The aim of this test method is to determine the thermal diffusivity of soils under quasi-steady-state conditions, using the finite-difference method. In chronological order, the experimental procedure is as follows:

Recommendation: The choice of time step must be made in accordance with a stability condition specific to the method chosen to solve the problem posed. Stability is the property that ensures that the difference between the numerical solution obtained and the exact solution of the discretized equations is bounded [9] . For temporal evolution problems, some schemes are stable provided that the time step is less than a certain critical value depending on the space step. In the present case, this is written as:

$\alpha \frac{\text{Δ}t}{\text{Δ}{x}^2}<0.5$ (1)

The calculation of thermal diffusivity is based on the solution of the heat equation in the one-dimensional case according to Mrawira and Luca [7] . The heat equation in the one-dimensional case is written:

$\frac{\partial T}{\partial t}=\alpha \frac{{\partial }^{2}T}{\partial x^2},0\le x\le L,t\ge 0$ (2)

with $T=T\left(x,t\right)$ the temperature at time $t$ and position $x$, and $\alpha$ the thermal diffusivity of the medium. Relation 1 is written for a semi-infinite medium of thickness $L$ ( Figure 4 ). Initial conditions (at $t=0$) and boundary conditions (for $x=0$ and $x=L$) are such that:

$T\left(0,t\right)=T_0,T\left(L,t\right)=T_L,T\left(x,0\right)=f_0\left(x\right)$ (3)

with $f_0\left(x\right)$ the initial relative temperature distribution as $f_0\left(x\right)=T\left(x,t\right)-$ $T\left(x,0\right)$. Equation (2) is a partial differential equation whose numerical solution requires discretization of the study domain under consideration, in both time and space. For this purpose, it is assumed that the time interval is divided into a grid with a finite number of points and intervals ( Figure 4 ). The solution is calculated only at these points. If we note $\Delta t$ the time interval (assumed constant) of the temporal discretization grid, and $\Delta x$ the space step (assumed constant) of the spatial discretization grid, then the calculation of the solution will take place at points $x=i\Delta x$ and $t=m\Delta t$ with $i=1,2,\cdots ,N$ and $m=1,2,\cdots ,MN$ et $M$ being respectively the total number of space and time nodes.

By replacing $\delta x$ by the time step $\Delta t$ in relation 2, and by the space step $\Delta x$ in relation 2, then Equation (2) becomes:

$\frac{T\left(x,t+\text{Δ}t\right)+T\left(x,t\right)}{\text{Δ}t}=\alpha \frac{T\left(x+\text{Δ}x,t\right)-2T\left(x,t\right)+T\left(x-\text{Δ}x,t\right)}{\text{Δ}{x}^2}$ (4)

Referring to Figure 4 , if:

$T\left(x,t\right)=T_i^m$ $T\left(x,t+\Delta t\right)=T_i^{m+1}$

$T\left(x-\Delta x,t\right)=T_{i-1}^m$ $T\left(x+\Delta x,t\right)=T_{i+1}^m$

then thermal diffusivity is given by the following relationship:

$\alpha =\frac{T_i^{m+1}-T_i^m}{T_{i-1}^m+T_{i+1}^m-2T_i^m}x\frac{\Delta x²}{\Delta t}$ (5)

This method takes into account certain physical and mechanical parameters such as density, water content, mineralogy, grain size and grain arrangement [11] [12] .

Obtain the thermal diffusivity by averaging the different thermal diffusivities obtained.

Figure 5 shows the device used to measure thermal diffusivity.

The aim of the calibration phase is to determine the correct configuration and materials to be used for the set-up [8] . Calibration tests were carried out on dry sand (gully sand) with a thermal diffusivity of 3.17 × 10⁻⁷ m²/s and a density of 1600 kg/m³.

The results obtained enabled us to evaluate and correct several parameters concerning the assembly, the most important being:

The relative error $\delta {\alpha }_r$, defined by relation 6, is used to assess the accuracy of the measurement.

$\delta {\alpha }_r=\left|\frac{{\alpha }_m-{\alpha }_c}{{\alpha }_c}\right|$ (6)

The calibration phase also made it possible to determine the correct approach to take when analyzing the results.

In order to assess the accuracy of measurements obtained using thermocouples, calibration tests were carried out on dry gully sand. Once the gully sand had been sampled, it was placed in an oven and heated to 105 ± 5˚C for over 48 hours, in order to dry it to constant mass. The target thermal diffusivity is. 3.17 × 10⁻⁷ m²/s.

Dry gully sand, being the material chosen as the reference for defining the relative error, was placed under the same conditions as the material under study.

Figure 6 shows the variation in temperature as a function of time for dry gully sand.

As described above, the discretization of space is given by the spacing between thermocouples. Hence, according to Figure 6 , we have:

$\Delta x=21\times {10}^{-3}\text{m}$

Thus, the condition expressed by equation 1 in §3.1 is verified in all the tests carried out as part of this research project. The choice of time step was therefore made by considering the evolution of the relative error of calibration test results as a function of time step, and the value chosen is:

$\text{Δ}t=100\text{s}$

Since the calculation of thermal diffusivity requires a variation in temperature at all three measurement points, it is natural to consider temperature values only after a certain time has elapsed. According to Mrawira and Luca [7] , the observation period to be used for thermal diffusivity evaluation is determined from the curves, which must be included in the ascending range common to all three curves. It is delimited on Figure 6 by the two vertical segments. This portion of the curve begins at $t_d=2500\text{s}$ and ends at $t_f=3500\text{s}$.

Given the delicacy of the temporary range to be retained when evaluating a material’s thermophysical parameter, Gustafsson [13] explains that for the theoretical model to be valid, the duration of the experiment must be chosen carefully. Thus, the time must be long enough:

This double constraint means we must select a “time window” for analysis that excludes the very first instants (dominated by contacts) and longer times (where the semi-infinite model is no longer valid).

Let’s take a look at $t=2800\text{s}$, then we have:

$T\left(x,t\right)=T\left(m,2800\right)=35.75˚\text{C}$

$T\left(x,t+\Delta t\right)=T\left(m,2900\right)=36˚\text{C}$

$T\left(x-\Delta x,2800\right)=T\left(m-1,2800\right)=33.25˚\text{C}$

$T\left(x+\Delta x,2800\right)=T\left(m+1,2800\right)=41.5˚\text{C}$

Thus, we have:

$\frac{36-35.75}{100}=\alpha \frac{41.5-2\times 35.75+33.25}{{\left(21\times {10}^{-3}\right)}^2}$

$\alpha =\frac{36-35.75}{41.5-2\times 35.75+33.25}\times \frac{{\left(21\times {10}^{-3}\right)}^2}{100}$

$\alpha =3.39\times {10}^{-7}{\text{m}}^{\text{2}}/\text{s}$

Keeping the same time step, the previous calculation can be repeated at different positions along the time axis. In our case, a calculation is performed every 100 seconds (let’s note this translation time $t_{dt}$). In other words, at:

These results are shown in Figure 7 , which also shows the target value 3.17 × 10⁻⁷ m²/s and the arithmetic mean of the values obtained. This figure, therefore, shows the thermal diffusivity values calculated over the time interval chosen previously $\left(\left[t_d;t_f\right]\right)$ with a space between each measurement, which is $t_{dt}=100\text{s}$.

In summary, the parameters chosen in the example calculation above are:

$\Delta x=21\times {10}^{-3}\text{m},t_d=2500\text{s},t_f=3500\text{s},\Delta t=100\text{s},t_{td}=100\text{s}.$

The mean thermal diffusivity value obtained is 3.35 × 10⁻⁷ m²/s, giving a relative error $\delta {\alpha }_r$ of 5.4%. The relative error obtained lies within the range 3.2% to 5.9% determined by Hay et al. [14] . In this study, Hay et al. [14] showed that it is difficult to obtain relative errors below 3% when measuring the thermal diffusivity of opaque, homogeneous, and isotropic materials using the partial time-moment method. It is generally accepted that measurements are accurate when the relative error is less than 10%.