## 2 Thermal Properties  
Thermal properties such as heat capacity, thermal conductivity, thermal expansion, and thermal stress are required to model the heat transfer in soils [12; 13]. Heat capacity is the measurable physical energy required to increase the temperature of a material by 1 \({}^{\circ}\)C. Thermal conductivity is the ability of a material to conduct heat from high- to low-temperature regions. Thermal expansion is the change in volume of a material in response to a change in temperature. Thermal stress is the change in stress of a material owing to a change in temperature. Thermal conductivity plays an important role in heat transfer; for example, heat transfers at a low rate in a material with low thermal conductivity. As such, a material with low thermal conductivity can be used for thermal insulation, whereas a material with high thermal conductivity can be used as a heat sink. Thermal conductivity is a property explaining the conduction, convection, and radiation in granular materials [14]. Conduction is affected by the properties of the mineral and the contact area between particles; convection is related to the properties of the fluid filling the sediment pores; and radiation is related to the propagation of heat without any medium [14]. Thus, thermal conductivity depends on the porosity, mineral type, effective stress, saturation, and properties of the pore fluid. For example, thermal conductivity increases with a decrease in porosity, an increase in effective stress, and an increase in degree of water saturation [15; 16; 17; 18; 19; 20]. The thermal conductivity of granular materials can be estimated using a steady state or transient method. In the steady state method, the thermal conductivity is evaluated based on [PERSON]'s law for heat conduction, according to Equation (1):  
\[\lambda=\frac{Q\cdot L}{A\cdot(T_{2}-T_{1})} \tag{1}\]  
where \(\lambda\) is the thermal conductivity (W/(m\(\cdot\)K)), \(Q\) is the amount of heat passing through a cross section of the specimen (W), \(A\) is the cross-sectional area of the specimen (m\({}^{2}\)), \(L\) is the length of the specimen (m), and \(T_{2}\) and \(T_{1}\) are the temperatures at two locations (K). In the transient method, the thermal conductivity of a material is evaluated based on the change in temperature over time (after heat generation). It normally takes less time to conduct a test using the transient method than to apply the steady state method; therefore, the moisture migration in the material during the test may be minimized [21]. The temperature change \(\Delta T\) (K) at time \(t\) (s), when heating starts at \(t\) = 0 and stops at \(t\) = \(t_{p}\), can be represented through Equations (2) and (3) (ASTM D5334-14):  
\[\Delta T=-\frac{Q}{4\pi\lambda}\text{Ei}\left(\frac{-r^{2}}{4 Dt}\right)\,0<t \leq t_{p} \tag{2}\]  
\[\Delta T=-\frac{Q}{4\pi\lambda}\left[-Ei\left(\frac{-r^{2}}{4 Dt}\right)+Ei \left(\frac{-r^{2}}{4D\left(t-t_{p}\right)}\right)\right]\,t>t_{p} \tag{3}\]  
where, \(Q\) is the heat applied per unit length of the heater (W/m), \(r\) is the distance from the heat source (m), and \(D\) is the thermal diffusivity (m\({}^{2}\)/s). Equations (2) and (3) can be simplified (using a non-linear least-squares inversion method) into Equations (4) and (5), which are the basis for the evaluation of thermal conductivity using the transient method (ASTM D5334-14):  
\[\Delta T=\frac{Q}{4\pi\lambda}ln(t)\,0<t\leq t_{p} \tag{4}\]  
\[\Delta T=\frac{Q}{4\pi\lambda}ln\left(\frac{t}{t-t_{p}}\right)\,t>t_{p} \tag{5}\]  
[PERSON] and [PERSON] [22] developed two theoretical models (the parallel and series models) for the thermal conductivity of a soil matrix composed of soil, water, and air.