## 4 Simulation of Transient Flow Wave Signals in Tubular Strings  
### Transient Flow Wave Signal Distribution along the Tubular String  
During the water injection process, the modulated sinusoidal transient flow wave signals need to be transmitted from the well to the ground. The main concern in this process is the attenuation of the signal; the pressure fluctuation amplitude at the transmitting start point is \(p_{i}\) and the pressure fluctuation amplitude at the receiving point is \(p_{l}\). The larger the \(p_{l}/p_{i}\) ratio, the stronger the received signal, and hence, the better signal transmission effect. The ratio of the head amplitude \(h_{l}/h_{i}\) and the ratio of the pressure amplitude \(p_{l}/p_{i}\) at the two ends of the tubular string are equal. If defining \(Z_{l}=h_{l}/q_{l}\) as a terminal impedance, according to Equations (16) and (17) it can be deduced:  
\[\frac{h_{l}}{h_{i}}=\frac{p_{l}}{p_{i}}=\frac{A(l)D(l)-B(l)C(l)}{D(l)-B(l) \frac{1}{Z_{l}}}=\frac{e^{\gamma_{l}l-\gamma_{2}l}}{D(l)-B(l)\frac{1}{Z_{l}}} \tag{30}\]  
Equation (30) represents the value of amplitude transmission losses in the wave signal transmission. It can also be seen that the amplitude of the transient flow wave in the tubular string is influenced not only by the initial pressure head \(h_{i}\), but also by the terminal impedance \(Z_{l}\). For a fixed frequency signal, it is assumed that the friction force is constant throughout the wave signal transmission process. The effect of the change in density of the gas content of the injection well fluid on the wave velocity is neglected. The effect of gas spillage on the signal is ignored, while the tubular string is considered rigid and its expansion and deformation during the transient flow wave transmission is ignored. Using the transfer matrix method derived earlier, the transfer process of the downhole continuous wave signal in the tubular string can be calculated. According to the previous analysis, if the amplitude and frequency of pressure and discharge fluctuations at the begin of fluctuating signal are known, the amplitude of pressure and discharge fluctuations at any point in the tubular string can be calculated for a determined terminal impedance. Assume that the tube diameter \(D\) is 100 mm. The length \(l\) of the tubular string is 1500 m. The fluid viscosity is 3.81 mPa\(\cdot\)s.  
First, calculate the frictional resistance \(R\), the transfer matrix \(\mathbf{M}_{l}\) and the characteristic impedance \(Z_{c}\). Set the terminal impedance value \(Z_{l}\), and define the pressure value at position \(x\) between the beginning and the terminal as \(p_{x}\); according to Equations (15) and (30), the following relationship can be obtained:  
\[\frac{h_{x}}{h_{i}}=\frac{p_{x}}{p_{i}}=\frac{A(x)D(l)-\frac{1}{Z_{l}}A(x)B(l) -B(x)C(l)+\frac{1}{Z_{l}}A(l)B(x)}{D(l)-B(l)\frac{1}{Z_{l}}} \tag{31}\]  
According to the matrix expression of Equation (12), when \(x=l\), there is \(A(l)D(l)-B(l)C(l)=e^{(\gamma_{1}-\gamma_{2})l}\). From Equation (31), the pressure wave amplitude distribution curve along the tubular string can be calculated, as shown in Figure 12.  
The wave frequency of the starting signal is 1 Hz. It can be seen that the wave amplitude of the pressure at any position in the tubular string is also fixed when the wave at the beginning is a periodic fluctuation state. When \(Z_{l}\
eq Z_{c}\), the amplitude of the pressure wave in the tubular string fluctuates with the increasing \(x\). Even zero can be observed at some positions, indicating that no fluctuation occurs at this position. Throughout the tubular string, the waves show a standing wave distribution. When the terminal impedance is smaller than the characteristic impedance, the ratio of the pressure head fluctuation is relatively small, while when the terminal impedance is larger than the characteristic impedance, the ratio of the pressure head fluctuation is relatively large.