## 5 Conclusions  
In this paper, a matrix transfer model of transient flow wave in tubular strings and its transmission characteristics in stratified water injection are investigated. The effects of tubular string parameters and fluid properties on the waves of pressure and discharge are studied. The results show that:  
1. The transfer process of transient flow waves in a tubular string is studied, and a continuous transient flow wave transmission model based on transfer coefficient is proposed. The model defines the transfer coefficient to reflect the influence of tube size and fluid properties on the wave transmission and considers that the wave at any position in the tubular string can be regarded as the superposition of the wave in both directions along the tubular string. The analysis of the model shows that when one end of the tubular string is in the forced vibration state, the fluctuation in the tubular string shows a standing wave state. When the terminal impedance of the tubular string is small, the fluctuation amplitude in the tubular string is small. The model provides a powerful tool for analyzing the fluctuation wave in the tubular string. This study can provide clear insights into the use of transient flow waves for intelligent measurement and regulation and improve accurate control of downhole cableless intelligent water injection.
2. An analytical method is proposed to calculate the transmission of fluctuations in the tubular string using a continuous transient flow wave transmission matrix model. The validity of the model based on the transfer matrix has been verified [32]. By analyzing the transmission of fluctuations, it is shown that the range of carrier frequencies suitable for communication is constantly changing as the well depth increases. Therefore, it is necessary to calculate the carrier frequency suitable for communication based on  
Figure 16: Amplitude–frequency space for different lengths of tubular strings.  
specific parameters. The transfer matrix method allows a simple expression of the complex transmission process of wave signals in a tubular string. It is applicable to the analysis of wave signals in series tubular strings with various materials and sizes.
3. The calculation methods of wave transmission velocity and wave attenuation are studied. The relationship between the influence of tubular string parameters and fluid characteristics on the wave transmission velocity and wave signal strength is obtained by simulation analysis. A two-layer stratified water injection test platform was built to study the fluctuation of discharge and pressure at monitoring points in the tubular string. The error of wave velocity was 0.69% and the error of wave signal amplitude was 3.85%, indicating the verifications of the calculation and simulation. The practical application of this work mainly provides theoretical support for the cable-free intelligent water injection technology of two-way signal transmission.  
Conceptualization, E.M.; data curation, L.Z.; formal analysis, J.Y.; funding acquisition, L.Z.; investigation, X.P.; methodology, E.M., C.L. and H.L.; project administration, H.L. and L.Z.; resources, L.Z. and X.P.; software, J.Y.; supervision, J.Y.; validation, J.Y. and X.P.; writing--original draft, C.L.; writing--review and editing, E.M., C.L. and H.L. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the China Institute of Petroleum Exploration and Development project (No. HYXY-ZB-2021-FW026).  
Not applicable.  
The data that support the findings of this study are available from the corresponding authors upon reasonable request.  
The authors sincerely thank the anonymous reviewers who made valuable comments on this paper.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.