## 2 Model of the Transient Flow Wave in Water Injection Tubular String  
### Transient Flow Wave Equations  
The classical water hammer equation consists of two governing differential equations [17], one is the momentum equation, and the other is the continuity equation. Expressed in terms of pressure head \(H\) and discharge \(Q\) as:  
\[\left\{\begin{array}{l}\frac{\partial Q}{\partial t}+\frac{\partial Q}{ \partial x}\frac{\partial Q}{\partial x}+g\frac{\partial H}{\partial x}+\frac{ \lambda Q|Q|}{2 gDA^{2}}=0\\ \frac{\partial H}{\partial t}+\frac{\partial^{2}}{\partial x}\frac{\partial Q }{\partial x}+\frac{Q}{\Lambda}\sin\beta+\frac{\beta}{gA}\frac{\partial Q}{ \partial x}=0\end{array}\right. \tag{1}\]  
This equation is a typical nonlinear hyperbolic partial differential equation. In many literatures, it uses various methods to find its numerical solution [31]. However, the numerical solution alone is obviously not enough for the transmission of continuous transient flow wave signals. Furthermore, it cannot be used to describe the transmission situation of continuous waves in the water injection tubular string, nor can they be used to choose the carrier frequency used by the transmitted transient flow wave signal communication.  
In most of the engineering applications, the convective acceleration terms, \(\frac{Q}{A}\frac{\partial H}{\partial x}\) and \(\frac{Q}{A^{2}}\frac{\partial Q}{\partial x}\), are small as compared to the other terms [32]. Therefore, ignoring these terms from the governing equations, we obtain  
\[\left\{\begin{array}{l}\frac{1}{gA}\frac{\partial Q}{\partial t}+\frac{ \partial H}{\partial x}+\frac{\lambda Q|Q|}{2 gDA^{2}}=0\\ \frac{\partial H}{\partial t}+\frac{g^{2}}{gA}\frac{\partial Q}{\partial x}+ \frac{\sin\beta}{A}Q=0\end{array}\right. \tag{2}\]  
In steady-oscillatory flow, the instantaneous discharge \(Q\) and the instantaneous pressure head \(H\), may be divided into two parts: \(Q=Q_{0}+Q^{\prime}\) and \(H=H_{0}+H^{\prime}\), where \(Q_{0}\) = mean discharge; \(Q^{\prime}\) = discharge deviation from the mean; \(H_{0}\) = mean pressure head; and \(H^{\prime}\) = pressure head deviation from the mean.  
Since the mean flow and pressure head are time-invariant (\(\frac{3\partial Q_{0}}{\partial t}=0\) and \(\frac{3H_{0}}{\partial t}=0\)) and the mean flow is constant along the tubular length (\(\frac{\partial Q_{0}}{\partial x}=0\)), \(\frac{\partial Q}{\partial x}=\frac{\partial Q^{\prime}}{\partial x}\), \(\frac{\partial Q}{\partial t}=\frac{\partial Q^{\prime}}{\partial t}\), \(\frac{\partial H}{\partial t}=\frac{\partial H^{\prime}}{\partial t}\)[32]. According to Equation (2), when the pressure head is stable, there is \(\frac{\partial H_{0}}{\partial x}=-\frac{\lambda Q_{0}|Q_{0}|}{2 gD\Lambda^{2}}\).  
Because \(Q^{\prime}\ll Q_{0}\), the higher-order term \({Q^{\prime}}^{2}\) can be neglected and \(\frac{\lambda(Q_{0}+Q^{\prime})^{2}}{2 gDA^{2}}\doteq\frac{\lambda Q_{0}^{2}} {2 gDA^{2}}+\frac{\lambda Q_{0}Q^{\prime}}{8 DA^{2}}\). It follows from Equation (2) that:  
\[\left\{\begin{array}{l}\frac{\partial H^{\prime}}{\partial x}+\frac{1}{gA} \frac{\partial Q^{\prime}}{\partial t}+RQ^{\prime}=0\\ \frac{\partial H^{\prime}}{\partial t}+\frac{a^{2}}{2g}\frac{\partial Q}{ \partial x}+\frac{\sin\beta}{A}Q^{\prime}=0\end{array}\right. \tag{3}\]  
where \(R=\frac{\lambda Q_{0}}{gDA^{2}}\) for turbulent flow. If \(\lambda\) is unchanged during wave transmission, \(R\) represents the average resistance of fluctuations in the tubular string, which is the resistance of the tube when the fluid stabilizes. Equation (3) can be called the continuous transient flow wave equations in the tubular string.