# A sigmoidal model for predicting soil thermal conductivity-water content function in room temperature

2 min read### Johansen model

The concept of normalized thermal conductivity, K_{e} (i.e., the Kersten number) was proposed by Johansen^{7} and he used a relationship between λ and K_{e} as follows:

$$ \lambda = \left( \lambda_sat – \lambda_dry \right)K_e + \lambda_dry $$

(1)

where, λ_{dry} and λ_{sat} are the thermal conductivity of dry and saturated soils (w m^{−1} K^{−1}), respectively. The values of λ_{dry} and λ_{sat} were determined as follows:

$$ \lambda_sat = \lambda_s^1 – n\lambda_w^n $$

(2)

and

$$ \lambda_dry = \left( 0.135\rho_b + 64.7 \right)/\left( 2700 – 0.947\rho_b \right) $$

(3)

where, n is the soil porosity, λ_{w} is the thermal conductivity of water (0.594 w m^{−1} K^{−1} at 20 °C), ρ_{b} is the soil bulk density (kg m^{−3}), and λ_{s} is the effective thermal conductivity of soil solid particles (w m^{−1} K^{−1}). The values of λ_{s} are calculated by the following equation as a geometric mean:

$$ \lambda_s = \lambda_q^q\lambda_o^1 – q $$

(4)

where, λ_{q} is the thermal conductivity of quartz (sand) (7.7 w m^{−1} K^{−1}), q is the quartz (sand) content (fraction), λ_{o} is the thermal conductivity of other soil particles content (2.0 w m^{−1} K^{−1} for soils with q > 0.2 and 3.0 w m^{−1} K^{−1} for soils with q ≤ 0.2). Furthermore, Johansen (1975) proposed empirical relationships between K_{e} and normalized soil water content for different soil textures as follows:

$$ K_e = 0.7logS_r + 1.0\quad \left( S_r > 0.05 \right)\,\textfor\,\textcoarse\,\text – textured\,\textsoils $$

(5)

$$ K_e = logS_r + 1.0\quad \left( S_r > 0.1 \right)\,\textfor fine – textured\,\textsoils $$

(6)

where, S_{r} is the normalized soil water content as θ/θ_{s} where θ and θ_{s} are the soil water content and saturated soil water content (cm^{3} cm^{−3}), respectively.

### Modified Cote and Konrad model

Although Cote and Konrad^{9} improved the Johansen^{7} model, however their modified model was not accurate in λ prediction for fine-textured soils at lower water contents. Therefore, Lu et al.^{10} proposed the following equation for K_{e} estimation across the entire range of soil water content:

$$ K_e = exp\left\ \alpha \left[ 1 – S_r^(\alpha – 1.33) \right] \right\ $$

(7)

where, α is the soil texture dependent parameter (α = 0.96 for coarse-textured soils and α = 0.27 for fine-textured soils, respectively) and 1.33 is a shape factor.

Furthermore, Lu et al.^{10} used a simple empirical linear relationship between λ_{dry} and soil porosity (n) as follows:

$$ \lambda_dry = – an + b $$

(8)

where a and b are constants as a = 0.56 and b = 0.51 for 0.0 < n < 0.6, and n is the soil porosity. The value of λ_{sat} was calculated according to Eq. (2) used in Johansen^{7} model.

### New Xiong et al. model

The proposed model by Xiong et al.^{5} is as follows:

$$ \lambda = P + QS_r^R + S\left\{ exp\left[ S_r\left( 1 – S_r \right) \right] \right\} $$

(9a)

$$ P = \, \lambda_dry = 0.51 – 0.6n $$

(9b)

$$ Q = \, \lambda_sat – \, \lambda_dry = \left( \, \lambda_q^q\lambda_o^1 – q \right)^1 – n\left( \lambda_w \right)^n\left( 0.51 – 0.6n \right) $$

(9c)

$$ S = 1.5\left( S_r – S_r^2 \right) $$

(9d)

where, n is the soil porosity (–), S_{r} is the degree of soil saturation as θ/θ_{s} (%), λ_{q} is the thermal conductivity of quartz (sand) (7.7 w m^{−1} K^{−1}), q is the quartz (sand) content (fraction), λ_{o} is the thermal conductivity of other soil particles content (2.0 w m^{−1} K^{−1} for soils with q > 0.2 and 3.0 w m^{−1} K^{−1} for soils with q ≤ 0.2), λ_{w} is the thermal conductivity of water (0.594 w m^{−1} K^{−1}) and the value of R is 1.2 for sand, 1.5 for the fine-textured soils, and 2.0 for clay. However, the values of R for fine-textured soils are not well designated for different fine-textured soils.

### Sigmoidal model

Sigmoidal model (i.e., logistic equation) to describe the soil thermal conductivity as a function of soil water content is as follows:

$$ \lambda = K/\left[ 1 + Aexp\left( – B\theta \right) \right] $$

(10)

where, λ is the soil thermal conductivity (w m^{−1} K^{−1}), θ is the soil volumetric water content (cm^{3} cm^{−3}), K is the upper most asymptote implies the upper limits of soil thermal conductivity, and A and B are coefficients as taken of initial stage and total accretion rate.

Measured soil thermal conductivities at different soil water contents for six soils with different textures were used in Eq. (10) to determine the values of K, A, and B by Solver tool in EXCEL software. Then, the values of K, A, and B were used in multiple linear regression analysis in EXCEL software to obtain an empirical model to estimate the values of K, A, and B based on soil physical parameters. These parameters were sand, silt, and clay contents, soil bulk density, soil porosity. Among these parameters sand content and soil bulk density entered in the empirical multiple linear regression as follows:

$$ K = a_o + a_1q + a_2\rho_b $$

(11)

$$ A = b_o + b_1q + b_2\rho_b $$

(12)

$$ B = c_o + c_1q + c_2\rho_b $$

(13)

where, a_{o}, a_{1}, a_{2}, b_{o}, b_{1}, b_{2}, c_{o}, c_{1}, and c_{2} are constants, q is the sand particle content (%), and ρ_{b} is the soil bulk density (g cm^{−3}). The entrance of soil bulk density and sand content is due to the fact that the thermal conductivity of sand (instead of quartz) is much higher than the clay and silt particles and it contributes much more to the λ value of soil, and value of bulk density indicates the compaction of soil, soil pores distribution and extend the contact of the soil particles to cause the soil thermal conductance.

Using the measured values of λ in sigmoidal model (Eq. 10) and Solver tool in EXCEL software, the values of K, A, and B were determined for six different soil textures (Table 2). These values were used in Eq. (10) to estimate λ(θ) for different soil water contents.

### Sigmoidal model validation

For validation of the new model, measured data from Lu et al.^{10} for three different soil textures are used (Table 1). By use of the empirical multi-linear equations (Eqs. 11, 12, and 13), the values of the empirical-logistic constants (K, A, and B) were estimated. Based on these coefficients the values of λ(θ) were estimated. Then, the estimated values of λ(θ) were compared with the measured values reported by Lu et al.^{10}.

### Statistical analysis

The outputs of the model were compared by the measured values using following statistical parameters:

$$ NRMSE = {\left( {1/N\Sigma^n_i = 1\left( X_i – Y_i \right)^2} \right)^0.5}/O $$

(14)

where, NRMSE is the normalized root mean square error, N is the number of observations, X is the measured values, Y is the estimated values and O is the mean values of measured data. The value of NRMSE approaches 0.0 for the accurate estimation. The closer the NRMSE is to 0, the model is more accurate. Linear relationship between the measured and predicted values is compared with 1:1 line with slope and intercept of 1.0 and 0, respectively by using Fisher F-test.

link