## 2 Model of the Transient Flow Wave in Water Injection Tubular String
### Transient Flow Wave Transmission Model  
The field matrix for a tube may be derived by using the separation-of-variable method [17]. Elimination of \(Q^{\prime}\) from Equation (3) yields  
\[\frac{1}{a^{2}}\frac{\partial^{2}H^{\prime}}{\partial t^{2}}-\frac{g\sin\beta }{a^{2}}\frac{\partial H^{\prime}}{\partial x}+\frac{RgA}{a^{2}}\frac{\partial H ^{\prime}}{\partial t}-\frac{\partial^{2}H^{\prime}}{\partial x^{2}}=0 \tag{4}\]  
As a result, the differential equation of the pressure head about time and position is obtained for the water injection tubular string. It can be seen from Equation (4) that there is a continuous fluctuation of discharge and pressure head in any position in the tube. The solution of Equation (4) is:  
\[H^{\prime}=\big{(}c_{1}e^{\gamma_{1}x}+c_{2}e^{-\gamma_{2}x}\big{)}e^{\mathrm{ j}\omega t} \tag{5}\]  
where \(H^{\prime}=h(x)e^{\mathrm{j}\omega t}\), \(h(x)\) represents the amplitude of the pressure head at the \(x\), \(c_{1}\) and \(c_{2}\) are arbitrary constants, and \(\omega\) = angular frequency in \(\mathrm{rad}/\mathrm{s};\dot{\mathrm{\mathrm{j}}}^{2}=-1\). Equation (5) indicates that the pressure head in any position in the water injection tube can be regarded as a superposition of pressure head fluctuations in two different directions. The parameters \(\gamma_{1}\) and \(\gamma_{2}\) (as the transfer coefficients in the tubular string) can be calculated as:  
\[\gamma_{1}=\frac{\sqrt{g^{2}\sin^{2}\beta-4(a^{2}\omega^{2}-\mathrm{j}\omega Rg Aa ^{2})}-g\sin\beta}{2a^{2}} \tag{6}\]  
\[\gamma_{2}=\frac{\sqrt{g^{2}\sin^{2}\beta-4(a^{2}\omega^{2}-\mathrm{j}\omega Rg Aa ^{2})}+g\sin\beta}{2a^{2}} \tag{7}\]  
The real part of \(\gamma_{1}\) and \(\gamma_{2}\) represents the attenuation of the amplitude during transmission, and the imaginary part represents the change of phase. \(g\sin\beta\) represents the effect of gravity on the transient flow wave. For a vertical downhole string, there is \(\sin\beta=1\). Derivative of Equation (5) for \(t\):  
\[\frac{\partial H^{\prime}}{\partial t}=\mathrm{j}\omega e^{\mathrm{j}\omega t }\big{(}c_{1}e^{\gamma_{1}x}+c_{2}e^{-\gamma_{2}x}\big{)} \tag{8}\]  
Substituting Equation (8) into the second formula of Equation (3) can obtain:  
\[\frac{\partial Q^{\prime}}{\partial x}+\frac{g\sin\beta}{c^{2}}Q^{\prime}=- \frac{\mathrm{j}\omega gA}{c^{2}}\big{(}c_{1}e^{\gamma_{1}x}+c_{2}e^{-\gamma_{2 }x}\big{)}e^{\mathrm{j}\omega t}\]Thus \(Q^{\prime}\) can be calculated:  
\[Q^{\prime}=\frac{\omega gA}{ja^{2}}\left(\frac{c_{1}e^{\gamma_{1}x}}{\gamma_{2}}- \frac{c_{2}e^{-\gamma_{2}x}}{\gamma_{1}}\right)e^{j\omega t} \tag{9}\]  
Equation (9) shows that there is also a fluctuation of discharge in any position in the tubular string. In which \(Q^{\prime}=q(x)e^{j\omega t}\), \(q(x)\) represents the amplitude of the pressure head at the \(x\). In the tubular string, fluctuations in pressure head cause fluctuations in discharge. The hydraulic impedance \(Z(x)\) in a water injection system is defined as the ratio of the complex head to the complex discharge at a particular point in the tube. \[Z(x)=\frac{h(x)}{q(x)}=-\frac{a^{2}\gamma_{1}\gamma_{2}}{j\omega gA}\frac{c_{1 }e^{\gamma_{1}x}+c_{2}e^{-\gamma_{2}x}}{(\gamma_{1}c_{1}e^{\gamma_{1}x}-\gamma _{2}c_{2}e^{-\gamma_{2}x})} \tag{10}\]  
The impedance of an infinitely long tubular string can be defined as the characteristic impedance, and the characteristic impedance can be written [17]:  
\[Z_{c}=\lim_{x\to 0}-\frac{a^{2}\gamma_{1}\gamma_{2}}{j\omega gA}\frac{(c_{1}e^{ \gamma_{1}x}+c_{2}e^{-\gamma_{2}x})}{(\gamma_{1}c_{1}e^{\gamma_{1}x}-\gamma_{ 2}c_{2}e^{-\gamma_{2}x})}=-\frac{a^{2}\gamma_{2}}{j\omega gA} \tag{11}\]  
Equation (11) shows that the characteristic impedance is constant for a fixed frequency signal and represents the inherent influence of the tube on the transient flow wave signal at a certain frequency. For an infinitely long tubular string, the fluctuations are transmitted in only one direction of the wave, without the presence of reflected waves.