# A Numerical Model to Estimate the Soil Thermal Conductivity Using Field Experimental Data
## 1 Introduction  
Soil thermal conductivity is an important thermal property that governs the transfer of the soil heat flux. The complexity of soil makes its thermal conductivity dependent on physical or chemical properties, such as soil water content, porosity, density, texture, and mineral composition [1]. Different empirical models have been suggested for the estimation of soil thermal conductivity, taking into account these properties [2; 3; 4; 5; 6; 7; 8]. However, these models are mainly obtained using data from controlled experiments in the laboratory by measurement techniques including steady-state methods such as the guarded hot plate method [9], transient methods including the single line heat source probe [10; 11] and the dual-probe heat-pulse method [12; 13; 14]. A challenge has been to obtain estimates of the soil thermal conductivity in situ (or in field experiments), in which soil properties are not well characterized as a result of technical limitations [15; 16] and/or climatic conditions. Theoretically, in situ experimental measurements of soil heat flux and soil temperature can be used to estimate the soil thermal conductivity by the Fourier Law of heat conduction, defined by [17]:  
\[G_{z}=\ -\lambda\frac{\partial T}{\partial z} \tag{1}\]where \(G_{z}\) (Wm\({}^{-2}\)) is the soil heat flux at a given depth \(z\) (m), \(T\) (K) is the soil temperature and \(\lambda\) (Wm\({}^{-1}\)K\({}^{-1}\)) is the thermal conductivity of the soil layer. However, because the data are discrete (measured in time intervals), the soil temperature gradient, \(\frac{\partial T}{\partial z}\), can be obtained by the variation rate between the soil temperature and the depth, \(\frac{\Delta T}{\Delta z}=\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\). This methodology is called the Gradient Method [4; 5; 16; 17; 18; 19] Therefore, if we know the measurement of the soil heat flux at a given depth \(z\) (\(G_{z}\)) and the soil temperature at depths relatively close to \(G_{z}\), Equation (1) can be written as:  
\[G_{z}=\ -\lambda\bigg{[}\frac{T_{2}-T_{1}}{z_{2}-z_{1}}\bigg{]} \tag{2}\]  
with \(z_{1}<z<z_{2}\), and where \(T_{1}=T(z_{1})\) and \(T_{2}=T(z_{2})\) represent the temperatures at depths \(z_{1}\) and \(z_{2}\), respectively. The experimental estimate of soil heat flux in \(z\) is often performed by the heat flux plate method. Soil heat flux plates are small, rigid, disc-shaped sensors of known and constant thermal properties that are placed horizontally in the soil near the surface. The heat flux through a calibrated plate is used to estimate \(G_{z}\) in the surrounding soil at the plate depth [20]. That is, Equation (2) is used to estimate the soil heat flux in a small soil layer if the \(\lambda\) of the plate is known. Heat flux plate manufacturers describe important factors to be considered in an experiment to estimate \(G_{z}\), especially in a field experiment in which several procedures are not controlled, such as the exact installation depth and the thermal contact resistance at the plate-soil interface [21; 22; 23]. Soil heat flux sensors are often installed at depths ranging from 0.05 m to 0.20 m, considered to be mechanically stable to ensure good measurement conditions besides not suffering significant time delays that can reduce the accuracy of the measurements [24; 25; 26]. [PERSON] et al. [15] concluded that the heat flux plate underestimates soil heat flux because of the low plate thermal conductivity, thermal contact resistance, and latent heat transfer effects. However, this is the main method to obtain the experimental soil heat flux and widely used in studies of surface energy balance closure [27; 28; 29]. Using experimental measures of soil heat flux and soil temperature, several authors estimated the soil thermal conductivity by isolating \(\lambda\) in Equation (2). However, this procedure can lead to possible numerical divergences, as shown by [PERSON] et al. [19; 30] that eliminated data when soil temperature gradient or soil heat flux is very close to zero to avoid this problem.