Canadian Field Soils IV: Modeling Thermal Conductivity at Dryness and Saturation

Abstract

The thermal conductivity data of 40 Canadian soils at dryness $(\lambda _{\mathrm{dry}})$ and at full saturation $(\lambda _{\mathrm{sat}})$ were used to verify 13 predictive models, i.e., four mechanistic, four semi-empirical and five empirical equations. The performance of each model, for $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$, was evaluated using a standard deviation (SD) formula. Among the mechanistic models applied to dry soils, the closest $\lambda _{\mathrm{dry}}$ estimates were obtained by MaxRTCM $(\textit{SD} = \pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$, followed by de Vries and a series-parallel model ($\hbox {S-}{\vert }{\vert }$). Among the semi-empirical equations (deVries-ave, Advanced Geometric Mean Model (A-GMM), Chaudhary and Bhandari (C–B) and Chen’s equation), the closest $\lambda _{\mathrm{dry}}$ estimates were obtained by the C–B model $(\pm ~0.022\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$. Among the empirical equations, the top $\lambda _{\mathrm{dry}}$ estimates were given by CDry-40 $(\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for18-coarse and 22-fine soils, respectively). In addition, $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ models were applied to the $\lambda _{\mathrm{sat}}$ database of 21 other soils. From all the models tested, only the maxRTCM and the CDry-40 models provided the closest $\lambda _{\mathrm{dry}}$ estimates for the 40 Canadian soils as well as the 21 soils. The best $\lambda _{\mathrm{sat}}$ estimates for the 40-Canadian soils and the 21 soils were given by the A-GMM and the $\hbox {S-}{\vert }{\vert }$ model.

1 Introduction

Thorough knowledge of soil thermal conductivity $(\lambda )$ is essential in modeling ground temperature (T) regimes in the vicinity of earth-contact facilities dissipating heat, such as buried power cables, ground heat exchangers, pipelines, building foundations, etc. [1]. For example, underground high voltage power cables release a substantial amount of heat that must be dissipated to the surrounding ground in order to avoid overheating and consequent undesirable failure. Therefore, knowledge of the soil ability to conduct heat plays a key role in the design and operation of buried power cables, nuclear waste disposal, underground soil heating, etc. Coarse soils are commonly used as backfill materials around ground-contact devices; this is due to a large content of quartz in these soils and their capability for denser compaction. In general, the flow of heat in soils is influenced by their porosity (n), soil fluid (air and/or water) in soil void space, inter-particle contact resistance and mineral composition of a solid phase. As a result, soil $\lambda$ is primarily influenced by thermal conductivities of each phase, air (a): $\lambda _{\mathrm{a}}\approx 0.026\,\hbox {Wm }^{-1} \cdot \hbox { K}^{-1}$; water (w): $\lambda _{\mathrm{w}} \approx 0.61\,\hbox {Wm }^{-1}\cdot \hbox {K }^{-1}$ [1]; solids (s): $\lambda _{\mathrm{s}} \approx$ 2.2$\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$–7.7 $\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$ [2]; and their volumetric fractions ($\theta _{\mathrm{a}}$, $\theta _{\mathrm{w}}$ and $\theta _{\mathrm{s}})$. In general, dry soils $(\theta _{\mathrm{w}} = 0)$ have a low thermal conductivity $(\lambda _{\mathrm{dry}})$ and can consequently act like an insulating medium causing a temperature increase in insulating applications; in addition, $\lambda _{\mathrm{dry}}$ decreases with increasing n. Due to the presence of these important factors, modeling a ground domain, with buried, heat releasing engineering devices, is a complex problem and the obtained results are often unreliable mainly due to a lack of consolidated soil $\lambda$ data. Measuring $\lambda$ is time consuming and expensive due to the enormous complexity and variability of soil structure, intricate heat and mass transfer phenomena [3, 4], and limitations of measuring techniques and manufactured devices. Consequently, a prevailing majority of research concentrates on $\lambda$ modeling rather than obtaining this property through measurements. One frequently used approach to $\lambda$ modeling is a normalized expression, $Ke={\left( {\lambda -\lambda _{\mathrm{dry}} } \right) }/{\left( {\lambda _{\mathrm{sat}} -\lambda _{\mathrm{dry}} } \right) }$ [2]; where Ke is an empirical, non-dimensional function $(0\le Ke\le 1)$ that depends on the degree of saturation $(S_{{r}})$ and soil texture. Subsequently, $\lambda$ estimates are obtained from a simple relation, $\lambda =\lambda _{\mathrm{dry}} +Ke(\lambda _{\mathrm{sat}} -\lambda _{\mathrm{dry}} )$, that requires knowledge of reliable $\lambda$ data at soil dryness $(\lambda _{\mathrm{dry}})$ and full saturation $(\lambda _{\mathrm{sat}})$. In dry soils, heat is mainly transmitted by conduction through the soil minerals, inter-particle contacts, and air. The path of heat transfer through inter-particle contacts is difficult to examine analytically due to unpredictable soil structure and complex particle shapes. However, inter-particle soil contacts have a strong impact on the amount of heat transferred and, therefore, should not be completely disregarded [5, 6]. Consequently, modeling $\lambda _{\mathrm{dry}}$ remains a challenging task. As a result, there is a continuous demand for more complete $\lambda _{\mathrm{dry}}$ models that should also include models that exclusively apply to standard pure quartz sands [5, 7]. The majority of existing $\lambda _{\mathrm{dry}}$ models are semi-empirical or purely empirical, e.g., Yun and Santamarina [5], Chaudhary and Bhandari [8], Campbell [9], Chen [10], Côté and Konrad [11], and Lu et al. [12]. The application of these models is usually limited to the soils and measuring conditions for which they were fitted to. In addition, these models do not use physical parameters associated with conduction heat flow in dry soils, such as, $\lambda _{\mathrm{s}},\lambda _{\mathrm{a}}$, shapes of soil particles, inter-particle thermal contact resistance. More versatile $\lambda _{\mathrm{dry}}$ models are those of a mechanistic nature such as, a weighted average $\lambda$ model, initially proposed by de Vries [13] and then simplified by Johansen [2]; a series-parallel ($\hbox {S-}{\vert }{\vert }$) model for two-phase soils, originally proposed by Woodside and Messmer [14] and later modified by Kasubuchi et al. [15]; a new $\hbox {S-}{\vert }{\vert }$ model applied to three-phase soils in a full $S_{{r}}$ range proposed by Tarnawski and Leong [16]; a model that uses an array of cubic cells representing three-phase granular materials developed by Gori and Corasaniti [17, 18]; and a framework $\lambda$ model (MaxRTCM) for porous planetary regolith by Wood [19] that includes an inter-particle contact factor in unconsolidated planetary dust, soil and mineral fragments covering bedrock.

Assessments of $\lambda _{\mathrm{sat}}$ are usually obtained by a geometric mean model (GMM), $\lambda _{\mathrm{sat}} =\lambda _{\mathrm{s}}^{1-n} \lambda _{\mathrm{w}}^n$, that usually produces acceptable $\lambda$ estimates of experimental data. Other models, also valid to saturated soils, are: Chaudhary and Bhandari [8], Campbell [9], de Vries [13], Gori and Corasaniti [17, 18], and Tarnawski et al. [16, 20].

In summary, a large majority of $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ models are of empirical or semi-empirical nature. These models rarely produce satisfactory estimates on soil data other than that they were fitted to. In addition, a majority of these models lack the inclusion of conduction heat transfer through inter-particle contacts [21, 22]. In turn, the mechanistic models appear to be, at first glance, more versatile as they include a simplified soil structure combined with some elements of heat flow. On the other hand, they are rather intricate to use and often lack accurate and unbiased predictions. For that reason, an independent validation of the published models, with respect to a complete soil $\lambda$ database, still remains an unachieved goal. Therefore, the objective of this paper is to review modeling approaches to $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ followed by a critical analysis of their estimates with respect to the published database of 40 Canadian soils [23].

2 Thermal Conductivity Database of Canadian Dry and Saturated Soils

Forty Canadian soil samples, ranging from coarse sands to fine clays, were tested in laboratory conditions for their ability to conduct heat using a non-stationary probe technique [23]. The measurements were taken on compacted soil samples, at room T of about $25~{}^{\circ }\hbox {C}$, and at a full range of $S_{{r}}$ from dryness $(S_{{r}} = 0)$ to full saturation $(S_{{r}} = 1)$. Due to a wide diversity in soil texture, the entire database was split into two soil textural groups, namely: coarse soils, with a mass fraction of sand $(m_{\mathrm{sa}} \ge 0.40)$, and fine soils $(m_{\mathrm{sa}} < 0.40)$. Physical characteristics of both soil textures are shown in Table 1 (Coarse Soils) and Table 2 (Fine Soils). Each record of $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ represents an average value of nine $\lambda$ measurements taken from three separate soil samples of approximately the same n and T. More details about Canadian field soils, measurements and $\lambda$ database were given by Tarnawski et al. [23]. A graphical display of $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ versus n is shown in Figs. 1 and 2, respectively.

Table 1 18 Canadian coarse soils—physical characteristics

Table 2 22 Canadian fine soils—physical characteristics

From Fig. 1, a declining trend of $\lambda _{\mathrm{dry}}$ with increasing n was noted and it was approximately linear; $\lambda _{\mathrm{dry}}$ varied approximately from $0.13\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$ to $0.30\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$. Also, for a large majority of saturated soils, a roughly linearly declining trend of $\lambda _{\mathrm{sat}}$ versus n is also observed (Fig. 2); however, $\lambda _{\mathrm{sat}}$ varies roughly from $1.0\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$ to $3.2\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$.

3 Specifics of Published Modeling Approaches

Several $\lambda$ models for soils were developed in the past with detailed reviews given by Farouki [1], Côté and Konrad [24], and Tarnawski et al. [25]. Mechanistic models are usually complex in form, but for dry $(\theta _{\mathrm{w}} = 0)$ and saturated ($\theta _{\mathrm{w}} = n)$ conditions, their forms are greatly simplified. Moreover, if the soil mineral composition is not available, the thermal conductivity of solids $(\lambda _{\mathrm{s}})$ is a fitting parameter in all the mechanistic and semi-empirical models.

Fig. 1

40 Canadian dry soils: variation of $\lambda _{\mathrm{dry}}$ versus n

Fig. 2

40 Canadian saturated soils: variation of $\lambda _{\mathrm{sat}}$ versus n

3.1 Assessment of the Mechanistic Models

Mechanistic models are based on the basic principles of conduction heat transfer in porous media comprised of solids, air and water. In theory, mechanistic models should not require any fitting parameters. However, in reality, some mechanistic models use several theoretical coefficients that are often cumbersome to determine. As a result, these coefficients are usually obtained by fitting the models to experimental data [13]. Therefore, these models are prone to a subjective validation, particularly if the $\lambda$ database used is incomplete. Furthermore, these models use a simplified soil structure; often, they do not consider heat transfer through inter-particle contacts [14,15,16,17,18].

3.1.1 deVries Model [13]

Soil $\lambda$ is modeled as the weighted average of soil components with their shape factors (k), $\lambda$ and corresponding volume fractions $(\theta )$.

$$\begin{aligned} \lambda =\frac{k_{\mathrm{w}} \cdot \lambda _{\mathrm{w}} \cdot \theta _{\mathrm{w}} +\sum \nolimits _1^{\mathrm{j}} {k_{\mathrm{i}} \cdot \lambda _{\mathrm{i}} \cdot \theta _{\mathrm{i}} +k_{\mathrm{a}} \cdot \lambda _{\mathrm{a}} \cdot \theta _{\mathrm{a}} } }{k_{\mathrm{w}} \cdot \theta _{\mathrm{w}} +\sum \nolimits _1^{\mathrm{j}} {k_{\mathrm{i}} \cdot \theta _{\mathrm{i}} +k_{\mathrm{a}} \cdot \theta _{\mathrm{a}} } } \end{aligned}$$

(1)

The soil structure is assumed to be composed of ellipsoidal grains freely floating in a continuous medium of air and/or water, i.e., the soil grains do not touch each other. If all soil grains are lumped into one solid phase (s), then Eq. 1 converts to the following form:

$$\begin{aligned} \lambda =\frac{k_{\mathrm{w}} \lambda _{\mathrm{w}} \theta _{\mathrm{w}} +k_{\mathrm{s}} \lambda _{\mathrm{s}} \theta _{\mathrm{s}} +k_{\mathrm{a}} \lambda _{\mathrm{a}} \theta _{\mathrm{a}} }{k_{\mathrm{w}} \theta _{\mathrm{w}} +k_{\mathrm{s}} \theta _{\mathrm{s}} +k_{\mathrm{a}} \theta _{\mathrm{a}} } \end{aligned}$$

(2)

where $\theta _{\mathrm{s}} ={\rho _{\mathrm{b}} }/{\rho _{\mathrm{s}} }$; $\theta _{\mathrm{s}} +\theta _{\mathrm{a}} +\theta _{\mathrm{w}} =1$; $\rho _{\mathrm{s}}$ is the solid (mineral) density; and $k_{\mathrm{s}}$, $k_{\mathrm{a}}$, $k_{\mathrm{w}}$ are the shape factors of solids, air and water.

For dry conditions ($\theta _{\mathrm{w}} = 0, k_{\mathrm{w}} = 0, \theta _{\mathrm{a}} = \hbox {n}, k_{\mathrm{a}} = 1$), Eq. 2 converts to the following form:

$$\begin{aligned} \lambda _{{dry}} =\lambda _{{a}} \frac{\rho _{{s}} +\rho _{{b}} \left( {k_{{s}} \frac{\lambda _{{s}} }{\lambda _{{a}} }-1} \right) }{\rho _{{s}} +\rho _{{b}} \left( {k_{{s}} -1} \right) }=\lambda _{{a}} \frac{n+\left( {1-n} \right) k_{{s}} \frac{\lambda _{{s}} }{\lambda _{{a}} }}{n+\left( {1-n} \right) k_{{s}} } \end{aligned}$$

(3)

where $k_{\mathrm{s}}$ is given by:

$$\begin{aligned} k_{\mathrm{s}} =\frac{1}{3}\left[ {\frac{2}{1+\left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{a}} }-1} \right) g_{\mathrm{s}} }+\frac{1}{1+\left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{a}} }-1} \right) \left( {1-2g_{\mathrm{s}} } \right) }} \right] \end{aligned}$$

(4)

where $g_{\mathrm{s}}$ is a soil particle shape value (a fitting parameter of 0.125 [13]) and $\lambda _{\mathrm{s}}$ is the thermal conductivity of solids.

According to de Vries [13], the use of Eq. 4 is restricted by the following conditions:

• 1. Soil grains are treated as rotated ellipsoids.

• 2. The ellipsoids are far apart, so that they do not impact each other.

The first condition is roughly acceptable for coarse soils, while it is objectionable for fine soils composed mainly of flat, tiny particles (i.e., lamellae). This could be a major reason of generally reduced the model performance when it is applied to fine soils. The second condition is also not fulfilled in soils and it is a major reason behind a correction factor introduced by de Vries [13]. The $\lambda _{\mathrm{s}}$ was estimated by the geometric mean of $\lambda$ of quartz $(\lambda _{\mathrm{qtz}})$ and other minerals $(\lambda _{\mathrm{o-min}})$ [2]: $\lambda _{\mathrm{s}} =\lambda _{\mathrm{qtz}}^{\theta _{\mathrm{qtz}} } \lambda _{o-\min }^{\theta _{\mathrm{o-min}} } =\lambda _{\mathrm{qtz}}^{\theta _{\mathrm{qtz}} } \lambda _{o-\min }^{1-\theta _{\mathrm{qtz}} }$. Where, $\lambda _{\mathrm{o-min}} \approx 2-2.2\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\lambda _{\mathrm{qtz}} \approx 7.7\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$.

Equation 3 produces a nearly linear increase of $\lambda _{\mathrm{dry}}$ with increasing $\rho _{\mathrm{b}}$ at large soil n; while at small n, a noticeable increase of $\lambda _{\mathrm{dry}}$ was noted [1]. The deVries Model, in its original version (Eqs. 2 and 3), uses $\lambda _{\mathrm{qtz}} = 8.8\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and in spite of that it underestimates $\lambda$; consequently, a multiplication factor of 1.25 was recommended [13].

For saturated soil conditions ($\theta _{\mathrm{w}} = n, \theta _{\mathrm{a}} = 0, k_{\mathrm{a}} = 0, k_{\mathrm{w}} = 1$), Eq. 2 converts to the following form:

$$\begin{aligned} \lambda _{\mathrm{sat}} =\lambda _{\mathrm{w}} \frac{1+(1-n)(k_{\mathrm{s}} {\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{w}} }-1)}{1+(1-n)(k_{\mathrm{s}} -1)} \end{aligned}$$

(5)

where $k_{\mathrm{s}}$ is given by:

$$\begin{aligned} k_{\mathrm{s}} =\frac{1}{3}\left[ {\frac{2}{1+\left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{w}} }-1} \right) g_{\mathrm{s}} }+\frac{1}{1+\left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{w}} }-1} \right) \left( {1-2g_{\mathrm{s}} } \right) }} \right] \end{aligned}$$

(6)

Further discussion regarding adaptation of this model to soil environment is given in Sect. 5.

3.1.2 Series-Parallel Models ($\hbox {S-}{\vert }{\vert }$) Models [14,15,16]

The basic soil constituents (solids, air and water) are arranged in either series or parallel with respect to heat flow. In fact, these arrangements establish the upper or lower bounds of effective thermal conductivity $(\lambda _{\mathrm{eff}})$, respectively. Woodside published the first $\hbox {S-}{\vert }{\vert }$ model for two-phase granular systems, and Messmer [14] assumed two parallel paths of heat flow. The first path, through soil air as a continuous medium, and the second path, along solid grains bridged with tiny volumes of soil air in series. However, the second path disregards heat flow through inter-particle contacts. Later, Kasubuchi et al. [15] applied this model (Eq. 7) to three dry Japanese soils.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\lambda _{\mathrm{a}} \left( {n-n_{\mathrm{wm}} } \right) +\frac{\left( {1-n+n_{\mathrm{wm}} } \right) ^{2}}{{(1-n)}/{\lambda _{\mathrm{s}} }+{n_{\mathrm{wm}} }/{\lambda _{\mathrm{a}} }} \end{aligned}$$

(7)

where $n_{\mathrm{a}}$ is the air fraction in a series path of heat flow. See above equation.

Then, Tarnawski and Leong [16] developed an $\hbox {S-}{\vert }{\vert }$ model applied to a full range of $S_{\mathrm{r}}$ (three-phase granular materials). In a partitioned cubic cell (Fig. 3), representing a piece of soil, heat is conducted through three pathways, namely a solid uniform passage $(\varTheta _{{sb}} )$, a series-parallel passage composed of solids bridged with a parallel path of minuscule portion of soil water $(n_{\mathrm{w}})$ and a minuscule portion of soil air $(n_{\mathrm{a}})$ and, finally, a path of water $(\varTheta _{{w}} )$ and air $(\varTheta _{{a}} )$ in a parallel arrangement. In fact, $\varTheta _{{sb}}$ and $n_{\mathrm{wm}} =n_{\mathrm{a}} +n_{\mathrm{w}}$ are fitting parameters.

Fig. 3

Schematic representation of three paths of heat flow in a partitioned soil cubic cell

By applying a classical resistor model to each heat pathway, the following expression was obtained:

$$\begin{aligned} \lambda _{{\mathrm{S}}-||-||}= & {} \lambda _{\mathrm{s}} \varTheta _{\mathrm{sb}} +\frac{(1-n-\varTheta _{\mathrm{sb}} +n_{\mathrm{wm}} )^{2}}{\frac{1-n-\varTheta _{\mathrm{sb}} }{\lambda _{\mathrm{s}} }+\frac{n_{\mathrm{wm}}}{\lambda _{\mathrm{w}} \frac{n_{\mathrm{w}} }{n_{\mathrm{wm}} }+\lambda _{\mathrm{a}} (1-\frac{n_{\mathrm{w}} }{n_{\mathrm{wm}} })}}+\lambda _{\mathrm{w}} \left( {nS_{\mathrm{r}} -n_{\mathrm{wm}} \frac{n_{\mathrm{w}} }{n_{\mathrm{wm}} }} \right) \nonumber \\&+\lambda _{\mathrm{a}} \left[ {(n-n_{\mathrm{wm}} -\left( nS_{\mathrm{r}} -n_{\mathrm{wm}} \frac{n_{\mathrm{w}} }{n_{\mathrm{wm}} }\right) } \right] \end{aligned}$$

(8)

where $n_{\mathrm{wm}} =n_{\mathrm{a}} +n_{\mathrm{w}}$ is the total fraction of minuscule soil pore space.

For dry conditions $(S_{\mathrm{r}} = 0)$:

$$\begin{aligned} \lambda _{\mathrm{dry}} =\lambda _{\mathrm{s}} \varTheta _{\mathrm{sb}} +\frac{(1-n-\varTheta _{\mathrm{sb}} +n_{\mathrm{wm}} )^{2}}{\frac{1-n-\varTheta _{\mathrm{sb}} }{\lambda _{\mathrm{s}} }+\frac{n_{\mathrm{wm}}}{\lambda _{\mathrm{a}} }}+\lambda _{\mathrm{a}} (n-n_{\mathrm{wm}} ) \end{aligned}$$

(9)

For saturated conditions $(S_{\mathrm{r}} = 1)$:

$$\begin{aligned} \lambda _{\mathrm{sat}} =\lambda _{\mathrm{s}} \varTheta _{\mathrm{sb}} +\frac{(1-n-\varTheta _{\mathrm{sb}} +n_{\mathrm{wm}} )^{2}}{\frac{1-n-\varTheta _{\mathrm{sb}} }{\lambda _{\mathrm{s}} }+\frac{n_{\mathrm{wm}}}{\lambda _{\mathrm{w}} }}+\lambda _{\mathrm{w}} (n-n_{\mathrm{wm}} ) \end{aligned}$$

(10)

The model was calibrated using $\lambda$ data of five soils representing coarse, medium, and fine textures. Two structural characteristics of the elemental soil cell, $n_{\mathrm{wm}}$ and $\varTheta _{{sb}}$, can be obtained from the following relations:

$$\begin{aligned} n_{\mathrm{wm}}= & {} 0.088-0.037m_{\mathrm{sa}}^3 \end{aligned}$$

(11)

$$\begin{aligned} \varTheta _{\mathrm{sb}}= & {} 0.0237-0.0175m_{\mathrm{sa}}^3 \end{aligned}$$

(12)

Further details regarding adaptation of this model to soil environment is given in Sect. 5.

3.1.3 Cubic Cell [17]

A structure of dry/saturated soils was assumed as an array of infinitesimal cubical cells; each containing a centrally positioned dice representing lumped soil solids, while the space surrounding the solid core was filled with air or water. The effective $\lambda _{\mathrm{eff}}$ was estimated from the following relation:

$$\begin{aligned} \frac{1}{\lambda _{\mathrm{eff}} }=\frac{\beta -1}{\lambda _{\mathrm{f}} \cdot \beta }+\frac{\beta }{\lambda _{\mathrm{s}} +\lambda _{\mathrm{f}} \cdot \left[ {\beta ^{2}-1} \right] } \end{aligned}$$

(13)

where $\beta =\root 3 \of {1/(1-n)}$; at $S_{\mathrm{r}} = 0$: $\lambda _{\mathrm{f}} = \lambda _{\mathrm{a}}$ and $\lambda _{\mathrm{eff}} = \lambda _{\mathrm{dry}}$; and at $S_{\mathrm{r}} = 1$: $\lambda _{\mathrm{f}} = \lambda _{\mathrm{w}}$ and $\lambda _{\mathrm{eff}} = \lambda _{\mathrm{sat}}$.

Recently, this model was revised [18] and a spherical shape of soil grains was introduced.

3.1.4 MaxRTCM [19]

It is an analytical model that estimates $\lambda _{\mathrm{eff}}$ of dry planetary regolith (fine-grained soil covering a solid bedrock). The model consisted of the thermal conductivity of solid material $(\lambda _{\mathrm{s}})$, system porosity (n), non-spherical particle size and shape ($\varPsi$), a factor representing an inter-particle cementation $(f_{\mathrm{sc}})$ and radiation heat transfer between particles surrounded by a vacuum $(\lambda _{\mathrm{r}})$. Particle shape was described by its spherical nature and roundness.

$$\begin{aligned} \lambda _{\mathrm{eff}} =\lambda _{\mathrm{s}} f_{\mathrm{sc}} \frac{\left( {\frac{3}{\varPsi }-1} \right) \left( {1-n} \right) }{\frac{3}{\varPsi }-(1-n)}+\lambda _{\mathrm{r}} (1-f_{\mathrm{sc}} )\frac{\frac{3}{\varPsi }\left( {1-n} \right) +n}{n} \end{aligned}$$

(14)

The inter-particle cementation, $f_{\mathrm{sc}}$, is an empirical factor that represents a fractional continuity of the high conductivity phase, ranging from 0 to $1:f_{\mathrm{sc}} =0.6\left( {{R_{\mathrm{con}} }/{R_{\mathrm{s}} }} \right)$, where $R_{\mathrm{con}}$ is the radius of an elastic contact region between particles and $R_{\mathrm{s}}$ is the mean radius of solid particles. In general, $\lambda _{\mathrm{s}}$, $\varPsi$ and $f_{\mathrm{sc}}$ are difficult to determine analytically; therefore, they are usually treated as fitting parameters. Further discussion regarding adaptation of this model to soil environment is given in Sect. 5.

3.2 Review of the Semi-empirical Models

Semi-empirical models are a transitional class of models between the mechanistic and empirical types. These models use some implicit principles of heat transfer combined with experimental data; no soil structure is considered in their function; usually, they contain at least one fitted parameter.

3.2.1 deVries-ave [13]

Johansen [2] simplified Eq. (3) to a more practical form for handling data of field soils. The unknown coefficients of the model by de Vries (Eq. 3) were obtained by fitting to $\lambda$ data of soils studied by Kersten [26], namely, soil particle shape values $g_{1} = g_{2} = 0.1$ and $g_{3} = 0.8$; thermal conductivity of solids $\lambda _{\mathrm{s}} = 3 \,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and the shape factor of solids $k_{\mathrm{s}} = 0.053$. Other fixed coefficients were also introduced, such as, $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$, $\lambda _{\mathrm{qtz}} = 7.69 \,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\lambda _{\mathrm{a}} = 0.024 \,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$. After substituting the above coefficients into Eq.(3), the following simplified expression was obtained:

$$\begin{aligned} \lambda _{\mathrm{dry}} =\frac{A\rho _{\mathrm{b}} +B}{\rho _{\mathrm{s}} -C\rho _{\mathrm{b}} }=\frac{A(1-n)\rho _{\mathrm{s}} +B}{\rho _{\mathrm{s}} -C(1-n)\rho _{\mathrm{s}} } \end{aligned}$$

(15)

where $A=\lambda _{\mathrm{a}} \cdot (k_{\mathrm{s}} \cdot \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{a}} }-1)=0.135$; $B=\lambda _{\mathrm{a}} \cdot \rho _{\mathrm{s}} =64.8$; and $\hbox {C} =(1-k_{\mathrm{s}} )=0.947$.

The additional fitting parameter was a soil particle shape value $g_{\mathrm{s}}$ upon which $k_{\mathrm{s}}$ is calculated (Eq. (6)). In spite of a fixed $\lambda _{\mathrm{s}}$ of $3\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, Eq. (15) has been commonly applied to field soils regardless of their texture and mineral composition [1]. It is also worth noting that some of Kersten’s soils were of volcanic origin (Alaska) and hence, their physical properties were usually different from those of mineral soils. Further discussion regarding adaptation of this model to Canadian Soils is provided in Sect 5.

3.2.2 Advanced Geometric Mean Model (A-GMM) [20]

When the classical geometric mean model (Eq. 16) is applied to dry soils, it largely over-predicts experimental data.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\lambda _{\mathrm{s}}^{1-n} \cdot \lambda _{\mathrm{a}}^n \end{aligned}$$

(16)

Therefore, in spite of its simplicity, this model is not used for dry soils due to a large ratio of ${\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{a}} \approx 100}-300$ and also due to the fact that the thermal contact resistance at the inter-particle level is missing. Recently, Tarnawski and Leong [20] successfully introduced inter-particle contact resistance factors ($\alpha _{\mathrm{dry}}$ and $\alpha _{\mathrm{sat}})$ in the classical geometric mean model.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\left[ {\alpha _{\mathrm{dry}} \cdot \lambda _{\mathrm{s}} } \right] ^{1-n}\cdot \left[ {\lambda _{\mathrm{a}} } \right] ^{n} \end{aligned}$$

(17)

For saturated conditions:

$$\begin{aligned} \lambda _{\mathrm{sat}} =\left[ {\alpha _{\mathrm{sat}} \lambda _{\mathrm{s}} } \right] ^{1-n}\cdot \left[ {\lambda _{\mathrm{w}} } \right] ^{n} \end{aligned}$$

(18)

where $\alpha =\left[ {\varepsilon +\left( {1-\varepsilon } \right) \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{f}} }} \right] ^{-1}$; at $S_{\mathrm{r}} = 0$: $\alpha = \alpha _{\mathrm{dry}}$ and $\lambda _{\mathrm{f}} = \lambda _{\mathrm{a}}$; and at $S_{\mathrm{r}} = 1$: $\alpha = \alpha _{\mathrm{sat}} = 1$ and $\lambda _{\mathrm{f}} = \lambda _{\mathrm{w}}$.

The advanced geometric mean model (A-GMM) also introduces a dimensionless inter-particle contact coefficient ($\varepsilon$). According to Tarnawski et al. [20], $\varepsilon$ mainly depends on soil texture. The soil specific area (SSA) is inversely proportional to the soil grain size, i.e., on average, small soil particles have a larger SSA than for the same mass made up of large particles. Also, the particle shape appears to be interrelated with SSA; flattened or elongated particles (fine soils) have a greater SSA than spherical or cubical particles (coarse soils). Therefore, fine soils have a larger $\varepsilon$ than coarse soils. However, due to irregular soil inter-particle structure, there is no analytical relation for $\varepsilon$. Therefore, $\varepsilon$ is a fitting parameter. Further details, regarding adaptation of this model to Canadian Soils is provided in Sect. 5.

3.2.3 Chaudhary–Bhandari (C–B) [8]

This model is another adaptation of the weighted geometric mean model to three-phase porous media. It combines $\lambda _{||}$ (in the parallel direction to heat flow), $\lambda _\bot$ (in the perpendicular direction to heat flow) and a phase weighting factor $\zeta$ [8].

$$\begin{aligned} \lambda _{\mathrm{eff}} =\lambda _{||}^{\upzeta } \lambda _\bot ^{1-\upzeta } \end{aligned}$$

(19)

where $\lambda _{||} =n\lambda _{\mathrm{f}} +(1-n)\lambda _{\mathrm{s}}$ and $\lambda _\bot =\left[ {n\lambda _{\mathrm{f}} +(1-n)\lambda _{\mathrm{s}} } \right] ^{-1}$

The model assumes that the $\zeta$ fraction of the porous media is oriented in parallel to the direction of heat flow, while the 1- $\zeta$ fraction is oriented in perpendicular to the heat flow.

For two-phase unconsolidated soils (i.e., at dry and/or saturated state), the above relation can be written as follows:

$$\begin{aligned} \lambda _{\mathrm{eff}} =\lambda _{\mathrm{s}}^{1+\upzeta } \lambda _{\mathrm{f}}^{1-\upzeta } \frac{\left[ {n\left( {1-n} \right) } \right] ^{\upzeta }}{n\lambda _{\mathrm{s}} +\left( {1-n} \right) \lambda _{\mathrm{f}} } \end{aligned}$$

(20)

where $\zeta$ is assessed from $\zeta ={u\left( {1-\log n} \right) }/{\log \left[ {n\left( {1-n} \right) \lambda _{\mathrm{s}} /\lambda _{\mathrm{a}} } \right] }$ with $u = 0.5$ as a fitting parameter.

Further details regarding adaptation of this model to Canadian Soils is given in Sect. 5.

3.2.4 Chen’s Equations [10]

The $\lambda$ data of four unsaturated sands (pure quartz) showed an exponential $\lambda$ dependence on n and the following equation was given:

$$\begin{aligned} \lambda =\lambda _{\mathrm{s}} ^{1-n}\lambda _{\mathrm{w}} ^{n}\left[ {\left( {1-0.0022} \right) \,S_{\mathrm{r}} +0.0022} \right] ^{0.78n} \end{aligned}$$

(21)

However, at $S_{\mathrm{r}} = 0$, the air component $(\lambda _{\mathrm{a}})$ is missing, while $\lambda _{\mathrm{w}}$ is present. It is likely that $\lambda _{\mathrm{a}}$ is hidden in the constant of $0.0022 \,(\approx 0.0846\cdot \lambda _{\mathrm{a}})$. The following equation was obtained by substituting $0.0846\cdot \lambda _{\mathrm{a}}$ and $\lambda _{\mathrm{w}} = 0.61 \,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\lambda _{\mathrm{s}} ^{1-n}\left[ {0.61\left( {0.0846\lambda _{\mathrm{a}} } \right) ^{0.78}} \right] ^{n}=\lambda _{\mathrm{s}} ^{1-n}\left[ {0.089\lambda _{\mathrm{a}} ^{0.78}} \right] ^{n} \end{aligned}$$

(22)

At $S_{\mathrm{r}} = 1$, Eq. (21) can be simplified to the geometric mean form:

$$\begin{aligned} \lambda _{\mathrm{sat}} =\lambda _{\mathrm{qtz}} ^{1-n}\lambda _{\mathrm{w}} ^{n}. \end{aligned}$$

(23)

3.3 Review of the Empirical Equations

These models are obtained directly by fitting experimental data of $\lambda$ versus n. The obtained equations are relatively simple, e.g., 1st and 2nd order polynomials with n as the input parameter.

3.3.1 GSC [9]

The following empirical equation was fitted to $\lambda$ data of five soils from Eastern Washington (USA).

$$\begin{aligned} \lambda =A+B\theta _{\mathrm{w}} -(A-D)\exp \left[ -(C\theta _{\mathrm{w}} )^{E}\right] \end{aligned}$$

(24)

where A, B, C, D and E are fitting coefficients and $\theta _{\mathrm{w}}$ is volumetric water content.

For dry soils, $\theta _{\mathrm{w}} = 0$, the following expression is applied:

$$\begin{aligned} \lambda _{\mathrm{dry}} =0.03+0.7(1-n)^{2} \end{aligned}$$

(25)

For saturated soils, $\theta _{\mathrm{w}} = n$, the last term in Eq. (24) is zero, so that

$$\begin{aligned} \lambda _{\mathrm{sat}} =A+Bn \end{aligned}$$

(26)

where $A=\frac{0.57/(1-n)+1.73\theta _{\mathrm{qtz}} +0.93(1-\theta _{\mathrm{qtz}} )}{1/(1-n)-0.74\theta _{\mathrm{qtz}} -0.49(1-\theta _{\mathrm{qtz}} )}-2.8n(1-n)$ and $B = 2.8(1-\hbox {n})$

3.3.2 Côté–Konrad (C–K) [11]

The thermal conductivity at dryness, $\lambda _{\mathrm{dry}}$, is estimated from a fitted relation dependent on n and two other parameters ($\lambda _{\mathrm{x}}$ and b) accounting for $\lambda _{\mathrm{s}}$ and soil grain shape.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\lambda _{\mathrm{x}} 10^{-z\cdot n} \end{aligned}$$

(27)

where $\lambda _{\mathrm{x}} = 0.75\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $z = 1.2$

3.3.3 LRGH [12]

The following n dependent empirical relation for $\lambda _{\mathrm{dry}}$ was proposed for 10 Chinese soils.

$$\begin{aligned} \lambda _{\mathrm{dry}} =-0.56n+0.51 \end{aligned}$$

(28)

3.3.4 Yun-San [5]

For pure quartz sands, the following equation was proposed.

$$\begin{aligned} \lambda _{{dry}} =-0.291\ln (n)+0.026 \end{aligned}$$

(29)

A quick reference summary for all the models is given in “Appendix A”.

4 Modeling $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ for 40 Canadian Soils

Using the Database of Canadian 40 Soils, the following empirical equations for $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ were obtained.

4.1 Canadian Soils at Dryness (CDry-40)

A simple $\lambda _{\mathrm{dry}}$ dependence on n was obtained, regardless of soil texture.

$$\begin{aligned} \lambda _{\mathrm{dry}} =0.55(1-n)^{1.4} \end{aligned}$$

(30)

However, this correlation required two fitting parameters (0.55 and 1.4), i.e., $M = 2$.

4.2 Canadian Coarse Soils at Saturation (CSat-18)

For $\lambda _{\mathrm{sat}}$ data of 18 coarse texture soils, the following correlation was obtained:

$$\begin{aligned} \lambda _{\mathrm{sat}} =1.147+0.007n^{-5.31} \end{aligned}$$

(31)

This model required three fitting parameters (1.147, 0.007, and $-5.31$), i.e., $M = 3$.

4.3 Canadian Fine Soils at Saturation (CSat-22)

For $\lambda _{\mathrm{sat}}$ data of 22 fine texture soils, the following correlation was obtained:

$$\begin{aligned} \lambda _{\mathrm{sat}} =1.284+13.36\cdot 10^{-6}n^{-17.484} \end{aligned}$$

(32)

This model required three fitting parameters ($1.284, 13.36\cdot 10^{-6}$, and $-17.484$), i.e., $M = 3$.

5 Adaptation of Published Models to 40 Canadian Soils

5.1 deVries-ave

Direct application of Eq. 15, originally developed for soils by Kersten [26], to $\lambda$ data of 40 Canadian soils [23] appeared to be uncertain due to noticeably different mineral compositions. Thermal conductivity of solids $(\lambda _{\mathrm{s}})$ was a key parameter as it is a lumped value of all minerals in a particular soil. Its numerical value is usually estimated by the geometric mean model [1] applied to all mineral soil components.

$$\begin{aligned} \lambda _{\mathrm{s}} =\prod _{i=1}^m {\lambda _i^{\varTheta _i } } \end{aligned}$$

(33)

where $\varTheta _{\mathrm{i}}$ is the volumetric fraction of the ith soil mineral in soil solids, m is the total number of solid soil minerals and $\lambda _{\mathrm{i}}$ is the thermal conductivity of the ith soil mineral (for more details, see Appendix).

Application of the above equation to the 40 Canadian soils revealed that $\lambda _{\mathrm{s}}$ varied from $3.3\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ to $7.7\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for coarse soils and from $2.5\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ to $4.5\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for fine soils. Consequently, most $\lambda _{\mathrm{s}}$ values are higher than $3\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ [2]. Also, the values of soil particle shapes $(g_{1}$, $g_{2}$ and $g_{3})$ are obtained by fitting calculated $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ to experimental $\lambda$ data of the 40 Canadian soils. For 18 coarse soils, the following average coefficients were obtained: $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$, $g_{1} = g_{2} = 0.1$, $g_{3} = 0.8$, $\lambda _{\mathrm{s}} = 4.46\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, $\lambda _{\mathrm{a}} = 0.026\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $k_{\mathrm{s}} = 0.04$. Substituting these coefficients into Eqs. 3 and 4, the following relation for 18 coarse soils was obtained:

$$\begin{aligned} \lambda _{\mathrm{dry}} =\frac{A\rho _{\mathrm{s}} \cdot (1-n)+B}{\rho _{\mathrm{s}} -C\rho _{\mathrm{s}} \cdot (1-n)} \end{aligned}$$

(34)

where $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$; $A=\lambda _{\mathrm{a}} \cdot (k_{\mathrm{s}} \cdot \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{a}} }-1)=0.145$; $B=\lambda _{\mathrm{a}} \cdot \rho _{\mathrm{s}} =70.2$ and $\hbox {C}=(1-k_{\mathrm{s}} )=0.96$.

For 22 fine soils, the following average coefficients were obtained: $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$, $g_{1} = g_{2} = 0.053$, $g_{3} = 0.894$, $\lambda _{\mathrm{s}} = 3.454\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, $\lambda _{\mathrm{air}} = 0.026\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $k_{\mathrm{s}} = 0.088$.

$$\begin{aligned} \lambda _{\mathrm{dry}} =\frac{A\rho _{\mathrm{s}} \cdot (1-n)+B}{\rho _{\mathrm{s}} -C\rho _{\mathrm{s}} \cdot (1-n)} \end{aligned}$$

(35)

where $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$, $A=\lambda _{\mathrm{a}} \cdot (k_{\mathrm{s}} \cdot \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{a}} }-1)=0.157$; $B=\lambda _{\mathrm{a}} \cdot \rho _{\mathrm{s}} =70.2$; and $\hbox {C}=(1-k_{\mathrm{s}} )=0.912$.

For saturated conditions and 18 coarse soils:

$$\begin{aligned} \lambda _{sat\hbox {-}coarse} =\frac{A\rho _{{s}} \cdot (1-n)+B}{\rho _{{s}} -C\cdot \rho _{{s}} \cdot (1-n)} \end{aligned}$$

(36)

where: $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$; $\lambda _{\mathrm{s}} = 4.46\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$; $A=\lambda _{\mathrm{w}} \cdot (k_{\mathrm{s}} \cdot \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{w}} }-1)=1.248$; $B=\lambda _{\mathrm{w}} \cdot \rho _{\mathrm{s}} =1643$; $\hbox {C}=(1-k_{\mathrm{s}} )=0.575$.

For saturated conditions and 22 fine soils:

$$\begin{aligned} \lambda _{\mathrm{sat\hbox {-}fine}} =\frac{A\rho _{\mathrm{s}} \cdot (1-n)+B}{\rho _{\mathrm{s}} -C\cdot \rho _{\mathrm{s}} \cdot (1-n)} \end{aligned}$$

(37)

where: $\rho _{\mathrm{s}} = 2700\,\hbox {kg}\cdot \hbox {m}^{-3}$; $\lambda _{\mathrm{s}} = 3.454\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$; $A=\lambda _{\mathrm{w}} \cdot (k_{\mathrm{s}} \cdot \frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{w}} }-1)=1.273$; $B=\lambda _{\mathrm{w}} \cdot \rho _{\mathrm{s}} =1646$; $\hbox {C}=(1-k_{\mathrm{s}} )=0.438$.

In general, the predictive performance of the model depends on the number of fitting parameters (M). This model requires one fitting parameter, $g_{1}$ (Eq. 6); i.e., $M = 1$.

5.2 C–B

Improved $\lambda _{\mathrm{dry}}$ and $\uplambda _{\mathrm{sat}}$ estimates were obtained when u values, applied to the phase weighting factor $\zeta$, were fitted to the 40 Canadian soil data. For dry soils, $u = 0.445$, while for saturated soils, $u = 0.925$. This model requires one fitting parameter, u; i.e., $M = 1$.

5.3 maxRTCM

Originally, this model was developed for dry, porous and airless planetary regolith, where the presence of radiation heat transfer $(\lambda _{\mathrm{r}})$ is dominant. Therefore, for field soils, the model was modified and the following assumptions were made:

• (a) For dry field soils $\lambda _{\mathrm{a}}>>\lambda _{\mathrm{r}}$, was replaced with $\lambda _{\mathrm{a}}$:

  $$\begin{aligned} \lambda _{\mathrm{dry}} =f_{\mathrm{sc}} \frac{\left( {\frac{3}{\varPsi }-1} \right) \left( {1-n} \right) }{\frac{3}{\varPsi }-\left( {1-n} \right) }\lambda _{\mathrm{s}} +\left( {1-f_{\mathrm{sc}} } \right) \frac{\frac{3}{\varPsi }\left( {1-n} \right) +n}{n}\lambda _{\mathrm{a}} \end{aligned}$$

  (38)

• (b) The sphericity of soil particles ($\varPsi$) depends on their roundness $(R^{\prime })$. According to Krumbein and Sloss [27]: $0.3< \varPsi < 0.9$, while $0.1< R^{\prime } < 0.9$. Due to an endless diversity of soil grain shapes, there is no analytic equation for $\varPsi$. Therefore, it has been defined as a weighted average based on volumetric fractions of sand ($\theta _{\mathrm{sa}})$, silt ($\theta _{\mathrm{si}})$ and clay ($\theta _{\mathrm{cl}})$ and an assumed constant, $\varPsi$, for each major soil fractions, namely, sand ($\varPsi _{\mathrm{sa}} = 0.865$), silt ($\varPsi _{\mathrm{si}} = 0.645$) and clay ($\varPsi _{\mathrm{cl}} = 0.085$).

  $$\begin{aligned} \varPsi =\varPsi _{\mathrm{sa}} \theta _{\mathrm{sa}} +\varPsi _{\mathrm{si}} \theta _{\mathrm{si}} +\varPsi _{\mathrm{cl}} \theta _{\mathrm{cl}} \approx 0.865m_{\mathrm{sa}} +0.645m_{\mathrm{si}} +0.085m_{\mathrm{cl}} \end{aligned}$$

  (39)

• (c) The solid/cement continuity factor $(f_{\mathrm{sc}})$ depends on the ratio of soil particle contact radius and mean radius of soil particles [19]. However, due to the complexity of soil structure, this specific data are generally not available; consequently, there is no analytical relation for $f_{\mathrm{sc}}$. The missing $f_{\mathrm{sc}}$ data were obtained by minimizing the RMSE between experimental $\lambda _{\mathrm{dry}}$ and Eqs. 38 with 39. Then, the obtained $f_{\mathrm{sc}}$ values were used to fit a correlation $f_{\mathrm{sc\hbox {-}cor}}$ as a function of $\varPsi$ and $\ln \left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{a}} }} \right) /\left( {{\lambda _{\mathrm{s}} }/{\lambda _{\mathrm{a}} }} \right)$.

  $$\begin{aligned} f_{\mathrm{sc\hbox {-}cor}} =\max \left( {0,-0.02220+\frac{0.05520}{\varPsi }-\frac{0.03024}{\varPsi ^{2}}+1.4784\frac{\ln \left( {\frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{a}} }} \right) }{\frac{\lambda _{\mathrm{s}} }{\lambda _{\mathrm{a}} }}} \right) \quad \end{aligned}$$

  (40)

It appears that introduction of correlated $f_{\mathrm{sc\hbox {-}cor}}$ might be very beneficial to the model performance when applied to dry soils. This model required one fitting parameter $(f_{\mathrm{sc\hbox {-}cor}})$, i.e., $M = 1$.

5.4 A-GMM

A dimensionless inter-particle contact coefficient $\varepsilon$ was obtained by fitting Eqs. 17 and 18 to $\lambda$ data of 18 coarse and 22 fine unsaturated soils [20] and values of 0.988 and 0.996 were obtained, respectively. For $\lambda _{\mathrm{dry}}$, the model requires one fitting parameter $\varepsilon$, i.e., $M = 1$; while for $\lambda _{\mathrm{sat}}$ the model does not require any fitting as $\alpha _{\mathrm{sat}} = 1$.

6 Assessment of Predictive Models

The predictive performance of each $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ model was evaluated with respect to experimental data $(\lambda _{\mathrm{exp}})$ using the standard deviation value (SD):

$$\begin{aligned} \textit{SD}=\sqrt{\frac{1}{N-M}\sum \nolimits _1^N {\left( {\lambda _{\mathrm{exp}} -\lambda _{\mathrm{cal}} } \right) ^{2}} } \end{aligned}$$

(41)

where N is the number of independent $\lambda _{\mathrm{dry}}$ or $\lambda _{\mathrm{sat}}$ records; M is the number of parameters in the model used for fitting to experimental $\lambda$ data.

A smaller SD means a better performance of the predictive model. Basically, the model predictive performance (i.e., SD) decreases with increasing M.

6.1 Dry Soils

Tables 3 and 4 summarize each model performance for 18 coarse and 22 fine soils, respectively, with both listed according to their performance (SD values).

Table 3 18 dry Canadian soils (coarse)

Table 4 22 Dry Canadian soils (fine)

6.1.1 Model Assessment

Overall, the closest $\lambda _{\mathrm{dry}}$ predictions to experimental data were made by MaxRTCM (Eqs. 38–40). The MaxRTCM average prediction error, ${\left| {\lambda _{\mathrm{dry\hbox {-}cal}} -\lambda _{\mathrm{dry\hbox {-}exp}} } \right| }/{\lambda _{\mathrm{dry\hbox {-}ave}} }$, for the 18 coarse soils, with respect to their average $\lambda _{\mathrm{dry\hbox {-}exp}}$ value of $0.247\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, was about 5.3 %, while, for the 22 fine soils, with respect to their average $\lambda _{\mathrm{dry\hbox {-}exp}}$ value of $0.199\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, was about 6.3 %. For coarse soils, the largest over-prediction of 14 % was noted for ON-05 soil and the worst under-prediction of 16 % was noted for PE-03 soil. For fine soils, the largest over-prediction was noted for NB-04 soil (18.7 %) and the worst under-prediction was for ON-01 soil at 28.3 %. For all 40 Canadian soils, the overall SD for MaxRTCM was $\pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ which is equivalent to about 8.2 % error in $\lambda _{\mathrm{dry}}$ predictions with respect to the average value of $\lambda _{\mathrm{dry\hbox {-}exp}}\,(0.220\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$. The obtained results closely followed experimental data and indicated a positive influence of the correlated $f_{\mathrm{sc\hbox {-}cor}}$ on the model performance, when applied to dry soils.

When the deVries Model (Eqs. 3 and 4) is applied separately to coarse and fine soils, it produces an SD of $\pm ~0.022\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.044\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for coarse and fine soils, respectively.

Hence, $\lambda _{\mathrm{dry}}$ estimates for fine soils by the deVries Model do not follow experimental data well (Figs. 4 and 5).

The $\hbox {S-}{\vert }{\vert }$ model (Eq. 9) delivers $\lambda _{\mathrm{dry}}$ estimates at $\pm ~0.034 \hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.042 \hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$ for coarse and fine soils, respectively. The Cubic Cell model (Eq.  13) under estimates $\lambda _{\mathrm{dry}}$ data for coarse and fine soils ($-0.102\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $-0.087\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively).

Among four semi-empirical (deVries-ave, C–B, A-GMM and Chen) and five empirical equations (GSC, C–K, LRGH, Yun, and CDry-40), the best estimates were obtained by CDry-40, i.e., Eq. 30$(\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.018 \hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for coarse and fine soils, respectively), closely followed by C–B and GSC, and A-GMM estimates. In turn, deVries-ave (Eqs. 34 and 35) gives $\pm ~0.023\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.055\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for coarse and fine soils, respectively; therefore, there is no significant difference in using the full deVries Model or its simplified version, deVries-ave.

Fig. 4

18 Canadian coarse dry soils: $\lambda _{\mathrm{dry\hbox {-}cal }}$ versus $\lambda _{\mathrm{exp}}$

Fig. 5

22 Canadian fine dry soils: $\lambda _{\mathrm{dry\hbox {-}cal }}$ versus $\lambda _{\mathrm{exp}}$

With respect to CDry-40 (Eq. 30), its predictive performance (Eq. 41) is degraded due to a higher number of fitted coefficients $(M = 2)$, while MaxRTCM and A-GMM require just one fitting coefficient $(M = 1)$. In addition, a brief review of Tables 3 and 4 revealed that the SD of the top 10 $\lambda _{\mathrm{dry}}$ models varied in a narrow range (from $\pm ~0.014$ to $\pm ~0.028\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$. This suggests that simplicity could be a dominating factor when choosing a model. In reality, the requirement of simplicity is not met by the MaxRTCM and deVries models, but it is met by CDry-40 (Eq. 30).

6.2 Saturated Soils

Tables 5 and 6 summarize the predictive performances of eight $\lambda _{\mathrm{sat}}$ models, for 18 coarse and 22 fine soils, respectively.

Table 5 Saturated 18 Canadian soils (coarse)

Table 6 Saturated 22 Canadian soils (fine)

With respect to CSat-18 and CSat-22 (Eqs. 31 and 32), it is worth to mention that their predictive performance (Eq. 41) is lowered due to the use of a high number of fitted coefficients $(M = 3)$, while the two best models, A-GMM and $\hbox {S-}{\vert }{\vert }$, do not require any fitting coefficients $(M = 0)$ to the 40-Canadian soil data.

Figures 6 and 7 display the $\lambda _{\mathrm{sat}}$ estimates of the two best models versus experimental data. It appears that the models estimate experimental $\lambda _{\mathrm{sat}}$ quite well. In reality, their SD values are bigger than those for $\lambda _{\mathrm{dry}}$ (Tables 5 and 6 vs. 3 and 4). However, due to much larger $\lambda _{\mathrm{sat}}$ values with respect to $\lambda _{\mathrm{dry}}$, the relative errors of $\lambda _{\mathrm{sat}}$ estimates are relatively small. For example, for coarse soils, the largest over-prediction by A-GMM of 6.4 % was noted for SK-04 soil and the worst under-prediction of 11.4 % was noted for NS-05 soil. For fine soils, the largest over-prediction by A-GMM was noted for NB-04 soil (14.8 %) and the worst under-prediction was for MN-02 soil at 15.6 %.

Fig. 6

18 Canadian saturated coarse soils: $\lambda _{\mathrm{sat\hbox {-}calc}}$ versus $\lambda _{\mathrm{exp}}$

Fig. 7

22 Canadian saturated fine soils: $\lambda _{\mathrm{sat\hbox {-}calc}}$ versus $\lambda _{\mathrm{exp}}$

7 Model Verification Versus Other Soils

The extension of each model under scrutiny to other soils was evaluated by comparison of their estimates against $\lambda$ data of 21 other soils, listed in Table 4. Among them, coarse soils were represented by Ottawa sands (C109 and C190), Toyoura sand, Pozzolana soil, five Chinese soils and Tottori sand, while fine soils were represented by five Chinese and two Japanese soils. The physical characteristics of these soils are displayed in Table 7.

Table 7 Physical characteristics of 21 other soils

The SD of the best six models of $\lambda _{\mathrm{dry}}$ are displayed in Table 8 (coarse soils) and Table 9 (fine soils), while the same for $\lambda _{\mathrm{sat}}$ is displayed in Table 10 (coarse soils) and Table 11 (fine soils). None of the models were fitted to $\lambda$ data of these 21 soils; consequently, $M = 0$.

Table 8 Ranking of $\lambda _{\mathrm{dry}}$ models (14 coarse other soils)

For dry coarse soils, the best $\lambda _{\mathrm{dry}}$ estimates are given by mechanistic models (MaxRTCM and deVries Model).

Table 9 Ranking of $\lambda _{\mathrm{dry}}$ models (7 fine other soils)

For fine soils, the top ranking models were LRGH (fitted to $\lambda _{\mathrm{dry}}$ data of 10 Chinese soils) and A-GMM. Tables 8 and 9 show a narrow range of SD which suggests that there is no significant advantage among any of the listed models for both coarse and fine soils tested. However, from Tables 8 and 9 in conjunction with Tables 3 and 4, it can be seen in MaxRTCM model consistently appears on or near the top in the ranking lists as the best model for estimating $\lambda _{\mathrm{dry}}$ of soils.

From Table 10 (saturated coarse other soils) and Table 11 (saturated fine other soils), the best $\lambda _{\mathrm{sat}}$ estimates were given by $\hbox {S-}{\vert }{\vert }$ or A-GMM models for both coarse and fine soils. However, unlike $\lambda _{\mathrm{dry}}$, Tables 10 and 11 in conjunction with Tables 5 and 6 show a wide range of SD which suggests that $\hbox {S-}{\vert }{\vert }$ and A-GMM models are in general the best models for estimating $\lambda _{\mathrm{sat}}$ of soils.

Table 10 Ranking of $\lambda _{\mathrm{sat}}$ models (14 coarse soils)

Table 11 Ranking of $\lambda _{\mathrm{sat}}$ models (7 fine soils)

8 Summary of Model Performance

The thermal conductivity data of 40 Canadian soils at dryness and full saturation were used to verify commonly applied $\lambda _{\mathrm{dry}}$ and $\lambda _{\mathrm{sat}}$ models of mechanistic, semi-empirical and empirical types. For mechanistic models, applied to dry soils, the closest $\lambda _{\mathrm{dry}}$ estimates were given by MaxRTCM ($\hbox {SD} = 0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for the 40 Canadian soils). The deVries Model also provided good estimates $(\pm ~0.022\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$, but only for 18 coarse Canadian soils. The $\hbox {S-}{\vert }{\vert }$ model gave satisfactory estimates $(\pm ~0.034\, \hbox { Wm}^{-1}\cdot \hbox {K}^{-1 }$ and $\pm ~0.042\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively). The Cubic Cell Model largely under-estimated experimental $\lambda$ (i.e., $-0.102\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $-0.087\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively). From four semi-empirical equations: deVries-ave, A-GMM, C–B and Chen’s equation, the best $\lambda _{\mathrm{dry}}$ estimates were given by C–B model $(\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively). The A-GMM model also produced acceptable predictions $(\pm ~0.029\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.028\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively). Finally, deVries-ave produced good estimates for coarse soils $(\pm ~0.023\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$. Among five empirical equations, the best estimates were given by CDry-40 $(\pm ~0.021\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.018\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively) and GSC $(\pm ~0.023 \hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.016\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, for coarse and fine soils, respectively). For 18 coarse Canadian soils at saturation, the best $\lambda _{\mathrm{sat}}$ estimates were given by A-GMM $(\pm ~0.085\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$ and $\hbox {S-}{\vert }{\vert }$ model $(\pm ~0.107\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$, while, for 22 fine Canadian soils at saturation, the best $\lambda _{\mathrm{sat}}$ data were also produced by A-GMM $(\pm ~0.113\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$ and $\hbox {S-}{\vert }{\vert }$ model $(\pm ~0.128\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1})$.

In addition, $\lambda _{\mathrm{sat}}$ and $\lambda _{\mathrm{dry}}$ models, without fitting parameters $(M = 0)$, were applied to $\lambda$ data of 21 other soils. Testing results for $\lambda _{\mathrm{dry}}$ models revealed a good performance for MaxRTCM $(\pm ~0.025\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for 14 coarse soils and $\pm ~0.027\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ for 7 fine soils). The MaxRTCM model performance was followed by CDry-40 $(\pm ~0.033\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.029\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively), LRGH $(\pm ~0.043\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.022\,\hbox { Wm}\cdot \hbox {K}^{-1}$, respectively), GSC $(\pm ~0.038\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.032\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively), $\hbox {S-}{\vert }{\vert }\,(\pm ~0.029\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.041\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively), and deVries $(\pm ~0.028\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.044\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively). The closest estimation of $\lambda _{\mathrm{sat}}$ for 14 coarse soils was given by A-GMM and $\hbox {S-}{\vert }{\vert }$ model $(\pm ~0.0747\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.0795\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively), while the closest estimation for 7 fine soils was given by $\hbox {S-}{\vert }{\vert }$ model and A-GMM $(\pm ~0.1354\,\hbox { Wm}^{-1}\cdot \hbox {K}^{-1}$ and $\pm ~0.1451\,\hbox {Wm}^{-1}\cdot \hbox {K}^{-1}$, respectively).

9 Conclusions

Among the mechanistic $\lambda _{\mathrm{dry}}$ models discussed, only MaxRTCM provided acceptable estimates for 40 Canadian soils as well as the 21 other soils. It appears that the use of $f_{\mathrm{sc\hbox {-}cor}}$ benefitted the model performance, when applied to dry soils. However, the complex structure of this model is a discouraging factor.

Also, there was no SD difference between the full model by de Vries and its simplified version (deVries-ave) when applied to 18 Canadian coarse soils.

A brief review of the ranking tables for dry soils revealed that for the majority of models, SD varied over a narrow range, whereas, for saturated soils, SD varied over a much wider range. This suggests that in real dry soil applications, simple models can be a preferable choice, rather than MaxRTCM. For example, CDry-40 offers close predictions against $\lambda _{\mathrm{dry}}$ data of the 40 Canadian soils and the 21 other soils. As far as $\lambda _{\mathrm{sat}}$ models are concerned, only A-GMM and $\hbox {S-}{\vert }{\vert }$ models provide good estimates when applied to 40 Canadian soils as well as the 21 other soils. There are also slight differences in the model ranking obtained for 40 Canadian versus 21 other soils. These small irregularities are probably due to missing mineral composition data for Chinese and some Japanese soils; consequently, rough $\lambda _{\mathrm{s}}$ estimates were used. In addition, for 21 other soils, often rough $m_{\mathrm{sa}}$ estimates were used to assess parameters such as the quartz content.

With respect to mechanistic (physics based) models, their advantage is in their hypothetical flexibility of application to any soil data. However, they are based on very simplified assumptions regarding soil structure and heat and mass transfer phenomena involved. They are complex in form and often are packed with coefficients that are hard to determine; consequently, they are fitted to soil experimental data. In conclusion, when applied to soils with incomplete characteristics (e.g., grain size distribution, n and mineral data), these models are likely prone to be bias. Therefore, there is a need for new models and more reliable experimental data of soils. Subsequently, more study is needed on soil systems made of solid minerals and air/water, with an inclusion of inter-particle thermal contact resistance.

Appendix

See Table 12.

Table 12 Selected soil minerals and their approximate $\lambda _{\mathrm{min}}$ values