## 3 Results and Discussion
### Influence of Fluid Salinity
#### 3.1.2 Influence of Borehole Fluid Salinity  
Salinity in the borehole mainly affects the fluid conductivity value. The relationship between the borehole fluid ion concentration \(N_{b}\) and the borehole fluid salinity is: \(N_{b}~{}=~{}Na~{}\times~{}10^{3}~{}\times~{}C_{b}\), where \(C_{b}\) is the borehole fluid salinity. Replacing \(N_{f}\) in Equation (17) by \(N_{b}\) and substituting it into Equation (16), we can get the conductivity of the borehole fluid \(\sigma_{b}\). Figure 4 shows the electric fields in the borehole for different \(C_{b}\) values; the formation parameters of this part are consistent with Table 1, and the pore fluid salinity is fixed at 0.0001 Mol/L. Similarly, the vertical source distance taken here is still z = 3 m.  
As shown in Figure 4, the influence of \(C_{b}\) on the electric field is different from that of the formation salinity level. When the fluid salinity \(C_{b}\) varies, the waveform of \(E_{\theta}^{b}\) hardly changes. According to Equation (9), the influence of \(C_{b}\). on the borehole electric field \(E_{\theta}^{b}\) is reflected on its influence on \(A_{H}\) and \(\eta_{e}\). Since \(\eta_{e}=\sqrt{k^{2}-k_{e}^{2}}\), the change of \(C_{b}\) directly leads to the change of \(k_{e}\) and thus influences \(\eta_{e}\). The borehole electromagnetic wave number \(k_{e}\) is given by Equation (22). \[k_{e}=\omega^{2}\mu\varepsilon_{b}\bigg{(}1+\frac{i\sigma_{b}}{\omega \varepsilon_{b}}\bigg{)} \tag{22}\]  
\(\varepsilon_{b}\) is the borehole fluid dielectric constant. Since \(\varepsilon_{b}\) and \(\mu\) are independent of salinity, the influence of \(C_{b}\) on \(k_{e}\) is only reflected on its influence on \(\sigma_{b}\). According to Equation (15) and the matrix elements given in Appendix A, the \(A_{H}\) form shown in Equation (23) can be obtained. \[A_{H}=\frac{Gn_{22}n_{33}-Gn_{23}n_{32}}{m_{21}(m_{13}m_{32}-m_{12}m_{33})+m_{ 31}(m_{23}m_{12}-m_{13}m_{22})} \tag{23}\]  
Figure 4: Electric fields for different levels of borehole fluid salinity. The solid line, dashed line, and dot-dashed line correspond to the salinity \(C_{b}=0.01\) Mol/L, 0.001 Mol/L, 0.0001 Mol/L, respectively. \(m_{ij}\) and \(n_{ij}\) are the matrix elements of \(\left[M_{ij}\right]_{3\times 3}\) and \(\left[N_{ij}\right]_{3\times 3}\) respectively, \(i,j=1,2,3\). The borehole fluid salinity \(C_{b}\) influences \(A_{H}\) only by influencing the borehole electromagnetic wave number \(k_{e}\). The numerator on the right side of Equation (23) does not include \(k_{e}\). The two matrix elements containing \(k_{e}\) in the denominator are \(m_{21}=\frac{i\omega\mu}{\eta_{e}}I_{1}(\eta_{e}a)\) and \(m_{31}=I_{0}(\eta_{e}a)\), where \(\eta_{e}=\sqrt{k^{2}-k_{e}^{2}}\). Therefore, it is only necessary to investigate the matrix elements \(m_{21}\) and \(m_{31}\) to analyze the influence of \(C_{b}\) on the electromagnetic response coefficient \(A_{H}\). Figure 5 is the schematic diagram of the first kind of zero-th order and first-order Bessel functions of imaginary argument, when the independent variable range is 0-0.2. From left to right are the absolute value of \(I_{0}(\eta_{e}a)\), the absolute value of \(I_{1}(\eta_{e}a)\), and the ratio of \(I_{1}(\eta_{e}a)\) to the independent variable. As shown in Figure 5, when the value range of the independent variable is 0-0.2, the value of \(I_{0}(\eta_{e}a)\) hardly varies with the variation of the independent variable; it is fixed at 1. Under the conditions of this article, the magnitude range of the independent variable \(\eta_{e}a\) of the Bessel function is \(10^{-4}\sim 10^{-2}\), which is in the independent variable range of Figure 5. Therefore, the value of the matrix element \(m_{31}\) does not change with the change of the independent variable \(\eta_{e}a\). The ratio of \(I_{1}(\eta_{e}a)\) to the independent variable is fixed at 0.5, that is, \(\frac{I_{1}(\eta_{e}a)}{\eta_{e}}\) in the matrix element \(m_{21}\) is a fixed value.