## 3 MWD Positioning Method Based on Dual-Electrode Current Excitation  
### Dual-Electrode Current Excitation Ranging Model  
The model for active magnetic positioning with dual-electrode current excitation is constructed, taking into account the key calculation parameters in the ranging system,  
Figure 1: Diagram of electromagnetic MWD model based on dual-electrode current excitation.  
including the formation characteristics, drilling well borehole, target well casing, BHA, and MWD.  
The schematic diagram of the electromagnetic MWD model based on dual-electrode current excitation is shown in Figure 1. The Q point represents the center position of the upper/lower electrodes in the drilling well. The M point represents the position of the ranging point corresponding to the casing of the target well. Appendix B summarizes the meaning of the manuscript parameters. The connection \(d\) of the Q-M two points is perpendicular to the casing axis, and \(d\) is the distance between the drilling well and the target well. \(\theta\) is the angle between the central axis of the casing and BHA; \(H_{u}\) and \(H_{d}\), respectively, represent the upper/lower frequency magnetic field detectors. The \(d_{u}\) and \(d_{d}\) indicate the vertical distance between the upper/lower low-frequency magnetic field detectors and the casing. \(L\) is the distance between the upper current electrodes and the lower current electrodes. \(L_{ch}\) and \(L_{cl}\), respectively, represent the distance between the measuring point of the casing M and the casing wellhead and well bottom.  
The (\(X\),\(Y\),\(Z\)) coordinate system to represent the relative distance between the drilling well and the target well is established. The point Q in the center of the upper/lower electrodes serves as the coordinate origin, with the Z-axis parallel to the target well axis \(\Phi_{2}\), the X-axis perpendicular to the target well axis \(\Phi_{2}\), and the Y-axis perpendicular to the X-Z plane. The coordinate system and cross section of \(H_{u}\) and \(H_{d}\) are illustrated in Figure 1.  
It is assumed that the drilling well axis aligns with the Y-axis of the (\(X\),\(Y\),\(Z\)) coordinate system and that both the upper and lower electrodes are equidistant from their origin. Additionally, the interface potential of the (\(X\),\(0\),\(Z\)) in the formation with a uniform medium is equal, and the target well casing at this interface will not gather current. Consequently, no consideration is given to any Y-axis direction component for the drilling well excitation electrode in this model. Therefore, both the drilling well axis \(\Phi_{1}\) and the target well casing axis \(\Phi_{2}\) are arranged on a common plane represented by (\(X\),\(0\),\(Z\)) in this model.  
The coordinates of the dual-electrode in the (\(X\),\(Y\),\(Z\)) coordinate system are as follows.  
\[\left\{\begin{array}{l}P_{ux}=-L\cdot\sin(\theta)/2\\ P_{uy}=0\\ P_{uz}=L\cdot\cos(\theta)/2\\ P_{dx}=L\cdot\sin(\theta)/2\\ P_{dy}=0\\ P_{dz}=-L\cdot\cos(\theta)/2\\ \end{array}\right. \tag{1}\]  
where \(L\) represents the distance between the upper and lower electrodes closest to the end, m; \(\theta\) is the included angle between the central axis of the target well and the central axis of the drilling well, rad; and (\(P_{ux}\),\(P_{uy}\),\(P_{uz}\)) and (\(P_{dx}\),\(P_{dy}\),\(P_{dz}\)) denote the coordinate positions of the upper and lower electrodes in the (\(X\),\(Y\),\(Z\)) coordinate system, m.