Estimating Soil Thermal Conductivity by Weighted Average Models with Soil Solids as a Continuous Medium

Abstract

In an attempt to further simplify and to refine the modeling of soil thermal conductivity (λ), two novel weighted average models (WAMs) were developed in which soil solids represent the continuous phase. In the first model, WAM_s-1, the continuous phase consists of two distinctive minerals groups (quartz and compounded remaining soil minerals), while air and water are treated as dispersed components. In the second model, WAM_s-2, all soil minerals are compounded and considered the continuous phase, while air and water are dispersed components. In contrast to de Vries’ original WAM with two continuous phases (soil air or soil water), the proposed models are very simple due to the following assumptions: using soil solids as a single continuous medium lead to eliminating the discontinuity of thermal conductivity when switching between soil air and soil water as continuous medium, and using the thermal conductivity of dry air simplifies a complex expression for an apparent thermal conductivity of humid soil air. Both models were successfully calibrated and validated using 39 Canadian Field Soil database and 3 Standard Sands and were successfully applied to 10 Chinese soils.

1 Introduction

A thorough knowledge of the soil’s thermal conductivity (λ) is essential for the design of earth-contact installations, such as high-voltage power cables, ground heat exchangers, vaults containing nuclear waste, and steam and hot water pipelines. Reliable data on the thermal conductivity of the soil are also necessary for assessing the long-term impact of underground facilities on the earth surface environment. Despite this, it is rare to find trustworthy and complete thermal conductivity data, covering a full range of the degree of saturation (S_r). This is due to the great diversity of soil texture, the unusual shape of soil particles, the soil water redistribution, and the labor-intensive measurement procedures that require unique and expensive equipment [1, 2]. In addition, thermal conductivity measurements are prone to error because the operating principle of thermal conductivity probes is based on the classical theory of conduction heat flow in solids. Extending this theory to porous soil systems raises several problems [3, 4] such as a complex and porous capillary system, contact resistance between the thermal conductivity probe and surrounding soil particles, and a simultaneous heat and soil moisture flow in combination with latent heat effects. Consequently, soil thermal conductivity measurements are prone to errors that are difficult to assess. For this reason, estimating soil thermal conductivity from general soil properties has attracted increasing attention. Several thermal conductivity models have been developed in the past [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Of these, six are based on semi-physical principles [6, 9, 10, 14, 16, 20], one is based on a statistical-physical principle [11] and the remaining eight are based on empirical principles [5, 7, 8, 12, 13, 15, 17, 19]. For more details on the previous thermal conductivity models see [21]. In general, the physical-based (mechanistic) models use a simplified theory of conduction heat flow in solids adapted to porous media. Consequently, they have added value in estimating thermal conductivity and extrapolation capability when applied to different soil textures. These models usually take into account the main characteristics of the soil system, such as porosity (n), density of solids (ρ_s), mineral composition, texture, water content, λ of water (λ_w), air (λ_a), and solids (λ_s) as well as some discrete parameters that cannot be observed directly (e.g., thermal conductivity of water vapor with latent heat effects, λ_v). The thermal conductivity of solids (λ_s) is also commonly used in the majority of models. However, due to the often-unknown mineralogy of the soils, it is very difficult to assess representative values of λ_s. For this reason, λ_s is usually treated as a fitting parameter. Therefore, complete soil mineralogy data are essential for the successful development and verification of thermal conductivity models. Tarnawski et al. [20] reviewed and analyzed de Vries’ model [6] using a complete thermal conductivity database of 39 Canadian Field Soils and 3 Standard Sands; each soil thermal conductivity data set was measured using the transient thermal probe technique at room temperature (T). In general, the de Vries model (deV-1) provides good estimation (λ_est) for fine and coarse textured soils. However, the wide application of this model is considerably limited by its considerable complexity and the obligatory use of numerous coefficients that are very difficult to determine. Originally, these coefficients were obtained from fitting to selected soil thermal conductivity data that may not be fully applicable to different field soils. As a result, these coefficients are often roughly estimated or taken from other published data. Furthermore, deV-1 is based on several unrealistic assumptions regarding soil structure: solid grains are considered as rotated oblate ellipsoids, all grains are identical in size, do not touch each other, and are uniformly dispersed in a homogeneous continuous medium (air or water). In fact, deV-1 uses two distinctly different continuous media (water or air: ${{\lambda_{w} } \mathord{\left/ {\vphantom {{\lambda_{w} } {\lambda_{a} }}} \right. \kern-\nulldelimiterspace} {\lambda_{a} }} \approx 25$) with a boundary point (at the critical soil water content, θ_cr) between them. As a result, there is an obvious step change in the calculated thermal conductivity (at θ_cr) which has no physical interpretation. Furthermore, there are no clear guidelines for θ_cr values when applied to different soil textures. Additionally, the use of deV-1 between dryness and θ_cr was not recommended and an alternative linear interpolation of thermal conductivity was recommended. For soils at elevated temperatures (30—90 °C), deV-1 obviously underestimates the experimental thermal conductivity. This issue was partially reduced by replacing the thermal conductivity of dry air with the thermal conductivity of moist air (λ_app = λ_a + λ_v), where λ_v represents the thermal conductivity of migrating water vapor carrying the latent heat [6]. Even though only a satisfactory agreement with the measured thermal conductivity data was obtained, the above issues are very difficult to solve within the framework of the deV-1 model, so the development of a new model is urgently needed. Perhaps, new research should consider a simplified, but still versatile, weighted average model for soils that includes only one physically based, continuous phase that is applicable to a full range of wetness. The new model should be far simpler to use compared to the deV-1 model, while still maintaining close λ_est to λ_exp.

The main objective of this paper is therefore to develop two weighted average models with soil solids (s) as the continuous medium (WAM_s). The first model (WAM_s-1) considers two distinctly different mineral groups (i.e., quartz and other minerals) as the continuous medium. Quartz is one of the most common minerals in the Earth’s crust and its λ is superior (7.6 W·m⁻¹·K⁻¹) with respect to other minerals (2.2 W·m⁻¹·K⁻¹); therefore, quartz is treated separately from other minerals. In turn, the second model (WAM_s-2) considers all soil minerals as consolidated and as a result forming the continuous medium. The fundamental principles of these two models originate from Maxwell’s electrical conductivity model (ECM) [22], for non-interacting solid spheres immersed in the continuous medium, and its extension to identical elongated ellipsoids by Fricke [23]. In contrast to the ECM, its current adaptation to conduction heat flow in unsaturated soils assumes that the continuous medium is formed from mineral soil components while air and water are dispersed components. This modeling approach reflects the actual soil structure more realistically than in the deV-1, where soil solids are compacted together to naturally form the continuous phase. Both new models are subject to calibration and thorough verification by comparing their λ_est with measured thermal conductivity values of 39 Canadian Field Soils and 3 Standard Sands. Then their predicted thermal conductivity is compared to the λ_est obtained with the original deV-1 model [20].

2 Materials and Methods

2.1 Brief Analysis of Weighted Average Model by de Vries

First, Maxwell [22] published details of the ECM for a two-phase dispersion system consisting of randomly distributed and non-interacting uniform spheres immersed in a homogeneous continuous medium. Then, Fricke [23] and Burger [24] extended the application of this model to elongated spheroids. Eight years later, this model was customized by Eucken [25] to conduction heat transfer through multi-component systems. Subsequently, in 1952, Eucken’s model [25] was further modified by de Vries [26] for estimating thermal conductivity of unsaturated soils. This specifically customized model was based on weighted average contributions of the basic components of the soil (quartz, other minerals, water, and air) which gave the first visible impression of a physical-based model. However, de Vries’ 1952 model [26] was very complex due to the need to deal with two different continuous media (air and water); and therefore, it introduced numerous coefficients that were difficult to determine. In addition, the use of a soil weighting factor (κ_i) introduced several implicit characteristics of soil structure that were not fulfilled in the soils. For example, it was assumed that all soil grains were of the same size and ellipsoidal in shape. It was also assumed that the soil grains are far apart from each other so that they have no contact with each other. These restrictions do not reflect real conditions in the field soils. However, apart from these limitations, it has been shown by de Vries [6] and Tarnawski [20] that κ_i can still be successfully applied to field soils after some modifications. This is probably the reason why, despite its complexity de Vries model, it is still frequently cited in the soil science literature as the top mechanistic model. However, the complexity of this model can be reduced considerably if a continuous phase of compacted soil minerals is considered. Indeed, the soil matrix is a key factor affecting the soil thermal conductivity [15, 21]. It is defined as a random collection of compacted solid particles, each with a unique shape and myriad irregular surfaces. A dominant heat pathway is via the soil grains, while the other potential heat pathways, via air and/or water account for a much smaller proportion of the total amount of heat transferred. Considering the above, the main objective of this paper is to develop a new weighted average model in which soil solids represent the continuous phase.

2.2 Weighted Average Models with Soil Solids as the Continuous Phase

Two weighted average models (deV-1 and deV-2), recently examined by Tarnawski [20], significantly underestimated soil thermal conductivity. As a result, they were customized by introducing complex expressions describing the latent heat effects due to migration of water vapor in soil air. For simplicity, this issue was disregarded in the development of new models with soil solids as the continuous medium.

2.2.1 Model with Continuous Phase made of Two Distinctive Mineral Groups: WAM_s -1

Like the de Vries’ model [26], WAM_s-1 includes weighted average contributions of the volumetric fractions (θ_i), thermal conductivities (λ_i), and weighting factors (κ_i) of the soil basic components, i.e., quartz, other minerals, water, and air. Then, the effective soil thermal conductivity (λ) is estimated from Eq. 1.

$$\lambda = \frac{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} \cdot \lambda_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} \cdot \lambda_{{\text{a}}} + \kappa_{{{\text{qtz}}}} \cdot \theta_{{{\text{qtz}}}} \cdot \lambda_{{{\text{qtz}}}} + \kappa_{{{{\text{o}} - {\text{min}}}}} \cdot \theta_{{{{\text{o}} - {\text{min}}}}} \cdot \lambda_{{{{\text{o}} - {\text{min}}}}} }}{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} + \kappa_{{{\text{qtz}}}} \cdot \theta_{{{\text{qtz}}}} + \kappa_{{{{\text{o}} - {\text{min}}}}} \cdot \theta_{{{{\text{o}} - {\text{min}}}}} }}$$

(1)

where κ_i is

$$\kappa_{{\text{i}}} = \frac{1}{3}\left[ {\frac{2}{{1 + \left( {\frac{{\lambda_{{\text{i}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)g_{{\text{i}}} }} + \frac{1}{{1 + \left( {\frac{{\lambda_{{\text{i}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)\left( {1 - 2g_{{\text{i}}} } \right)}}} \right]$$

(2)

where g is control factor and the subscript s represents continuous medium, i.e., soil solids.

The main innovation of this model is based on the hypothesis that the continuous phase consists of soil minerals, divided into two distinctive groups; namely, quartz and other minerals while soil water and air were considered as the dispersed phase.

The heat flux through the continuous phase was characterized by its thermal conductivity, i.e., soil solids (λ_s), which can be evaluated by a geometric mean model [15].

$$\lambda_{{\text{s}}} = \lambda_{{{\text{qtz}}}}^{{\Theta_{{{\text{qtz}}}} }} \cdot \left( {\lambda_{{{\text{o}} - \min }} } \right)^{{1 - \Theta_{{{\text{qtz}}}} }}$$

(3)

where Θ_qtz is the volumetric fraction of quartz in the soil solids.

Then, thermal conductivity of other minerals (λ_o−min) was obtained from [15], i.e., the geometric mean relation applied to the thermal conductivity of fully saturated soils (λ_sat).

$$\lambda_{o -- \min } = \left[ {{{\left( {{{\lambda_{sat} } \mathord{\left/ {\vphantom {{\lambda_{sat} } {\lambda_{w}^{{\text{n}}} }}} \right. \kern-\nulldelimiterspace} {\lambda_{w}^{{\text{n}}} }}} \right)^{{\frac{1}{{1 - {\text{n}}}}}} } \mathord{\left/ {\vphantom {{\left( {{{\lambda_{sat} } \mathord{\left/ {\vphantom {{\lambda_{sat} } {\lambda_{w}^{{\text{n}}} }}} \right. \kern-\nulldelimiterspace} {\lambda_{w}^{{\text{n}}} }}} \right)^{{\frac{1}{{1 - {\text{n}}}}}} } {\lambda_{qtz}^{{\Theta_{qtz} }} }}} \right. \kern-\nulldelimiterspace} {\lambda_{qtz}^{{\Theta_{qtz} }} }}} \right]^{{1/(1 - \Theta_{qtz} )}}$$

(4)

The weighting factor κ, for basic soil components made of ellipsoidal particles, has the following forms:

$$\kappa_{{{\text{qtz}}}} = \frac{1}{3}\left[ {\frac{2}{{1 + \left( {\frac{{\lambda_{{{\text{qtz}}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)g_{{{\text{qtz}}}} }} + \frac{1}{{1 + \left( {\frac{{\lambda_{{{\text{qtz}}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)\left( {1 - 2g_{{\text{qtz}}} } \right)}}} \right]$$

(5)

$$\kappa_{{{{\text{o}} - {\text{min}}}}} = \frac{1}{3}\left[ {\frac{2}{{1 + \left( {\frac{{\lambda_{{{{\text{o}} - {\text{min}}}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)g_{{{{\text{o}} - {\text{min}}}}} }} + \frac{1}{{1 + \left( {\frac{{\lambda_{o - \min } }}{{\lambda_{{\text{s}}} }} - 1} \right)\left( {1 - 2g_{{{{\text{o}} - {\text{min}}}}} } \right)}}} \right]$$

(6)

$$\kappa_{{\text{w}}} = \frac{1}{3}\left[ {\frac{2}{{1 + \left( {\frac{{\lambda_{{\text{w}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)g_{{\text{w}}} }} + \frac{1}{{1 + \left( {\frac{{\lambda_{{\text{w}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)\left( {1 - 2g_{{\text{w}}} } \right)}}} \right]$$

(7)

$$\kappa_{{\text{a}}} = \frac{1}{3}\left[ {\frac{2}{{1 + \left( {\frac{{\lambda_{{\text{a}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)g_{{\text{a}}} }} + \frac{1}{{1 + \left( {\frac{{\lambda_{{\text{a}}} }}{{\lambda_{{\text{s}}} }} - 1} \right)\left( {1 - 2g_{{\text{a}}} } \right)}}} \right]$$

(8)

where the control factor for other minerals is assessed as follows [20]:

$$g_{{{\text{o}} - \min }} = m_{{{\text{sa}}}} \cdot g_{{{\text{sa}}}} + m_{{{\text{si}}}} \cdot g_{{{\text{si}}}} + m_{{{\text{cl}}}} \cdot g_{{{\text{cl}}}}$$

(9)

The WAM_s-1 offers a noticeable simplicity with respect to deV-1, i.e., a lack of complex and controversial expressions for migration of water vapor in soil air carrying latent heat, an absence of a controversial switching point θ_cr between air and water as a continuous phase, and a complete elimination of discontinuity in λ as a function of S_r, λ(S_r), at the switching point. Consequently, the WAM_s-1 structure is simple and the model is straightforward to follow and apply.

2.2.2 Model with Continuous Phase Considering all Soil Minerals: WAM_s -2

The effective soil thermal conductivity can also be estimated by considering all soil solids (minerals) as the continuous phase. Hence, the weighting factor for the soil solids becomes unity (κ_s = 1). Therefore, Eq. 1 is further simplified to the following form:

$$\lambda = \frac{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} \cdot \lambda_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} \cdot \lambda_{{\text{a}}} + \theta_{{\text{s}}} \cdot \lambda_{{\text{s}}} }}{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} + \theta_{{\text{s}}} }}$$

(10)

After substituting Eq. 3 into Eq. 10, the following expression was obtained:

$$\lambda = \frac{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} \cdot \lambda_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} \cdot \lambda_{{\text{a}}} + \theta_{{\text{s}}} \cdot \lambda_{{{\text{qtz}}}}^{{\Theta_{{{\text{qtz}}}} }} \cdot \left( {\lambda_{{{\text{o}} - \min }} } \right)^{{1 - \Theta_{{{\text{qtz}}}} }} }}{{\kappa_{{\text{w}}} \cdot \theta_{{\text{w}}} + \kappa_{{\text{a}}} \cdot \theta_{{\text{a}}} + \theta_{{\text{s}}} }}$$

(11)

Only two weighting factors are required, κ_w and κ_a, as defined by Eqs. 7 and 8. With this regard, the WAM_s-2 is noticeably simpler than WAM_s-1 as it does not require soil control factors for quartz (g_qtz) and other minerals (g_o-min). The only control parameters required are g_w and g_a, for evaluating the weighting factors κ_w and κ_a.

2.3 Soil Thermal Conductivity Database

A comprehensive soil database is needed to assess two new models for soil thermal conductivity. This type of database should include the following information: fractions of grain size distribution for clay, silt, and sand (i.e., m_cl, m_si, m_sa), soil porosity (n), mineral composition (Θ_min), density of soil solids (ρ_s), volumetric water content (θ_w) or degree of saturation (S_r), and experimentally measured thermal conductivity data. Currently, the above requirements are solely met by the Canadian soil database [17, 20, 27]. The other databases on thermal conductivity of soils usually do not contain complete information on mineral composition and thermal conductivity data at dryness and saturation are often not available. The next subsection provides a summary of the 40 Canadian Soils thermal conductivity database, i.e., 39 field soils and one pure quartz sand from Sable Island (Atlantic Canada). For details of the 40 Canadian Soils database and the experimental methods see Tarnawski [17, 27].

2.3.1 Canadian Soils Database

Forty field soil samples, from nine Canadian provinces, were subjected to non-stationary thermal conductivity tests under laboratory conditions. The texture of seven soil samples from Nova Scotia (NS) varied from coarse to silty. Three soil samples from Prince Edward Island (PE) were mainly loamy sand. Five soil samples from New Brunswick (NB) were mainly silty loam or silty clay loam. Two soil samples from Quebec (QC) were coarse-grained (sand and loamy sand). The texture of seven soil samples from Ontario (ON) ranged from sand to silt loam. Four soil samples from Manitoba (MN) were loamy sand to silt loam. Five soil samples from Saskatchewan (SK) were loamy sand or silt loam. One soil sample from Alberta (AB) was silty loam. All six samples from British Columbia (BC) were generally fine soils, ranging from silty loam to silty clay loam. Each soil sample was tested nine times, with an overall uncertainty of ± 6.3% at 95% confidence level, at each S_r value (i.e., S_r = 0, 0.1, 0.25, 0.5, 0.7 and 1), under laboratory conditions, to obtain the one average thermal conductivity value. The basic data on the grain size distribution of the soil and the complete thermal conductivity data were summarized in Tarnawski [17, 27]. A summary of the mineral occurrence in the 39 Canadian Field Soils and a corresponding summary for λ_o-min and λ_s can be found in Tarnawski [17, 27].

2.3.2 Standard Sands Database

The complete physical data of the standard sands (C-109, C-190, and NS-04) were given by Tarnawski [28, 29]. C-109 and C-190 are natural silica sands (99.8% quartz) with rounded or sub-rounded shape grains, while NS-04 is pure quartz sand from Sable Island (Atlantic Canada). A summary of the complete thermal conductivity data of three Standard Sands can be found in [28, 29].

2.4 Model Calibration Procedure

The model estimates (λ_est) were compared with experimental data (λ_exp) and the standard deviation (SD) was used as a measure of the model performance, defined as follows:

$${\text{SD}} = \sqrt {\frac{1}{N - M}\sum\limits_{1}^{N} {\left( {\lambda_{\exp } - \lambda_{{{\text{est}}}} } \right)^{2} } }$$

(12)

where M is the number of presently fitted coefficients in the WAMs and N is the total number of sampled data.

2.4.1 WAM_s -1

The model control parameters (g_qtz, g_sa, g_si, g_cl, and g_w) were assumed to be the same as for deV-1, i.e., they were already previously determined by fitting to λ data from 39 Canadian Field Soils [20]. So, consequently, they do not presently contribute to M because they are already known.

Table 1 summarizes the previously fitted factors for soil grains (g_i) and water (g_w). The fitted factor g_qtz = 0.15 was used only to evaluate κ_qtz, while the other g_i values, corresponding to soil texture: sandy (0.5 < m_sa ≤ 1), silty (0.5 ≤ m_sa ≤ 0.1), and clayey (m_sa < 0.1), were used to evaluate κ_o-min.

Table 1 Fitted control factors: grains (g_i) [20]

The remaining model control factor is the soil air coefficient (g_a) which is particularly important due to the elimination of the latent heat effects of water vapor in soil air. Therefore, it is useful to reveal details on the procedure for fitting and calculating g_a. First, it was assumed that the g_a values were the same as in deV-1. Next, λ_est was calculated for all S_r values (S_r = 0, 0.1, 0.25, 0.5, 0.7, and 1) and the corresponding average SDs were obtained. Then, g_a values were adjusted at each S_r value, keeping the smallest total SD. Next, a dynamic plot of g_a versus S_r was made for 17 coarse and 22 fine soils and the final g_a adjustment, combined with a g_a(S_r) curve fit, was performed. Finally, the following g_a equations were obtained for 17 coarse and 22 fine soils, minimizing the SD between λ_est and λ_exp.

Coarse soils (m_sa > 0.5):

$$\begin{aligned} g_{{\text{a}}} &= 0.0035 - 0.0267 \cdot S_{{\text{r}}} + 1.1186 \cdot S_{{\text{r}}}^{2} \\ & \quad - 3.3481 \cdot S_{{\text{r}}}^{3} + 3.8429 \cdot S_{{\text{r}}}^{4} - 1.5169 \cdot S_{{\text{r}}}^{5} \;R^{2} = 1.0\end{aligned}$$

(13)

Fine soils (m_sa ≤ 0.5):

$$\begin{aligned} g_{{\text{a}}} & = 0.005 - 0.0194 \cdot S_{{\text{r}}} + 0.7168 \cdot S_{{\text{r}}}^{2} \\ &\quad - 1.9042 \cdot S_{{\text{r}}}^{3} + 2.0573 \cdot S_{{\text{r}}}^{4} - 0.7986 \cdot S_{{\text{r}}}^{5} \;R^{2} = 1.0 \hfill \\ \end{aligned}$$

(14)

2.4.2 Standard Sands (NS-04, C-109, and C-190)

The grains of pure quartz sands (NS-04, C-109, and C-190) are larger and more uniform in size, compared to ordinary field soils; therefore, they require different g_a and g_w values. Below is a function of g_a versus S_r obtained by fitting to the thermal conductivity data of the Standard Sands, while minimizing SD.

Pure quartz sands:

$$g_{{\text{a}}} = 0.05674 \cdot S_{{\text{r}}}^{0.3053} \quad R^{2} = 0.9904$$

(15)

For g_w = 0.0001 and g_a as above, underestimates of thermal conductivity were observed at S_r > 0.5. To minimize this problem, a new function g_w(S_r) was introduced:

Pure quartz sands:

$$g_{{\text{w}}} = 0.001 + 0.02252 \cdot S_{{\text{r}}} \quad R^{2} = 0.995$$

(16)

2.4.3 WAM_s -2

The WAM_s-2 assumes that all soil minerals are consolidated and form the continuous phase. For this reason, the model structure (Eq. 1) is further simplified by κ_s = 1 (according to Eq. 2). Consequently, the control parameters for the soil solids (g_qtz, g_sa, g_si, and g_cl) were eliminated. Then, the control parameter for soil water (g_w) was assumed to be the same as for deV-1 and for WAM_s-1, leaving g_a as the only control parameter which was assumed to be the same as for WAM_s-1, since both the models are based on the same continuous phase of soil solids (i.e., soil minerals).

3 Results

The following guiding principles for the model performance (SD values) were recently established [17]:

• Superior performance: SD < 0.1 W·m⁻¹·K.⁻¹

• Good performance: 0.1 ≤ SD ≤ 0.15 W·m⁻¹·K.⁻¹

• Satisfactory performance: 0.15 < SD ≤ 0.25 W·m⁻¹·K.⁻¹

• Poor performance: SD > 0.25 W·m⁻¹·K.⁻¹

3.1 Canadian Field Soils

Table 2 summarizes the model predictive performance of WAM_s-1, WAM_s-2, and deV-1 for 17 coarse soils, 22 fine soils, and all 39 Canadian Field Soils. The SD data for WAM_s-1 (M = 6, due to six fitted polynomial coefficients of Eq. 13 or 14) were compared with SD records for WAM_s-2 (M = 6, due to six fitted polynomial coefficients of Eq. 13 or 14) and deV-1 (M = 7).

Table 2 Canadian soils: model performance (SD in W·m⁻¹·K⁻¹)

For 17 coarse soils, 22 fine soils, and all 39 soils, the λ_est by the two new models (with λ_qtz = 7.6 W·m⁻¹·K⁻¹ and λ_a = 0.026 W·m⁻¹·K⁻¹) are approximately in the SD range which corresponds to good/superior model performance. This result confirms that the superior values of SD can be achieved even without considering the latent heat effects due to water vapor migration in soils. For WAM_s-1 and WAM_s-2, the average SDs are ± 0.110 and ± 0.135 W·m⁻¹·K⁻¹ for 17 coarse soils, ± 0.054 and ± 0.069 W·m⁻¹·K⁻¹ for 22 fine soils, and ± 0.078 and ± 0.098 W·m⁻¹·K⁻¹ for all 39 soils. For deV-1 (with λ_qtz = 7.6 W·m⁻¹·K⁻¹ and λ_a-app = λ_a + λ_v, where λ_a-app is the apparent thermal conductivity of humid soil air [20]), average SDs of ± 0.103 W·m⁻¹·K⁻¹ were obtained for 17 coarse soils, ± 0.087 W·m⁻¹·K⁻¹ for 22 fine soils, and ± 0.094 W·m⁻¹·K⁻¹ for all 39 soils; these SDs correspond to superior/good levels.

Figures 1, 2, 3, and 4 show some examples of the worst λ_est(S_r) produced by deV-1, WAM_s-1, and WAM_s-2. The Nova Scotia soil (NS-05), Fig. 1, belongs to the group of coarse texture with a high content of quartz (Θ_qtz = 0.72); consequently, a high value of λ_exp ≈ 2.4 W·m⁻¹·K⁻¹ is observed at S_r = 1. The following SDs were obtained: deV-1: ± 0.073 W·m⁻¹·K⁻¹, WAM_s-1: ± 0.182 W·m⁻¹·K⁻¹, and WAM_s-2: ± 0.177 W·m⁻¹·K⁻¹. In general, the obtained λ_est closely followed the trend of λ_exp(S_r) with an overestimation in the low S_r range (S_r ≈ 0.1) and an underestimation at S_r ≈ 1.

Fig. 1

Nova Scotia soil (NS-05): λ vs. S_r, m_sa = 0.85, Θ_qtz = 0.72, n = 0.40, (SD_deV-1 = 0.073 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.182 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.177 W·m⁻¹·K.⁻¹)

The Ontario soil (ON-04), Fig. 2, is a coarse sand (m_sa = 0.89) with a relatively low quartz content (Θ_qtz = 0.38). All three models provide good/acceptable estimates. The λ_est of all models agree well with the experimental data at S_r = 0 and S_r > 0.7. Underestimates of the thermal conductivity were observed in the remaining S_r range.

Fig. 2

Ontario soil (ON-04): λ vs. S_r, m_sa = 0.89, Θ_qtz = 0.38, n = 0.39, (SD_deV-1 = 0.144 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.157 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.205 W·m⁻¹·K.⁻¹)

Figure 3 shows the modeling results for MN-02 (silty loam: m_sa = 0.217). Despite a low content of quartz (Θ_qtz = 0.20), its λ_sat was about 2.2 W·m⁻¹·K⁻¹. The high values of thermal conductivity are due to a high content of calcite and dolomite (Θ_cal = 0.28, Θ_dol = 0.37); both minerals have thermal conductivity values of 3.03 W·m⁻¹·K⁻¹ and 5.07 W·m⁻¹·K⁻¹, respectively. The two WAM_s models had slightly underestimated thermal conductivity values throughout the S_r range, except at S_r = 0 and 0.7, which were slightly overestimated; SD values were ± 0.105 W·m⁻¹·K⁻¹ for WAM_s-1 and ± 0.110 W·m⁻¹·K⁻¹ for WAM_s-2. The SD value of ± 0.252 W·m⁻¹·K⁻¹ was obtained for deV-1, which falls within the poor performance range.

Fig. 3

Manitoba soil (MN-02): λ vs. S_r, m_sa = 0.217, Θ_qtz = 0.20, Θ_cal = 0.28, Θ_dol = 0.37, n = 0.41, (SD_deV-1 = 0.252 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.105 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.110 W·m⁻¹·K.⁻¹)

Figure 4 shows the λ_est for SK-02 (silt loam) which achieved the worst result among the 39 Canadian soils. All three models performed poorly at S_r = 0.25 and S_r = 0.50, while good λ_est were obtained for soil dryness and soil saturation.

Fig. 4

Saskatchewan soil (SK-02): λ vs. S_r, m_sa = 0.67, Θ_qtz = 0.61, n = 0.45, (SD_deV-1 = 0.209 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.164 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.236 W·m⁻¹·K.⁻¹)

Figures 5 and 6 show two examples of the superior performance of the models. The New Brunswick soil (NB-05) is a fine-textured soil (m_sa = 0) for which the thermal conductivity is best estimated by all three models (Fig. 5), although the thermal conductivity is slightly underestimated at soil dryness. In particular, the λ_est of WAM_s-1 and WAM_s-2 followed the experimental data very closely over the whole S_r range, while the estimates of deV-1 showed increasing overestimates for S_r > 0.5.

Fig. 5

New Brunswick soil (NB-05): λ vs. S_r, m_cl = 0.33, m_si = 0.67, Θ_qtz = 0.39, n = 0.54, (SD_deV-1 = 0.080 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.022 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.037 W·m⁻¹·K.⁻¹)

Fig. 6

British Columbia soil (BC-04): λ vs. S_r, m_cl = 0.41, m_si = 0.59, Θ_qtz = 0.17, n = 0.52, (SD_deV-1 = 0.088 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.035 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.043 W·m⁻¹·K.⁻¹)

Figure 6 shows the close agreement of λ_est and λ_exp of all models for clayey soil (BC-04), which also has a fine texture. Again, the predictive performance (SD) of all three models falls into the superior category, i.e., SD < 0.1 W·m⁻¹·K⁻¹. In particular, λ_est of WAM_s-1 and WAM_s-2 followed the trend of experimental data very closely throughout the S_r range with slight underestimates of thermal conductivity observed at soil dryness.

3.2 Standard Sands

Table 3 summarizes average SD for deV-1 (θ_cr = 0.0625 n; λ_a-app = λ_a + λ_v), WAM_s-1 and WAM_s-2; the SD comparison was made using λ_qtz = 7.6 W·m⁻¹·K⁻¹. It is obvious that the closest λ_est were given by WAM_s-1 and WAM_s-2, while λ_est by deV-1 were noticeably worse in comparison. At soil dryness, obtained estimates of λ consistently showed inflated values above λ_dry-exp.

Table 3 Standard sands: model performance (SD in W·m⁻¹·K⁻¹)

Figure 7 shows the modeling results compared to the experimental data for C-109. The obtained average SDs were ± 0.109 W·m⁻¹·K⁻¹ for deV-1, ± 0.098 W·m⁻¹·K⁻¹ for WAM_s-1 and ± 0.096 W·m⁻¹·K⁻¹ for WAM_s-2, which are good/superior λ estimates.

Fig. 7

Ottawa sand (C-109): λ vs. S_r, m_sa = 1.00, Θ_qtz = 1.00, n = 0.32, (SD_deV-1 = 0.109 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.098 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.096 W·m⁻¹·K.⁻¹)

Figure 8 shows the modeling results compared to the experimental data for NS-04 (100% pure quartz). Again, the WAM_s-1 and WAM_s-2 models offered similar SD data and their thermal conductivity trends closely followed the experimental data. In general, the average SDs were ± 0.180 W·m⁻¹·K⁻¹ for deV-1, ± 0.139 W·m⁻¹·K⁻¹ for WAM_s-1 and ± 0.135 W·m⁻¹·K⁻¹ for WAM_s-2, which is a good predictive performance.

Fig. 8

Sable Island sand (NS-04): λ vs. S_r, m_sa = 1.0, Θ_qtz = 1.0, n = 0.36, (SD_deV-1 = 0.180 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.139 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.135 W·m⁻¹·K.⁻¹)

3.3 Application to 10 Chinese Soils

Both WAM_s with fixed parameters (g_qtz, g_sa, g_si, g_cl, g_w, and g_a) were applied to the thermal conductivity data of 10 Chinese soils [13], divided into two groups: coarse (m_sa > 0.5) and fine (m_sa ≤ 0.5). The quartz content, Θ_qtz, was estimated by applying an iterative fit of λ_est to experimental λ data [30]. The basic physical characteristics of soil were summarized by Tarnawski [20]. The λ_est were determined by assuming the following data: λ_qtz = 7.6 W·m⁻¹·K⁻¹ and λ_o-min = 2.12 W·m⁻¹·K⁻¹. For deV-1, the following constraints were applied: M = 7 and θ_cr values according to m_sa: 0.023 (m_sa = 1.0); 0.05 (0.50 < m_sa < 1.0); 0.10 (0.15 < m_sa < 0.5); and 0.15 (m_sa < 0.15). A summary of average SD values for deV-1, WAM_s-1, and WAM_s-2 can be found in Table 4.

Table 4 Chinese soils: model performance (SD in W·m⁻¹·K⁻¹)

Typical trends of λ_est versus S_r are shown in Fig. 9 and 10. The WAM_s-1 model underestimated the experimental data at 0.06 < S_r < 0.20 (Fig. 9), while λ_est of deV-1 closely follows the experimental data, with an exception at S_r = 0.06, where the largest underestimation is observed. For both WAM_s models, the average SDs are within acceptable predictive performance. The WAM_s-1 estimates closely follow the experimental data (Fig. 10). The average SDs, for both models, are within good predictive performance, while deV-1 underestimates λ_exp within 0.15 < S_r < 0.3.

Fig. 9

Chinese soil (S-Ren-001): λ vs. S_r, m_sa = 0.94, Θ_qtz = 0.74, n = 0.408, (SD_deV-1 = 0.186 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.236 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.277 W·m⁻¹·K.⁻¹)

Fig. 10

Chinese soil (S-Ren-005): λ vs. S_r, m_sa = 0.27, Θ_qtz = 0.55, n = 0.51, (SD_deV-1 = 0.132 W·m⁻¹·K⁻¹, SD_WAMs-1 = 0.069 W·m⁻¹·K⁻¹, SD_WAMs-2 = 0.080 W·m⁻¹·K.⁻¹)

4 Discussion

The two novel models were compared with deV-1 in terms of simplicity, quality of thermal conductivity estimates, and extension to other soils of different genesis and location. Concerning the model simplicity: the WAM_s-2, where all soil minerals are consolidated and form the continuous phase, offers the simplest structure and is therefore, very straightforward to use, i.e., only one soil air factor (g_a) required fitting to experimental thermal conductivity data. The WAM_s-1 offered a very similar structure to the WAM_s-2, but it considered soil solids divided into two different groups: quartz and other minerals and is therefore slightly more complex in structure than WAM_s-2. Furthermore, it does not require any adjustments for g_qtz, g_sa, g_si, g_cl, and g_w, as they were assumed to be the same as in the deV-1 model. Therefore, both WAM_s-1 and WAM_s-2 were more straightforward than the original de Vries model (deV-1), which is more complex, more difficult to apply and more prone to manipulation. In deV-1, the final adjustment of the critical moisture content (θ_cr), corresponding to the switching point of the continuous medium (i.e., from air to water), is done a posteriori, when preliminary λ_est have been obtained; i.e., a value of θ_cr is shifted up or down to obtain a minimum SD average. However, for soils without complete λ_exp data, guessed values of θ_cr would have to be used. In this case, the deV-1 model might not perform well because of the uncertainty in θ_cr. In contrary to deV-1, this problem does not exist for WAM_s-1 and WAM_s-2. Furthermore, the deV-1 model required the use of apparent thermal conductivity of humid soil air, i.e., enhanced λ_a that includes the latent heat effects of water vapor. In contrast to deV-1 model, both WAM_s models use only λ_a, i.e., without latent heat effects due to the migration of soil water vapor. Regarding the model estimates quality, WAM_s-1 provided more accurate λ_est than WAM_s-2 and deV-1, mainly due to a more appropriate representation of the soil solid phase which consist of quartz and other soil minerals. When both WAM_s models were applied to 39 Canadian Soils, their average standard deviations (SDs) were as follows: ± 0.110/0.135 W·m⁻¹·K⁻¹ for 17 coarse soils, ± 0.054/0.069 W·m⁻¹·K⁻¹ for 22 fine soils, and ± 0.078/0.098 W·m⁻¹·K⁻¹ for all 39 soils. That outcome falls into the “good/superior” category. When applied to three Standard Sands, the structures of WAM_s-1 and WAM_s-2 were identical because the quartz was the only soil mineral.

Consequently, the λ_est of both models followed the experimental data closely; their average SDs were in a good performance range. Essentially, the WAM_s-1, WAM_s-2, and deV-1 models offered similar SD data (SD $<$ 0.14 W·m⁻¹·K⁻¹) and their thermal conductivity trends closely followed experimental data.

When the models were applied to the thermal conductivity database of 10 Chinese soils, good λ_est were obtained with the deV-1 model, while both WAM_s (without latent heat) provided only satisfactory thermal conductivity estimates. For four coarse soils (S-Ren: 001–002-003–011), the deV-1 model (with the apparent thermal conductivity of humid soil air) provided closer λ_est than WAM_s-1 and WAM_s-2 (without latent heat transfer). In contrast, for six fine soils (S-Ren: 004–005-006–007-008–009) WAM_s-1 provided comparable λ_est value to deV-1. In general, both WAM_s models underestimated the experimental data; therefore, additional model improvement by applying λ_a-app should perhaps be considered. Consequently, further investigations are needed to include the apparent thermal conductivity of humid soil air.

5 Conclusions and Recommendations

Two innovative weighted average models (WAM_s-1 and WAM_s-2) were developed, calibrated, and successfully verified using thermal conductivity data of 39 Canadian Field Soils and three, differently compacted, pure quartz sands (Standard Sands). The SD data, for 39 Canadian Field Soils, confirm that WAM_s-1 produced better λ_est than WAM_s-2. This outcome was likely due to a more realistic representation of the solid phase (qtz/o-min) in WAM_s-1 by highlighting quartz as the most unique and dominant soil mineral. However, at soil dryness, both models, with some exceptions, generally underestimate the experimental data. In turn, the WAM_s-2 offers outstanding simplicity, i.e., the simple structure of model, the least number of adjustable parameters, and easy tractability, while maintaining a good/superior λ_est. However, its representation of the soil solid phase does not fully reflect the dominant role of quartz, in terms of Θ_qtz and λ_qtz. Consequently, its estimates of thermal conductivity were slightly poorer for coarse sands.

In short, the overall predictive performance (SD) of all three models (WAM_s-1, WAM_s-2, and deV-1) falls into the superior category, i.e., SD < 0.1 W·m⁻¹·K⁻¹, for fine soils. Despite the superior/good λ_est by the three models, future work on modeling soil should be continued to further improve λ estimations, especially for coarse soils.

When the three models were applied to Standard Sands, the obtained λ_est followed the experimental data closely; their average SDs were in a good performance range (SD $<$ 0.14 W·m⁻¹·K⁻¹) and their thermal conductivity trends closely followed experimental data.

When the three models were applied to the thermal conductivity database of 10 Chinese soils of unknown mineralogy, good λ_est were obtained with the deV-1 model (with apparent thermal conductivity of humid soil air), while both WAM_s (without latent heat) provided only satisfactory/acceptable estimates. In general, both WAM_s models underestimated the experimental data; therefore, additional model improvement by applying λ_a-app should perhaps be considered.