## 3 Numerical Simulation and Verification for Hydraulic Fracturing  
For the frequency-domain borehole-to-surface electromagnetic imaging observation system, our study needs to discuss the influence degree of the placement mode of the B electrode on surface observations in practical observations. Numerical simulations are carried out for different azimuth placement cases of the B electrode to obtain surface responses. The placement mode of the B electrode is determined by comparing the magnitudes of the simulated surface response results so as to guide the actual monitoring construction operations. The simulation transmission frequency is 2.5 Hz, the transmission current is set to 10 A, and the model is a half-space model, which is consistent with the model shown in Figure 5. The relative magnetic permeability is 1. Figure 6 is a schematic diagram of the transmitter source (B electrode) position settings. In the three cases, the B electrode is set in the west direction (Figure 6a), east direction (Figure 6b), and north direction (Figure 6c), all at a distance of 2500 m from the target wellhead. In this model, the transmitter source (A electrode) is decomposed into a vertical line source and a horizontal line source to simplify the simulation of actual high-angle wells and horizontal wells. The calculation speed for the model shown in Figure 5 is approximately one frequency of 10-20 min. Figure 5: Half-space model: (**a**) formation resistivity setting; (**b**) fracturing anomaly body configuration. Figure 6: _Cont_. Simulations were conducted for surface monitoring of \(Ex\) and \(Ey\) components in three scenarios. \(Ex\) and \(Hy\) share similarities as do \(Ey\) and \(Hx\). This similarity in form fundamentally originates from the orthogonality of plane electromagnetic waves. In a homogeneous medium, \(Ex\) and \(Hy\) are components of the same electromagnetic wave and thus have precisely identical propagation characteristics. Consequently, the \(Ex\), \(Ey\) and \(Hx\) components were chosen for presentation and analysis. Figure 7 shows the contour maps of surface-monitored \(Ex\) and \(Ey\) components. Figure 7a presents the amplitude values of the \(Ex\) component, and Figure 7b shows the amplitude values of the \(Ey\) component. The left column represents the prefructuring stage, the middle column the post-fracturing stage, and the right column the differential anomaly field. By comparing Figure 7a,b, it can be concluded that under the same model, the amplitude values of surface electromagnetic responses (\(Ex\), \(Ey\)) for the three placement methods are similar, and placement method (b) exhibits a higher anomaly field amplitude compared to the other two methods. With this observation method, the target area is in the transition zone, demonstrating good monitoring effects. Based on the model in Figure 5 and observation mode (b) in Figure 6, the surface response simulation calculations of \(Ex\) and \(Hx\) components were carried out. Figure 8 shows the comparison of Ex component responses received on the surface before and after the simulated fracturing. Figure 8a is the real part of the \(Ex\) component of the background  
Figure 6: Schematic diagram of transmitter source (B electrode) positioning: the B electrode is set in (**a**) the west direction, (**b**) the east direction, and (**c**) the north direction. Figure 7: Comparison of contour maps of surface electric field components: (**a**) \(Ex\); (**b**) \(Ey\). field before fracturing, Figure 8b is the real part of the \(Ex\) component after fracturing, and Figure 8c is the real part of the \(Ex\) component of the differential anomaly field. Figure 8d, Figure 8e, and Figure 8f are the imaginary parts of the \(Ex\) component in the three cases (before and after fracturing, and the difference), respectively. And Figure 8g, Figure 8h, and Figure 8i are the total fields of the \(Ex\) component in the three cases, respectively.