# Propagation Mechanism of Pressure Waves during Pulse Hydraulic Fracturing in Horizontal Wells
## 1 Introduction  
With the continuous development of social economy, the demand for oil and gas resources has been increasing exponentially. However, the exploration of oil and gas resources has entered the middle and late stages, with increasingly high costs and difficulties. In particular, in recent years, low-permeability oilfields have accounted for more than half of the newly discovered reservoirs [1; 2; 3]. The reservoir environment of low-permeability reservoirs is complex, and traditional oil recovery methods have little effect; therefore, hydraulic fracturing technology is often used for reservoir transformation [4; 5]. Hydraulic fracturing is a commonly used method for increasing production in low-permeability reservoirs. During the operation, high-pressure fracturing fluid is first injected into the wellbore, causing the rock in the reservoir to form cracks, making it easier for oil or natural gas to flow out of the reservoir [6; 7; 8]. In traditional hydraulic fracturing operations, multiple sets of fracturing pumps and pressure-resistant equipment are usually installed at the wellhead, allowing the fracturing fluid to be injected into the wellbore at a high pressure. However, due to the pressure-resistant equipment, the injection pressure at the wellhead cannot be too high, resulting in failure to achieve the expected fracturing effect. As a result, variable-flow hydraulic fracturing technology has emerged. Unlike traditional hydraulic fracturing technology, variable-flow hydraulic fracturing technology uses variable-flow circulation to inject fracturing fluid, generating fluctuating pressure at the bottom of the well, causing fatiguedamage to the rock in the reservoir, and increasing the complexity of the cracks [9; 10]. Thus, the wellhead injection pressure can be significantly reduced while ensuring the fracturing effect [11; 12; 13; 14]. A comparison between constant-flow and variable-flow hydraulic fracturing is shown in Figure 1. Refs. [15; 16; 17; 18; 19; 20] demonstrated in their 2019 study that the use of a variable-flow injection scheme could reduce the fracturing pressure by about 20%. In variable-flow hydraulic fracturing operations, the length of the fracturing string typically extends several kilometers, and the key research questions revolve around the injection pressure of the fracturing fluid and the ability to generate fluctuating pressure at the well bottom. In traditional constant-flow hydraulic fracturing research, the pressure at the wellhead is essentially constant, and there are no significant fluctuations in the fluid flow velocity within the wellbore, which is considered steady-state flow [21]. The shear stress between the fracturing fluid and the pipe wall can be calculated using [PERSON]'s formula, and the pressure transmission characteristics within the wellbore are relatively simple. [PERSON] et al. [22] revised the parameters in the formula of the drag reduction ratio method in 2010 and used the drag reduction ratio method to calculate the friction along the fracturing string. In variable-flow hydraulic fracturing, the velocity of the fracturing fluid within the wellbore is constantly changing, and it is not possible to calculate the shear stress based on steady-state flow; instead, the impact of non-constant friction must be considered. Among the existing research, non-constant friction models mainly include the Zielke weighted function model and the Brunone model [23; 24]. [PERSON] [25] proposed a time-domain numerical solution algorithm based on the characteristic line method in 2018, which has been instrumental in solving transient fluid flow within the pipe column. In 2020, [PERSON] and [PERSON] [26] established a pressure fluctuation model in a vertical wellbore under variable-flow injection, considering the effects of fluid gravity and non-steady friction. However, the research objects in these papers were all conventional vertical wells [27], which are generally perpendicular to the reservoir and have a relatively short length of reservoir penetration.