## 3 MWD Positioning Method Based on Dual-Electrode Current Excitation
### Calculation Method for Casing Current Distribution in the Target Well  
Assuming that the formation medium is uniform and the casing of the target well is of infinite length, the conductive properties of the casing in the formation can be determined by a formation cylinder with a radius of \(r_{e}\), and the equivalent resistance per unit length of the cylinder is the same as that of the casing, as shown in Figure 1.  
A current source is placed on the upper/lower electrodes, with the lower electrode set as positive and the upper electrode set as negative. In this configuration, the current source generates an electric field in the form of an electric dipole within a homogeneous formation [15; 16]. In accordance with the (\(X\),\(Y\),\(Z\)) coordinate system, the intensity of current density in the X-direction perpendicular to the casing is as follows [16; 17]:  
\[J_{X}=\frac{I_{s}}{4\pi}\left( \begin{array}{c}\frac{x-P_{dx}}{\left[\left(x -P_{dx}\right)^{2}+\left(y-P_{dy}\right)^{2}+\left(z-P_{dz}\right)^{2}\right]^ {\frac{2}{2}}}\\ -\frac{x-P_{xx}}{\left[\left(x-P_{xx}\right)^{2}+\left(y-P_{xy}\right)^{2}+ \left(z-P_{xx}\right)^{2}\right]^{\frac{2}{2}}}\end{array} \right) \tag{4}\]  
where \(I_{s}\) is the excitation current intensity, A. Due to the high conductivity of the casing, the current of the vertical casing in the X-direction enters the casing and flows along the casing axis. Therefore, the current density distribution along the casing axis is equivalent to the integration of the current density intensity in the X-direction along the Z-axis of the casing. The integration results from the bottom of the casing along the Z-axis [\(-L_{cl}\): \(L_{ch}\)] can be expressed as follows:  
\[J_{c}(x,y,z)=\int_{-L_{cl}}^{z}J_{x}(x,y,z)dz \tag{5}\]By substituting Formula (4) into Formula (5), the axial equivalent current density of the casing can be calculated using the following, Equation (6). The derivation of Equation (6) is shown in Appendix A. \[J_{c}(x,y,z)=\frac{I_{s}}{4\pi}\left( \begin{array}{c}\frac{(x-P_{dx})\cdot(z-P _{dz})}{\left((x-P_{dx})^{2}+(y-P_{dy})^{2}\right)\sqrt{(x-P_{dx})^{2}+\left(y- P_{dy}\right)^{2}+\left(z-P_{dz}\right)^{2}}}+\\ \frac{(x-P_{dx})\cdot(L_{td}-P_{dz})}{\left((x-P_{dx})^{2}+(y-P_{dy})^{2} \right)\sqrt{(x-P_{dz})^{2}+\left(y-P_{dy}\right)^{2}+\left(L_{td}+P_{dz} \right)^{2}}}-\\ \frac{(x-P_{dx})\cdot(z-P_{dz})}{\left((x-P_{dx})^{2}+(y-P_{dy})^{2} \right)\sqrt{(x-P_{dx})^{2}+\left(y-P_{dy}\right)^{2}+\left(z-P_{dz}\right)^{2 }}}-\\ \frac{(x-P_{xx})\cdot(L_{td}+P_{dz})}{\left((x-P_{dx})^{2}+\left(y-P_{dy} \right)^{2}\right)\sqrt{(x-P_{dx})^{2}+\left(y-P_{dy}\right)^{2}+\left(z+P_{ dz}\right)^{2}}}\end{array} \right) \tag{6}\]  
where \(L_{ch}\) represents the distance between the casing measurement point M and the well-head of the target well, m; and \(L_{cl}\) is the distance between the casing measurement point M on the casing and the bottom hole of the target well, m.  
According to the law of the conservation of charge, the current flowing into the surface of a cylinder with radius \(r_{e}\) in the X-axis direction can be equivalent to the current flowing through the cylinder on the projected cross section of the Y-Z plane. The casing current can be equivalent to the current density \(J_{c}\) in the Y-Z plane integrated in the [\(-r_{e}\): \(r_{e}\)] range. Combined with the current attenuation factor \(k_{c}\), influenced by the mud channel and formation distribution, the casing axial current \(l_{c}\) along the Z-axis can be equivalent to that expressed as Formula (7). \[l_{c}(d,z)=\int_{-r_{e}}^{r_{e}}2k_{c}\cdot J_{c}(d,y,z)dy \tag{7}\]  
If \(L_{cl}\) and \(L_{ch}\) are much longer than the distance \(d\), the simplified formula can be obtained by combining Formulas (6) and (7). \[l_{c}(d,z)\approx\frac{k_{c}\cdot I_{s}}{2\pi}\cdot\int_{-r_{e}}^{r_{e}}\left(