## 3 Simulation and Analysis  
Based on the structure model of the CCS in Figure 3, the electric field can be formed between exciting and measuring electrodes through voltage excitation. Because of the complicated nature, it is difficult to establish a function to describe the relationship between sensor output and sensor parameters. Thus, finite element analysis [21; 22; 23] was used to calculate the sensor responses. For a given dielectric distribution, electrode configuration, and boundary conditions, the potential of the sensor can be calculated by solving [PERSON]'s equation:  
\[\
abla^{2}\phi=-\frac{\rho}{\epsilon}. \tag{6}\]  
For the three-dimensional electric field:  
\[\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2 }}+\frac{\partial^{2}\phi}{\partial z^{2}}=-\frac{\rho(x,y,z)}{\epsilon(x,y,z )}\quad\phi=\phi(x,y,z) \tag{7}\]  
where \(\phi\) is the space potential distribution function, \(\rho\) refers to the space charge density, \(\epsilon\) represents the space permittivity distribution function. If there is no free charge in the measurement field (\(\rho=0\)), Equation (6) can be calculated by \(\
abla^{2}\phi=0\). Thus, the capacitance values can be obtained by the finite element method. The normalized capacitance values denoted by \(\rm C_{N}\) were used to analyze the static response characteristics of the CCS, and can be calculated by  
\[\rm C_{N}=\frac{C_{w}-C}{C_{w}-C_{o}} \tag{8}\]  
where \(\rm C_{w}\) and \(\rm C_{o}\) are the capacitance response values of sensors in the pure water phase and the pure oil phase, and \(\rm C\) is the capacitance response values of sensors under different flow patterns. The response resolution of the sensor was analyzed by obtaining the curve of normalized capacitance value \(\rm C_{N}\) with the water cut. Obviously, the greater the curve slope, the better the resolution of the sensor. The employed geometry of the CCS was: \(\rm R_{0}=4\) mm, \(\rm R_{1}=5\) mm, \(\rm R_{2}=13\) mm, \(\rm R_{3}=14\) mm, \(\rm R_{4}=15\) mm, \(\rm R_{5}=19\) mm, \(L=100\) mm. The CCS was modeled and meshed by COMSOL simulation software, and the tetrahedron mesh was used to divide it. The relative dielectric constant of oil and insulating material was set as 3.5. The relative dielectric constant of water was set as 80. The relative dielectric constant of the metal material was set as 1000. The voltage of the center electrode and the metal electrode was set to 10 V, and the shell and water were grounded. Furthermore, the dependency of the CCS model on the number of the meshed grids in the finite model was investigated. Under the conditions that the CCS was in a pure oil phase, the simulation results of the CCS for different mesh numbers are listed in Table 1. As shown in Table 1, it is obvious that the capacitance values of the CCS have very small differences in different grids, which indicates the relative independence of the calculated results on the meshed grids. Considering that the calculation speed and accuracy were not affected, the number 228,374 was selected to be the optimal number of elements for CCS simulation in this paper. \begin{table}
\begin{tabular}{c c c c c} \hline
**No.** & **Total Grids** & **Maximum Unit Size (mm)** & **Minimum Unit Size (mm)** & **Capacitance Values (F)** \\ \hline
1 & 43,952 & 10 & 3.5 & 4.3235 \(\times\) 10\({}^{-10}\) \\
2 & 60,352 & 10 & 2.5 & 4.3237 \(\times\) 10\({}^{-10}\) \\
3 & 145,827 & 9 & 2 & 4.3238 \(\times\) 10\({}^{-10}\) \\
4 & 228,374 & 8 & 1.5 & 4.3237 \(\times\) 10\({}^{-10}\) \\
5 & 229,413 & 7 & 0.3 & 4.3237 \(\times\) 10\({}^{-10}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Capacitance values of the combined capacitance sensor (CCS) with different grids.