# Each-Phase Metering with Gas-Liquid Stratified Flow Based on the Multi-Frequency Coriolis Principle
## 3 Methodology  
In this section, the Finite Element Analysis (FEA) method is used to simulate measurement characteristics of single-frequency Coriolis flowmeters for the gas-containing fluids. In the stratified flow and bubbly flow with low gas content (GVF less than 10%), the influence of gas content and flow patterns on the vibration frequencies of each mode of the measuring tube is analyzed. Then the effect of different working conditions on the density and mass flow is fully understood.  
In order to simulate actual conditions and observe flow patterns in gas-containing fluids, the DN25 micro-bend measuring tube is connected with a long straight tube with a length of 10D in the front and a short tube with a length of 2D to the back. The test device is shown in Figure 2, wherein the micro-bend pipe refers to the size of Promass F single-frequency Coriolis flowmeter by E+H company [3].  
According to the wellhead production data of Shengli Oilfield China, shown in Table 1, temperature and pressure are 20-60 \({}^{\circ}\)C and 4-25 bar respectively, liquid flowrate is smaller than 100 t/d.  
According to the Mandhane flow pattern map [21], the superficial cross-section velocities of gas and liquid phases are taken as inlet conditions in simulation. The stratified and bubble flow are set under terms of temperature and pressure (i.e., 20 \({}^{\circ}\)C and 6 bar). The setting conditions of numerical simulation are shown in Table 2. The gas and liquid phases enter from the long straight pipe with different cross-section superficial velocities and flow through the straight pipe to the micro-bend measuring pipe.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & \(\rho\) **(kg/m\({}^{3}\))** & \(f_{1}\) **(Hz)** & **Viscosity (P)** & **t \({}^{\circ}\)C** & **P (bar)** & **Liquid** \\  & & & & & & **Flowrate (t/d)** \\ \hline Water & 998.2 & 342.0 & 0.001003 & 20–60 & 4–25 & \textless{}100 \\ Methane & 4.3084 & 497.6 & \(1.087\times 10^{-5}\) & 20–60 & 4–25 & / \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physical parameters of pure water and gas.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & \(\dot{m}_{0}\) **(kg/s)** & \(\dot{m}_{sl}\) **(kg/s)** & \(\dot{m}_{sq}\) **(10\({}^{-5}\) kg/s)** & **Flow Patterns** \\ \hline
1 & 0.0910 & 0.0904–0.0822 & 0.3940–3.941 & Stratified \\
2 & 0.1370 & 0.1356–0.1232 & 0.5910–5.911 & Stratified \\
3 & 0.2283 & 0.2279–0.2278 & 0.9950–10.939 & Stratified \\
4 & 0.3420 & 0.3389–0.3081 & 1.478–14.78 & Stratified \\
5 & 2.281 & 2.280–2.278 & 9.949–109.4 & Bubble \\ \hline \hline \end{tabular}
\end{table}
Table 2: Inlet setting ranges of gas-liquid two-phase flow simulation.  
Figure 2: The geometric model of numerical simulation.  
In stratified flow and bubble flow, the influences of different working conditions on the frequencies and phase difference output deviations of the vibration response of the measuring tube are analyzed to obtain the deviations of the actual density and mass flow with a single-frequency Coriolis flowmeter. Therefore, the measured fluid density and mass flow (Equations (10) and (11)) measured by a single-frequency Coriolis flowmeter are obtained by numerical simulation. The density coefficients can be determined by fitting densities and the corresponding frequencies; the mass flow apparent values of the measured fluid can be obtained by using the output value of the time difference generated by deformation of measuring tube, and \(K_{f}\) can be determined by fitting the time differences and the corresponding mass flow values for water.  
\[\rho_{\text{app1}}=-885.90+2.204\times 10^{8}\frac{1}{{f_{1}}^{2}} \tag{10}\]  
\[\dot{m}_{app}=K_{f}\cdot\Delta t=1212.77\Delta t \tag{11}\]  
where \(f_{1}\) is the first-order vibration frequency of measuring tube obtained by modal analysis considering Fluid-Structure Interactions.