## 3 MWD Positioning Method Based on Dual-Electrode Current Excitation
### Analysis of the Influence of Excitation Electrode Length on Casing Current Distribution  
The calculation method mentioned above is based on the assumption that the upper/lower electrodes function as point electrodes. In practical engineering applications, if the electrode has a certain length, the drill assembly connected to the ranging tool can be used as the upper electrode, or the drill bit combination can act as the lower electrode. In such cases, it is necessary to adjust the calculation method for determining the casing current distribution. Firstly, the electrode lengths \(l_{u}\) and \(l_{d}\) are divided into \(N\) equal parts. In order to simplify the calculation method, the current density on the electrode surface in contact with the formation is assumed to be uniform, and the current amplitude of \(N\) equal-lengthelectrodes injected into the formation is consistent. In the model, \(N\) upper/lower differential electrode combinations are established, so that the total casing current intensity is calculated as the sum of the individual current excitation of multiple differential electrodes. The coordinate values of \(N\) differential electrode combinations in the coordinate system (\(X\),\(Y\),\(Z\)) are calculated as follows:  
\[\left\{ \begin{array}{l}P_{nxx}=-[L/2+n\cdot(l_{u}/N)]\cdot\sin(\theta)\\ P_{nuz}=[L/2+n\cdot(l_{u}/N)]\cdot\cos(\theta)\\ P_{ndx}=[L/2+(N-n-1)\cdot(l_{d}/N)]\cdot\sin(\theta)\\ P_{ndz}=-[L/2+(N-n-1)\cdot(l_{d}/N)]\cdot\cos(\theta)\\ n=0:1:N-1\end{array} \right. \tag{9}\]  
where \(N\) represents the number of upper/lower electrode subdivision combinations. The distance \(L\) of the upper/lower electrodes is defined as the closest distance between the two electrodes, excluding the length of the electrodes. Furthermore, with an increase in electrode length, there is a decrease in the formation equivalent resistance (\(R_{\textit{off}}\)) and a reduction in the formation resistance between the connecting electrode and casing, leading to a decrease in the equivalent resistance (\(R_{\textit{ec}}\)) of the target well casing loop circuit. Therefore, it is essential to establish correction coefficients for \(R_{\textit{off}}\) and \(R_{\textit{ec}}\) within the calculation methods and to obtain the corrected resistance values (\(R_{\textit{off}}\) and \(R_{\textit{ec}}\)), as demonstrated below:  
\[\left\{ \begin{array}{l}R_{\textit{efc}}\approx\frac{R_{f}}{4\pi}\cdot\left( \left|\frac{L-2l_{u}}{L\cdot l_{u}}\right|+\left|\frac{L-2l_{d}}{L\cdot l_{d} }\right|\right)\\ R_{\textit{ecc}}\approx\left(\frac{R_{f}}{2\pi}\cdot\left(P_{\textit{uc}} \cdot\frac{d+L\cdot\sin(\theta)/2-2r_{d}}{(d+L\cdot\sin(\theta)/2)\cdot r_{d} }+P_{\textit{dc}}\cdot\frac{d-L\cdot\sin(\theta)/2-2r_{d}}{(d-L\cdot\sin( \theta)/2)\cdot r_{d}}\right)+L\cdot\cos(\theta)/(2\pi r_{c}h_{c}\sigma_{c}) \right)\\ P_{\textit{uc}}=(d+L\cdot\sin(\theta)/2)/(l_{u}\cdot\cos(\theta)+d+L\cdot \sin(\theta)/2)\\ P_{\textit{dc}}=(d-L\cdot\sin(\theta)/2)/(l_{u}\cdot\cos(\theta)+d-L\cdot \sin(\theta)/2)\\ \end{array} \right. \tag{10}\]  
where \(l_{u}\) represents the effective length of the upper electrode, m; \(l_{d}\) represents the effective length of the lower electrode, m; \(P_{\textit{uc}}\) is the correction coefficient for the formation equivalent resistance from the upper electrode to the casing, and \(P_{\textit{dc}}\) is the correction coefficient for the formation equivalent resistance from the lower electrode to the casing. The values \(R_{\textit{ecc}}\) and \(R_{\textit{offc}}\) from Equation (10) are substituted into Equation (3) to obtain the corrected current attenuation coefficient, \(k_{\textit{cc}}\). Finally, the differential electrode combination coordinates represented by Equation (9) are then substituted back into Equation (6).