## 2 Materials and Methods  
The numerical analysis used in the authors' research is based mainly on the quantitative examination of the thermal states of power cables (cable lines) placed in the ground with and without the use of stabilized backfill. The power cable lines consisting of three single-core copper conductors were analyzed (three-phase system--three cables in a flat, touching formation). The cross-sectional area of the cables was taken into account in the model tests. The tested power cable lines consisted of cables with a cross-sectional area of 3 \(\times\) 35 mm\({}^{2}\), 3 \(\times\) 120 mm\({}^{2}\), and 3 \(\times\) 240 mm\({}^{2}\), respectively. The cross-sectional view of the analyzed cable type is shown in Figure 1.  
The characteristic dimensions of a given type of power cable are included in Table 1.  
The introduction to detailed analyses related to the influence of stabilized backfill parameters on the ampacity of power cable lines is conditional to the verification of the numerical model. Verification of the numerical model can be performed on a relatively simple model--in this case, it will concern a power cable line buried directly in the ground without the use of stabilized backfill. The illustrative model is presented in Figure 2. The results of numerical calculations for the simple model (without stabilized backfill) will be compared with the known values contained in the standard [19]. The simplified model shown in Figure 2 was implemented in the ANSYS Fluent software(version 2020 R2 and 19.1). According to the presented model, the power cable line is arranged in a flat, touching arrangement and is embedded in the ground to a depth \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cross-Sectional Area,** & **Copper Conductor** & **External Diameter \(D_{2}\),** \\
**mm\({}^{2}\)** & **Diameter \(D_{1}\), mm** & **mm** \\ \hline
35 & 7.2 & 12.4 \\
120 & 13.2 & 19.4 \\
240 & 18.8 & 26.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of dimensions of the analyzed power cables. Figure 1: Illustrative drawing of the analyzed power cable type. Figure 2: A simple version of the cable system (three cables in a flat, touching formation, without stabilized backfill) to verify the correctness of the numerical model. of \(h\) = 0.7 m. The assumption made, as well as a certain simplification, is the homogeneity of materials in the analyzed geometric system. A steady-state simulation was conducted to analyze the heat transfer between power cables and their surrounding material. During the simulation, the energy equation was enabled and the physical properties of various materials, such as density and thermal characteristics (including thermal conductivity and specific heat), were implemented into the model. The energy equation in ANSYS Fluent is complex; however, for the specified computational case, thermal conduction in the solid body is the only heat transfer mechanism that needs to be resolved. The temperature boundary condition in the lower part of the model was set to 20 \({}^{\circ}\)C, which aligns with Polish geological and thermal conditions. The design air temperature \(t_{\text{air}}\) is 20 \({}^{\circ}\)C and the convective heat coefficient of 15 W/(m\({}^{2}\)K) is set on the upper boundary of the domain. These parameters are consistent with the assumptions of the standard [22] and possible unfavourable conditions of heat exchange in the environment. Figure 3 shows an example of the distribution of computational mesh elements for a selected part of the model. The computational domain encompasses approximately 400,000 computational cells in total. Within this domain, calculations were conducted to determine the ampacity (\(I_{\text{A}}\)) of the power cable line under investigation, which was modelled using varying values of soil thermal resistivity \(\rho_{\text{g}}\): 0.5, 1.0, and 2.0 (K\(\cdot\)m)/W.