assessmentforthermalconductivityoffrozensoilbasedon...

Research ArticleAssessment for Thermal Conductivity of Frozen Soil Based onNonlinear Regression and Support Vector Regression Methods

Fu-Qing Cui 12 Wei Zhang 1 Zhi-Yun Liu 1 Wei Wang 1 Jian-bing Chen 2

Long Jin 2 and Hui Peng 2

1College of Geological Engineering and Geomatics Changrsquoan University Xirsquoan Shaanxi 710054 China2State Key Laboratory of Road Engineering Safety and Health in Cold and High-Altitude RegionsCCCC First Highway Consultants Co Ltd Xirsquoan Shaanxi 710065 China

Correspondence should be addressed to Zhi-Yun Liu dcdgx33chdeducn

Received 19 July 2020 Revised 5 August 2020 Accepted 19 August 2020 Published 28 August 2020

Academic Editor Yanjun Shen

Copyright copy 2020 Fu-Qing Cui et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e comprehensive understanding of the variation law of soil thermal conductivity is the prerequisite of design and con-struction of engineering applications in permafrost regions Compared with the unfrozen soil the specimen preparation andexperimental procedures of frozen soil thermal conductivity testing are more complex and challengeable In this workconsidering for essentially multiphase and porous structural characteristic information reflection of unfrozen soil thermalconductivity prediction models of frozen soil thermal conductivity using nonlinear regression and Support Vector Regression(SVR) methods have been developed -ermal conductivity of multiple types of soil samples which are sampled from theQinghai-Tibet Engineering Corridor (QTEC) are tested by the transient plane source (TPS) method Correlations of thermalconductivity between unfrozen and frozen soil has been analyzed and recognized Based on the measurement data of unfrozensoil thermal conductivity the prediction models of frozen soil thermal conductivity for 7 typical soils in the QTEC areproposed To further facilitate engineering applications the prediction models of two soil categories (coarse and fine-grainedsoil) have also been proposed -e results demonstrate that compared with nonideal prediction accuracy of using watercontent and dry density as the fitting parameter the ternary fitting model has a higher thermal conductivity prediction accuracyfor 7 types of frozen soils (more than 98 of the soil specimensrsquo relative error are within 20) -e SVR model can furtherimprove the frozen soil thermal conductivity prediction accuracy and more than 98 of the soil specimensrsquo relative error arewithin 15 For coarse and fine-grained soil categories the above two models still have reliable prediction accuracy anddetermine coefficient (R2) ranges from 08 to 091 which validates the applicability for small sample soils -is study providesfeasible prediction models for frozen soil thermal conductivity and guidelines of the thermal design and freeze-thaw damageprevention for engineering structures in cold regions

1 Introduction

With the increasing frequency of human activities andthe trend of global warming the temperature sensitivepermafrost is changing significantly most of which are incritical equilibrium state or even gradual degradation[1ndash3] -e widely distributed permafrost in the Qinghai-Tibet Plateau exhibits stronger temperature sensitivityand faster temperature rise rate than other permafrostregions with the same latitude [4ndash6] -e dynamic deg-radation process of frozen soil has a great impact on the

stability of buildings in cold regions [7ndash10] In themeanwhile it will also lead to local environmentaldegradation or further affect the global climate carboncycle and climate change [11ndash14] -erefore many re-searchers had investigated the distribution stability andthermal regime of permafrost [15ndash17] -e thermalconductivity is one of the important thermophysicalparameters for reconstructing the past and predicting thetemperature status of permafrost under climate changeconditions and also determining a parameter for engi-neering design in cold regions [18 19] -erefore the

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8898126 12 pageshttpsdoiorg10115520208898126

comprehensive understanding of variation law of soilthermal conductivity is one of the important tasks inpermafrost research

-ermal conductivity is an inherent parameter tocharacterize the heat transfer performance of soil which canusually be obtained by means of experimental test andprediction models [20ndash22] -e test of thermal conductivitycan generally be divided into the steady-state method andtransient method [23 24] Specifically it includes steady-state comparison method steady-state heat flow metermethod transient hot wire method transient heat pulsemethod and TPS method [25] However the disadvantagesof the experimental test are usually strict technical re-quirements long time consumption and high cost andsometimes limited test results cannot satisfy the needs ofpractical applications -erefore as reported [26 27] manysoil thermal conductivity prediction models had beenproposed Empirical models are usually based on statisticalanalysis of thermal conductivity experimental results anduse water content dry density porosity and other relatedsoil properties as fitting parameters [28ndash31] Kersten [28]analyzed the test results of 19 soil types and established theempirical formula of thermal conductivity using watercontent and dry density Johansen [29] proposed the conceptof normalized thermal conductivity and given the inter-polation calculation model of soil thermal conductivitybased on the relationship between normalized coefficientand soil saturation Lu et al [30] performed a series ofthermo-TDR tests on twelve natural soils and proposed alinear prediction model across a wide range of soil moisturecondition Yan et al [31] developed a generalized effectivesoil thermal conductivity model for soils of various texturesfrom dry to saturation -en many scholars improved themodel for wider applicability and higher prediction accuracy[32ndash35] At the same time many scholars have establishedtheoretical predictive models of soil thermal conductivitybased on specific physical theory and models [36ndash38]Farouki [39] and Xu et al [40] respectively established theweighted calculation model and geometric average calcu-lation formula based on the volume ratio and thermalconductivity of each component of the frozen soil on thebasis of predecessors Considering for the interactionsamong soil particle water and air of a soil unit cell Haigh[41] derived a theoretical thermal conductivity model forsand soil In recent years with the maturity of machinelearning and its advantages in dealing with complex non-linear problems many researchers had adopted it to theprediction of soil thermal conductivity [42 43] For instanceBang et al [44] investigated the application effect of linearregression and various machine learning methods in theprediction of thermal conductivity of compacted bentoniteand verified the feasibility and superiority of the machinelearning method in the prediction of the soil thermalconductivity

Compared to unfrozen soil the specimen preparationand experimental procedures of frozen soil thermal con-ductivity testing are more complex and challengeable And itcan be found that the prediction accuracy of the frozen soilthermal conductivity predictive model is lower than that of

unfrozen soil [40] Meanwhile as a complex multiphasecomposition previous research indicated that the thermalconductivity of frozen soil was associated with many factors[26 27 45]-e existing frozen soil thermal conductivityempirical model (usually using dry density water contentporosity etc as the fitting parameters) cannot compre-hensively consider the influences of complex multiphase andporous structural characteristics of soil -erefore in thepresent work prediction models of frozen soil thermalconductivity using nonlinear regression and SVR methodshave been developed considering for essentially multiphaseand porous structural characteristic information reflectionof unfrozen soil thermal conductivity -ermal conductivityof various types of soil samples which are sampled from theQTEC are tested by the TPS method Correlation analysis ofthermal conductivity of unfrozen and frozen soil has beenadopted Based on the measurement data of unfrozen soilthermal conductivity the ternary fitting and SVR predictionmodels of frozen soil thermal conductivity for the typicalsoils in the QTEC are proposed To further facilitate engi-neering applications the prediction models of coarse andfine-grained soil categories have been proposed Further-more the prediction models of two soil categories have alsobeen used to predict the frozen soil thermal conductivity ofsmall size soil samples in the QTEC

2 Materials and Methods

21 Soil Sample

211 Collection of Soil Samples -e soil samples of thermalconductivity measurement were collected from the drillingspecimens of Qinghai-Tibet expressway geological explo-ration project which was implemented by the CCCC FirstHighway Consultants Co Ltd and Eco-Environment andResources Northwest Institute of CAS from September 2017to June 2018 As shown in Figure 1 the sampling spot mainlylocates in the permafrost regions between Xidatan andTanggula Mountain (the corresponding Qinghai-TibetHighway mileage is K2870simK3307) Every drilling spot issampled for different depths and the sampling depth variesfrom 05m to 40m Total number of 638 unfrozen soilsamples and 860 frozen soil samples are tested

Soil types of testing specimens are determined by theclassification criteria of geotechnical engineering Figure 2shows the statistic results of different types of soil samples(only numbers more than 15 are given) It can be seen thatsilty clay is the most widely distributed frozen soil type in theQTEC and the following soil types are sandy and gravel soilBecause some soil samples with large water (ice) content areimpossible to implement unfrozen soil thermal conductivitytesting a total of 609 specimens tested for both frozen andunfrozen are selected as the research objects in this workand the statistical distribution of various soil samples isshown in Table 1 It can be seen that the proportions of siltyclay silt fine sand gravel sand boulder breccia and allweathered rock soils accounting for 862 are defined as thetypical analytical soil types in the following predictive modelresearch

2 Advances in Civil Engineering

212 Water Content and Dry Density Distribution of SoilSamples -e dry density (ρd) and water content (w) of soilsamples are statistically calculated and the accumulativeproportion distribution is shown in Figure 3 It shows thatthe water content and dry density distributions are signif-icantly correlated with soil properties -e dry density isnegatively correlated with soil particle size and basic ordersof dry density of the samples are silt silty clay weatheredrock fine sand gravel sand breccia and boulder soils -e

average dry density of above soil samples are 164 169 179183 187 192 and 206 gcm3 and the main distributionintervals (proportion within 10sim90) are 140sim181147sim191 153sim204 161sim203 166sim215 166sim220 and181sim231 gcm3 Likewise the orders of water content aresilty clay silt weathered rock fine sand breccia gravel sandand boulder -e corresponding average values are 19331905 1604 1327 1088 1066 and 928 andmain distribution interval are 1168sim267 1103sim

80degE 90degE 100degE

90degE 100degE

20degN

25degN

30degN

35degN

91deg0prime0PrimeE 92deg0prime0PrimeE 93deg0prime0PrimeE 94deg0prime0PrimeE

92deg0prime0PrimeE 93deg0prime0PrimeE 94deg0prime0PrimeE

Permafrost regionHighwayProvincial boundary

Qinghai-tibet highwayPermafrostTalik

Exploration pointsPlace nameQinghai-tibet highway

25degN

30degN

35degN

40degN

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

N

S

W E

Figure 1 Schematic of the sampling spot along the QTEC [16]

32

311

71

43

82

94

82

69

23

42

43

15

Clay

Silty caly

Silt

Silty sand

Medium fine sand

Gravel sand

Breccia

Boulder

Gravel

Weathered phyllite

Weathered mudstone

Weathered slate

50 100 150 200 250 300 3500Number of specimens

Figure 2 Statistic numbers of test samples for certain soil type

Advances in Civil Engineering 3

2662 396sim259 47sim2016 46sim174 466sim1864 and 4sim152 respectively which basically coversall the water content ranges from dry to saturated

22 ExperimentalMethods andApparatus -e TPS methodis utilized as the thermal conductivity testing measure forall soil samples-e experimental apparatus is the Hot Disk1500s -ermal Conductivity Analyzer (as shown inFigure 4) which has plusmn3 measurement accuracy -eprinciple of the TPS method for measuring the thermalproperties of materials is based on the transient temper-ature response of a step-heated disc-shaped heat source inan infinite medium No 4922 Kapton film probe with

radius of 294mm is used to make specimen as large aspossible and the thermal conductivity (λ) is calculated byline equation of temperature variation values and di-mensionless time constant which is measured by the filmsensor

ΔT(τ) P0

π32rλ

D(τ) (1)

where ΔT(τ) is the temperature variation value P0 is thetotal output of power and r is the radius of Kapton filmsensor and D(τ) is the dimensionless time constant Duringthe test experimental soil samples are formed into 3 cmheight and 8 cm diameters column and the film probe is

Table 1 Statistical distribution of various soil specimens

Soil type Numbers Proportion ()Sampling depth (m)

le1 1sim5 5sim10 10sim20 ge20

Fine-grained soil

Clay 24 394 1 6 3 9 5Silty clay 202 3317 10 63 40 69 20

Silt 52 854 6 23 10 8 5Weathered slate 10 164 2 4 1 2 1

Weathered phyllite 29 476 0 7 10 10 2Weathered sandstone 6 099 0 4 0 1 1

Weathered marl 5 082 0 1 2 1 1Weathered mudstone 23 378 0 7 5 11 0

Coarse-grained soil

Silty sand 23 378 1 10 4 6 2Fine sand 49 805 14 19 9 2 5

Medium sand 5 082 0 1 3 1 0Coarse sand 3 049 0 3 0 0 0Gravel sand 45 739 5 28 5 5 2Boulder 49 805 3 31 5 5 5Breccia 55 903 7 32 6 8 2

Gravel + pebble 10 164 0 4 1 3 2Gravel soil 19 312 1 12 1 5 0

Dry

den

sity

(gmiddotcm

ndash3)

SiltGravelBreccia

Silty clayFine sandBoulder Weathered rock

14

16

18

20

22

24

10 20 30 40 50 60 70 80 90 1000Cumulative proportion ()

(a)

SiltGravelBreccia


Wat

er co

nten

t (

)

0

5

10

15

20

25

30


(b)

Figure 3 (a) Dry density and (b) water content probability distribution of soil samples


sandwiched between two soil samples and fixed with astainless steel sample holder -e detailed experimentalprocedures can be obtained in our previous work [46]

23 Nonlinear Regression Model Previous studies haveshown that dry density and water content have significantinfluence on soilrsquos thermal conductivity Partial correlationanalysis between frozen soil thermal conductivity of drydensity water content and unfrozen soil thermal conduc-tivity for two soil categories have been performed and theresults are shown in Table 2 -e results exhibit that drydensity and water content as well as unfrozen soil thermalconductivity (λu) have significant positive correlations withfrozen soil thermal conductivity (λf) In particular it shouldbe noted that the correlation between the frozen and un-frozen soil thermal conductivity for both coarse and fine-grained soils are very high (088 and 093 respectively) Itcan be inferred that considering for the essential soilproperties information reflection of unfrozen soil thermalconductivity it would be possible to utilize the unfrozen soilthermal conductivity to predict frozen soil thermalconductivity

Curve estimation of the fitting relationship between thefrozen soil thermal conductivity and three factors has beentaken and it is found that the thermal conductivity of thefrozen soil is linear with the unfrozen soil thermal con-ductivity while its relation with dry density and watercontent are in the form of logarithmic functions-e ternaryfitting formulas of frozen soil thermal conductivity for 7typical soils in the QTEC are given as follows

λf a + bλu + cln(w) + dln ρd( 1113857 (2)

where a b c and d are the fitting coefficients of the equation-e detail fitting results of 7 typical soils are listed in Table 3It can be seen that the determine coefficients (R2) of 7 typicalsoils range from 076sim093 while the analog effect of sandysoils are better than other soil types

In order to analysis the predictive effect of the ternaryfitting model the estimated frozen soil thermal conductivitybased on the Kerstenmodel [28] Gangadhara Rao and Singhmodel [47] and ternary fitting model are plotted against themeasured value of testing dataset -e empirical formulasproposed by Kersten and Gangadhara Rao and Singh used

dry density and water content as fitting parameters while theapplicable soil types are silt fine sand and gravel sand soilsetc -e comparison results of the estimation values of threemodels and the experimental values are shown in Figure 5 Itcan be seen that the proposed ternary fitting model performsbest among all three models as most of the predictive valuesare within the plusmn10 relative error line However the pre-dictive values of Kersten and Gangadhara Rao and Singhmodels are generally overestimated for all soil types and theirprediction errors reach up to 40 -erefore it can beconcluded that compared with nonideal prediction accu-racy of binary fitting models the ternary fitting model usingthe unfrozen soil thermal conductivity as the fitting pa-rameter has a higher thermal conductivity predictionaccuracy

3 Development of SVR Models

31 7eory and Performance Evaluation of SVR ModelSVR method is a type of supervised machine learningmethod and proposed by Vapnik et al [48 49] which isbased on statistical learning theory -e SVR method useskernel functions to map low-dimensional nonlinear prob-lems to high-dimensional space to achieve linear separabilityand then to seek linear regression equations to fit sampledata (as shown in Figure 6) -e regression function can beexpressed as follows

f(x) KTφ(x) + m (3)

where x is the input vector K is the weight vector m is theoffset vector and φ(x) is eigenvector that maps input data tothe high-dimensional space -e Gaussian radial basisfunction is chosen as the kernel function in the calculation

-e SVRmethod is based on the principle of minimizingstructural risk transforming the linear regression probleminto the following optimization problem and then the valuesof K and m can be determined [48]

Minimize12K

TK + t 1113944

n

i1ξi + ξ lowasti( 1113857

Subject to

yi minus KTφ(x) minus mle ε + ξi

KTφ(x) + m minus yi le ε + ξ lowasti

ξi ξlowasti ge 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where t is the penalty parameter which is greater than 0 ε isthe insensitive loss coefficient and ξi and ξ lowasti are slackvariables

-e predictive results of SVR models for silt and gravelsand soils are shown in Figure 7 It can be seen that thepredictive results of the SVR model are in good agreementwith the experimental results (R2 086 for silt and R2 094for gravel sand) -e SVR model exhibits remarkable pre-diction accuracy and most of predictive values are within theplusmn10 error bars which validates their predictive availability

Figure 4 -ermal conductivity test system


KerstenTernary fitting

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)

09 12 15 18 21 24 27 30 33 36 3906λf-experimental Wmiddot(mndash1middotKndash1)

Gangadhara Raoplusmn10

(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


KerstenTernary fitting Gangadhara Rao

plusmn10

(b)

Figure 5 Continued

Table 2 Partial correlation analysis between frozen soil thermal conductivity dry density water content and unfrozen soil thermalconductivity

Soil categories Related variables Control variables Unfrozen soil thermal conductivity Frozen soil thermalconductivity

Fine-grained soil

Water content Dry density 0312 0450Unfrozen soil thermal conductivity mdash 0880

Dry density Water content 0456 0385Unfrozen soil thermal conductivity mdash 0891

Coarse-grained soil



Table 3 Ternary fitting result of frozen soil thermal conductivity for 7 typical soils in the QTEC

Soil typeFitting coefficients

R2

a b c dSilty clay minus0703 1291 0238 0007 0759Silt minus1358 1283 0424 0248 0792Weathered rock minus0102 1370 0183 minus0854 0895Fine sand minus0220 1290 0146 minus0429 0908Gravel sand minus0617 1272 0254 minus0138 0933Boulder minus1257 1049 0497 0702 0865Breccia minus1282 0884 0459 1239 078


06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)

09 12 15 18 21 24 27 30 3306λf-experimental Wmiddot(mndash1middotKndash1)

KerstenTernary fitting plusmn10

(d)

Figure 5 A comparison ofmeasured and estimated frozen soil thermal conductivity (a) Slit (b) Fine sand (c) Gravel sand (d) Fine-grained sand

f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m

Figure 6 Schematic diagram of the SVR method

Ternary fittingSVR (R2 = 086)plusmn10

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)

09 12 15 18 21 24 27 30 33 3606λf-experimental Wmiddot(mndash1middotKndash1)

(b)

Figure 7 Predictive results of SVR models for (a) slit and (b) gravel sand


and engineering suitability Furthermore it also should benoted that evaluation spots in Figure 7 is distributed uni-formly throughout the thermal conductivity interval (λf

ranges 06sim33W(mmiddotK)) which proves the broader appli-cation range of the SVR model

32 Comparison of SVR Model with Ternary Fitting Model-e comparison of predictive results of the SVR model andternary fitting model are shown in Figure 8 It can be seenthat the fitting degree of the SVR model is much higher thanthat of the ternary fitting model Especially for the silty clayand breccia soils the predictive improvement effect of theSVRmethod is more obvious For improvement at both high

and low thermal conductivity intervals the R2 increases to084 and 085 respectively It can be considered that for thecomplex multiphase and porous structural characteristics ofsoil the SVRmodel has more advantages than the traditionalempirical formula model

-e fitting results of the ternary fitting method and SVRmethod are statistically calculated -e distributions of R2sample probability of mean absolute percent error (MAPE)less than 10 and maximum relative error (δmax) are shownin Table 4 It can be seen that compared with ternary fittingmethod the minimum R2 of the SVR method is 084 (siltyclay) and the R2 of 4 soil types is above 09 -e 7 typicalsoilsrsquo average δmax of the SVR model is 181 and ternaryfitting model is 219 Moreover the proportion of MAPE

Ternary fitting (R2 = 076)SVR (R2 = 084)plusmn10

06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)

09 12 15 18 21 24 27 3006λf-experimental Wmiddot(mndash1middotKndash1)

(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)

Ternary fitting ((R2 = 078)SVR (R2 = 085)plusmn10

06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)

Figure 8 Comparison of predictive results of the SVR model and ternary fitting model (a) Slit (b) Fine sand (c) Boulder (d) Breccia


less than 10 of SVR model are larger than that of ternaryfitting model for all 7 typical soils which further indicatethat the SVR model has better performance than the ternaryfitting model -e sample probability of the MAPE within20 of the ternary fitting method is more than 98 whilethe sample probability of the MAPE within 15 of the SVRmethod is more than 98 -erefore it can be consideredthat the ternary fitting method has a predictive accuracy ofplusmn20 and the SVR method has a predictive accuracy ofplusmn15

4 Generalized PredictionModel for Coarse andFine-Grained Soil

Based on the previous analysis it can be found that some soiltypes such as clay silty sand and gravel soil cannot beanalyzed by the nonlinear regression or SVR method for

their small sample sizes Furthermore the overly detailedprediction models of subdivided soil types will also induce acertain degree of inconvenience in engineering application-us generalized frozen soil thermal conductivity predic-tion models for coarse and fine-grained soil categories havebeen developed -e R2 and fitting coefficient values of theternary fitting model and SVR model are listed in Table 5and the comparison of predictive results of the above twomodels for coarse and fine-grained soil categories are shownin Figure 9 It shows that both the ternary fitting model andSVR model have acceptable fitting effect for two soil cate-gories and most evaluation spots of two models are withinor near the plusmn10 error -e R2 of the ternary fitting modelfor the coarse and fine-grained soil categories is 080 and089 respectively while the SVR model is 085 and 091which proves the feasibility of soil categories predictivemodels Nevertheless it also can be found that the SVR

Table 4 Statistics of prediction results of the ternary fitting model and SVR model

Predictive model Statistical values Silty clay Silt Weathered rock Fine sand Gravel sand Boulder Breccia

Ternary fitting methodR2 076 079 09 091 093 087 078

MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004

Table 5 Fitting results of coarse and fine-grained soil categories

Soil typeTernary fitting SVR

a b c d R2 R2

Fine-grained soil minus0770 1355 0253 minus0102 080 085Coarse-grained soil minus0788 1190 0293 0225 089 091

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)

Figure 9 Comparison of predictive results of ternary fitting and SVR models for (a) fine-grained sand and (b) coarse-grained sand


method still has a better performance despite the increase ofvariety and complexity of soil samples

-e MAPE distribution of two prediction models ofcoarse- and fine-grained soil categories is shown inFigure 10 It can be seen that the prediction accuracy ofcoarse-grained soil category is generally higher than thatof fine-grained soil category -e MAPE of coarse-grained soil category distributes uniformly in eachthermal conductivity range and its average value is lowerthan 10 which means that the predictive results of twoprediction models for coarse-grained soil category aremore reliable in the entire frozen soil thermal

conductivity distribution interval Additionally itshould be mentioned that the large error intervals of fine-grained soil category are mainly concentrated in0sim12W(mmiddotK) and 27sim36W(mmiddotK) ranges and theproportion of the two ranges are 667 and 26 (total of927) which is roughly excluded from its main thermalconductivity distribution range -erefore consideringfor applicability scope and convenience the fine-grainedsoil category prediction models definitely have certainapplication values in engineering

-e prediction models of two soil categories are appliedto small sample soils (clay silty sand and gravel soil) andthe results are shown in Figure 11 It can be seen that bothternary fitting model and SVRmodel exhibit good predictiveaccuracy for all three small sample soils which supportsapplicability of prediction models Furthermore it can beclearly noted that the SVR model has better predictiveperformance for clay soil which testify that the machinelearning method has broad application prospects for itscapability of nonlinear and complex relationship capturingand imitating

5 Conclusions

In present work a large-scale soil thermal conductivity testhas been conducted by the TPSmethod Correlation analysisof thermal conductivity of unfrozen and frozen soil has beenadopted Based on the measurement data of unfrozen soilthermal conductivity the ternary fitting and SVR predictionmodels of frozen soil thermal conductivity for the typicalsoils in the QTEC are proposed considering for the essentialsoil properties information reflection of unfrozen soilthermal conductivity Furthermore to facilitate engineeringapplications the prediction models of coarse and fine-grained soil categories have also been proposed and com-pared -e results show that

(1) Compared with nonideal prediction accuracy ofusing water content and dry density as the fittingparameter the ternary fitting model has a higherthermal conductivity prediction accuracy for typicalsoil types in QTEC and more than 98 of soilspecimensrsquo relative error are within 20

(2) With the capability of nonlinear and complex rela-tionship capturing and imitating the SVRmodel canfurther improve the frozen soil thermal conductivityprediction accuracy and more than 98 of the soilspecimensrsquo relative error are within 15

(3) For coarse- and fine-grained soil categories theabove ternary fitting and SVR models still have re-liable prediction accuracy and their R2 ranges from08 to 091 which validates the applicability for smallsample soils

Data Availability

All the data used to support the findings of this study areavailable from the corresponding author upon request

Fine-grained soil(ternary fitting)Coarse-grained soil(ternary fitting)

Fine-grained soil (SVR)Coarse-grained soil (SVR)

0

3

6

9

12

15

18

MA

PE (

)

12 16 20 24 28 3208Thermal conductivity Wmiddot(mndash1middotKndash1)

Figure 10 MAPE distribution of ternary fitting and SVR modelsfor coarse- and fine-grained soil categories

Silty sand (SVR)Gravel soil (SVR)

Clay (SVR)Silty sand (ternary fitting)Gravel soil (ternary fitting)

Clay (ternary fitting)plusmn10


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)

Figure 11 Prediction results of small sample soils


Conflicts of Interest

-e authors declare that they have no conflicts of interest

Acknowledgments

-is research was supported by the National ScienceFoundation of China (Grant nos 41502292 and 51574037)Applied Fundamental Research Project of China Commu-nications Construction Co Ltd (nos 2018-ZJKJ-PTJS03and 2016-ZJKJ-02) and project funded by China Postdoc-toral Science Foundation (no 2014M560739)

References

[1] G Dore F Niu and H Brooks ldquoAdaptation methods fortransportation infrastructure built on degrading permafrostrdquoPermafrost and Periglacial Processes vol 27 no 4 pp 352ndash364 2016

[2] Y Ding S Zhang L Zhao Z Li and S Kang ldquoGlobalwarming weakening the inherent stability of glaciers andpermafrostrdquo Science Bulletin vol 64 no 4 pp 245ndash253 2019

[3] Y Ran X Li and G Cheng ldquoClimate warming over the pasthalf century has led to thermal degradation of permafrost onthe Qinghai-Tibet Plateaurdquo 7e Cryosphere vol 12 no 2pp 595ndash608 2018

[4] S Chen X F Li and T H Wu ldquoSoil thermal regime al-teration under experimental warming in permafrost regionsof the central Tibetan Plateaurdquo Geoderma vol 372 Article ID114397 2020

[5] Z Q Zhang Q B Wu G L Jiang S R Gao J Chen andY Z Liu ldquoChanges in the permafrost temperatures from 2003to 2015 in the Qinghai-Tibet Plateaurdquo Cold Regions Scienceand Technology vol 169 Article ID 102904 2020

[6] T Wang T Wu P Wang R Li C Xie and D Zou ldquoSpatialdistribution and changes of permafrost on the Qinghai-TibetPlateau revealed by statistical models during the period of1980 to 2010rdquo Science of 7e Total Environment vol 650no 1 pp 661ndash670 2019

[7] F E Nelson O A Anisimov and N I ShiklomanovldquoSubsidence risk from thawing permafrostrdquo Nature vol 410no 6831 pp 889-890 2001

[8] J Hjort O Karjalainen J Aalto et al ldquoDegrading permafrostputs Arctic infrastructure at risk by mid-centuryrdquo NatureCommunications vol 9 no 1 Article ID 5147 2018

[9] Y J Shen Y Z Wang X Wei H L Jia and R X YanldquoInvestigation on meso-debonding process of the sand-stonendashconcrete interface induced by freezendashthaw cycles usingNMR technologyrdquo Construction and Building Materialsvol 252 Article ID 118962 2020

[10] Y J Shen X Hou J Q Z H Yuan et al ldquo-ermal dete-rioration of high-temperature granite after cooling shockmultiple-identification and damage mechanismrdquo Bulletin ofEngineering Geology and the Environment vol 7 pp 1ndash142020

[11] M X Yang F E Nelson N I Shiklomanov D L Dong andG N Wan ldquoPermafrost degradation and its environmentaleffects on the Tibetan Plateau a review of recent researchrdquoEarth-Science Reviews vol 103 no 1-2 pp 31ndash44 2010

[12] M R Turetsky B W Abbott M C Jones et al ldquoPermafrostcollapse is accelerating carbon releaserdquo Nature vol 569no 7754 pp 32ndash34 2019

[13] E A G Schuur A D Mcguire C Schadel et al ldquoClimatechange and the permafrost carbon feedbackrdquoNature vol 520no 7546 pp 171ndash179 2015

[14] M Yang X Wang G Pang G Wan and Z Liu ldquo-e TibetanPlateau cryosphere observations and model simulations forcurrent status and recent changesrdquo Earth-Science Reviewsvol 190 pp 353ndash369 2019

[15] Q Z Wang J H Fang X Q Zhao and K Hu ldquo-e influenceof pavement type on the thermal stability of block-stoneembankments in the warm permafrost regionrdquo Trans-portation Geotechnics vol 23 Article ID 100334 2020

[16] M Zhang Y Lai QWu et al ldquoA full-scale field experiment toevaluate the cooling performance of a novel composite em-bankment in permafrost regionsrdquo International Journal ofHeat and Mass Transfer vol 95 pp 1047ndash1056 2016

[17] Z Y Liu F Q Cui J B Chen L Jin W Wang andW Zhang ldquoStudy on the permafrost heat transfer mechanismand reasonable interval of separate embankment for theQinghai-Tibet expresswayrdquo Cold Regions Science and Tech-nology vol 170 Article ID 102952 2020

[18] Y J Shen H W Yang J M Xi Y Yang Y Z Wang andX Wei ldquoA novel shearing fracture morphology method toassess the influence of freezendashthaw actions on con-cretendashgranite interfacerdquo Cold Regions Science and Technologyvol 169 Article ID 102900 2020

[19] Y Zhang W Chen and D W Riseborough ldquoTransientprojections of permafrost distribution in Canada during the21st century under scenarios of climate changerdquo Global andPlanetary Change vol 60 no 3-4 pp 443ndash456 2008

[20] A Alrtimi M Rouainia and S Haigh ldquo-ermal conductivityof a sandy soilrdquo Applied 7ermal Engineering vol 106pp 551ndash560 2016

[21] R Li L Zhao T Wu et al ldquoSoil thermal conductivity and itsinfluencing factors at the Tanggula permafrost region on theQinghai-Tibet Plateaurdquo Agricultural and Forest Meteorologyvol 264 pp 235ndash246 2019

[22] Y Z Du R Li L Zhao et al ldquoEvaluation of 11 soil thermalconductivity schemes for the permafrost region of the centralQinghai-Tibet Plateaurdquo Catena vol 193 Article ID 104608 2020

[23] D Barry-Macaulay A Bouazza R M Singh B Wang andP G Ranjith ldquo-ermal conductivity of soils and rocks fromthe Melbourne (Australia) regionrdquo Engineering Geologyvol 164 pp 131ndash138 2013

[24] G Cai T Zhang A J Puppala and S Liu ldquo-ermal char-acterization and prediction model of typical soils in Nanjingarea of Chinardquo Engineering Geology vol 191 pp 23ndash30 2015

[25] X Zhao G Zhou and X Jiang ldquoMeasurement of thermalconductivity for frozen soil at temperatures close to 0degCrdquoMeasurement vol 140 pp 504ndash510 2019

[26] N Zhang Z Wang S Q Xia and X Y Hou ldquoReview of soilthermal conductivity and predictive modelsrdquo InternationalJournal of 7ermal Sciences vol 117 pp 172ndash183 2017

[27] Y Dong J S McCartney and N Lu ldquoCritical review ofthermal conductivity models for unsaturated soilsrdquo Geo-technical and Geological Engineering vol 33 no 2 pp 207ndash221 2015

[28] M S Kersten Laboratory Research for the Determination ofthe 7ermal Properties of Soils University of MinnesotaMinneapolis MN USA 1949

[29] O Johansen 7ermal Conductivity of Soils TrondheimUniversity Trondheim Norway 1975

[30] S Lu T Ren Y Gong and R Horton ldquoAn improved modelfor predicting soil thermal conductivity from water content at


room temperaturerdquo Soil Science Society of America Journalvol 71 no 1 pp 8ndash14 2007

[31] H Yan H He M Dyck et al ldquoA generalized model forestimating effective soil thermal conductivity based on theKasubuchi algorithmrdquoGeoderma vol 353 pp 227ndash242 2019

[32] J Cote and J-M Konrad ldquoA generalized thermal conductivitymodel for soils and construction materialsrdquo Canadian Geo-technical Journal vol 42 no 2 pp 443ndash458 2005

[33] S Q Luo S H Lu Y Zhang et al ldquoSoil thermal conductivityparameterization establishment and application in numericalmodel of central Tibetan Plateaurdquo Chinese Journal of Geo-physics vol 52 no 4 pp 338-339 2009

[34] N Zhang X B Yu A Pradhan and A J Puppala ldquo-ermalconductivity of quartz sands by thermo-time domain re-flectometry probe and model predictionrdquo Journal of Materialsin Civil Engineering vol 27 no 12 pp 1ndash10 Article ID04015059 2015

[35] V Balland and P A Arp ldquoModeling soil thermal conduc-tivities over a wide range of conditionsrdquo Journal of Envi-ronmental Engineering and Science vol 4 no 6 pp 549ndash5582005

[36] D A De Vries ldquo-ermal properties of soilsrdquo Physics of thePlant Environment pp 210ndash235 North-Holland PublishingCo Amsterdam Netherlands 1963

[37] F Gori ldquoA theoretical model for predicting the effectivethermal conductivity of unsaturated frozen soilsrdquo in Pro-ceedings of the 4th International Conference on PermafrostFairbanks pp 363ndash368 Fairbanks AK USA July 1983

[38] F Tong L Jing and R W Zimmerman ldquoA fully coupledthermo-hydro-mechanical model for simulating multiphaseflow deformation and heat transfer in buffer material androck massesrdquo International Journal of Rock Mechanics andMining Sciences vol 47 no 2 pp 205ndash217 2010

[39] O T Farouki ldquo-e thermal properties of soils in cold re-gionsrdquo Cold Regions Science and Technology vol 5 no 1pp 67ndash75 1981

[40] X Z Xu J C Wang and L X Zhang Frozen Soil PhysicsScience Press Beijing China 2010

[41] S K Haigh ldquo-ermal conductivity of sandsrdquo Geotechniquevol 62 no 7 pp 617ndash625 2012

[42] T Zhang C J Wang S Y Liu N Zhang and T W ZhangldquoAssessment of soil thermal conduction using artificial neuralnetwork modelsrdquo Cold Regions Science and Technologyvol 169 Article ID 102907 2019

[43] N Zhang H Zou L Zhang A J Puppala S Liu and G CaildquoA unified soil thermal conductivity model based on artificialneural networkrdquo International Journal of 7ermal Sciencesvol 155 Article ID 106414 2020

[44] H T Bang S Yoon and H Jeon ldquoApplication of machinelearning methods to predict a thermal conductivity model forcompacted bentoniterdquo Annals of Nuclear Energy vol 142Article ID 107395 2020

[45] J Bi M Zhang W Chen J Lu and Y Lai ldquoA new model todetermine the thermal conductivity of fine-grained soilsrdquoInternational Journal of Heat and Mass Transfer vol 123pp 407ndash417 2018

[46] F Q Cui Z Y Liu J B Chen Y H Dong L Jin andH PengldquoExperimental test and prediction model of soil thermalconductivity in permafrost regionsrdquo Applied Sciences vol 10no 7 Article ID 2476 2020

[47] M Gangadhara Rao and D N Singh ldquoA generalized rela-tionship to estimate thermal resistivity of soilsrdquo CanadianGeotechnical Journal vol 36 no 4 pp 767ndash773 1999

[48] V Vapnik S E Golowich and A J Smola ldquoSupport vectormethod for function approximation regression estimationand signal processingrdquo Advances in Neural InformationProcessing Systems pp 281ndash287 MIT Press Cambridge MAUSA 1997

[49] R G Brereton and G R Lloyd ldquoSupport vector machines forclassification and regressionrdquo 7e Analyst vol 135 no 2pp 230ndash267 2010


comprehensive understanding of variation law of soilthermal conductivity is one of the important tasks inpermafrost research

-ermal conductivity is an inherent parameter tocharacterize the heat transfer performance of soil which canusually be obtained by means of experimental test andprediction models [20ndash22] -e test of thermal conductivitycan generally be divided into the steady-state method andtransient method [23 24] Specifically it includes steady-state comparison method steady-state heat flow metermethod transient hot wire method transient heat pulsemethod and TPS method [25] However the disadvantagesof the experimental test are usually strict technical re-quirements long time consumption and high cost andsometimes limited test results cannot satisfy the needs ofpractical applications -erefore as reported [26 27] manysoil thermal conductivity prediction models had beenproposed Empirical models are usually based on statisticalanalysis of thermal conductivity experimental results anduse water content dry density porosity and other relatedsoil properties as fitting parameters [28ndash31] Kersten [28]analyzed the test results of 19 soil types and established theempirical formula of thermal conductivity using watercontent and dry density Johansen [29] proposed the conceptof normalized thermal conductivity and given the inter-polation calculation model of soil thermal conductivitybased on the relationship between normalized coefficientand soil saturation Lu et al [30] performed a series ofthermo-TDR tests on twelve natural soils and proposed alinear prediction model across a wide range of soil moisturecondition Yan et al [31] developed a generalized effectivesoil thermal conductivity model for soils of various texturesfrom dry to saturation -en many scholars improved themodel for wider applicability and higher prediction accuracy[32ndash35] At the same time many scholars have establishedtheoretical predictive models of soil thermal conductivitybased on specific physical theory and models [36ndash38]Farouki [39] and Xu et al [40] respectively established theweighted calculation model and geometric average calcu-lation formula based on the volume ratio and thermalconductivity of each component of the frozen soil on thebasis of predecessors Considering for the interactionsamong soil particle water and air of a soil unit cell Haigh[41] derived a theoretical thermal conductivity model forsand soil In recent years with the maturity of machinelearning and its advantages in dealing with complex non-linear problems many researchers had adopted it to theprediction of soil thermal conductivity [42 43] For instanceBang et al [44] investigated the application effect of linearregression and various machine learning methods in theprediction of thermal conductivity of compacted bentoniteand verified the feasibility and superiority of the machinelearning method in the prediction of the soil thermalconductivity

Compared to unfrozen soil the specimen preparationand experimental procedures of frozen soil thermal con-ductivity testing are more complex and challengeable And itcan be found that the prediction accuracy of the frozen soilthermal conductivity predictive model is lower than that of

unfrozen soil [40] Meanwhile as a complex multiphasecomposition previous research indicated that the thermalconductivity of frozen soil was associated with many factors[26 27 45]-e existing frozen soil thermal conductivityempirical model (usually using dry density water contentporosity etc as the fitting parameters) cannot compre-hensively consider the influences of complex multiphase andporous structural characteristics of soil -erefore in thepresent work prediction models of frozen soil thermalconductivity using nonlinear regression and SVR methodshave been developed considering for essentially multiphaseand porous structural characteristic information reflectionof unfrozen soil thermal conductivity -ermal conductivityof various types of soil samples which are sampled from theQTEC are tested by the TPS method Correlation analysis ofthermal conductivity of unfrozen and frozen soil has beenadopted Based on the measurement data of unfrozen soilthermal conductivity the ternary fitting and SVR predictionmodels of frozen soil thermal conductivity for the typicalsoils in the QTEC are proposed To further facilitate engi-neering applications the prediction models of coarse andfine-grained soil categories have been proposed Further-more the prediction models of two soil categories have alsobeen used to predict the frozen soil thermal conductivity ofsmall size soil samples in the QTEC

2 Materials and Methods

21 Soil Sample

211 Collection of Soil Samples -e soil samples of thermalconductivity measurement were collected from the drillingspecimens of Qinghai-Tibet expressway geological explo-ration project which was implemented by the CCCC FirstHighway Consultants Co Ltd and Eco-Environment andResources Northwest Institute of CAS from September 2017to June 2018 As shown in Figure 1 the sampling spot mainlylocates in the permafrost regions between Xidatan andTanggula Mountain (the corresponding Qinghai-TibetHighway mileage is K2870simK3307) Every drilling spot issampled for different depths and the sampling depth variesfrom 05m to 40m Total number of 638 unfrozen soilsamples and 860 frozen soil samples are tested

Soil types of testing specimens are determined by theclassification criteria of geotechnical engineering Figure 2shows the statistic results of different types of soil samples(only numbers more than 15 are given) It can be seen thatsilty clay is the most widely distributed frozen soil type in theQTEC and the following soil types are sandy and gravel soilBecause some soil samples with large water (ice) content areimpossible to implement unfrozen soil thermal conductivitytesting a total of 609 specimens tested for both frozen andunfrozen are selected as the research objects in this workand the statistical distribution of various soil samples isshown in Table 1 It can be seen that the proportions of siltyclay silt fine sand gravel sand boulder breccia and allweathered rock soils accounting for 862 are defined as thetypical analytical soil types in the following predictive modelresearch





90degE 100degE

20degN

25degN

30degN

35degN






25degN

30degN

35degN

40degN

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

N

S

W E


32

311

71

43

82

94

82

69

23

42

43

15

Clay

Silty caly

Silt

Silty sand

Medium fine sand

Gravel sand

Breccia

Boulder

Gravel

Weathered phyllite

Weathered mudstone

Weathered slate







ΔT(τ) P0

π32rλ

D(τ) (1)





Fine-grained soil

Clay 24 394 1 6 3 9 5Silty clay 202 3317 10 63 40 69 20




Coarse-grained soil




Dry

den

sity

(gmiddotcm

ndash3)

SiltGravelBreccia


14

16

18

20

22

24


(a)

SiltGravelBreccia


Wat

er co

nten

t (

)

0

5

10

15

20

25

30


(b)












f(x) KTφ(x) + m (3)



Minimize12K

TK + t 1113944

n


Subject to



ξi ξlowasti ge 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)






06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



plusmn10

(b)

Figure 5 Continued



Fine-grained soil



Coarse-grained soil





R2



06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References
























































90degE 100degE

20degN

25degN

30degN

35degN






25degN

30degN

35degN

40degN

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

35deg0prime0Prime

N34

deg0prime0Prime

N33

deg0prime0Prime

N

N

S

W E


32

311

71

43

82

94

82

69

23

42

43

15

Clay

Silty caly

Silt

Silty sand

Medium fine sand

Gravel sand

Breccia

Boulder

Gravel

Weathered phyllite

Weathered mudstone

Weathered slate







ΔT(τ) P0

π32rλ

D(τ) (1)





Fine-grained soil

Clay 24 394 1 6 3 9 5Silty clay 202 3317 10 63 40 69 20




Coarse-grained soil




Dry

den

sity

(gmiddotcm

ndash3)

SiltGravelBreccia


14

16

18

20

22

24


(a)

SiltGravelBreccia


Wat

er co

nten

t (

)

0

5

10

15

20

25

30


(b)












f(x) KTφ(x) + m (3)



Minimize12K

TK + t 1113944

n


Subject to



ξi ξlowasti ge 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)






06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



plusmn10

(b)

Figure 5 Continued



Fine-grained soil



Coarse-grained soil





R2



06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References
























































ΔT(τ) P0

π32rλ

D(τ) (1)





Fine-grained soil

Clay 24 394 1 6 3 9 5Silty clay 202 3317 10 63 40 69 20




Coarse-grained soil




Dry

den

sity

(gmiddotcm

ndash3)

SiltGravelBreccia


14

16

18

20

22

24


(a)

SiltGravelBreccia


Wat

er co

nten

t (

)

0

5

10

15

20

25

30


(b)












f(x) KTφ(x) + m (3)



Minimize12K

TK + t 1113944

n


Subject to



ξi ξlowasti ge 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)






06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



plusmn10

(b)

Figure 5 Continued



Fine-grained soil



Coarse-grained soil





R2



06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References






























































f(x) KTφ(x) + m (3)



Minimize12K

TK + t 1113944

n


Subject to



ξi ξlowasti ge 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(4)






06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



plusmn10

(b)

Figure 5 Continued



Fine-grained soil



Coarse-grained soil





R2



06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References






















































06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

39

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



plusmn10

(b)

Figure 5 Continued



Fine-grained soil



Coarse-grained soil





R2



06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References





















































06

09

12

15

18

21

24

27

30

33

36

39λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)



plusmn10

(c)

06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(d)


f (x) + ε

f (x) ndash ε

f (x) = KT φ (x) + m



06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)









06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References



























































06

09

12

15

18

21

24

27

30λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(a)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(b)


06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)


(c)


06

09

12

15

18

21

24

27

30

33

36λ f

-pre

dict

ion

Wmiddot(m

ndash1middotK

ndash1)


(d)










MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References




























































MAPEle 10 6330 68 7636 76 7674 7708 6538δmax 266 2288 1678 1962 2211 1973 253

SVR modelR2 084 086 091 092 094 094 085

MAPEle 10 7474 8364 7255 7843 7907 8333 6981δmax 2047 1989 1962 1749 1562 1323 2004



a b c d R2 R2


06

09

12

15

18

21

24

27

30

33

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(a)

06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)



(b)







5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References

























































5 Conclusions





Data Availability




0

3

6

9

12

15

18

MA

PE (

)







06

09

12

15

18

21

24

27

30

33

36

λ f-p

redi

ctio

n W

middot(mndash1

middotKndash1

)





Acknowledgments


References























































Acknowledgments


References





















































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