Introduction

A debris flow is a suddenly occurring natural disaster phenomenon in mountainous areas (Liu et al. 2013a). The existence of vulnerable geological structures with heavy and concentrated rainfall typically leads to the occurrence of a debris flow disaster (Turconi et al. 2013). Generally, a debris flow can cause loss of human lives, destruction of houses and facilities, etc. (Tang et al. 2012). Consequently, assessing the runout of potential debris flows in these places is essential to prevent or minimize the hazard. In northern China, debris flows occur infrequently, and the land planning in the alluvial fan is poor, which is likely to produce a large debris flow hazard (Shi et al. 2016). Currently, it is difficult to quantitatively assess the effect of a potential debris flow using the traditional method. To solve the problem, a numerical simulation method was typically used to quantitatively assess the effect, which could intuitively determine the runout of a potential debris flow (Nakatani et al. 2016).

The Panshan area, located in northern China, is a national 5A level scenic spot. Many tourists visit the area during the summer each year. Currently, many houses and transport infrastructures exist in the alluvial fan of the Dongwopu gully (Fig. 1). Historical records show that two major debris flows have occurred in the gully, with the first in the Guangxu years of the Qing Dynasty (1871–1908) and the second in 1958 (Zhang et al. 2014). The two debris flows both occurred as a result of heavy rainstorms. Thus, it was very necessary to study the runoff effect of potential debris flows in the Dongwopu gully induced by extraordinarily heavy rainfall. In this paper, the runoff of potential debris flows in the Dongwopu gully was studied based on a well-balanced finite volume scheme with first-order accuracy in time and second-order accuracy in space, which had the potential to adapt to the complex topography and reasonably simulate the runout of debris flow events. Though simulation, the runout distance and inundation path, as well as flow depths and flow velocities, for the potential debris flows were obtained, which contribute to the analysis of the runout characteristics of potential debris flows to improve risk management.

Fig. 1
figure 1

Remote-sensing image of study area

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Study area

The Dongwopu gully (40°05′ ∼ 40°06′ N and 117°15′ ∼ 117°18′ E) is situated at the border of Tianjin and Beijing City, which belongs administratively to the Panshan area in Tianjin, China (Fig. 2). The Panshan area is characterized by a low-medium relief landscape with rugged mountains that belong to the southern branch of the Yanshan Mountains. The highest elevation of the Panshan area is approximately 864 m. Over 60% of the Panshan area has a topography that is steeper than 30°, and over 30% is steeper than 55°.

Fig. 2
figure 2

Location map of study area in Tianjin City, China

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Formation conditions of potential debris flow

Topography condition

The Dongwopu gully has a watershed area of 3.20 km2. The main drainage channel length is approximately 2.67 km, which has an average gradient of 150‰. The altitude is approximately 590 m with the highest point at 855 m to the northeast of the gully and the lowest point at 265 m in the southern channel. The shape of the watershed is elongated with a length approximately three times the width, and its average slope is approximately 35°. Moreover, in the field investigation, the elevation of the northern channel is 5 ∼ 10 m greater than that of the southern channel (Fig. 3). The potential debris flow will run through the southern channel.

Fig. 3
figure 3

The southern channel and northern channel in the Dongwopu gully

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Geology condition

The geological structure of the Panshan area is controlled by the Panshan dome (Yang et al. 2007). The Panshan pluton is primarily composed of granite and adamellite (Fig. 4). Moreover, the Jixian reverse faults run though the southwestern part of the study area. During the field investigation, the physical weathering in the study area was extremely strong, and thick unconsolidated regolith was widely distributed in the hillsides (Fig. 5). This characteristic can create significant amounts of loose solid materials in the Dongwopu gully. In the field, the volume of loose solid materials that may supply a debris flow when enduring heavy rainfall was approximately 4.5 × 105 m3 according to the “Specification of geological investigation for debris flow stabilization” (the Chinese geological mineral industry standard, DZ/T0220–2006). Moreover, the fine materials in the hillsides were primarily silty sand, which were of low viscosity and high permeability.

Fig. 4
figure 4

Simplified geological map of study area (1 The Quaternary, 2 The Granite, 3 The Adamellite, 4 Yangzhuang Limestone, 5 Gaoyuzhuang Limestone and 6 Wumishan Limestone)

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Fig. 5
figure 5

Strong physical weathering in study area (top: the northern hillside and bottom: the southern hillside)

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Rainfall condition

The study area has a warm-temperate sub-humid monsoon climate. According to the data from the weather station in Jixian County, the average annual rainfall is approximately 719.8 mm, the maximum annual rainfall is 1216.6 mm, occurring in 1978, and the minimum annual rainfall is 351.7 mm, occurring in 1978. Moreover, the maximum daily rainfall of 353.5 mm occurred on the 25th of July 1978, and the maximum hourly rainfall of 71.2 mm occurred on the 24th of July 1985. The annual precipitation is unevenly distributed, primarily concentrated from June to August, which accounts for 73.7% of the total annual precipitation (Fig. 6).

Fig. 6
figure 6

The average monthly rainfall data of the Jixian weather station

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Numerical model

Recent debris flow research primarily focuses on the physical behaviour of the debris flow process (Iverson 1997; Jakob and Hungr 2005; Frank et al. 2015). Physically based numerical models have been developed to determine runoff paths and inundation regions as well as runoff heights and runoff velocities (Hungr and McDougall 2009; Frank et al. 2015). Runoff numerical models have become viable tools to determine the potential debris flow in a mountain basin and for back analysis of the debris flow event in history. However, one main challenge in applying runoff numerical models is developing a kind of robust numerical scheme that is suitable for simulating free-surface shallow flow (e.g., floods and debris flows) over highly irregular topography with complex geometry (Song et al. 2011; Liu et al. 2013b). To ensure a robust runoff numerical model for runout analysis of a potential debris flow in the Dongwopu gully, a well-balanced Godunov-type finite volume algorithm was employed for modelling the debris flow over highly irregular topography with complex geometry in this study.

Model description

The movement of debris mixtures is typically simulated by solving the following shallow water equations, which include the depth-averaged mass conservation equation and the momentum conservation equations (Ouyang et al. 2013; Wu et al. 2016). The vector format is expressed as.

$$ \frac{\partial \mathbf{q}}{\partial t}+\frac{\partial \mathbf{f}}{\partial x}+\frac{\partial \mathbf{g}}{\partial y}=\mathbf{s} $$
(1)

where t denotes time, x and y are Cartesian coordinates, and q, f, g and s are vectors representing the conserved variables, fluxes in the x- and y-directions, and source terms. These vectors are given by

$$ \mathbf{q}=\left[\begin{array}{c}\hfill h\hfill \\ {}\hfill uh\hfill \\ {}\hfill vh\hfill \end{array}\right];\kern0.5em \mathbf{f}=\left[\begin{array}{c}\hfill uh\hfill \\ {}\hfill {u}^2 h+\frac{1}{2}{gh}^2\hfill \\ {}\hfill uvh\hfill \end{array}\right];\kern0.5em \mathbf{g}=\left[\begin{array}{c}\hfill vh\hfill \\ {}\hfill uvh\hfill \\ {}\hfill {v}^2 h+\frac{1}{2}{gh}^2\hfill \end{array}\right];\kern0.5em \mathbf{s}=\left[\begin{array}{c}\hfill 0\hfill \\ {}\hfill - gh\frac{\partial {z}_b}{\partial x}-{S}_{fx}\hfill \\ {}\hfill - gh\frac{\partial {z}_b}{\partial y}-{S}_{fy}\hfill \end{array}\right] $$
(2)

Herein, h is the water depth; z b is defined as the bed elevation above the datum; g refers to the acceleration due to gravity; uh and vh denote the unit width discharge in the x- and y-directions, respectively; and S fx and S fy represent the frictional resistance in the x- and y-directions, respectively. Currently, various friction models, such as the Coulomb friction model, the Voellmy friction model and the quadratic rheologic friction model, have been used in debris flow simulation (Rickenmann et al. 2006). In this study, the quadratic rheologic friction model, which has been most widely applied to natural debris flows, was adopted and the fictional resistances (S fx and S fy ) were expressed as follows (Li et al. 2011 and Zhang et al. 2015):

$$ {S}_{fx}=\frac{\tau}{\rho_m}+\frac{K\eta u}{8{\rho}_m h}+\frac{gn^2 u\sqrt{u^2+{v}^2}}{h^{1/3}};\kern0.5em {S}_{fy}=\frac{\tau}{\rho_m}+\frac{K\eta v}{8{\rho}_m h}+\frac{gn^2 v\sqrt{u^2+{v}^2}}{h^{1/3}} $$
(3)

where τ is the yield stress parameterized as τ = α 2 ⋅ exp(β 2 ⋅ C v ); ρ m denotes the solid density of debris flow mixture; K refers to the resistance coefficient; η is the viscosity of the sediment mixture parameterized as η = α 1 ⋅ exp(β 1 ⋅ C v ); and n represents the pseudo-Manning’s resistance coefficient, which accounts for both turbulent boundary friction and internal collisional stresses. α 1 , β 1 , α 2 and β 2 are empirical coefficients; and C v is sediment concentration by volume.

These governing equations are solved based on digital elevation data with a finite volume Godunov-type numerical scheme. The sketch for the numerical calculation is shown in Fig. 7. To obtain a second-order accuracy Godunov-type scheme in space, a linear slope-limited reconstruction method introduced by Liang (2010) was adopted, which could better control numerical oscillation in solving these governing equations. In the numerical calculation, the computational domain and the boundary conditions are first allocated based on the actual conditions in the study area following the method introduced by Hu et al. (2000) and Ouyang et al. (2013). Then, the inflow hydrology of a debris flow entering a research region is set as the inflow boundary condition. The process of model implementation is depicted in the flowchart illustrated in Fig. 8.

Fig. 7
figure 7

Sketch of the numerical calculation with the numerical model

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Fig. 8
figure 8

Flowchart of the numerical model implementation

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Model test and validation

An irregular topography with channel bulge was used to verify that the present numerical model had a strong ability to adapt to these conditions (Fig. 9). The computational domain for this problem consists of a 94 m × 98 m (width × height) region and was divided into 47 × 49 rectangular meshes. For the initial inflow conditions, the debris flow inflow flux was set to 30 m3/s. The inflow duration was set to 55 s. The current computational results show a good potential for adapting to the complex topography.

Fig. 9
figure 9

Debris flow over an irregular topography with a bulge in the channel

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The numerical model also requires validation prior to engineering and scientific application. Presently, the FLO-2D model has been widely used in debris flow research. In this study, an available simulation research result of a catastrophic debris flow event that occurred in the Xiaojia Ravine with a FLO-2D model was used to compare our simulated result with the established numerical model in the paper. The comparison result for the two models is shown in Fig. 10. Figure 10A and B show the flow depth and velocity of the debris flow obtained with the FLO-2D model, which emanated from Chen et al. (2013). Figure 10C and D describe the corresponding calculated result of our numerical model, in which the simulation domain and the parameters for simulation calculation were the same as the design parameters of the high yield stress and high viscosity in their paper. Through the comparison analysis, differences in the calculation results between the two models were discovered. The maximum flow height and velocity determined using the numerical model in this paper were greater than those of the FLO-2D model. The flow depths larger than 7.5 m were primarily distributed in the upper and lower runoff paths in Fig. 10C, while the flow depths larger than 7.5 m in Fig. 10A generally occupied the total runoff path. The larger flow velocities were primarily distributed in the middle part of the runoff path in Fig. 10D, while they were predominantly distributed in the upper and middle part of the runoff path in Fig. 10B. A similarity between the two models was that the larger flow depths and formative inundation regions were consistent with the field in which the average deposition depth of the debris flow fan was approximately 15 m, and the runoff distance was approximately 593 m (Chen et al. 2013; Chen et al. 2016).

Fig. 10
figure 10

Comparison between the numerical model and FLO-2D model (A and B FLO-2D model, C and D the numerical model in this study)

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Model advantages and disadvantages

The numerical model for the dynamic simulation of debris flows was established with a well-balanced finite volume scheme with first-order accuracy in time and second-order accuracy in space. As the FLO-2D model, the model could reasonably simulate the result of a catastrophic debris flow event that occurred in the Xiaojia Ravine, which is located in Yingxiu Town, Sichuan Province, China. Again, the model test indicated that it had a good potential to adapt to the complex topography. However, the numerical analysis model based on the fixed gully bed elevation and the bed entrainment could not be included in the model. In many actual events, the bed entrainments of the gullies or channels could affect the debris flow volume and rheological behaviour during water and sediment movement to the mountain basin outlet, especially while flowing through the steep channel with loose materials (Frank et al. 2015). In the future, when meeting steep channels with loose materials in the study region, relevant improvements of the model considering the erosion/deposition process should be carried out.

Runout simulation of potential debris flow

The simulation was conducted as depicted in the flowchart in Fig. 11. To conduct the simulation of the potential debris flow in the Dongwopu gully based on the aforementioned numerical model, relevant parameters and hydrographs of the potential debris flow and digital elevation data of the calculation domain were required in the simulation.

Fig. 11
figure 11

Flowchart of runout analysis for the potential debris flow

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Supplied parameters

Identification of the physical and rheological properties of potential debris flows usually depends on the research and knowledge of sediment mixture exported by recent debris flows (Fei and Shu 2004). In this study, the density and rheological parameters of the potential debris flow were inferred using sediment mixture exported by debris flows that occurred in 1958. The size components of the two sediment mixture samples with a size of less than 200 mm were tested in the field and laboratory following the method proposed by Ge et al. (2013), and the grain-size distributions are shown in Fig. 12. The results indicated that the clay content (<0.005 mm) was approximately equal to 2.6%, considering that the debris flow was of low viscosity (Ni et al. 2011). Moreover, the result was also consistent with the loose solid material property in the field investigation. Based on this fact, the potential debris flow density was assumed to be 1800 kg/m3 (Fei and Shu 2004). The rheological parameters for debris mixtures and pseudo-Manning’s resistance coefficient for surface roughness in the calculation domain were determined based on the suggestions introduced by Lin et al. (2005) and Li et al. (2011) (Table 1).

Fig. 12
figure 12

Grain-size distribution of residual debris in the southern channel exported by the debris flow that occurred in 1958

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Table 1 Empirical parameters used in the debris flow simulation
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Supplied hydrograph

Since there were no observed data on debris flow peak discharge for the Dongwopu gully, the empirical method was used to obtain the debris flow peak discharge for the simulation. Based on the history records, two debris flows occurred in the Dongwopu gully, and both were the result of heavy rainstorms. Thus, the maximum hourly rainfall of 71.2 mm, which occurred on the 24th of July 1985 in the study region, was selected to determine peak discharge from the drainage basin runoff based on the following rational formula (Nakatani et al. 2016):

$$ {Q}_p=1/3.6\cdot f\cdot i\cdot A\kern0.1em $$
(4)

where Q p is the peak discharge (m3/s); f denotes the coefficient of runoff (here, 0.3 was selected considering the coarse-grained soil slope based on the Specifications of Drainage Design for Highways (JTJ 018–97), published by the China Ministry of Communications); i refers to the rainfall intensity (=71.2 mm/h in this study); and A represents the catchment area (km2) and is equal to 2.3 km2 in the simulation (watershed area upstream of point A for the southern channel). After calculation, the peak flood discharge, not considering the sediment mixture, was equal to 13.7 m3/s.

Then, considering the debris flow characteristics and the supply conditions of unconsolidated soil in the source area, the peak debris flow discharge can be estimated from the formula (Cui et al. 2011)

$$ {Q}_c=\left(1+\varphi \right)\kern0.2em {Q}_p\kern0.2em {D}_c $$
(5)

where Q c is the debris flow discharge (m3/s); D c denotes the debris flow amplification coefficient, which is verified based on the simulation of channel blocks (here, 1.5 was selected based on the Specifications for Geological Investigation of Debris Flows Stabilization (DZ/T0020–2006), published by the China Ministry of Lands and Resources); and Φ refers to the debris flow correction coefficient, defined as

$$ \begin{array}{c}\hfill \varphi ={C}_v/1-{C}_v\Big)\hfill \\ {}\hfill {C}_v=\left({\rho}_c-{\rho}_w\right)/\left({\rho}_m-{\rho}_w\right)\hfill \end{array} $$
(6)

where ρ c is the debris flow density (kg/m3); ρ w denotes the density of water (=1000 kg/m3); and ρ m refers to the sediment mixture density (=2650 kg/m3). Based on Eqs. (5) and (6), the peak debris flow discharge was calculated to be 30.5 m3/s.

The duration of a single debris flow event is typically short due to the sudden increase-decrease characteristics of the debris flow discharge (Liu et al. 2013a). Based on the information of the debris flow event that occurred in 1958 provided by many local eyewitnesses, the duration of the potential debris flow of the Dongwopu gully was designed to be 15, 30, 45 and 60 min. Then, the supplied hydrographs of the potential debris flow were generalized as an isosceles triangle type for simplicity (Chen et al. 2016).

Supplied topography

The topography data of the Dongwopu gully from upstream (Fig. 1 point A) to the alluvial fan could be obtained from the 1:10,000 topography map. Then, through the Kriging interpolation algorithm, the digital elevation points in the calculation domain could be selected using an interval scale of 8 m in this study.

Runout analysis of potential debris flow

Based on the above work, four kinds of simulation results for flow depth and flow velocity were calculated and are shown in the terrain sensing image of the study region (Fig. 13). The flow depths for the four designed durations all clustered predominantly in the range of 1.5 m ∼ 3.0 m. Correspondingly, the sediment deposition depths were in the range of 0.73 m ∼ 1.46 m. This result was in accordance with data from the field survey (Fig. 12). For the flow velocities of the potential debris flow in the Dongwopu gully, the simulation results showed small values (mainly, 0.3 ∼ 0.7 m/s), which were likely to cause channel siltation. In the field, the debris exported by the debris flow that occurred in 1958 covered all parts of the southern channel (Fig. 12). The inundation regions of the potential debris flow covered houses in low-lying areas of the alluvial fan, which impacts safety. Thus, relevant spatial planning and risk mitigation should be considered in the region.

Fig. 13
figure 13

Flow depths and corresponding flow velocities with four designed hydrographs (A and B 15 min, C and D 30 min, E and F 45 min, G and H 60 min)

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Simulation result discussion

During model validation with the debris flow event that occurred in the Xiaojia Ravine, the maximum debris flow discharge reached approximately 2300 m3/s. The simulation results showed larger flow depth and velocity for the debris flow event, which were consistent with the field data. In the study of the runout effect of the potential debris flow in the Dongwopu gully, the maximum debris flow discharge entering our reach was only 30.5 m3/s. The simulation results showed small flow depth and velocity for the potential debris flow. Based on this, the input debris flow discharge had an important influence on the flow depth and velocity of the debris flow. In the study region, the fine materials in the hillsides were primarily silty sand, and they had low viscosity and high permeability. While enduring rainfall, the characteristics of the materials were not helpful for forming large flood discharge and debris flow discharge in the study region. Again, the channel gradient of the Dongwopu gully was very small, and the width of the gully bed was larger. This also prevented the potential debris flow from easily developing into a rapid flow.

Conclusion

A good free-surface shallow flow model over highly irregular topography with complex geometry was adopted in the study. Runout characteristics of the potential debris flow in the Dongwopu gully were incorporated into the numerical model analysis. The simulation results showed a small flow depth and velocity for the potential debris flow. This was due to a small debris flow discharge entering our reach and a wide and gentle channel, in accordance with the knowledge from the field survey. In addition, inundation regions of the potential debris flow covered the houses in low-lying areas of the alluvial fan, which has safety impacts. Thus, relevant spatial planning and risk mitigation should be considered in the region.