

 Figure 6 Correlation between frost heave and time elapsed using data from (a) 0~24 hours; (b) 0~48 hours; (c) 0~72 hours; (d) 0~96 hours; (e) 0~120 hours; (f) 0~144 hours; and (g) 0~144 hours

In Cases FT1 to FT5, identical thermal conditions, i.e., Stage 1 to Stage 4 (Figure 7), were applied to the samples. The top of each sample was successively cooled through one copper plate to various freezing temperatures of 0 °C, −5 °C, −10 °C, −15 °C, and −20 °C, stage by stage; and each freezing stage had a 72-hour duration.

3 Mathematical procedures 3.1 Analysis of experimental results

Figure 8 comprises the frost-heave time histories (i.e., SFT0(t) and SFT1(t)) in cases FT0 and FT1. It was found that the maximum frost heave was not attained by the end of every freezing stage in Case FT1. The recorded frost heave, S(t), versus time for Cases FT1 to FT5 is illustrated in Figure 9, in which each freezing interval is 72 hours. For each freezing stage, a similar trend of the frost heave under various overburden stresses was observed, at first accumulating rapidly and then more slowly until freezing ended. As Tc was decreased from one stage to the next, frost heave demonstrated a slight but abrupt increase.

 Figure 8 Variation of frost heave with time elapsed in Cases FT0 and FT1

 Figure 9 Variations of measured frost heave and freezing front of soil columns with time elapsed in Cases FT1~FT5

Frost-heave amounts under overburden stresses (σv) of 0.000 MPa, 0.015 MPa, 0.028 MPa, 0.041 MPa, and 0.054 MPa attained 18.1 mm, 15.1 mm, 13.0 mm, 11.4 mm, and 10.0 mm, respectively, by the end of freezing (i.e., t=288 hours). Less frost heave occurred as the overburden stress (σv) increased, indicating that the reduction in frost heave maybe due to increasing σv at the sample surface. It may be that water was inhibited from entering into the soil due to the increase of σv.

3.2 Formulation of equation for frost-heave increment due to change of Tc

Maximum frost heave is a critical parameter for the design of foundations in frozen regions (Singh and Chaudhary, 1995; Peppin et al., 2011 ; Sheng et al., 2013 ). A new approach is firstly proposed to evaluate maximum frost heave by determining the appropriate equation for the relationship between incremental frost heave (ΔS) induced by the change of Tc and t, utilizing information obtained during the each stage.

From Figure 8, variations of ΔS1,2(t) (i.e., ΔS1,2(t)=SFT0(t)−SFT1(t)) and the associated frost-heave increment rate (i.e., dΔS1,2(t)/dt) are obtained. These variations are presented in Figure 10, where the t varies in the range of 72~144 hours. Using data for ΔS1,2(t) (Figure 10), a straight line is obtained by plotting 1/ΔS1,2(t1) versus 1/t1 (i.e., t1=t−72, t varies over the range of 72~144 hours), as presented in Figure 11. We have

 Figure 10 Variation of frost heave increment and frost heave increment rate with time elapsed (t varies in the range of 72~144 hours)

 Figure 11 Variation of 1/ΔS1,2(t1) versus 1/t1 (i.e., t1=t−72, t varies in range of 72~144 hours)
 $\frac{1}{{\Delta {S_{1,2}}({t_1})}} = a + b\frac{1}{{{t_1}}}$ (1)

Which can be expressed as

 $\Delta {S_{1,2}}({t_1}) = \frac{{{t_1}}}{{a + b{t_1}}}$ (2)

where the parameters a and b are undetermined coefficients. The values of a and b can be obtained from the slope and intercept of the straight line. Even with sufficient time, the ΔS1,2(t1) induced by the change of Tc from −5 °C to −10 °C can be fitted with the hyperbolic curve.

3.3 Estimation method for maximum frost heave

In this section, the proposed method for estimation of maximum frost heave is outlined. The above investigations have clarified that a hyperbolic curve can be used to fit the frost-heave increment (Figures 6c and 10) induced by the change of Tc.

It should be noted that ΔS0,1(t) (ΔS0,1(t)=S(t)−S(0), Figure 12a) is a known quantity, where t varies from 0 to 72 hours. After careful inspection, we have

 Figure 12 Frost heave increase caused by the change of freezing temperature at the cooling end under the tested specimens. (a) Case FT1; (b) FT2; (c) FT3; (d) FT4; and (e) FT5
 $\Delta {f_0}(t) = \frac{t}{{{a_0} + {b_0}t}}$ (3)

where the parameters a0 and b0 are undetermined coefficients. A straight line can be obtained by plotting 1/Δf0(t) versus 1/t. The value of a0 and b0 can be obtained from the slope and intercept of the straight line. This approach may also be used to calculate the undetermined coefficients in the hyperbolic curves' equations defining the other stages.

Theoretically, as time t→∞, the maximum frost heave, Smax,1, approaches that attained at Tc of −5 °C. Thus, it could be assessed from Equation (3) as

 ${S_{\max ,1}} = \mathop {\lim }\limits_{t \to \infty } \Delta {f_0}(t) = \frac{1}{{{b_0}}}$ (4)

The predicted frost heave continues to grow at the Tc of −5 °C where t exceeds 72 hours, and the contribution of relevant frost heave can be extracted using Equation (3). Therefore, to remove this contribution, the ΔS1,2(t) occurring during the change of Tc from −5 °C to −10 °C can be obtained from Equation (5) as follows:

 $\Delta {S_{1,2}}(t) = S(t) - \Delta {f_0}(t)$ (5)

where t varies in the range 72~144 hours. Further, ΔS1,2(t) is a known quantity (Figure 12a).

Therefore, a similar hyperbolic equation is utilized to fit the discrete data of ΔS1,2(t) below:

 $\Delta {f_1}({t_1}) = \frac{{{t_1}}}{{{a_1} + {b_1}{t_1}}}$ (6)

where t1 (i.e., t1=t−72) is t at the beginning of Stage 2, and parameters a1 and b1 are undetermined coefficients.

Combined with Equations (3) and (6), maximum frost heave, Smax,2, at Tc of −10 °C is described in the following form:

 ${S_{\max ,2}} = \mathop {\lim }\limits_{t \to \infty } \Delta {f_0}(t) + \mathop {\lim }\limits_{t \to \infty } \Delta {f_1}(t) = \sum\limits_{j = 0}^1 {\frac{1}{{{b_j}}}}$ (7)

Maximum frost heave, Smax,3 and Smax,4 at Tc of −15 °C and −20 °C, respectively, can be similarly calculated.

To remove the overall effect of the change of Tc on the frost heave, we can separate out the effect of the change in Tc during Stage 1 on the increase of frost-heave amount by extrapolating the curve obtained from Stage 1.

Similarly, the effect of Stage 2 is extrapolated to isolate the effect of the change in Tc on the next freezing stage. In this analysis, in addition to the assumptions made in the extrapolation, we have also supposed that the effect of Tc on frost heave is additive. On this basis, Figures 12a to 12e, present the recorded and predicted frost-heave increment at each freezing stage under various overburden stresses.

3.4 Empirical equation between Smax,n and Tc

Finally, frost heave can be assumed as a function of σv and Tc. By using the above-mentioned mathematical procedure, data for Smax,n (n=1, 2, 3, and 4) can be obtained; and eventually, the relationship between Smax,n and Tc under various overburden stresses, as shown in Figure 13, may be derived. The hyperbolic curve is still chosen to fit data for Smax,n during each freezing stage in this uniform form:

 Figure 13 Relationship between Smax,n and freezing temperature (Tc) at the cooling end under various overburden stresses
 ${S_{\max }}({T_{\rm c}}) = \frac{{{T_{\rm c}}}}{{m + n{T_{\rm c}}}}$ (8)

where Smax(Tc) is the fitting formula and represents the maximum frost heave at Tc. Parameters m and n are undetermined coefficients, which as the function of the σv, are expressed by Equations (9) and (10) as follows:

 $m = - {\rm{2}}{\rm{.05}}{{\rm{e}}^{{\rm{0}}{\rm{.02}}{\sigma _v}}}$ (9)
 $n = {\rm{0}}{\rm{.42 + }}\frac{{{\sigma _v}}}{{{\rm{205}}{\rm{.4 + 0}}{\rm{.5}}9{\sigma _v}}}$ (10)
4 Conclusions

A series of laboratory tests was conducted to investigate the frost heave of Harbin silty clay with free access to an external water source under one-sided, multistage freezing conditions. The results and conclusions are summarized as follows.

(1) Frost heave at the ground surface decreases due to increased overburden pressure, which may act as a constraint on frost heave. Frost heave can have more abrupt increases from one stage to the next. Overall, the evolution of frost-heave amount goes through two phases: rapid accumulation at the beginning of the freezing stage and then slow increase during the remainder of it.

(2) Maximum frost heave is not attained under test conditions with a 72-hour freezing period for each stage. Sufficient freezing time is required to reach maximum frost heave under a given thermal boundary setting. Further, a hyperbolic equation can be used to approximate the relations between incremental frost heave and freezing time, as well as those between frost-heave amount and freezing time at each freezing stage.

(3) Based on the assumption of the effect of the freezing temperature at the cooling end and its change on frost heave being an isolated additive, a simple method considering the effect of overburden pressure is proposed to estimate maximum frost heave. Meanwhile, the magnitude of maximum surface frost heave is found to be a function of cooling temperature and overburden stresses.

(4) This investigation provides a practical method to estimate maximum frost heave with accuracy sufficient for engineering. Future studies should address other important factors and introduce them into the theoretical model and framework. Including these additional factors can achieve a more general understanding of the frost-heave mechanism in unsaturated frost-susceptible soils.

Acknowledgments:

We gratefully acknowledge support for this research from the State Key Program of National Natural Science of China (Grant No. 41430634), the National Natural Science Foundation of China (Grant Nos. 41702382, 51578195, 51378161, and 51308547), the Foundation Project Program 973 of China (No. 2012CB026104), and the State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining & Technology (Grant No. SKLGDUEK1209).

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