## 1 Introduction
However, because the signal transmission in the tubular string is affected by many factors, these influences need to be further studied for propagation mechanisms for transient flow wave signals. It is necessary to find better mathematical models to describe the transmission process and amplitude-frequency characteristics of transient flow waves. The model proposed in this paper can more accurately describe the transient flow wave transmission characteristics in the tubular string of water injection wells. The transient flow wave characteristic is very important in long-distance transportation and valve operation. Therefore, many experimental and numerical studies were conducted to prevent mistaken operation of valve switches and to ensure the safety of pipelines and pumps [23; 24; 25]. Based on the different pressure transient responses, corresponding algorithms are developed and applied for blockage or leak detection [26; 27]. All the studies can help to understand the transient flow wave transmission behavior. However, the water hammer response characteristics of the wellbore system are different from those in the pipeline system. Thus, experiment and numerical studies are needed to explain the transient flow wave response characteristics in the wellbore system. [PERSON] et al. [28] studied the transient flow wave signal in the water injector and verified the water hammer propagation model through experiments. [PERSON] et al. [29] conducted a comprehensive study of water hammer effects in injection wells under different design parameters and operating parameters using OLGA simulations. In addition, signal analysis of transient flow waves mentioned above is only considered in time or frequency domain. The transient flow wave transmission behavior is more complicated. Combining methods in both time and frequency domains can help to better understand the transient flow wave transmission in a complex system. The gap between this paper and previous studies is the analysis of transient flow waves using time and frequency domains in two dimensions. This combination is needed because wave attenuation mechanisms are frequency- and time-dependent [30]. In this paper, a two-dimensional transient flow wave analysis method in time and frequency domains is proposed. Meanwhile, the transient flow transmission characteristics of the wellbore system are sufficiently presented to accurately investigate the transient flow wave transmission in the wellbore system. First, based on the theory of transient flow, the matrix calculation model of the transient flow wave signal transmitted in the water injection tubular string is established. The transient flow wave signal distribution in the water injection tubular string under the average friction is analyzed. When the continuous transient flow wave signal is transmitted in the tubular string, if the starting pressure head amplitude and discharge amplitude are known, the amplitude of the pressure head and discharge at any position can be calculated. Second, through experiments and calculations, the transmission characteristics of the transient flow wave signal are verified in the water injection tubular string. The calculation method of wave velocity and attenuation are studied. Third, the frequency domain characteristics of the water injection tubular string is also analyzed. It provides theoretical support for the selection of signal frequency and signal transmission in the layered water injection process. This study can provide clear insights into the use of transient flow waves for intelligent measurement and regulation and improve accurate control of downhole water injection.