# The Electromagnetic Field Analytical Solution Analysis to Downhole Current Injection [PERSON]
## 3 Simulation Experiments  
To verify the correctness of the analytical expression, a simulation environment was constructed, as shown in Figure 3, and the electromagnetic field distribution was calculated by the analytical expressions derived in Section 2. The multiphysics simulation software COMSOL Multiphysics (version: 6.0) was utilized to simulate the electromagnetic field distribution in the same environment. In Figure 3, the upper layer of the column model is the air domain, whose conductivity is basically 0, and the lower layer is the stratum domain, whose conductivity is 0.02 S/m. Assume that the accident well is a straight well with a depth of 1000 m, and its casing conductivity is \(10^{7}\) S/m. The wellhead of the relief well and the wellhead of the accident well are separated by 200 m, and the surface electrode is deployed near the wellhead of the relief well. Additionally, the excitation electrode is located 500 m below the relief well and 30 m horizontally from the accident well. The simulation parameters are presented in Table 1. Based on the parameters, the COMSOL simulation results and the analytical calculation (run in MATLAB R2023a) results will be compared in terms of different variables. First, the convergent current on the target well surface was simulated by COMSOL and analytical calculation. The results are illustrated in Figure 4. It can be observed that the variation trends of the convergent current density simulated by the two methods are consistent, and their results are in the same order of magnitude. The convergent current reaches its maximum at the depth position corresponding to the excitation electrode (i.e., near \(-500\) m). As the depth continues to increase, the convergent current gradually decreases. This observation aligns with the basic principle of current injection, thereby validating the rationality of the analytical expression. Additionally, it is noteworthy that since the simplified two-layer model used in the analytical calculation ignores the relief well, which is considered In the COMSOL simulation, the converge current obtained by the analytical calculation is higher than that derived from the COMSOL simulation. For further comparison, the magnetic field distribution was simulated, and the corresponding placements of the magnetic field detection point are demonstrated in Figure 5. To minimize interference from the primary field generated by the excitation electrode on the effective received signals, three observation lines were set up 100 m below the excitation electrode at inclination angles of \(0^{\circ}\), \(5^{\circ}\), and \(10^{\circ}\), respectively. In each observation line, 10 observation points are distributed uniformly at a spacing of 0.5 m. The observed magnetic field at each point was analyzed with the parameter settings listed in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Position of Downhole Electrode & 500 m underground, 30 m away from the target well \\ Surface Electrode & On the ground, 200 m from the target well \\ Target Well Length & 1000 m \\ Stratum Conductivity & 0.02 S/m \\ Target Well Conductivity & \(10^{7}\) S/m \\ Current intensity & 100 mA \\ Range of \(\rho\) & 0–30 m \\ Range of \(\varphi\) & 0–2\(\pi\) \\ Range of \(z\) & \(-1000\)–0 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the simulation experiment. Figure 3: The simulation environment of downhole current injection ranging built by COMSOL. The magnetic fields at the observation positions obtained with the two calculation methods are illustrated in Figure 6. Figure 6a,b show the analytical calculation results and the COMSOL simulation results, respectively. It can be found that the magnetic field results obtained by analytical calculation and COMSOL simulation are also consistent.