# Improved Calculation Method for Siphon Drainage with Extended Horizontal Sections
## 1 Introduction  
Landslide disasters frequently occur in hilly mountainous areas of China, seriously endangering the safety of life and property of residents and road traffic. In addition, the stability of many landslides is closely related to the action of groundwater and is vulnerable to solid rainfall. Therefore, adopting methods such as slope drainage to reduce and control the groundwater level is a necessary technical means to manage landslides. Under atmospheric pressure, the imbalance of forces caused by the difference in the height of the inlet and outlet causes the physical phenomenon of liquid moving from a lower to a higher and then to a lower level, which is called \"siphoning\". Based on this principle, slope siphon drainage technology cleverly uses the ubiquitous atmospheric pressure and the topographic conditions of the slope, which leads to the deeply buried groundwater discharging from the slope, as shown in Figure 1.  
Siphon drainage technology is used worldwide, with pipe diameters ranging from millimeters to meters. As early as 1988, [PERSON] designed an automatic siphon drainage system for slopes in a patent [1], and based on this improved patent, he proposed a better choice of 6-10 mm diameter pipes [2]. The team of [PERSON] and [PERSON] explored the use of 10 cm diameter siphons for siphoning agricultural water-saving irrigation and reservoir slope drainage management, while the team of [PERSON] and [PERSON] [3; 4; 5; 6] concentrated on the application of millimeter siphon pipe diameters for potential landslide water table precipitation. In the early stage of siphoning technology applied to slope drainage, air bubble accumulation and consolidation occurred at the top of the tube due to the long-term stoppage of siphoning caused by the dry water period and under the action of several factors. The accumulation of air led to the destruction of the vacuum characteristics at thetop of the pipe, which eventually reduced the siphon restarting capacity or even made the siphon device ineffective and reduced the safety of the landslide management project. In a study by [PERSON] et al. [5], reducing the pipe diameter to form a segmental slug flow in the pipe allowed the air bubbles to be discharged from the pipe in time to prevent air accumulation during the siphoning action. The application of siphon technology has significant advantages, such as not requiring power and low construction cost. However, topographic constraints lead to the need for most projects to use siphons longer than 100 m. The increased length of the horizontal section poses a more significant challenge for siphon flow calculation and subsequent engineering application scheduling. The flow pattern plays a crucial role in the gas-liquid two-phase flow; its flow characteristics affect the pressure drop and energy loss. The models explored in this paper are all air-water two-phase flows in a circular tube. Due to the self-driven nature of siphon flow, the velocity of the gas phase within the fluid does not appear much larger than the velocity of the liquid phase, so the distribution of siphon flow type is relatively single. The Lockhart gas-liquid two-phase flow pattern diagram shows that the siphon flow occurs in millimeter diameter siphons at high-lift conditions. The segmental slug flow and the elastomeric flow are the most significant and long-term applicable flow patterns for the whole pipe section. Segmental slug flow is a complex flow pattern with non-homogeneous, periodic, and random unit length characteristics. At the same time, different types of slug flow in a circular tube will show different characteristics, and the different forms of gas present in the liquid will directly affect the energy and momentum exchange rate. Therefore, it is essential to distinguish the different flow types, especially the segmental slug flow, and to choose an appropriate gas-liquid two-phase motion model to accurately describe the siphoning process and predict the siphoning flow rate.