## 3 Transient Flow Wave Transmission Characteristics in Tubular String
### Transmission Characteristics Verification Experiment  
In order to verify the accuracy of the transient flow wave velocity and wave attenuation calculation formula, an experiment was carried out, as shown in Figures 8 and 9.  
The experimental conditions were as follows: the depth of the injection well was 1400 m with a two-layered water injection, the length of the simulated tubular string from the water distribution room (injection room) to the first layer section was 1200 m, the length of the tubular string between the first layer and the second layer was 200 m. The test conditions are shown in Table 1. The gas density could be obtained from the equation of state as \(\rho_{g}=34.43\) kg/m\({}^{3}\). The specific heat ratio of the gas was \(m=1.2\); the Poisson's ratio of the tube was 0.3; only axial restraint was applied to the tube at the well  
Figure 8: Diagram of two-layer stratified water injection test.  
Figure 9: Two-layer stratified water injection test platform.  
The procedure was as follows: Install the wave generator D1 at the beginning of the tube. It represents the transient flow wave signal at the wellhead generated by the variation of the ground valve opening. Install pressure sensor A1.2 near the wave generator D1 and record the measured pressure as \(p_{0}\). Install flowmeter C1 near the wave generator and record the measured discharge as \(q_{0}\). Similarly, install pressure sensor A3.1 near the wave generator D3 and record the measured pressure as \(p_{2}\). Install flowmeter C3 near the wave generator D3 and record the measured discharge as \(q_{2}\).  
The control valve produced a continuous 'on-off' signal with a flow rate of 30 m\({}^{3}\)/d. Stable transient flow fluctuations were generated in the pipeline and the values of \(p_{0}\), \(p_{2}\), \(q_{0}\) and \(q_{2}\) were recorded. The pressure and discharge fluctuation curves were plotted, and the test results are shown in Figures 10 and 11.  
The rise and fall attenuation time of the transient flow wave was about 1.9 s, which can be obtained experimentally. The wave transmission velocity can be obtained by Equation (25) as \(a\) = 725.27 m/s. Therefore, the transfer attenuation time was \(t=1/a\)  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Tube 1 Length (m)** & **Tube 2 Length (m)** & \
\begin{tabular}{c} **Average** \\ **Pressure (MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Average** \\ **Temperature (°C)** \\ \end{tabular} & **Gas Content (\%)** \\ \hline
1200 & 200 & 3 & 30 & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Two-layer stratified water injection test conditions.  
Figure 11: Transient pressure variation curves at upstream and downstream monitoring points.  
Figure 10: Transient discharge variation curves at upstream and downstream monitoring points.  
\(1400/725.27\doteq 1.93\) s. Compared to the wave velocity test results with the calculated results, the error was 0.69%, indicating that the formula for calculating the fluctuation velocity of transient flow is correct.  
The amplitude ratio of the monitoring point at 1400 m can be calculated by Equation (29) as \(P(x)/P_{0}=e^{(-x/s)}=e^{(-1400/5696.2)}\doteq 0.78\). The pressure values of the stable section at the upstream and downstream monitoring points were approximately 0.47 MPa and 0.58 MPa, in that order, and the ratio was \(P_{2}/P_{0}=0.47/0.58\doteq 0.81\). The error between the experimental value and the calculated value was 3.85%, which proves the correctness of the formula for calculating the attenuation of transient flow fluctuations.