## 2 Model of the Transient Flow Wave in Water Injection Tubular String
### Transient Flow Wave Transfer Matrix in a Water Injection Tubular String  
The transfer matrix method is used for the analysis of steady oscillatory flows and the determination of the frequency response of hydraulic systems [17]. According to Equations (13) and (14), the transfer matrix of transient flow wave fluctuations in the tubular string can be defined, and the fluctuations of pressure head \(h_{i}\) and \(q_{i}\) at the beginning are used to define the fluctuations of pressure head \(h_{x}\) and discharge \(q_{x}\) at position \(x\) in the tubular string:  
\[\begin{bmatrix}h_{x}\\ q_{x}\end{bmatrix}=\begin{bmatrix}A(x)&B(x)\\ C(x)&D(x)\end{bmatrix}\begin{bmatrix}h_{i}\\ q_{i}\end{bmatrix} \tag{15}\]where  
\[A(x) =\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}e^{\gamma_{1}x}+\frac{\gamma_ {1}}{\gamma_{1}+\gamma_{2}}e^{-\gamma_{2}x},\;B(x)=Z_{c}\frac{\gamma_{1}}{ \gamma_{1}+\gamma_{2}}(e^{\gamma_{1}x}-e^{-\gamma_{2}x}) \tag{16}\] \[C(x) =\frac{1}{Z_{c}}\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}\big{(}e^ {\gamma_{1}x}-e^{-\gamma_{2}x}\big{)},\;D(x)=\frac{\gamma_{1}}{\gamma_{1}+ \gamma_{2}}e^{\gamma_{1}x}+\frac{\gamma_{2}}{\gamma_{1}+\gamma_{2}}e^{-\gamma _{2}x} \tag{17}\]  
Equation (15) illustrates that matrix \(\mathbf{M}_{l}=\begin{bmatrix}A(x)&B(x)\\ C(x)&D(x)\end{bmatrix}\) can be defined as the transfer matrix of a single tubular string. From the definition of \(\mathbf{M}_{l}\), it is known that the determinant value is \(|\mathbf{M}_{l}|=e^{(\gamma_{1}-\gamma_{2})x}\
eq 0\). So \(\mathbf{M}_{l}\) must be reversible, indicating that if the end pressure head amplitude \(h_{l}\), the discharge amplitude \(q_{l}\) and length \(l\) of a tubular string are known, the pressure head amplitude and discharge amplitude at any position can be calculated. The relationship among inputs \(h_{i}\) and \(q_{i}\) at the beginning and outputs of the tubular string, \(h_{i}\) and \(q_{i}\) at the end can be represented by a four-port model shown in Figure 1.  
A junction of two tubes that have different diameters, wall thicknesses, wall materials or any combination of these variables is called a series junction. It follows from the continuity equation with \(q_{l}=q_{i}\) and \(h_{l}=h_{i}\). The point transfer matrix for a series junction can be expressed as:  
\[\mathbf{M}_{D}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix} \tag{18}\]  
Since Equation (18) is a unit matrix, for a series connection of multiple tubes, the transfer matrix of the whole tubular string can be expressed as:  
\[\mathbf{M}^{*}=\mathbf{M}_{l1}\mathbf{M}_{l2}\dots\mathbf{M}_{ln} \tag{19}\]  
The relationship between these series tubes can be represented by the multiple series four-port model shown in Figure 2.  
When the continuous transient flow wave is transmitted in a series of tubes, according to the model shown in Figure 2 and Equation (19), the transmission matrix when multiple tubes are connected in series can be simplified to an equivalent transmission matrix \(\mathbf{M}^{*}\) according to the matrix multiplication method, so that the transmission process can be expressed as:  
\[\begin{bmatrix}p_{l}\\ q_{l}\end{bmatrix}=\mathbf{M}^{*}\begin{bmatrix}p_{i}\\ q_{i}\end{bmatrix}=\begin{bmatrix}A^{*}&B^{*}\\ C^{*}&D^{*}\end{bmatrix}\begin{bmatrix}p_{i}\\ q_{i}\end{bmatrix} \tag{20}\]  
Figure 1: Single pipe four-port equivalence model.  
Figure 2: Four-port equivalent model when multiple tubes are connected in series.  
During the water injection process, if the downhole distributor is considered as the signal source, the transmission of the transient flow wave signal in the tubular string and the amplitude reaching the wellhead can be calculated by Equation (20).