Assessment of soil thermal conduction using artificial neural network models

Cold Regions Science and Technology 169 (2020) 102907

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Assessment of soil thermal conduction using artificial neural network models

T

Tao Zhanga,b, , Cai-jin Wanga, Song-yu Liub, Nan Zhangc, , Tong-wei Zhangd ⁎⁎

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a

Faculty of Engineering, China University of Geosciences, Wuhan 430074, China Institute of Geotechnical Engineering, Southeast University, Nanjing 210096, China c Department of Civil Engineering, The University of Texas at Arlington, Arlington, TX 76019, United States of America d School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China b

ARTICLE INFO

ABSTRACT

Keywords: Soil Heat transfer Thermal conductivity Prediction model

Thermal conductivity is a fundamental engineering property governing heat transfer process in soils. It depends on mineral component, compaction moisture content, dry density, soil gradation and temperature, and usually varies over an order of magnitude in soils with different status. The present study investigated the mechanism of heat transfer in soils and developed new models for thermal conductivity prediction via artificial neural network (ANN) technology. The performance of the proposed models (individual ANN model and generalized ANN model) were evaluated by comparing with three empirical models. Based on the results presented in this study, it is revealed that heat flow through soil was a multi-field coupled process (i.e., thermal-hydro-mechanical process) and was closely related to the intrinsic properties of three phases those constitute a soil. The cross validation was conducted to validate the reliability of the proposed models. It was concluded that both individual and generalized ANN models were able to provide good matching with laboratory measured thermal conductivity values. For each proposed model, the coefficient of correlation (R2) and variance account for (VAF) values were close to 1, and the mean absolute error (MAE) and root mean square error (RMSE) were lower than 0.360 W/K m and 1.000 W/K m, respectively. Results taken from the comparison of various models showed that the generalized ANN model, with RMSE value lower than 0.100 W/K m, exhibited highest accuracy in thermal conductivity prediction of all types of soil, followed by individual ANN models and the empirical models. A good performance of the proposed models in frozen soils was observed with a limited size of thermal conductivity data.

1. Introduction

geomaterials where they built. In addition, heat transfer in soils is an important issue in many of pressing engineering practices, such as lying of high voltage buried cable, radioactive waste disposal, and configuration of ground heat pumps (Bansal et al., 2013; Rad et al., 2013; Salata et al., 2015). Therefore, knowledge of the soil thermal properties is essential for understanding the process of heat transfer in soils and optimizing engineering design. Soil thermal properties mainly include thermal conductivity (inverse of thermal resistivity), thermal diffusivity, and thermal capacity (Ghuman and Lal, 1985; Bansal et al., 2013; Zhang et al., 2017b) among which thermal conductivity is one of the most crucial parameters in heat transfer analyzing and modeling (Lu and Dong, 2015). Thermal conductivity of soils is known to be affected by inherent factors such as moisture content, density, mineralogy, gradation and

Population growth and rapid urbanization have resulted in a great consumption of fossil energy and a corresponding increased emission of greenhouse gas, which poses risks to global climate and the surrounding environment (Horpibulsuk et al., 2013; Latifi et al., 2016; Zhang et al., 2017c; Yang et al., 2018a,b). Due to their great potential usage as an aid in solving climate challenges and saving fossil fuels, thermo-active ground structures such as energy piles and geothermal energy foundations are of a great interest to government planners and engineering designers (Zarrella et al., 2013; Nam and Chae, 2014; Zhang et al., 2017a; Guo et al., 2018; Han and Yu, 2018). Service performance and efficiency of these thermo-active ground structures are mainly dependent on the thermal conduction and heat storage capacity of the

Corresponding author. Correspondence to: T. Zhang, Faculty of Engineering, China University of Geosciences, Wuhan 430074, China. E-mail addresses: [email protected] (T. Zhang), [email protected] (C.-j. Wang), [email protected] (S.-y. Liu), [email protected] (N. Zhang), [email protected] (T.-w. Zhang). ⁎

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https://doi.org/10.1016/j.coldregions.2019.102907 Received 11 April 2019; Received in revised form 31 August 2019; Accepted 29 September 2019 Available online 17 October 2019 0165-232X/ © 2019 Published by Elsevier B.V.

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other external environmental factors (Zhang and Wang, 2017). In the past few decades, many studies have been conducted to quantify the effects of various factors on thermal conductivity of different soils, and to correlate the thermal conductivity with those influence factors (Von Herzen and Maxwell, 1959; Woodside and Messmer, 1961; Slusarchuk and Watson, 1975; Abu-Hamdeh and Reeder, 2000; Abu-Hamdeh et al., 2001; Tarnawski et al., 2009; Haigh, 2012; Zhang et al., 2015b; Bi et al., 2018). For example, Cai et al. (2015) conducted a series of laboratory tests using thermal needle probe to investigate the effects of moisture content, dry density and particle size on thermal conductivity of four types of Nanjing soils and proposed an improved thermal conductivity prediction model. Mickley (1951) measured in-situ thermal conductivity of the moist soils using a cast bronze sphere and reported that soil thermal conductivity can be estimated reasonably using its moisture content and dry density. In addition, Côté and Konrad (2005a) indicated that effects of the grain mineralogy and fabric should be taken into account when predicting thermal conductivity of the soils and construction materials. All the above mentioned studies were conducted at the room temperature (20 ± 2 °C) where soil thermal conductivity is relatively insensitive to temperature changes. For the soils deposited in a temperature below 0 °C, pore water (liquid phase) gradually turns into ice (solid phase) that is a temperature dependent material. The constant thermal conductivity cannot be used in heat flow calculations of frozen soils, because frozen soils possess a different heat transfer mechanism from that of soils at room temperature (Farouki, 1981; Wang et al., 2017). Consequently, temperature is an important factor that needs to be accounted for in the analysis of thermal conduction behaviors of the frozen soils. A considerable number of studies on thermal conductivity of frozen soils and soils experienced the freezing-thawing process have been conducted by earlier researchers (Overduin et al., 2006; Pei et al., 2013; Li et al. 2019a,b). These studies have provided a comprehensive assessment on the evolution of thermal conductivity of the frozen soils from a theoretical and experimental point of view (Wang et al. 2018a,b). Some useful prediction models also have been proposed to estimate thermal conductivity with consideration of temperature and phase transition. Earlier researchers have conducted great efforts to predict soil thermal conductivity on the basis of its basic geotechnical index properties. Subsequently, many calculation models were proposed for this purpose (Campbell et al., 1994; Balland and Arp, 2005; Gori and Corasaniti, 2002). However, a unified correlation between thermal conductivity and its influence factors has not been established. For example, Dong et al., 2014thoroughly reviewed the existing thermal conductivity predicting models and divided them into three categories according to their principles, viz., empirical models, mathematical models, and mixing models. The ultimate goal of mathematical models is to provide a close-form equation for all types of soils. These mathematical models are commonly adopted from other physical models those established for describing other similar properties of soils or for materials having similar heat transfer phenomena as that in soils. This deductive process requires specifical knowledge on the equivalence principles transferring from other physical properties, which is often complicated and involves many calculation parameters. In fact, most current mathematical models have been modified empirically for the convenience of application, thus making them semi-empirical models (i.e., mixing models). In contrast, empirical models and mixing models are mostly developed from the data regression or graphical/numerical techniques (De Vries, 1987). The empirical models have obvious superiority in calculating process with comparison to the mathematical models, but the accuracy of the results it predicts is usually low. The continually emerging new models for soil thermal conductivity prediction indicate a generalized model that is simple in application, accurate in prediction result, and applicable for all types of soils has not yet been developed. Therefore, a novel approach is urgently needed to be developed for assessing the thermal conductivity of soils effectively. The objective of this study is to clearly understand the mechanism of

heat transfer in soils and to develop a fully new thermal conductivity prediction model by resorting to artificial neural network (ANN) technology. The effects of moisture content, dry density, mineralogy, and other factors on soil thermal conductivity are thoroughly analyzed and critically reviewed. A detailed modeling work is performed to develop individual ANN models and generalized ANN model. The predicted results derived from each ANN model are verified via the cross validation method. The superiority and applicability of the proposed ANN models are manifested by comparing their prediction results with those of three traditional empirical models. The performance of the proposed ANN model in the prediction of frozen soils at different temperatures is evaluated. The results obtained in this investigation are quite useful to explore the heat transfer mechanism of porous materials and provide new ideas for predicting soil thermal conductivity with high accuracy. 2. Soil thermal conduction properties Heat flow through soil is almost entirely by conduction, with radiation unimportant, except for the case of surface soils. Heat flow rate depends on the soil thermal conductivity (inverse of thermal resistivity). Brandon and Mitchell (1989) had summarized that soil thermal conduction property is influenced by the following factors: moisture content, dry density, mineralogy, particle size distribution, time, and temperature. Heat conduction in soils is usually accompanied by changes in the hydro-mechanical properties of soils (Zhang et al., 2015a). For partly saturated soils, the heat transfer may cause an increase in temperature, and subsequence a migration of moisture which can alter the status of the soil and in turn affects its thermal conduction. Furthermore, the mechanical behavior of the soils changes as the heat transfer varies the soil suction. This coupled thermal-hydro-mechanical process has been extensively studied and a series of valuable results have been reported by earlier researchers (Lai et al., 2014). In this study, major efforts are made to investigate the variation characteristics of thermal conduction with various influence factors and to predict the thermal conductivity of soils, therefore, the changes in hydro-mechanical behaviors during heat flow through soil are not included here. Influences of geotechnical properties of the soil on its thermal conductivity are presented in the following sections to better understand the mechanism of heat transfer in soils. 2.1. Effect of moisture content and dry density on thermal conductivity Four types of soils collected in Nanjing area were tested by the authors in a previous study for investigating the relationships between thermal conductivity and geotechnical properties. These soils are clay, silt, fine sand, and coarse sand and their physicochemical properties can be found in literature (Cai et al., 2015). In this study, the testing results of clay and fine sand are chosen for exploring the effect of moisture content and dry density on thermal conductivity (see Fig. 1). It is shown in Fig. 1 that the increase in moisture content would result in an increase in thermal conductivity with thermal conductivity increases dramatically when moisture content is below 20% for clay and below 5% for fine sand and then it increases moderately as moisture content further increases. These change tendencies in thermal conductivity are consistent with those reported in previous studies work with thermal conductivity of other soil types and can be described using power functions. The dramatical increase of thermal conductivity is attributed to the fact that addition of water (increasing moisture content) will inevitably expel a portion of air from the void in soils and generate water films and water bridges among soil particles which is beneficial for thermal conduction because of the relatively higher thermal conductivity of water as compared to that of air. After most of the air is replaced by water and the soil particles are fully connected with water bridges, further increase in moisture content generates an insignificant increase in thermal conductivity and the maximum conductivity value is presented. The moisture content corresponding to this state is 2

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and saturation are not discussed here for the sake of concise. 2.2. Effect of mineralogy on thermal conductivity The solid phase of soil consists of different minerals, for instance, feldspar, calcite, mica, quartz, etc. Therefore, mineral composition has a control influence on the thermal conductivity performance of the solid particles (Lu et al., 2007; Chen, 2008). Of particular interest is often quartz, because thermal conductivity of the quartz (7.69 W/K·m) is much higher than that of other minerals which usually ranges from 1.25 W/K m to 4.00 W/K m. A soil with high quartz content would present high value of thermal conductivity (Horai and Simmons, 1969). Johansen (1975) has proposed a generalized geometric mean method for estimating thermal conductivity of soil particles and simplified this method by dividing all the minerals into two categories, namely quartz mineral and another mineral. It can be observed from Fig. 1 that the discrepancy between clay and fine sand in terms of thermal conductivity is obvious under the same moisture content and the same dry density condition. For instance, thermal conductivity of clay with moisture content of 5% and dry density of 1.4 g/cm3 is 0.270 W/K m, while the thermal conductivity value of the fine sand with the same condition is about 1.032 W/K m. This phenomenon reveals that the thermal conductivity of soil particle depends on the type and proportion of minerals in which quartz plays a controlling role. 2.3. Effect of anisotropy on thermal conductivity The influence of anisotropy on thermal conduction of geomaterials has attracted the attention of earlier researchers. Midttomme and Roaldset (1999) reported that for sedimentary rocks, thermal conductivity perpendicular to the solid grain was in some cases half of that measured parallel to the grain. However, Beardsmore and Cull (2001) indicated a negligible variation in thermal conduction of the shale with an increase in density, which was quite different from that found in other rocks. Popov et al. (2003) employed a parameter that defined as the ratio of parallel tensor component to the perpendicular one, namely thermal anisotropy coefficient, K, to evaluate the influence of heat transfer direction on rock thermal conductivity and reported that the thermal conductivity parallel to the bedding was higher than that perpendicular to the bedding. In addition, the parallel conductivity was approximately 1.5 times of perpendicular conductivity in most cases, which was consistent with that reported for sedimentary rocks. BarryMacaulay et al. (2013) tested the thermal conductivity of a number of Australian siltstones using divided bar apparatus and quantitatively evaluated the influence of bedding orientation on rock thermal conduction. The correlation between thermal conductivity and bedding orientation is plotted in Fig. 2. It is evident from Fig. 2 that thermal conductivity of siltstone presents an increasing trend with an increase in the bedding orientation angle. For example, when the bedding

Fig. 1. Effects of moisture content and dry density on soil thermal conductivity: (a) clay and (b) fine sand.

commonly referred as critical moisture content by previous researchers (Salomone and Kovacs, 1984). It is also found in Fig. 1 that the thermal conductivity increases with increasing dry density for a given moisture content regardless of the soil type. Natural soil is mainly composed of three phases including solid phase, liquid phase, and gas phase. It should be noted that in some cold regions, ice might exist in the soil as a solid phase. However, this situation is not included in this study because the testing work has been conducted at room temperature (about 20 ± 2 °C). The solid phase possesses the best heat-transfer capability, that is the highest thermal conductivity, as compared to both liquid and gas phases. Dry density is one of the engineering properties used to characterize the degree of density of soils. It can be easily deduced that the increase in dry density will increase the contact points and contact areas among solid particles. Theoretically, the improvement in solid particle contact will facilitate the heat transfer process and hence raise thermal conductivity of the soils. The variation in thermal conductivity with dry density presented in Fig. 1 is consistent with the above relevant description. In addition, the discrepancy in thermal conductivity induced by variation in dry density is found to be steady when moisture content increases, especially for the clay case. For example, thermal conductivity difference of 0.150 W/K·m is found for clay samples with dry density of 1.0 g/cm3 and 1.4 g/cm3 under moisture content of 5%, and then this difference is shown as 0.123 W/K·m at moisture content of 25%. This phenomenon probably reveals that the effect of dry density on soil thermal conductivity is independent of moisture content. It is noteworthy that although porosity, n, and degree of saturation, Sr, have an obvious influence on soil thermal conductivity, their influences can be presented directly or indirectly through moisture content and dry density. Consequently, the relationships between thermal conductivity and porosity

Fig. 2. Measured thermal conductivity of siltstone with different bedding orientation angles. 3

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orientation angle increases from 10° to 70°, thermal conductivity increases from 1.735 W/K m to 2.521 W/K m. It can be extrapolated from the presented data points that thermal conductivity parallel to the bedding is approximately 1.5 times as that measured perpendicular to it. This is fully consistent with that reported by Popov et al. (2003) for the sedimentary rocks. Due to the difficulty in quantitative evaluating anisotropy of the fine-grained soils, there is a limited number of studies focused on the influence of anisotropy on thermal conductivity of the clayey soils. 2.4. Effect of other factors on thermal conductivity

Fig. 3. Schematic diagram of artificial neural network architecture.

The effects of some other factors including particle size and distribution, particle shape, and temperature on soil thermal conduction are discussed in this section. Within a unit space, the number of contact points would be increased with a decrease in particle size. Thermal resistance of these contact points is known to be much higher than that of the internal soil particles. Therefore, heat flow through soil becomes more difficult and the soil thermal conductivity decreases. This comment gives a probable reason for the higher thermal conductivity of sandy soils as compared to that of clayey soils as shown in Fig. 1. It should be noted that heat transfer among soil particles is mainly relied on the contact points, especially in dry soils, because thermal conductivity of air is extremely lower than that of soil particles. In this case, an increasing number of contact points is obviously beneficial to the heat transfer in soils. Moreover, the addition of binders will improve thermal conduction of the soils due to the generation of more stable soil structure. Particle size and shape also have an effect on the spatial configuration of the soils at a micro level. For instance, the finer particles included in soils are commonly aggregated into large secondary aggregations with different morphology. Once the orientation of these large secondary aggregations which exist among the primary aggregations is parallel to the direction of heat flow, soil thermal conduction performance would be improved dramatically because the solid phase is the main channel for heat transfer. When investigating the transfer of heat in soil, especially in fine-grained soils, the effects of particle size and shape cannot be ignored. Temperature changes will affect the thermal movement of molecules and then influence the thermal conduction of soils. Generally, an increase in temperature may accelerate the thermal movement of molecules, which facilitates heat transfer through the materials. Mitchell and Soga (2005) indicated that all crystalline minerals in soils exhibit decreasing thermal conductivity with increasing temperature. In addition, liquid phase and gas phase have different responses to temperature increase in terms of thermal conduction. Specifically, thermal conductivity of saturated air increases markedly with an increase in temperature, whereas thermal conductivity of water increases slightly under the same condition. The net effect is that thermal conductivity of unsaturated sandy soils increases somewhat with increasing temperature. However, some researchers reported contradictive finding with the above mentioned comment. For example, Smits et al. (2013) and Hiraiwa and Kasubuchi (2000) measured thermal conductivity of sandy soils over a wide range of temperature (5 °C–75 °C) and found an obvious increment in thermal conductivity with increasing temperature of the tested samples. As explained earlier, the comment on the evolution of thermal conduction of soils with increasing temperature has not been unified to date, thus, further investigation is needed to explore the correlations between thermal conduction and environmental factors.

processing elements which are commonly referred as neurons (Schalkoff, 1997). Fig. 3 shows a typical architecture of feed-forward multi-layer artificial neural network with h1, h2, and hi represent the input parameters, yi and yj are the neurons in the hidden layer, and R is the output in the output layer which is also the target of the network calculation. Each neuron in the previous layer is connected to all the neurons in the next layer and the connection medium between each of two neurons is usually called weight. This connection possesses the ability to deliver the calculation results to the next layer completely. The quantity of the hidden layer and neurons in the network depends on the complexity level of the problem being identified. Earlier studies indicate that one or two hidden layers are sufficient for solving most civil engineering problems (Habibagahi and Bamdad, 2003). The backpropagation learning algorithm, one of the most popular algorithms in ANN, has been used extensively and successfully to model many engineering issues such as landslide stabilization, tunnel monitoring, and foundation pit supporting (Boubou et al., 2010). This algorithm usually consists of two parts, namely excitation propagation and weight updating. The input parameters are processed in neurons by the following steps: (1) multiplying each input parameter by its weight; (2) summing the product and then processing the summation using an activation function; (3) repeating above steps and updating the weights until the desired results are reached. A well-trained ANN model should have a coefficient of determination, R2, that is very close to 1 and all of its error terms should be in very small value. 3.2. Database By conducting a literature review, laboratory-measured data of five types of soils including clay, silt, silty sand, fine sand, and coarse sand were collected in this study for ANN model development. The tested soil samples were prepared with different dry densities and moisture contents and subjected to thermal conductivity measurement as reported by previous studies (Johansen, 1975; Cai et al., 2015; Lu and Dong, 2015). Basic physicochemical properties of these soil samples are listed in Table 1. As discussed earlier, soil thermal conduction is affected by many factors, viz., moisture content, dry density, type (corresponding to mineralogy), particle gradation and other environmental factors. In this study, the prediction models based on the artificial neural network were developed independently for each type of soil, which were denoted as PM-C for clay, PM-S for silt, PM-SS for silty sand, PM-FS for fine sand and PM-CS for coarse sand. Furthermore, a generalized model denoted as PM-G was also developed which accounts for five types of soils with various status. Fig. 4 shows the flow chart of the calculation process for the proposed ANN models. Two input parameters (i.e., moisture content, w, and dry density, γd) were set for individual models, while four input parameters (moisture content, w, dry density, γd, clay content, c, and quartz content, qc) were utilized for the generalized model. It should be noted that clay content and quartz content were selected as input parameters for the PM-G model because these two parameters represent variations in the gradation and the mineralogy, respectively. While, the soil type of each individual model

3. Development of ANN models 3.1. Background of ANN Artificial neural network (ANN) is a kind of information processing model, which is generally composed of three layers: an input layer, a hidden layer, and an output layer. Each layer is connected by 4

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maximum value, respectively. The boundary values for input-output parameters of each model are listed in Table 2 and Table 3. Moisture content of 0% represents that the soil sample was prepared with a dry status. The minimum and maximum values of predicted thermal conductivity from models are 0.08 W/K m and 2.95 W/K m, respectively, as shown in Table 3.

Table 1 Basic physical properties of five different types of soil (Johansen, 1975; Cai et al., 2015; Lu and Dong, 2015). Engineering property

Clay

Silt

Silty sand

Fine sand

Coarse sand

Specific gravity, Gs Grain size distribution (%) Clay (< 0.002 mm) Silt (0.002–0.074 mm) Sand (0.074–2 mm) Liquid limit, wL (%)b Plasticity limit, wP (%)b Maximum void ratio, emax Minimum void ratio, emin

2.74

2.71

2.70

2.66

2.65

89.3a 7.6a 3.1a 59.8 32.6 NA NA

11.5a 80.1a 8.4a 32.2 23.4 NA NA

0 35.8 64.2 21.4 15.3 NA NA

0 0 100 NA NA 0.791 0.550

0 0 100 NA NA 0.754 0.610

3.3. Parameter setting The model database is commonly divided into two subsets: (i) training set that used to construct the initial ANN model; (ii) validation set that used to verify the predicted results and assess the performance of models. However, this practice may easily result in over fitting problem which makes the model cannot generalize well in new environmental conditions. To solve this problem, the cross validation technique, the most appropriate method for developing ANN models, was employed as the stopping criterion in this study (Kersten, 1949). In the cross validation technique, the database is divided into three subsets: training set, validation set, and testing set. Training data is utilized to update the weights of networks and the updating process is monitored by the error of validation data. Model training will not stop until the error of validation data in the validation set begins to increase, at which point the model generalization is considered to reach its best stage. Finally, the performance of well-constructed networks is evaluated by feeding the testing data into it. In this study, training set, validation set, and testing set account for 55%, 25%, and 20% of the database in total, respectively. The networks with one hidden layer were selected in this study for

Note: NA = not available. a Measured using a laser particle size analyzer Mastersize 2000. b Measured as per China MOT JTG E40-2007.

is specified, and thus the influence of mineralogy and gradation on soil thermal conduction is very limited. All the developed models have only one output parameter, namely predicted thermal conductivity, kp. For the convenience of model calculation, both input and output parameters of each model were normalized using following equation as suggested by earlier researchers to create value that falls between 0 and 1 (Erzin et al., 2008):

xN =

(x x min ) (x max x min )

(1)

where xN is the normalized value for model calculation; x is the actual input-output value; xmin and xmax are the actual minimum and

Fig. 4. Flow chart of the calculation procedure for ANN models. 5

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Table 2 Boundary values of both input and output parameters employed for developing individual models. Model symbol

Input parameters

Output parameter 3

Moisture content, w (%)

PM-C PM-S PM-SS PM-FS PM-CS

Dry density, ρd (g/cm )

Minimum

Maximum

Minimum

Maximum

Minimum

Maximum

0 0 0 0 0

39.0 45.5 36.9 11.0 6.0

1.0 1.2 1.0 1.5 1.5

1.4 1.5 1.4 1.7 1.7

0.09 0.10 0.14 0.31 0.36

2.26 2.17 2.78 2.93 1.64

3.4. Model verification

Table 3 Boundary value setting of both input and output parameters for the generalized model (PM-G). Model parameters

Parameter name

Minimum value

Maximum value

Input parameters

Moisture content, w (%) Dry density, ρd (g/cm3) Clay content, c (%) Quartz content, qc (%) Thermal conductivity, kp (W/K·m)

0 1.0 0 2.7 0.08

45.5 1.7 89.3 89.7 2.95

Output parameter

Thermal conductivity, kp (W/K m)

Fig. 5 shows the comparison of predicted thermal conductivity obtained from each individual ANN model with the laboratory measured conductivity that divided into training, validation, and testing data. It can be observed that, for each type of soils, predicted thermal conductivity values, kp, from individual ANN models are quite close to the measured thermal conductivity values, km, indicating the high quality of the ANN architecture. Another noticeable characteristic is that the comparison results of clayey soils (i.e., clay and silt) possess much more scatter points as compared to that of sandy soils (i.e., silty sand, fine sand, and coarse sand), which could be attributed to the complexity of mineral component and variety of particle morphology. The developed individual models have an ability to predict thermal conductivity of various types of soils efficiently. In order to reinforce the applicability of the proposed ANN models, similar comparison also has been conducted for the kp derived from the generalized model (PM-G) with four input parameters and the km, and the result is shown in Fig. 6. In general, more related input parameters feed into the ANN model will make the predicted value much closer to the target value, whereas the increase in data size and discrepancy would probably have a negative effect on the predicted results. It is evident from Fig. 6 that PM-G performs excellently in predicting thermal conductivity for different types of soils. In each subset (i.e., training data, validation data, and testing data), predicted thermal conductivity values agree well with the measured ones. This indicates that the cross validation technique is effective and feasible for developing these ANN prediction models. Four error indicators, i.e., coefficient of correlation (R2), mean absolute error (MAE), variance account for (VAF), and root mean square error (RMSE), are employed in this study for further quantitatively checking the reliability of the proposed models. The expressions for VAF and RMSE are listed in the following:

model construction since one hidden layer has the ability to approximate any continuous functions under sufficient connection weights condition in geotechnical engineering. The optimum number of neurons in the hidden layer was determined by the following steps: first, setting one neuron in the layer at the starting time; then increasing the size of network in steps by adding one neuron each step; finally, terminating the increasing process when the network reaches its welltrained point. The training and testing processes of multi-layer perceptrons (MLPs) was conducted in the artificial neural network toolbox in Matlab 2017, which is an extensively used software for data analysis and visualization. Two momentum factors (μ1 = 0.01 and μ2 = 0.001) were set for the ANN models to pursue the most efficient structure, and the maximum training cycle, cmax, was selected as 1000 times in this study. The detailed information on parameter setting for each prediction model is shown in Table 4. It is acknowledged that different types of soils have different levels of sensitivity to the variation of input parameters. The transfer functions employed in this study were determined from the calculation results of several preliminary trials that have the best accuracy. Tan-sigmoid and log-sigmoid functions were selected as the transfer functions, f, for clayey soils (i.e., clay and silt in this study) for hidden layer and output layer, respectively. For other individual ANN models and the generalized model, log-sigmoid function was employed for both hidden layer and output layer. Tables 5 and 6 present the value of connection weights and biases for all ANN models. It should be noted that the number of neurons listed in the tables is obtained after the training and testing processes.

var (x x ) var (x )

VAF = 1

RMSE =

1 N

× 100%

(2)

N

(x i

x i )2

(3)

i=1

Table 4 Key parameters selection of each ANN model. Model symbol

PM-C PM-S PM-SS PM-FS PM-CS PM-G

Number of data utilized for

Hidden neurons

Training

Testing

Validation

72 21 33 17 18 161

26 8 12 6 6 58

32 14 19 11 12 88

7 8 9 6 5 8

Note: μ is momentum factor, which was selected for training process. 6

Transfer functions utilized in Hidden layer

Output layer

Tan-sigmoid Tan-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid

Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid Log-sigmoid

μ

0.01 0.01 0.001 0.001 0.001 0.001

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Table 5 Variation of both connection weights and biases for different individual models. Model symbol

Hidden neuron

PM-C

Weight values

1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 1 2 3 4 5

PM-S

PM-SS

PM-FS

PM-CS

Bias values

Moisture content, w

Dry density, ρd

Thermal conductivity, k

Hidden layer

Output layer

−0.422 −0.217 3.041 −3.369 2.503 0.123 −2.355 −3.909 0.711 0.874 0.641 −0.856 −1.869 −4.483 3.786 2.543 −1.736 −5.176 −0.884 9.550 −6.839 0.409 −2.036 7.944 −1.565 −2.700 −0.900 −3.159 0.064 −2.445 1.956 3.489 −2.580 3.633 −3.169

3.844 2.162 0.438 −2.035 0.263 −3.341 −1.285 4.118 1.826 −4.323 −5.093 −1.364 0.234 2.904 −1.467 2.620 5.548 −0.566 3.233 2.191 −0.762 2.591 7.371 0.291 2.855 −2.688 −2.559 0.206 3.549 −2.626 2.601 0.787 4.180 −5.252 −2.364

0.403 −1.031 −0.290 −0.700 −1.692 0.445 1.003 −0.081 0.608 0.216 −0.143 0.078 1.321 −0.159 0.759 −1.064 0.148 0.304 −0.195 0.295 0.045 −0.058 0.016 −2.997 0.067 −0.132 0.700 1.571 0.460 0.028 −0.412 0.151 −0.376 −0.606 0.556

−4.740 −4.513 −3.305 −1.203 3.298 −3.926 −1.146 0.839 −4.022 −1.584 0.960 0.023 −1.421 −1.871 4.073 −5.045 −1.135 1.234 −0.738 0.140 −1.436 2.921 −7.075 8.655 3.640 1.183 0.983 −3.336 −2.265 −4.435 −2.185 −2.895 −0.571 0.927 −3.251

0.752

Weight values w

ρd

c

qc

k

Hidden layer

Output layer

1 2 3 4 5 6 7 8

0.082 −0.094 −0.224 −0.386 2.541 −0.046 −0.309 0.413

2.277 −1.443 1.286 −5.326 −2.531 2.057 −1.646 1.239

0.377 −0.926 −0.922 −0.004 −0.094 −1.041 −1.261 −0.423

0.027 −0.574 −0.287 6.838 0.566 −0.209 −7.909 −1.322

−2.262 −1.024 1.377 −0.139 −0.732 −1.108 0.533 1.778

−1.690 0.821 −0.001 1.871 4.051 0.471 −9.243 −2.479

1.151

1.035

0.690

−0.317

individual ANN models, which reveals that the prediction level of the PM-G model is comparable to that of other individual models. Although the new data environment brings difficulties in the prediction process, the PM-G model is able to solve this problem well by increasing input parameters and optimizing architecture of the networks. Testing data, specially used for verifying the validity of the ANN models, was fed into the models and its prediction results were also compared with the laboratory-measured results, as shown in Figs. 7, 8 and Fig. 7 presents the comparison results for clayey soils, and R2 values of PM-G model are almost equal to those of individual models (i.e., PMC and PM-S) and they are very close to 1. Similar comparison results are observed for sandy soils as presented in Fig. 8. PM-G model has a superior performance in predicting sandy soil and its R2 values are slightly higher than those of individual models (i.e., PM-SS, PM-FS, and PM-CS). This observation could be partly attributed to the limitation on size of testing data used here. In a word, the proposed individual models and the generalized model predict thermal conductivity of different types of soil quite accurately.

Table 6 Connection weights and biases of the generalized model (PM-G). Hidden neuron

0.239

Bias values

Note: w is moisture content; ρd is dry density; c is clay content of soils; qc is quartz content of soils; and k is soil thermal conductivity.

where var is an abbreviation which denotes the variance; x is the actually measured value; x is the predicted value; xi and x i are the measured value and predicted value labeled i, respectively; N is the sample size of data. An ANN model is considered to be trained excellent when the value of R2 and VAF are close to 1 and the errors in terms of MAE and RMSE are almost 0. Table 7 lists the statistical results of these error indicators for each prediction model. It is evident that both individual models and generalized model are effective in estimating soil thermal conductivity since their R2 values are higher than 0.950 and VAF values are larger than 85%. In addition, the values of RMSE and MAE of each model are lower than 0.360 W/K·m and 1.000 W/K·m, respectively. The values of R2 and VAF of the generalized model (PM-G) are almost equivalent or slightly higher as compared to those of

4. Performance assessment of the proposed models In order to reinforce the applicability and the superiority of the proposed prediction models, a comparison between ANN models and traditional models has been conducted in this study. The existing prediction models of soil thermal conductivity can be roughly classified into two types: (1) theoretical models, which simplify the soil structure into an ideal state and usually have a complex calculation process; (2) empirical models, which are originated from the regression of the experimental data. The theoretical models are seldom employed in 7

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Fig. 5. Comparison of predicted thermal conductivity values with measured ones for the individual ANN models: (a) PM-C model; (b) PM-S model; (c) PM-SS model; (d) PM-FS model; and (e) PM-CS model.

engineering design, due to their much input parameters and complex calculation. Three typically empirical models (Kersten model, Gangadhara model, and Côté model) which have been extensively used in estimating soil thermal conductivity/resistivity in engineering design were selected as the comparison models and their background will be briefly depicted in the following section. Detailed information regarding the calculation process of these three models can be found in previous studies (Kersten, 1949; Gangadhara Rao and Singh, 1999; Côté and Konrad, 2005b).

temperature, degree of saturation, and etc. The influence of these factors on soil thermal conduction was studied and the correlations among them were also analyzed. He expressed the thermal conductivity of unfrozen soils in terms of moisture content w and dry weight γd. The proposed empirical model for estimating thermal conductivity is given as:

k = 0.1442[0.9 × log w

0.2] × 10 0.6243 d (for silt and clay soils, w

7%) (4)

k = 0.1442[0.7 × log w + 0.4] ×

4.1. Three empirical prediction models

10 0.6243 d (for

sandy soils, w

1%)

(5)

where k is the soil thermal conductivity in W/K m; w is the prepared moisture content of samples in %; and γd is the dry unit weight in lb./ ft3.

4.1.1. Kersten model Kersten (1949) tested thermal conductivity of nineteen types of soils, including crushed stones, gravels, sands, clayey soils and organic soil in the laboratory using a thermal probe. The testing samples were prepared in different states, i.e., moisture content, density,

4.1.2. Gangadhara model Gangadhara Rao and Singh (1999) measured thermal conductivity 8

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Fig. 6. Comparison of predicted thermal conductivity from the generalized model (PM-G) with the laboratory-measured conductivity. Table 7 Calculation results of performance indexes for each ANN model. Model symbol

Data set

R2

RMSE (W/K m)

MAE (W/K m)

VAF (%)

PM-C

Training Testing Validation Training Testing Validation Training Testing Validation Training Testing Validation Training Testing Validation Training Testing Validation

0.9605 0.9729 0.9738 0.9985 0.8441 0.9757 0.9995 0.9964 0.9980 0.9944 0.9752 0.9871 0.9961 0.9761 0.9561 0.9842 0.9665 0.9769

0.098 0.132 0.104 0.235 0.322 0.134 0.109 0.178 0.151 0.193 0.107 0.108 0.062 0.022 0.092 0.162 0.246 0.244

0.323 0.413 0.239 0.409 0.345 0.552 0.572 0.612 0.751 0.415 0.441 0.953 0.240 0.266 0.437 0.446 0.733 0.576

97.354 93.307 91.348 96.573 89.864 80.712 99.776 99.681 99.208 88.565 96.512 99.604 84.686 99.762 95.574 97.655 97.943 98.877

PM-S PM-SS PM-FS PM-CS PM-G

Fig. 7. Comparison of predicted thermal conductivity with laboratory-measured conductivity for clayey soils: (a) PM-C and PM-S models and (b) PM-G model.

is still used in the new model, but the formulas utilized to calculate the Kersten number (kr) and kdry have been modified. The effects of soil type and particle morphology on thermal conduction are incorporated in the new model as shown in the following equations:

of five different soils deposited in India areas via using a laboratory thermal probe which operates on the transient method. They developed an empirical relationship as expressed in Eq. (6).

1/ k = [1.07 × log w + b] ; for sandy soils, w

1

× 10(3 1%)

0.01 × d ) (for

clayey soils, w

kr =

10% (6)

kdry =

where b is a dimensionless parameter which depends on soil type. For clay, silt, silty sand, fine sand, and coarse sand, the b values are −0.73, −0.54, 0.12, 0.70, and 0.73, respectively, as recommended by Gangadhara Rao and Singh (1999). It should be noted that this prediction model is only valid for unfrozen soils since all the samples were measured at room temperature of 27.5 °C.

kdry ) × kr + kdry

(10

1) Sr n)

(8) (9)

where κ is an empirical parameter used to account for the effect of soil type and frozen or unfrozen status; χ and η are material parameters accounting for the particle shape effect; n is the porosity of soils. Detailed information on the values of these empirical parameters for various types of soils can be found in literature (Côté and Konrad, 2005b; Zhang et al., 2017d).

4.1.3. Côté model For the first time, Johansen (1975) proposed the conception of normalized thermal conductivity (kr), which is also known as Kersten number, to depict the correlations between the thermal conductivity of the soils in nature moisture content and the thermal conductivity of the soils under dry and saturated status (i.e., kdry and ksat), which was expressed by the following equation:

k = (ksat

Sr 1+(

4.2. Comparison of ANN models with empirical models A comparison of prediction performance between proposed ANN models and empirical models has been made in measured thermal conductivity-predicted thermal conductivity coordinations, and the results are shown in Figs. 9, 10 and Fig. 9 presents the comparison of the predicted thermal conductivity kp obtained from the ANN and empirical models with the laboratory measured thermal conductivity km for clayey soils. It is evident that the proposed ANN models yield much better matching with measured values as compared to those calculated from empirical models. The predicted results of both Kersten model and Gangadhara model have an obvious characteristic of

(7)

where kr is the Kersten number, which is a function of the saturation degree (Sr). Based on the Eq. (7), Côté and Konrad (2005a) developed an improved model to predict thermal conductivity of the soils. Eq. (7) 9

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Fig. 8. Comparison of predicted thermal conductivity with laboratory-measured conductivity for sandy soils: (a) PM-SS, PM-FS, and PM-CS models and (b) PM-G model.

Fig. 9. Comparison of calculation performance between proposed ANN models and empirical relationships for clayey soils: (a) PM-C and PM-S models and (b) PM-G model.

underestimation. Côté model provides a good fit to the experimental data, but some scattered data points are also found in Fig. 8. The comparison results of sandy soils are presented in Fig. 10. Similar to phenomena shown in Fig. 9 for clayey soils, it is also observed in Fig. 10 that both Kersten model and Gangadhara model lead to an underestimate in the thermal conductivity of soils. In addition, the predicted results of the Côté model are acceptable when the thermal conductivity values of the sandy soils are between 0.5 W/K m and 1.3 W/K m, whereas the Côté model possesses an obvious characteristic of low accuracy when the conductivity falls out of this range. In contrast, the ANN models possess a superior prediction performance than the three empirical models. Statistical analysis was conducted to quantitatively assess the accuracy of the concerned models in calculating thermal conductivity of soils. The root mean square error (RMSE), representing the data standard deviation as shown in Eq. (3), was employed here to determine the prediction accuracy of each model. The statistical analysis results of each prediction model are summarized in Table 8. The analysis attests that the generalized model (PM-G) provides the best matching to the measured data of the clayey and sandy soils because it possesses the lowest RMSE value. Individual ANN models also exert a good performance and their RMSE values are slightly higher as compared to that of the PM-G model. It is certain that the Côté model performs best in three empirical models, while predicted results of other empirical models are unacceptable since their RMSE values are too high (> 0.800 W/K m). Thus, two input parameters (i.e., w and γd) including in Kersten model and Gangadhara model are insufficient to predict the thermal

conductivity accurately. This is because the effects of the soil type and particle morphology on thermal conduction are incorporated in the Côté model, whereas these effects are absent in the Kersten model and Gangadhara model. In summary, the generalized model (PM-G) provides the best matching to the laboratory measured data, followed by the individual ANN models. The prediction performance of the three empirical models, especially the Kersten model and Gangadhara model, is incomparable with that of the ANN models. 4.3. Validation of ANN model for frozen soils To further verify the applicability of the proposed ANN models in frozen soils, the thermal conductivity data of northern Manitoba soils deposited in different temperatures (i.e., −10 °C, 3 °C, and 10 °C) is collected from the literature as the reference value (Kurz et al. 2017). The generalized model (PM-G) with four input parameters is employed here to predict the soil thermal conductivity. It is acknowledged that thermal conductivity of frozen soils is closely related to the level of temperature. Because the thermal conductivity of ice is more than four times higher than that of water at room temperature. Therefore, temperature, moisture content, dry density, and degree of saturation were selected as the input parameters in this validation. The configuration of the prediction model is consistent with the previous one. Fig. 11 presents the comparison result between measured and predicted thermal conductivity at different temperatures. It can be observed that the predicted thermal conductivity values have a good agreement with the 10

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Fig. 11. Comparison between measured and predicted thermal conductivity for northern Manitoba soils deposited in different temperatures.

models have been verified and compared with three empirical models to reinforce their applicability and superiority. The following conclusions can be advanced from this study: (1) Soil thermal conduction is affected by many factors, such as moisture content, dry density, mineralogy, and particle morphology, and it is difficult to be estimated accurately by existing models. Two types of ANN models, individual ANN model and generalized ANN model, were developed in this study and they were confirmed to describe the laboratory measured data well. The individual ANN models with moisture content and dry density as input parameters are only suitable for a given type of soil. However, after adding the clay content and the quartz content into the input parameters, the resultant generalized ANN model is applicable to all types of soils. (2) The cross validation results demonstrate that the developed ANN models are quite efficient in predicting thermal conductivity, as their R2 and VAF are very close to 1, and RMSE and MAE are lower than 0.360 W/K m and 1.000 W/K m, respectively. The prediction accuracy of generalized ANN model is almost equivalent to that of the individual ANN models in terms of four error indicators and it could be improved via adding relevant input parameters. (3) The generalized ANN model provides the highest accuracy in predicting the thermal conductivity of both clayey soils and sandy soils, with RMSE value of 0.095 W/K m and 0.040 W/K m, respectively. Three empirical models have the noticeable characteristic of either overestimation or underestimation because of the considerable effect of mineralogy and particle gradation on actual thermal conduction. Their RMSE values range from 0.424 W/K m to 1.356 W/K m, which is unacceptable in engineering design. (4) The individual ANN models are suggested to be used for predicting thermal conductivity when the physicochemical properties of targeted soil are unclear, whilst the generalized ANN model is preferable for soils with the significant discrepancy in engineering properties. The performance of the proposed ANN model in frozen soils is evaluated and verified by the limited size of data. Further improvement on the proposed ANN models with the larger size of database including frozen soils at different temperatures is warranted to reinforce the applicability of these models.

Fig. 10. Comparison of calculation performance between proposed ANN models and empirical relationships for sandy soils: (a) PM-SS, PM-FS, and PMCS models and (b) PM-G model. Table 8 RMSE values of each model for clayey and sandy soils. Soil type

Kersten model

Gangadhara model

Côté mode

individual ANN models

PM-G model

Clayey soils Sandy soils

1.332 0.858

1.356 0.875

0.759 0.424

0.114 0.099

0.095 0.040

Note: Individual ANN models include PM-C, PM-S, PM-SS, PM-FS, and PM-CS model; Bold values denote best fit to the laboratory-measured values; All the presented values are in W/K·m.

laboratory-measured ones for all temperature levels. The absolute percent error is almost < 15% that is acceptable for engineering design, as the dash lines marked in the figure. The values of MAE and RMSE were calculated and presented in the figure where the frozen soils (at −10 °C) has the lowest MAE value of 0.298 W/K m and RMSE value of 0.185 W/K m. This statistical analysis result reinforces that the proposed model has good applicability in frozen soils. 5. Summary and conclusions

Acknowledgements

Thermal conductivity is a key engineering property parameter for heat transfer through soils. This study provides a comprehensive assessment of soil thermal conduction characteristics and its prediction models to further understand the mechanism of heat transfer. The new prediction models for various types of soils are developed on the basis of artificial neural network (ANN) technology. The proposed ANN

Financial support for this project was provided by the National Natural Science Foundation of China (Grant No. 41807260) and the central universities, China University of Geosciences (Wuhan) (Grant Nos.CUG170636 and CUGL170807). The first author (Tao Zhang) 11

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acknowledges the China Scholarship Council for supporting his study at University College London and would like to thank Dr. Yu-Ling Yang at Southeast University, for her assistance in laboratory testing and technical writing check.

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List of notations Notations: the following symbols and abbreviation are used in this paper: κ, χ, η: empirical parameters in Côté's model

12

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T. Zhang, et al. μ1, μ2: momentum factors ANN: artificial neural network PM-C: individual model for clay PM-CS: individual model for coarse sand PM-FS: individual model for fine sand PM-G: generalized model PM-S: individual model for silt PM-SS: individual model for silty sand c: clay content cmax: maximum training cycle f: transfer functions h1, h2, hi: input parameters K: thermal anisotropy coefficient k: soil thermal conductivity km: measured thermal conductivity kp: predicted thermal conductivity ksat, kdry: thermal conductivity of saturated and dry soils, respectively

MAE: mean absolute error MLPs: multi-layer perceptrons N: sample size of data n: porosity qc: quartz content R: output parameter R2: coefficient of correlation RMSE: root mean square error Sr: degree of saturation var: variance w: moisture content γd: dry unit weight xN: normalized parameter value x, xi: actual input-output values x , xi : predicted values xmin, xmax: actual minimum and maximum value, respectively yi, yj: hidden neurons

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