## 3 Transient Flow Wave Transmission Characteristics in Tubular String  
### Transient Flow Wave Velocity in Tubular String  
The wave velocity is related to the components of the fluid, the tube characteristic and the environmental parameters [13]. The wave velocity has been estimated from [PERSON]'s formula [33]; the expression of wave velocity can be derived from the continuity equation as well as the equation of motion.  
\[a=\sqrt{\frac{K_{e}}{\rho}} \tag{21}\]  
From the above theoretical study, it is clear that the main parameter that now determines the wave velocity is the apparent bulk modulus of elasticity of the system \(K_{e}\).  
\[\frac{1}{K_{e}}=\frac{1}{K_{l}}\left[1+\psi\frac{K_{l}d}{Ee}+\beta_{g}\left( \frac{K_{l}}{K_{g}}-1\right)\right] \tag{22}\]  
Based on the total volume of the mixed fluid, the formula for the density of the transport fluid can be obtained. The solid content in the tubular string of the water injection well is negligible, and the density formula of the mixed fluid is as follows:  
\[\rho=\left(1-\beta_{g}\right)\rho_{l}+\beta_{g}\rho_{g} \tag{23}\]  
The gas density is determined from the equation of state as:  
\[\rho_{g}=\frac{P}{ZRT} \tag{24}\]  
where \(P\) is the absolute pressure, Pa; \(Z\) is the compression coefficient of the actual gas; \(T\) is the temperature, \(K\); and \(R\) is the gas constant, \(R\doteq 287.4\,\,\mathrm{N}\cdot\mathrm{m}/\mathrm{K_{g}}\cdot\mathrm{K}\).  
The transient flow wave velocity in the tubular string is as follows:  
\[a=\sqrt{\frac{K_{e}}{\rho}}=\sqrt{\frac{K_{l}/\rho}{1+\psi\frac{K_{l}d}{Ee}+ \beta_{g}\left(\frac{K_{l}}{K_{g}}-1\right)}}=\sqrt{\frac{\frac{K_{l}}{\left( 1-\beta_{g}\right)\rho_{l}+\beta_{g}\rho_{g}}}{1+\psi\frac{K_{l}d}{Ee}+\beta_ {g}\left(\frac{K_{l}}{K_{g}}-1\right)}} \tag{25}\]  
MATLAB software was used to analyze the relationship between the effects of different parameters on the signal of transient flow wave. Taking data from a water injection well in Daqing Oilfield as an example, the depth of a water injection well is 1400 m, the inner diameter of injection tube \(d\) is 25 mm, the wall thickness \(e\) is 3.5 mm, the average pressure in the injection well is 3 MPa, the average temperature is \(t=30\,\,^{\circ}\mathrm{C}\) and the gas content is \(\beta_{g}=0.5\%\). The modulus of elasticity of the tube is \(E=2.1\times 10^{5}\) MPa and the modulus of elasticity of water is \(K_{l}=2.04\times 10^{3}\) MPa. The density of water is \(\rho=997.87\,\,\mathrm{kg}/\mathrm{m^{3}}\). The specific heat ratio of the gas is \(m=1.2\), \(K_{g}=mP=1.2\,\times 30=36\) MPa; the Poisson's ratio of the tube is 0.3. Only axial restraint is applied to the tube at the wellhead, \(\psi=1.1096\). The liquid density \(\rho_{l}\) and the gas density \(\rho_{g}\) depend on the pressure and temperature in the tubular string.  
As shown in Figure 3, the pressure in the well has a large impact on the transmission rate. The higher the pressure, the greater the transmission rate. At constant pressure, the density of water and gas decreases with the increasing temperature, which in turn leads to a decrease in the density of the mixed water, resulting in an increase in wave velocity. The gas content has a significant impact on the compressibility of the water. A large gas content will have a large attenuation pulse amplitude. As shown in Figure 4, the wave velocity tends to decrease as the gas content increases. The diameter-thickness ratio only affects the characteristics and support of the tube. As the diameter-thickness ratio increases, the wave velocity decreases.