## 4 Example Verification
### Two-Dimensional Anisotropic Reservoir Model  
To validate the reliability and effectiveness of the borehole-to-surface electromagnetic three-dimensional unstructured vector finite element anisotropic algorithm, a two-dimensional borehole electromagnetic model was established for the algorithmic solution, and the results were compared with those obtained from [PERSON]'s 2.5-dimensional adaptive finite element method (MARE2 DEM, [https://mare2 dem.bitbucket.io/master/download.html#](https://mare2 dem.bitbucket.io/master/download.html#)). As shown in Figure 7, the two-dimensional anomaly is rectangular, with a length of 1000 m and a thickness of 500 m (in the three-dimensional model, the width is set to 16,000 m). The center of the anomaly is located at coordinates (600, 0, \(-\)600) m, with  
Figure 5: Comparison of numerical and analytical solutions for layered model component. (**a**) Amplitude (\(\mid\!E_{x}\!\mid\)) comparison; (**b**) phase (\(\Phi_{x}\)) comparison. Figure 6: Relative errors of amplitude (\(\mid\!E_{x}\!\mid\)) and phase (\(\Phi_{x}\)) for layered model. (**a**) Relative error of the amplitude; (**b**) relative error of the phase. resistivities of [120; 110; 200] \(\Omega\cdot\)m. The resistivity of the air layer is set to \(10^{8}\)\(\Omega\cdot\)m, and the surrounding rock is modeled as a mixture of loose shale and metamorphic rock with resistivities set to [30; 20; 10] \(\Omega\cdot\)m. The model was simulated using both the borehole-to-surface electromagnetic three-dimensional unstructured finite element anisotropic algorithm (with the computational domain set to \([-8000,\,8000]\)\([-8000,\,8000]\)\([-6000,\,6000]\) m) and MARE2 DEM. A 10 m long vertical finite-length-wire transmitter was placed at a depth of 600 m, with the field transmitter coordinates set at (0, 0, \(-600\)) m. The measurement line was arranged along the x-direction, with receivers placed every 100 m from \(-2000\) m to 2000 m, resulting in a total of 41 receivers. An observation frequency of 1 Hz was used. Figure 8 presents a schematic of the two-dimensional reservoir model generated using MARE2D, which employs an integrated mesh adaptation strategy for high-quality discretization. Figure 9a shows the mesh obtained using Gmsh, and Figure 9b illustrates the mesh refinement performed around the transmitter, receivers, and anomaly. The mesh consists of 624,664 elements, 1,261,287 faces, and 743,127 edges, with a total of 4,008,828 degrees of freedom. In this study, simulations were carried out using both [PERSON]'s 2.5-dimensional adaptive finite element method for controlled-transmitter electromagnetic modeling with anisotropy (MARE2 DEM) and the borehole-to-surface electromagnetic method based on the unstructured-mesh 3D edge-based finite element anisotropic algorithm (BSEM) proposed in this paper. The response curves obtained from both methods were compared. As  
Figure 8: MARE2 DEM two-dimensional reservoir model. Figure 7: Two-dimensional reservoir model. shown in Figure 10, the numerical simulation results from our algorithm are in excellent agreement with those from MARE2 DEM. Figure 10a displays the amplitude (\(\mid E_{x}\mid\)) of the electrical response along the receiver line, clearly demonstrating that the responses from the proposed method and MARE2 DEM are nearly identical; Figure 10b shows the corresponding phase comparison, with similarly high consistency. Furthermore, the relative errors in amplitude (\(\mid E_{x}\mid\)) and phase (\(\Phi_{x}\)) were calculated, as illustrated in Figure 11. The absolute values of the relative errors for the magnitude and phase of the Ex component are less than 5% and 1%, respectively, confirming the reliability of the unstructured-mesh 3D edge-based finite element algorithm under anisotropic conditions. Figure 11: Relative errors of amplitude (\(\mid E_{x}\mid\)) and phase (\(\Phi_{x}\)) for the 2D anisotropic reservoir model. (**a**) Relative error of the amplitude; (**b**) relative error of the phase.