# Study on Siphon Drainage Capacity of Slopes with Long-Horizontal Pipe Sections
## 4 Discussion
### Calculation of Slope Drainage Capacity Requirements in Combination with Single-Hole Gushing Water  
In order to improve the applicability of slope siphon drainage methods in actual projects, a reasonable arrangement of siphon drainage hole clusters is required to improve the efficiency of lowering the groundwater table, which requires quantifying the slope drainage requirements and the drainage volume of siphon boreholes. The French scholar [PERSON] proposed his famous steady well flow equation. He proposed the idea of the round island model: in a homogeneous, isotropic, submerged aquifer with a horizontal water barrier floor, with a complete well as the center and a constant head on the outside, without precipitation infiltration recharge and groundwater evaporation, the seepage flow is stable. Then, the flow rate or drop depth for a fixed value of water pumping from the well is controlled and at a particular time, seepage will reach a steady state, aquifer diving surface from the horizontal surface to funnel-shaped--[PERSON] called it a water level landing funnel. According to the principle of seepage continuity, the cross-sectional flow within the radius of the influence of the pumping well is equal to the flow of the pumping well. Dupuitwell flow is radial flow, and the hydraulic gradient at each point decreases as the distance between that point and the wellbore increases, while the hydraulic gradient is greatest near the well bore. The flow line near the bottom plate of the water barrier is straight, and the infiltration curve is convex, i.e., the flow line gradually transitions from a convex curve on the diving surface to a straight line. The iso-head surface in the well flow model is a series of surfaces formed by the rotation of the iso-head line on the profile around the well axis. In order to obtain these complex surface equations, [PERSON] ignores the flow velocity in the vertical direction and approximates the iso-head line on the profile as a plumb line, which reduces the three-dimensional well flow problem to a two-dimensional flow problem. The water barrier floor is used as the reference surface for the derivation of the equation, and the head value at the diving surface is equal to the seepage thickness h. According to the polar axis, the well axis is taken as the h (seepage thickness) axis, which is positive upward; the r axis is taken along the water barrier floor, which is positive outward, as shown in Figure 12.  
According to [PERSON]'s law and [PERSON]'s assumption, the flow rate of an arbitrary seepage cross section is as follows:  
\[Q_{0}=KA\frac{dh}{dr} \tag{5}\]  
Since \(h\) increases with \(r\), and thus \(\frac{dh}{dr}>0\), considering the seepage section as a cylindrical surface, so \(A=2\pi rh\), then  
\[Q_{0}=2\pi rhK\frac{dh}{dr} \tag{6}\]  
Figure 12: Schematic of Dupuit stabilized submerged well flow. Integrate over \(r\) and \(h\), respectively, where \(r\) is from \(r_{w}\) to \(R;h\) is from \(h_{w}\) to \(h_{0}\). According to the formula of the surging volume of Dupuit stabilized submerged well flow, it is known that. \[Q_{0}=\pi K\frac{h_{0}^{2}-h_{w}^{2}}{In\frac{R}{r_{w}}} \tag{7}\]  
where \(Q_{0}\) is the pumping flow rate, \(h_{0}\) is the water level (from the water barrier bottom) or seepage thickness at the outer boundary of the water content, \(h_{w}\) is the water level (from the water barrier bottom) or water layer thickness in the well, \(R\) is the radius of the cylindrical aquifer, \(r_{w}\) is the radius of the well, and \(K\) is the aquifer permeability coefficient. [PERSON] [23] used this formula to analyze and calculate the single-hole seepage volume, combining the good cluster hole precipitation capacity with the siphon inclined hole drainage capacity and introducing the concept of drainage interception ratio.