## 4 Examples of Numerical Simulation Calculations  
The geometric model and computational grid of the plunger lift model are shown in Figure 4. The dimensions of the plunger and tubing are based on the experimental setup. The distance from the inlet to the plunger is 70 mm, and the distance from the outlet to the plunger is 150 mm. Considering the complexity of the flow field around the plunger, a Poly-HexCore mesh is used in this study. The Poly-HexCore mesh is an advanced mesh type designed to provide flexibility and efficiency compared to the traditional hexahedral core mesh. This meshing technique combines the advantages of polyhedral mesh and hexa-core mesh, thus optimizing mesh quality and computational efficiency, especially for flow problems with complex structures or large size spans. After verifying mesh independence,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Plunger Type**} & \multirow{2}{*}{**Groups**} & **Flowmeter** & **Bottomhole** & **Wellhead** & **Air Flow** \\  & & **Pressure** & **Pressure** & **Pressure** & **Rate** \\  & & **(KPa)** & **(KPa)** & **(KPa)** & **(m\({}^{3}\)/h)** \\ \hline \multirow{8}{*}{Rectangular slot} & 1 & 163 & 145 & 131 & 77.19 \\  & 2 & 184 & 164 & 149 & 79.48 \\  & 3  
a physical model with a mesh size of 6 million cells is used to improve computational efficiency while ensuring accuracy. The physical model is shown in Figure 5.  
In this study, the boundary conditions were determined based on the experimental conditions. The fourth set of experimental data was randomly selected for data assimilation, while the remaining sets were used to validate the assimilation results. The SIMPLE algorithm was adopted for pressure-velocity coupling, and the convective terms were discretized using the first-order upwind scheme. The simulation was considered converged when either the inlet flow rate remained unchanged for more than 2000 iterations or the maximum residual of all monitored variables dropped below \(1\times 10^{-5}\). The detailed settings are provided in Table 3.  
DA using the EnKF method requires the configuration of several key hyperparameters as follows: the number of ensemble samples, the range of adjustment coefficients, and the variance of specific Gaussian noise. Properly setting these hyperparameters is critical, as inappropriate values may lead to issues such as excessive computational costs, significant deviations of calibration results from experimental values, and instability in data assimilation outcomes. Therefore, hyperparameters must be carefully selected to ensure the accuracy and stability of the data assimilation process. In this study, the optimized Latin Hypercube Sampling (OLHS) method was employed to generate random samples of four turbulence model constants in the RNG k-\(\epsilon\) model. The sampling range for each parameter was set to \(\pm 20\%\) of its nominal value, specifically \(0.0676\leq C_{u}\leq 0.1014\), \(1.136\leq C_{1\epsilon}\leq 1.704\), \(1.344\leq C_{2\epsilon}\leq 2.016\), \(0.68\leq P_{rt}\leq 1.02\). By evenly dividing each parameter range and optimizing the sample distribution, a total of 100 representative turbulence model sample sets were constructed. As illustrated in Figure 6, the optimized samples were evenly distributed across the high-dimensional parameter space, effectively avoiding clustering and ensuring both the uniformity and representativeness of the sample set. Figure 5: Plunger Modeling Model. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Turbulence** & **Entry** & **Export Boundary** & **Flow** & **Temp** & **Work** & **Wall Function** \\
**Model** & **Boundary** & **Export Boundary** & **Medium** & **Medium** & **Pressure** & **Wall Function** \\ \hline RNG k-\(\epsilon\) & Pressure inlet & Pressure outlet & Air & 15 & \
\begin{tabular}{c} Standard \\ atmosphere \\ \end{tabular} &
\begin{tabular}{c} Menter- \\ Lechner \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model and boundary settings.