## 3 Transient Flow Wave Transmission Characteristics in Tubular String
### Transient Flow Wave Amplitude Attenuation in the Tubular String  
The attenuation of the wave signal amplitude is related to the transmission distance of the signal in the tubular string and the characteristics of the transmission fluid. Most of the loss of transient flow wave transmission along the tubular string comes from the friction of the tube wall; the wave signal in the tubular string also conforms to the exponential attenuation law. According to [PERSON]'s law [34], the quantitative relationship between the transmission characteristics of the wave signal in the tube, the amplitude of signal attenuation and the transmission distance in a tube filled with a mixture of fluids is expressed as:  
\[P(x)=P_{0}e^{-x/s} \tag{26}\]  
Figure 4: The effect of gas content and diameter–thickness ratio on wave velocity.  
Figure 3: The effect of fluid temperature and pressure on wave velocity.  
Therefore, the transfer function is as follows:  
\[H=\frac{P(x)}{P_{0}}=\exp(-x/s) \tag{27}\]  
In which  
\[S=\frac{aD}{2}\sqrt{\frac{\rho}{\pi f\omega}} \tag{28}\]  
It follows from Equations (25) and (28) that  
\[S=\frac{D}{2}\sqrt{\frac{K_{l}}{\pi f\omega\left[1+\varphi\frac{K_{l}D}{Ee}+ \beta_{g}\left(\frac{K_{l}}{E_{g}}-1\right)\right]}} \tag{29}\]  
where \(P(x)\) is the pressure head at the transmission distance \(x\), Pa; \(P_{0}\) is the pressure head at the beginning, Pa.  
This paper mainly analyzed the effects of temperature, signal frequency and gas content on the signal attenuation for transient flow waves, and \(H=P(x)/P_{0}\) was used as an evaluation index of amplitude attenuation for signal waves. Three different temperatures 20 \({}^{\circ}\)C, 30 \({}^{\circ}\)C and 40 \({}^{\circ}\)C were selected for the transmission fluid (water). As can be seen from Figure 5, at a given transmission distance, the higher the temperature, the lower the signal attenuation. At the same temperature, as the transmission distance increases, the transmitted signal amplitude decreases exponentially with the transmission distance in a non-linear relationship. Because more energy is consumed the further the distance of pulse signal transmission, the received signal amplitude is relatively small.  
Three signals with different frequency 1 Hz, 2 Hz and 3 Hz were selected. From Figure 6, it can be concluded that the frequency causes a nonlinear near-exponential decrease in wave transmission amplitude with the transmission distance at the same distance. The higher the frequency, the smaller the signal amplitude.  
Figure 5: Effect of temperature on wave signal attenuation.  
Three gas contents of 0.5%, 1.0% and 1.5% were selected for analysis. As shown in Figure 7, the increase in gas content dramatically decreases wave amplitude. The main reason is that when the wave encounters the gas in the dispersed phase, scattering loss is caused by the diffuse reflection at the interface due to the great difference of gas and liquid impedance.  
Attenuation of the wave amplitude is caused by two frequency-dependent mechanisms, fluid-related and scattering. The fluid-related mechanism is caused by the wave-induced frictional movement between fluid and tubular wall, known by wave-induced fluid flow (WIFF) [35]. Scattering, which is an elastic mechanism, is caused by heterogeneities in media; for instance, the heterogeneity caused by gas content in this case [36]. The gas content causes the two mechanisms since the compressibility of gas stimulates frictional losses and increases the heterogeneity. However, several studies have shown that when the gas content exceeds 5% the effect of gas on attenuation stagnates [37].  
Figure 6: Effect of frequency on wave signal attenuation.  
Figure 7: Effect of gas content on wave signal attenuation.