1 Introduction

Most landslides are triggered or influenced by the rising of groundwater levels because of rainfall infiltration (Sassa, 2005; Sun et al., 2012; Shrestha et al., 2013). Positive pore pressure is the main factor governing the evolution of the entire landslide body. This is particularly true in the case of earth flows or earth slides where clay content plays a significant role. Consequently, reducing pressure is usually a main goal in achieving engineering safety. Therefore, the study of effective drainage techniques, such as siphon drainage, is essential for engineering geology and disaster prevention.

Siphon drainage can be used in slope drainage for its powerful water delivery capacity. During the siphon process, flow velocity can be controlled by automatically changing water levels. Govi (1989) and Cambiaghi and Schuster (1989) introduced a system using a siphon technique as an emergency drainage treatment for landslides. Currently, siphon drainage combined with electropneumatic drainage is a key method for discharging ground water in Europe, especially in France (Bertolini et al., 2005; Clark et al., 2007; Bomont, 2008; Gillarduzzi, 2008; Mrvik and Bomont, 2012). Studies have been conducted on slope siphon drainage since the 1990s, such as those on the initial siphon forming method, the maximum height limit of siphon drainage, the maintenance of siphon and management requirements. Associated engineering applications have also been developed (Zhang and Zhang, 1999a; 1999b; Yang et al., 2009; Shu and Zhang, 2005). However, the management of long-term operation of siphon techniques has not been fully explored yet.

Due to their non-persistence, siphon techniques have not yet been used widely. Generally, the solubility of a gas dissolved in liquid is proportional to the pressure acting on the liquid surface (Mao, 1987). In siphon drainage, the pressure of liquid gradually decreases with the increasing elevation over the liquid surface of the nozzle. Thus, during siphon processes, the air dissolved in the liquid separates out and generates bubbles, as a consequence of which, air accumulates at the tops of hoses leading to interruption of the siphon process. According to previous experiments (Sun et al., 2013), bubbles can be easily observed when the water-lift is higher than 5.4 m, which can be defined as high-lift siphon drainage. In high-lift siphon drainage, the dramatic pressure change brings intensive air precipitation. As a result, we conclude that there are fundamental differences in physical conditions between high and low lift siphons. The water lift of an on-site low-lift siphon is a great deal lower than its maximum level. Currently, most siphon applications are associated with low-lift siphons. For high-lift drainage, exhaust devices are usually applied at the tops of the drains to prevent air accumulation (Shu and Zhang, 2005; Yang et al., 2009).

High groundwater levels and lasting intense rainfall usually lead to rapid flow rates. Previous experimental investigations (Sun et al., 2013) indicate that no air accumulates within the hoses during rapid flow rates. Therefore, continuous rainfall results in a large flow velocity and smooth drainage, whereas during long-term rainfall shortage, flow velocity slows down and bubbles accumulate. Slope drainage is a long-term process in need of enduring maintenance, which is not easily achievable.

The present paper reports an experimental investigation of the flow pattern variables and relevant influencing factors at the tops of siphon hoses. A threshold of hose diameter to maintain siphon drainage is recommended for further application.

2 Test of siphon drainage

Fig. 1 shows the scheme of a practical siphon drainage application. The geometric scale of sloping projects, usually in meters, contrasts starkly with that of hose cross-sections, which are usually in millimeters. Thus, the effect of the radius of curvature at the siphon hose vertex can be ignored, and the flow patterns are determined by the diameter of the hoses. Therefore, a simplified set-up was used to investigate the effect of hose diameter on flow characteristics (Fig. 2). The hose segment approaching the vertex of the siphon hose should be as straight as possible to avoid the negative impact of curvature radius.

Fig. 1
figure 1

Scheme of a practical siphon drain application

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Fig. 2
figure 2

Schematic diagram of the experimental apparatus

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2.1 Test set-up

The experimental apparatus consisted of a standpipe, an upper hose, a lower hose, and a storage tank (Fig. 2). The standpipe, with an elevation of 2.0 m and installed at the bottom of the upper hose, simulates the end of a borehole in practical engineering. The water level of the upper hose represents the actual slope water level. A siphon can reach a maximum rising water head of about 10.24 m under standard atmospheric pressure. Therefore, about 1.0 m of water in the standpipe remains once the elevation of the upper hose is set at 11.23 m (Fig. 2). The difference in elevation between the siphon vertex and the hose outlet was 20.46 m. The outlet was placed in the storage tank. Siphon hoses with a series of diameters were used during the experiments. Based on observation, three typical kinds of polyurethane (PU) hoses with diameters of 4.0, 5.0, and 8.0 mm, respectively, were tested.

2.2 Siphon phenomena

At first, the siphon hose was filled full of water and sealed at both ends. The two ends of the hose were dipped into the standpipe and the storage tank, respectively, to start the siphon drainage. Water was added to the standpipe to maintain a continuous siphon process, and then stopped after a few minutes, as a consequence of which, the water level decreased in the standpipe until the lift head reached the highest limit. Finally, water was added to the standpipe at different time intervals and different water-flow rates to investigate the flow characteristics.

The test showed that

  1. 1.

    In the upper hoses, we observed no hose-size bubbles (bubbles filling the entire section of the hose) in 8.0 mm siphon hoses, a few in 5.0 mm hoses, and more in 4.0 mm hoses (Fig. 3a).

  2. 2.

    In the lower hoses, slug flow with different degrees of stability was found in hoses with diameters of 4.0, 5.0, and 8.0 mm. Fig. 3b shows the slug flow in a 4.0 mm lower hose.

  3. 3.

    After the siphon process, similar moisture distribution characteristics in both upper and lower hoses were observed. A small length of no-water section was found near the vertex of the upper hose, and a much longer no-water section in the lower hose. These were caused by the existence of elevation differences of over 20 m. The heights of full-water sections in both hoses were nearly the same.

  4. 4.

    After the siphon process, the bubbles in 8.0 mm hoses rose to the vertex and accumulated in less than 5.0 min. In 5.0 mm hoses, the bubbles remained for about 24 h before becoming entirely accumulated at the top. However, in the 4.0 mm hoses, full-hose bubbles were sealed between the water columns after 7 d.

Fig. 3
figure 3

Bubbles in 4.0 mm PU hoses

(a) Bubbles in upper hose; (b) Bubbles in lower hose

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2.3 Flow characteristics at vertex of siphon hoses

In slope engineering, the intensity and duration of rainfall determine the rising rate of the slope groundwater level and the flow velocity in the siphon drains. Therefore, in lab testing, different rainfall conditions can be simulated by adding water to the standpipe at different rates. Heavy rainfall can be simulated by adding water rapidly, while slowly dripping water at a rate of 700 ml/h and 400 ml/h simulates weak groundwater replenishment in dry seasons. In the case of rapid addition, 5.0 L water was added in 30 s, while in the case of slow dripping, rates of 700 ml/h and 400 ml/h were adopted over a duration of 5.0 h.

Tests were conducted using hoses of 4.0, 5.0 and 8.0 mm diameter. Table 1 and Figs. 4a–4c show in detail the distinct differences in flow characteristics at the vertex of the different siphon hoses during drainage. Rapid water addition led to a fast water level rise in the standpipe and a large flow rate in the siphon hoses at the vertex. In the 4.0 and 5.0 mm siphon hoses, a section of the water column passed through the vertex quickly and then fell down under gravity to generate slug flow, producing gas columns alternating with water columns (Fig. 4b). The slug flow was able to push the bubbles together with water to the outlet to prevent air accumulation in the siphon drainage. However, in the 8 mm hoses, most cases showed incomplete slug flows, with water channeling beneath the bubbles (Fig. 4c), which could not achieve an associated movement of air and water. Further increases in the flow rate could also lead to complete slug flow in 8.0 mm hoses.

Table 1 Flow characteristics at the vertex of siphon hoses during drainage
Full size table
Fig. 4
figure 4

Three types of flow at the vertex of hoses

(a) Wall pressing flow; (b) Slug flow; (c) Incomplete slug flow

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When the flow rate of water in the standpipe decreased to 700 ml/h, the water level remained around the siphon launching level and the flow velocity in the upper hose remained low. For a hose with a diameter of 8.0 mm, the slow increase in water level in the upper hose finally led to wall pressing flow, after water passed through the vertex into the lower hose (Fig. 4a). Wall pressing flow is more likely to discharge water without air. In this situation, the supplementary water separates increasingly more air in the low pressure area without water flow, and reduces the degree of vacuum within the hose. If the situation lasts long, it will ultimately lead to a complete stop of the siphon process. For 5.0 mm hoses, the 700 ml/h flow rate sometimes produced wall pressing flow and slug flow, demonstrating its threshold status. With a further decrease in the flow rate to about 400 ml/h, slug flow appears. For 4.0 mm hoses, both 700 ml/h and 400 ml/h flow rates could form slug flow in the lower hose and achieve air discharge.

2.4 Surface shape for wall pressing flow

Tests showed that the surface shapes that tended to form wall pressing flow could be divided into two categories (Figs. 5a and 5b).

Fig. 5
figure 5

Two surface shape categories leading to wall pressing flow

(a) Surface shape at the vertex of the 8.0 mm siphon hose; (b) Surface shape when there is a water film at the bottom of the hose

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In the case of 8.0 mm hoses, the water level in the upper hose rose slowly to the vertex of the siphon hose and a stable stratified surface was observed. This shape was easily collapsed by water flowing along the hose wall when dropping into the lower hose in the form of wall pressing flow (Fig. 5a).

Fig. 5b shows another case of wall pressing flow. When the siphon drainage process came to an end, the flow rate at the vertex became slower. In such a situation, a water film at the hose bottom may degenerate wall pressing flow. However, for slope drainage engineering, the siphon interval is long enough for the water film to evaporate and for some other physical interactions to occur. Therefore, siphon flow is less influenced by this category of wall pressing flow.

Thus, it can be seen from the tests that the water flow in larger diameter siphon drains is more likely to generate wall pressing flow and cause air accumulation at the top of the hose under a long period of inadequate water supply. The increasing air pressure at the top is associated with a decreasing siphon delivery lift, and finally terminates the entire siphon process. So it is important to conduct theoretical research on the causes of wall pressing flow and the threshold status of surface shapes (Fig. 5a).

3 Relationship between flow pattern and hose diameter

A hose with a large diameter generates effective siphon drainage (Kong, 2004; Taha and Cui, 2004). So it is better to choose hoses with a diameter as large as possible under the premise of a long-term effective siphon. From previous sections, we conclude that siphon hoses are more likely to generate slug flow and show no air accumulation when the siphon drainage has high flow velocity (Wallis, 1969; Jiao et al., 2009). However, the flow velocity in siphon drainage for slope engineering is generally very low. Therefore, attention should be paid to the siphon flow at low velocity in the current study.

3.1 Analytical model

For a siphon drain with a low flow rate, the velocity of the flow in the upper hose is low enough to ignore its dynamic influence during a short period (Bretherton, 1961). Therefore, it is reasonable to adopt a hydrostatic analysis method.

Note that the flow characteristics at the top are also influenced by the radius of curvature at the vertex of the siphon hose. The actual elbow at the top of the hose is an arc-shaped curve. However, as addressed in previous sections, the slope in engineering scale is in practice much larger than the geometrical size of the hose cross-section. Therefore, the radius of curvature at the vertex of the siphon hose was assumed to be infinite in the current study. Thus, we can simplify the calculation model to the derivation of gas-liquid-sectional shape in a horizontal hose.

Set the flow direction as y, vertical downward direction as x, and the perpendicular direction into the paper as z (Figs. 6a and 6b). The original point is located at the center of the hose section. Every infinitesimal element dz in the z direction has a similar surface shape, and thus has less impact on the water surface than those in the other two directions. Take a dz element near the original point, and simplify the 3D model to a 2D one, which is equivalent to the derivation of the gas-liquid-sectional shape between two flats.

Fig. 6
figure 6

Surface shapes for hoses with different diameters

(a) r>r c; (b) r≤r c

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3.2 Water surface

The Gibbs free energy of a curved surface liquid is (Wandelt 2012)

$$G = - TS + PV + \sigma A,$$
(1)

where σ denotes the surface tension coefficient, and G, T, S, P, V, and A denote the Gibbs free energy, temperature, entropy, pressure, volume, and area, respectively.

Taking gravitational potential energy into consideration, the total energy expression is (Tang et al., 2009)

$${E_t} = G + {E_g} = - TS + PV + \sigma A + mgh,$$
(2)

where E t and E g denote the total energy and gravitational potential energy, respectively; m, g, and h denote mass, gravity acceleration, and height, respectively.

Static water head is isothermal, isentropic and isobaric and incompressible. So we can neglect the invariants in Eq. (2) to obtain

$${E_v} = \sigma A + mgh = {\sigma_{lg}}{A_{lg}} + {\sigma_{sl}}{A_{sl}} + {\sigma_{sg}}{A_{sg}} + \rho gVh,$$
(3)

where E v denotes the variable energy, σ lg, σ sl, and σ sg respectively denote the surface tension coefficients of gas-liquid, solid-liquid, and solid-gas (Zhu et al., 2007; Jumars, 2013). Similarly, A lg, A sl, and A sg denote the contact area between gas and liquid, solid and liquid, solid and gas, respectively; ρ denotes the density of water.

According to the principle of minimum potential energy, when the value of E v in Eq. (3) approaches the minimum value, the water surface keeps stable.

By ignoring the influence from the z direction, we simplify the 3D model into a 2D model (Fig. 7). For a section with a length of 2a, the area is s=2ar, where r denotes the radius of the hose. Assuming the functional expression as y=f(x), the surface area and the volume can be simplified to be ldz and sdz, respectively. After dividing by dz, Eq. (3) can be written as

$$\eqalign{ {E_v}{\rm{/d}}z = \sigma l + \rho gsh = {\sigma_{lg}}{l_{lg}} + {\sigma_{sl}}{l_{sl}} + {\sigma_{sg}}{l_{sg}} + \rho gsh \cr & \quad \;\;\;\;\; = \int_{- r}^r {\left[ {{\sigma_{lg}}\sqrt {1 + y{'^2}} + \rho g(r - x)(y + a)} \right]} {\rm{d}}x \cr \quad \;\;\;\;\;\;\;\; + \int_{- r}^r {\left[ {{1 \over r}({\sigma_{sl}} - {\sigma_{sg}})(y + xy')} \right]} {\rm{d}}x + 2a({\sigma_{sl}} + {\sigma_{sg}}), \cr}$$
(4)

where l lg, l sl, and l sg denote the contact lengths between gas and liquid, solid and liquid, and solid and gas, respectively, and s denotes the water area within the section.

Fig. 7
figure 7

Relationship between contact angle θ and critical radius r c

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The boundary conditions satisfy

$$s = \int_{- r}^r {(y + a){\rm{d}}x} = 2ar \Rightarrow \int_{- r}^r {y{\rm{d}}x} = 0.$$
(5)

When x=±r, the moveable boundary condition can be expressed as (Lao, 2007)

$${\left. {\left[ {F + (\psi ' - y'){F_{y'}}} \right]} \right|_{x = \pm r}} = 0,$$
(6)

where F(x) is the function within the integral shown in Eq. (7), and ψ(x) is the known curve corresponding to the horizontal line of the hose wall under the actual conditions.

According to Eqs. (4) and (5), we obtain:

$$\eqalign{ F(x,y,y',\lambda) = {\sigma_{lg}}\sqrt {1 + {{y'}^2}} + \rho g(r - x)(y + a) \cr \quad \;\;\quad \;\;\; + {1 \over r}({\sigma_{sl}} - {\sigma_{sg}})(y + xy') + \lambda y, \cr}$$
(7)

where λ is the Lagrange multiplier.

So the unsolved question changes to find the extremal curve of the function \(J = \int_{-r}^{r}\). Its Euler equation follows:

$${F_y} - {{\rm{d}} \over {{\rm{d}}x}}{F_{y'}} = 0.$$
(8)

Substituting F(x, y, y′, λ) into Eq. (8), we can obtain:

$$\eqalign{ \rho g(r - x) + {1 \over r}({\sigma_{sl}} - {\sigma_{sg}}) + \lambda \cr \;\;\;\;\;\;\;\; = {{\rm{d}} \over {{\rm{d}}x}}\left[ {{\sigma_{lg}}{{y'} \over {\sqrt {1 + {{y'}^2}}}} + {x \over r}({\sigma_{sl}} - {\sigma_{sg}})} \right], \cr}$$
(9)
$${\sigma_{lg}}{{y'} \over {\sqrt {1 + y{'^2}}}} = - {1 \over 2}\rho g{x^2} + (\rho gr + \lambda)x + {C_1},$$
(10)
$${\sigma_{lg}}{{y} \over {{{(1 + y{'^2})}^{3/2}}}} = \rho g(r - x) + \lambda,$$
(11)

where σ lg denotes the gas-liquid surface tension coefficient, and C 1 is a constant number.

The above equations can be considered as the curvature expressions of y=f(x). Take y′=tanφ as the slope at a certain point on the curve, where φ is the intersection angle between the tangent line and the x axis. Substituting this into Eq. (10) leads to

$${\sigma_{lg}}\sin \varphi = - {1 \over 2}\rho g{x^2} + (\rho gr + \lambda)x + {C_1}.$$
(12)

The moveable boundary condition shown in Eq. (6) is satisfied when x=±r, where ψ(x) is the known curve corresponding to the horizontal line of the hose wall under the actual conditions. Therefore, we have shown that ψ′=∞. When x=r, Eq. (6) can be written as

$$\eqalign{ {\sigma_{lg}}\sqrt {1 + {{y'}^2}} + {1 \over r}({\sigma_{sl}} - {\sigma_{sg}})(y + ry') + \lambda y \cr \;\;\;\; + (\psi ' - y')\left[ {{\sigma_{lg}}{{y'} \over {\sqrt {1 + y{'^2}}}} + {\sigma_{sl}} - {\sigma_{sg}}} \right] = 0. \cr}$$
(13)

Since ψ′=∞, we have

$${\sigma_{lg}}{{y'} \over {\sqrt {1 + y{'^2}}}} + {\sigma_{sl}} - {\sigma_{sg}} = 0,$$
(14)
$$\sin \varphi (r) = {{{\sigma_{sg}} - {\sigma_{sl}}} \over {{\sigma_{lg}}}} = \cos \theta,$$
(15)

where θ is the contact angle of the gas-liquid-particle three-phase system according to the Young equation (Wandelt, 2012).

Similarly, when x=−r, we have

$$\eqalign{{{{\sigma_{lg}}} \over {\sqrt {1 + y{'^2}}}} + 2\rho gr(y + a) + {1 \over r}({\sigma_{sl}} - {\sigma_{sg}})y \cr \;\;\;\;\;\; + \lambda y + \psi '\left[ {{\sigma_{lg}}{{y'} \over {\sqrt {1 + y{'^2}}}} - {\sigma_{sl}} + {\sigma_{sg}}} \right] = 0, \cr}$$
(16)
$$\sin \varphi (- r) = {{{\sigma_{{sl}}} - {\sigma_{sg}}} \over {{\sigma_{lg}}}} = - \cos \theta.$$
(17)

Substituting Eqs. (15) and (17) into Eq. (12) shows

$$\eqalign{ {\sigma_{lg}}\cos \theta = - {1 \over 2}\rho g{r^2} + (\rho gr + \lambda)r + {C_1}, \cr - {\sigma_{lg}}\cos \theta = - {1 \over 2}\rho g{r^2} - (\rho gr + \lambda)r + {C_1}. \cr}$$
(18)

Therefore,

$$\lambda = {{{\sigma_{lg}}} \over r}\cos \theta - \rho gr\quad {\rm{and}}\quad {C_1} = {1 \over 2}\rho g{r^2}.$$
(19)

Subsequently, substituting Eq. (19) into the curvature expressions Eqs. (10) and (11) yields

$$\eqalign{ {\sigma_{lg}}{{y'} \over {\sqrt {1 + y{'^2}}}} = - {1 \over 2}\rho g{x^2} + {{{\sigma_{lg}}\cos \theta} \over r}x + {1 \over 2}\rho g{r^2}, \cr {\sigma_{lg}}{{y''} \over {{{(1 + y{'^2})}^{3/2}}}} = - \rho gx + {{{\sigma_{lg}}} \over r}\cos \theta. \cr}$$
(20)

Base on the previous derivations, when

$$x = {\lambda \over {\rho g}} + r = {{{\sigma_{lg}}} \over {\rho gr}}\cos \theta,$$
(21)

the curvature becomes zero and y″=0. Thus, this point is the inflection point of the curve. When \({{{\sigma_{1g}}} \over {\rho gr}}\cos \theta \in [0,r]\), there must be an inflection point within the range of the hose diameter. Otherwise, when \({{{\sigma_{1g}}} \over {\rho gr}}\cos \theta > r,y^{\prime\prime}\), y″ is always above zero in regard to the hose, thus a concave function within the diameter range. Thus, to define the curve shape in the hose, it is necessary to take the critical diameter and the critical radius into consideration.

The critical radius r c shows that the curvature comes to zero when x=r c, therefore

$$- \rho g{r_c} + {{{\sigma_{lg}}} \over {{r_c}}}\cos \theta = 0,$$
(22)
$${r_c} = \sqrt {{{{\sigma_{lg}}\cos \theta} \mathord{\left/{\vphantom {{{\sigma_{lg}}\cos \theta} {(\rho g)}}} \right.\kern-\nulldelimiterspace} {(\rho g)}}}.$$
(23)

Figs. 6a and 6b show the surface shapes when r>r c and r≤r c, respectively.

3.3 Threshold diameter of siphon hose

According to Eq. (23), it is possible to obtain the relationship between the contact angle θ and the threshold radius r c (Fig. 7), where the surface tension coefficient between air and water is σ lg=7.28×10−2 N/m under the normal temperature of 21 °C. The water surface shapes (Fig. 6a) may collapse and flow along the wall of the hose when dropping into the lower hose as wall pressing flow (Fig. 4a), unable to discharge the accumulated air at the top of the hose. However, the water surface shown in Fig. 6b has good integrity and is more likely to generate columnar water flow into the lower hose. Therefore, it will finally form slug flow and show gas columns alternating with water columns.

A typical PU siphon hose with a contact angle of 63° was used in the present study (Barajas and Panton, 1993; Tretinnikov and Ikada, 1994; Hilpert, 2009). Therefore, the threshold radius from the formation of slug flow to wall pressing flow can be calculated as r c=1.8 mm. Analytically, the smaller the hose, the better the siphon process performs. Hence, smaller hoses are better. However, smaller pipes more easily become silted up by sand and soil, and may even cause air embolisms. The purpose of present study was to find an optimal value satisfying these two conflicting considerations. Therefore, hoses of 3.6 mm diameter are recommended.

4 Conclusions

In the present study, we investigated experimentally the flow pattern variables at the top of siphon hoses. The threshold conditions for terminating siphon process and automatically restarting were theoretically studied. The following conclusions can be made from the present experiments and theoretical analyses:

  1. 1.

    Low flow velocity leads to different flow patterns, such as slug flow and wall pressing flow, in siphon hoses of different diameter when water passes through the vertex of the hose. Slug flow can discharge bubbles together with water while wall pressing flow cannot.

  2. 2.

    Thermodynamic derivation revealed that hose diameter is the main factor influencing the flow pattern, and the threshold of hose diameter can be theoretically obtained. Wall pressing flow is more likely to be generated in the lower hose when r>r c, while complete slug flow can be generated when r<r c.

  3. 3.

    PU hoses with a diameter of 3.6 mm are recommended, to guarantee the long-term operation of siphon drainage.

Note that both the hose diameter and the hose number influence the drainage flow. The present study focused on how to obtain continuous siphon drainage, which is significantly affected by hose diameter. Once the diameter has been decided, we can adjust the hose number to satisfy the requirement of the drainage flow. Further research will be conducted in more complex situations and considering more influencing factors.