## 5 Results and Discussion
### Measurement Reproducibility  
To verify the stability of the IM-WIDIM-DPIV algorithm, the oil-water two-phase immiscible flow in the inner diameter 125 mm vertical upward pipe was repeatedly measured under different water cut and flow velocity of \(9.4\times 10^{-3}\) m/s - \(51.9\times 10^{-3}\) m/s with intergroup interval of \(4.7\times 10^{-3}\) m/s. The measurement results are shown in Figure 11 and Table 4.  
Figure 11: Repeated measurement results at \(51.9\times 10^{-3}\) m/s-80%.  
As shown in Figure 11, the measured curve is effectively consistent with the theory curve, despite fluctuating slightly. The calculated maximum fluctuation was \(2.93\times 10^{-3}\) m/s, with a relative deviation of \(5.64\%\), standard deviation of \(1.50\times 10^{-3}\) m/s, and average value of \(51.74\times 10^{-3}\) m/s. This is close to the theoretical value of \(51.90\times 10^{-3}\) m/s, so the measured results have good stability.  
Table 4 shows that the overall average maximum deviation of repeated measurements under various working conditions was \(5.46\times 10^{-3}\) m/s; the highest was \(10.13\times 10^{-3}\) m/s and the lowest was \(2.12\times 10^{-3}\) m/s. The average reproducibility was \(1.98\%\); the highest was \(5.81\%\) and the lowest was \(0.74\%\), all of which did not exceed \(10\%\). When the water cut was \(70\%\), the average maximum deviation and average reproducibility were the minimum. They were \(4.85\times 10^{-3}\) m/s and \(1.26\%\), respectively. Therefore, the IM-WIDIM-DPIV algorithm has good reproducibility and stability.  
Figure 12 shows the velocity distribution of oil-water two-phase immiscible flow in the inner diameter \(125\) mm vertical upward pipe under different working conditions obtained by the IM-WIDIM-DPIV algorithm.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Water Cut 90\%**} & \multicolumn{2}{c}{**Water Cut 80\%**} & \multicolumn{2}{c}{**Water Cut 70\%**} \\ \hline
**Flow** & **Max-** & **Repeatability** & **Max-** & **Repeatability** & **Max-** & **Deviation** & **Repeatability** \\
**Velocity** & **Deviation** & **(\%)** & **(10\({}^{-3}\) m/s)** & **(10\({}^{-3}\) m/s)** & **(\%)** & **(10\({}^{-3}\) m/s)** & **(\%)** \\ \hline
9.4 & 4.64 & 1.51 & 2.79 & 3.02 & 3.87 & 1.89 \\
14.1 & 2.19 & 1.03 & 5.72 & 2.78 & 7.36 & 1.26 \\
18.9 & 8.50 & 5.81 & 8.44 & 2.98 & 8.13 & 1.91 \\
23.6 & 5.35 & 5.07 & 6.14 & 1.15 & 4.27 & 1.32 \\
28.3 & 10.13 & 4.34 & 4.61 & 0.91 & 4.33 & 0.83 \\
33.0 & 3.63 & 2.77 & 5.13 & 1.67 & 6.53 & 0.74 \\
37.7 & 9.54 & 2.61 & 5.49 & 0.97 & 2.12 & 0.88 \\
42.4 & 7.42 & 1.65 & 3.35 & 1.94 & 3.24 & 1.04 \\
47.2 & 9.48 & 1.57 & 4.76 & 1.19 & 6.24 & 1.93 \\
51.9 & 5.86 & 1.08 & 2.22 & 2.93 & 2.44 & 0.76 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Repeated velocity measurement results.  
Figure 12: _Cont._  
Figure 12: Cont.  
Figure 12 shows that the axial velocity of the flow in the pipe is clearly greater than the radial velocity, so the fluid flows upward as a whole, and the average velocity at the center of the vertical pipe is significantly higher than that near the boundary; this is consistent with the actual bubble flow in vertical wells [40,41].