## 2 Measuring Principle
### Measurement Principle of the Combined Capacitance Sensor  
The designed CCS in this paper is a combination of the cylindrical capacitance sensor and the coaxial capacitance sensor. As shown in Figure 3, the center electrode of the coaxial capacitance sensor is placed at the center of the cylindrical capacitance sensor, which is in parallel with the metal layer of the cylindrical capacitance sensor. Both of them are electrodes. The shell of the sensor is grounded and the fluid flows in the annular region. For the CCS, the cylindrical capacitance sensor and the coaxial capacitance sensor act together and compensate each other, which increases the response range of the sensor in the state of stratified flow. The effective contact area between the oil bubble and the sensor is increased and the adhesion rate of the oil bubble is increased under the condition of non-stratified  
Figure 2: Structure diagram of the cylindrical capacitance sensor. flow, thus improving the resolution of the sensor. In Figure 3, R\({}_{0}\), R\({}_{1}\), R\({}_{2}\), R\({}_{3}\), R\({}_{4}\), and R\({}_{5}\) are center electrode radius, insulating layer radius, flow runner radius, inner insulating layer radius, metal electrode radius, and outer insulating layer radius, respectively. \(L\) is the length of the sensor. \(\alpha\), \(\beta\) is the center angle of the circle with the radius of R\({}_{1}\), R\({}_{2}\) and oil-water interface, respectively. When the fluid is in the oil-water two-phase segregated flow, the total capacitance of the CCS is made up of five-part capacitors in parallel. The capacitance formed by the center electrode, metal electrode, and outer insulating layer is recorded as C0. The capacitance formed by the central electrode, the inner insulating layer, and the water in contact with the inner insulating layer is recorded as C1. The capacitance produced by the edge effect of C1 near the interface is denoted by C2. The capacitance formed by the central electrode, insulating layer, and water in contact with the insulating layer is recorded as C3. The capacitance produced by the edge effect of C3 near the interface is denoted by C4. \[\mathrm{C0}=\frac{2\pi\epsilon\,\varepsilon_{0}\mathrm{L}}{\mathrm{ln}( \mathrm{R_{5}}/\mathrm{R_{4}})} \tag{1}\]  
\[\mathrm{C1}=\frac{\beta\epsilon\,\varepsilon_{0}\mathrm{L}}{\mathrm{ln}( \mathrm{R_{3}}/\mathrm{R_{2}})} \tag{2}\]  
\[\mathrm{C2}=\frac{4\beta^{\prime}\,\epsilon\,\varepsilon_{0}\mathrm{L}}{ \mathrm{ln}(\mathrm{R^{\prime}}/\mathrm{R_{0}})}(\mathrm{Y_{w}}-\mathrm{Y_{w}}^ {2}) \tag{3}\]  
\[\mathrm{C3}= \begin{cases}0&\beta\leq 2\mathrm{arccos}(\mathrm{R_{1}}/ \mathrm{R_{2}})\\ \frac{\alpha\,\varepsilon\,\varepsilon_{0}\mathrm{L}}{\mathrm{ln}(\mathrm{R_{ 1}}/\mathrm{R_{0}})}&2\mathrm{arccos}(\mathrm{R_{1}}/\mathrm{R_{2}})<\beta<2 \pi-2\mathrm{arccos}(\mathrm{R_{1}}/\mathrm{R_{2}})\\ \frac{2\pi\epsilon\,\varepsilon_{0}\mathrm{L}}{\mathrm{ln}(\mathrm{R_{1}}/ \mathrm{R_{0}})}&\beta\geq 2\pi-2\mathrm{arccos}(\mathrm{R_{1}}/\mathrm{R_{2}}) \end{cases}  \tag{4}\]  
\[\mathrm{C4}= \begin{cases}\frac{\beta\,\varepsilon\,\varepsilon_{0}\mathrm{L}}{ \mathrm{ln}(\mathrm{R^{\prime}}/\mathrm{R_{0}})}&\beta\leq 2\mathrm{arccos}( \mathrm{R_{1}}/\mathrm{R_{2}})\\ \frac{\beta\,\varepsilon\,\varepsilon_{0}\mathrm{L}(1-\alpha/2\pi)}{\mathrm{ln }(\mathrm{R^{\prime}}/\mathrm{R_{0}})}&2\mathrm{arccos}(\mathrm{R_{1}}/ \mathrm{R_{2}})<\beta<2\pi-2\mathrm{arccos}(\mathrm{R_{1}}/\mathrm{R_{2}})\\ \frac{\beta\,\varepsilon\,\varepsilon_{0}\mathrm{L}(2\pi-\alpha)}{2\,\mathrm{Tr }(\mathrm{R^{\prime}}/\mathrm{R_{0}})}&\beta\geq 2\pi-2\mathrm{arccos}(\mathrm{R_{1}}/ \mathrm{R_{2}})\end{cases}  \tag{5}\]  
where, \(\varepsilon\) is the dielectric constant of the vacuum state, and \(\varepsilon_{0}\) is the dielectric constant of the insulating layer, the inner and outer insulating layer, and oil. \(\mathrm{Y_{w}}\) is the water cut of fluid. \(\mathrm{R^{\prime}}=(\mathrm{R_{1}}+\mathrm{R_{2}})/2\). \(\mathrm{R^{\prime\prime}}=[\mathrm{R_{2}}+\mathrm{R_{2}}\cos(\beta/2)]/2\).