# The Electromagnetic Field Analytical Solution Analysis to Downhole Current Injection [PERSON]
## 2 Signal Model and Electromagnetic Field Distribution Analysis
### The Electromagnetic Field Analysis  
Recalling the two-layer model established in 2.1, the region where the point source is located is defined as the first layer while the target well is defined as the second layer. Therefore, \(K_{0}(0)\rightarrow\infty\) when \(\rho\to 0\) and \(I_{0}(\infty)\rightarrow\infty\) when \(\rho\rightarrow\infty\). The results indicate that \(A_{1}=0\) and \(B_{2}=0\) can be obtained. As a result, the electric potential of the target well (located at the passive region) can be represented as follows:  
\[V_{t}=\frac{I}{4\pi^{2}\sigma}\int_{-\infty}^{+\infty}K_{0}(\lambda\rho_{0})A _{2}I_{0}(\lambda\rho)e^{i\lambda(z-z_{0})}\;d\lambda. \tag{17}\]  
According to the gradient relationship between the electric potential and the electric field, the electric field located at the target well is given by the following:  
\[E_{t}=-\
abla V_{t}=\frac{-I}{4\pi^{2}\sigma_{1}}\int_{-\infty}^{+\infty}K_{0 }(\lambda\rho_{0})I_{0}(\lambda\rho)\lambda e^{i\lambda(z-z_{0})}\;d\lambda, \tag{18}\]  
where \(\
abla\) represents the Del operator. The Del operator is a differential operator with directional properties [34]. Correspondingly, the current converging on target well can be expressed as follows:  
\[I_{t}=\pi{r_{e}}^{2}\sigma_{1}E_{t}=\frac{-{r_{e}}^{2}I}{4\pi}\int_{-\infty}^{ +\infty}K_{0}(\lambda\rho_{0})I_{0}(\lambda\rho)\lambda e^{i\lambda(z-z_{0})} \;d\lambda, \tag{19}\]  
where \(r_{e}\) denotes the equivalent radius after transforming the target well into a cylinder with the same conductivity as the stratum. By utilizing the boundary condition, the parameter \(A_{2}\) can be calculated as follows:  
\[A_{2}=\frac{1}{\lambda{r_{e}}\Big{[}\frac{\sigma_{2}}{\sigma_{1}}I_{1}( \lambda{r_{c}})K_{0}(\lambda{r_{c}})+I_{0}(\lambda{r_{c}})K_{1}(\lambda{r_{c }})\Big{]}}. \tag{20}\]The derivation of (20) is shown in Appendix A. After substituting \(A_{2}\) into Equations (17) and (18), the electric potential and the electric field along the axial direction on the target well can be expressed as follows:  
\[V_{t}(z)=\frac{I}{4\pi^{2}\sigma_{1}r_{c}}\int_{-\infty}^{+\infty}\frac{K_{0}( \lambda\rho_{0})I_{0}(\lambda r_{c})}{\lambda\left[\frac{\sigma_{2}}{\sigma_{1 }}I_{1}(\lambda r_{c})K_{0}(\lambda r_{c})+I_{0}(\lambda r_{c})K_{1}(\lambda r _{c})\right]}e^{i\lambda(z-z_{0})}\;d\lambda, \tag{21}\]  
\[E_{t}(z)=\frac{I}{4\pi^{2}\sigma_{1}r_{c}}\int_{-\infty}^{+\infty}\frac{K_{0}( \lambda\rho_{0})I_{0}(\lambda r_{c})\sin\lambda(z-z_{0})}{\left[\frac{\sigma_ {2}}{\sigma_{1}}I_{1}(\lambda r_{c})K_{0}(\lambda r_{c})+I_{0}(\lambda r_{c}) K_{1}(\lambda r_{c})\right]}d\lambda, \tag{22}\]  
\[I(z)=\pi r_{c}{}^{2}E(z)\sigma_{1}=\frac{I\sigma_{2}r_{c}}{4\pi\sigma_{1}} \int_{-\infty}^{+\infty}\frac{K_{0}(\lambda\rho_{0})I_{0}(\lambda r_{c})\sin \lambda(z-z_{0})}{\left[\frac{\sigma_{2}}{\sigma_{1}}I_{1}(\lambda r_{c})K_{ 0}(\lambda r_{c})+I_{0}(\lambda r_{c})K_{1}(\lambda r_{c})\right]}d\lambda. \tag{23}\]  
Assuming that the length of the target well is \(L\), then the magnetic field at a point \((\rho_{r},\varphi_{r},z_{r})\) around the target well can be expressed as follows:  
\[ \begin{array}{rl}H(z_{r})&=\frac{I(z_{r})}{4\pi\rho_{r}}(cos \theta_{1}-cos\theta_{2})\\ &=\frac{I\sigma_{2}r_{c}(cos\theta_{1}-cos\theta_{2})}{16\pi^{2} \sigma_{1}\rho_{r}}\int_{-\infty}^{+\infty}\frac{K_{0}(\lambda\rho_{0})I_{0}( \lambda r_{c})sin\lambda(z-z_{0})}{\left[\frac{\sigma_{2}}{\sigma_{1}}I_{1}( \lambda r_{c})K_{0}(\lambda r_{c})+I_{0}(\lambda r_{c})K_{1}(\lambda r_{c}) \right]}d\lambda,\end{array}  \tag{24}\]  
where \(cos\theta_{1}=\frac{L+z_{r}}{\sqrt{\rho^{2}+\left(L+z_{r}\right)^{2}}}\) and \(cos\theta_{2}=\frac{z_{r}-L}{\sqrt{\rho^{2}+\left(L-z_{r}\right)^{2}}}\). Generally speaking, the length of the target well usually reaches several kilometers and can be regarded as infinitely long.