## 3 Channel Modeling of the Ground Electrode Current Field in a Tunnel Environment
### Simulation Analysis of the Relationship between the Receiving Voltage and Collapse Distance  
Figure 3 shows the relationship between the received voltage and the collapse distance. It can be seen from the simulation that under different conductivities, the receiving voltage decreases sharply with the distance, and at a distance of 100 m, the receiving voltage is only \(\mu\)V. This puts forward the requirement for the receiving ability of the system to detect the \(\mu\)V signal. The path loss based on the ground electrode current field can be expressed as:  
\[PL=-20\log_{10}\left|\frac{l_{b}l_{o}}{4\pi d_{e}^{3}\sigma Z_{c}}\Big{(}1+ \gamma d_{e}+\gamma^{2}d_{e}^{2}\Big{)}e^{-\gamma d_{e}}\right| \tag{3}\]  
Assuming the operating frequency f = 10 Hz, the other conditions remain unchanged except for the penetration distance, the simulation is carried out according to Formula (3), and the simulation results are shown in Figure 4.  
Figure 3: Relation between the received voltage and collapse distance. It can be seen from the simulation that the path loss increases significantly with the increase in the penetration distance under different conductivities. According to the relationship between the path loss and the penetration distance, the matching and selection of the power of the transmitter can be guided by the sensitivity of the designed receiving unit [25]. The collapsed medium is not strictly an insulator, so electromagnetic waves can still transmit information. In oil drilling, information transmission based on the ground electrode current field is widely used in MWD. The transmission characteristics of the signal in the ground and in the air are very different. The transmission attenuation in the ground is extremely fast, and the transmission depth of the electromagnetic wave in the ground is at the skin depth of the electromagnetic wave signal in the ground. The skin depth in a homogeneous medium can be expressed as:  
\[\delta=\frac{1}{\sqrt{\pi f\mu/\rho}} \tag{4}\]  
Among them, \(\mu=4\pi\times 10^{-7}\Omega\cdot\) s/m, \(f\) is the frequency, the unit is Hz, \(\rho\) is the formation resistivity, and the unit is \(\Omega\cdot\) m. We select some typical resistivity values (5 \(\Omega\cdot\)m, 20 \(\Omega\cdot\)m, 30 \(\Omega\cdot\)m) according to Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{3}{*}{**Category**} & \multirow{3}{*}{**Name**} & \multirow{3}{*}{
\begin{tabular}{c} **Approximate Value** \\ **of Resistance** \\ **\(\Omega\cdot\)m** \\ \end{tabular} } & \multicolumn{3}{c}{**Variation Range of Resistivity under Different Conditions \(\Omega\cdot\)m**} \\ \cline{3-6}  & & & **Wet (General Areas,** & **When It Is Drier** & **When the** \\  & & **\(\Omega\cdot\)m** & **Rainy Areas)** & **(Less Rain Area,** & **Groundwater** \\  & & & & **Desert Area)** & **is Saline** \\ \hline \multirow{6}{*}{earth} & pottery clay & 10 & 5–20 & 10–100 & 3–10 \\  & peat, marl, marsh & 20 & 10–30 & 50–300 & 3–30 \\ \cline{1-1}  & sandy clay & 100 & 30–300 & 80–1000 & 10–80 \\ \cline{1-1}  & loes & 200 & 100–200 & 250 & 30 \\ \cline{1-1}  & clay, sandy soil & 300 & 100–1000 & 1000 or more & 30–100 \\ \cline{1-1}  & coal & — & 350 & — & — \\ \cline{1-1}  & rocky soil & 400 & — & — & — \\ \cline{1-1}  & topsoil intercalated, lower gravel & 600 (15\% humidity) & — & — & — \\ \hline \end{tabular}
\end{table} 
Table 2: Resistivity of different media. Figure 4: Relationship between the path loss and penetration distance. The corresponding transmission depth obtained using the simulation is shown in Figure 5[26; 27]. It can be seen from the simulation that, affected by the skin effect, if the resistivity is too small, the penetration ability of the electromagnetic wave signal will deteriorate, and if the resistivity is too high, the radiation power of the ground electrode will be low. When the resistivity is constant, the smaller the frequency, the stronger the signal penetration ability, but the information transmission rate is extremely low [28].