Model tests for surge height of rock avalanche–debris flows based on momentum balance
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Abstract
Rock avalanche–debris flows triggered by earthquakes commonly take place in mountainous areas. When entering a body of water, due to good fluidity they can move for some time instead of halting in water. In this study, we proposed a method for calculating the surge height of rock avalanche–debris flows based on momentum balance and designed a series of model tests to validate this method. The experimental variables include the initial water depth, landslide velocity, and landslide volume. According to the experimental results, we analyzed the maximum wave height in sliding zone based on momentum balance. In addition, we investigated the surge height and proposed the calculation method in propagating zone and running up zone. In this way, we can find out the surge height in different areas when a rock avalanche–debris flow impacts into the water, which could provide a basis for analyzing the burst of barrier lakes.
Keywords
Surge Rock avalanche–debris flow Momentum balance Barrier lake1 Introduction
Landslide surges are a major cause of barrier lake collapse, which can threaten highways, railways, and key facilities in mountainous regions [1, 2, 3]. One typical example is the ice lake collapse on July 15, 1988, in Midui, China. It was reported that a glacier with a total volume of 3.6 × 10^{5} m^{3} collapsed into the Midui Ice Lake. The glacier caused a 1.4-m surge, eventually leading to the collapse of the glacial lake. Subsequent burst floods destroyed a nearly 30-km section of the Sichuan–Tibet Highway, which took 6 months to repair [4].
The study of landslide surge has always been of great interest [5, 6, 7, 8, 9]. Noda [3] suggested a linear relationship between the height of a landslide surge and the Froude number of the landslide based on a piston model experiment. Huber and Hager [10] carried out model experiments on granular landslides, which took into consideration the impact angle, density, and geometric size of a landslide. By assuming that landslide velocity and thickness were the dominant factors, Fritz et al. [6] evaluated the maximum landslide surge height within the generated surge field. On the basis of the work of Fritz et al. [6], Zweifel et al. [9] investigated the effect of landslide density on surge height using different densities—including ice landslides—in their experiments. Ataie-Ashtiani and Najafi-Jilani [5] studied the effects of the underwater movement of solid landslides, granular landslides, and finite deformation granular landslides on wave height by setting the initial position immediately beneath the still water surface. Zitti et al. [11] assumed that the avalanche was a suspended particle after it entered a body of water and established a theoretical model to describe the momentum transfer between the particle and fluid when an avalanche enters a two-dimensional water body. The independent and dependent variables in the model were then reconstructed into a dimensionless form for scale analysis, and the theoretical approximate solution of the near-field wave amplitude of the surge was obtained. Mulligan and Take [12] studied the impact of a landslide on a water body. They determined that the momentum flux is the main driving force of a surge induced by a two-dimensional granular landslide and established the idealized formula of the maximum amplitude of the surge in the near field. Following the analysis of Mulligan and Take [12], Han and Wang [13] established a three-dimensional physical model of reservoir landslide surges and deduced the theoretical expression of the maximum near-field amplitude of a surge under the background effects of a three-dimensional bulk landslide.
Numerous landslide surge model experiments have been conducted with solid blocks, but clastic material is seldom used. In addition, the movement after the landslide enters the water is ignored. Due to water pressure and the friction at the bottom of the block, a solid block cannot move long distance after entering a body of water. However, as to a rock avalanche–debris flow, this is not the case. The distance a rock avalanche–debris flow travels within a body of water is much longer than that of a block.
In this study, we propose a method for calculating the landslide surge height based on momentum balance. Firstly, we analyze the surge height based on momentum balance and proposed a theoretical formula to calculate the surge height near the impact pit. Then, we conducted a series of model tests in order to verify the results of the theoretical analysis. Meanwhile, we investigated the surge height and proposed the calculation method in propagating zone and running up zone. Finally, we analyzed the experimental results and discussed the further research.
2 Theoretical analysis of surge height based on momentum balance
3 Physical model tests of rock avalanche–debris flows
3.1 Experimental setup
Parameter comparison between prototype lake and model
Parameter | Physical quantity | Model | Prototype | Scaling ratio | Reference ratio |
---|---|---|---|---|---|
Model water tank | Length L (m) | 3.76 | 950 | 1/252 | 1/300 |
Width W (m) | 1.76 | 550 | 1/313 | 1/300 | |
Water density ρ_{w} (g/cm^{3}) | 1 | 1 | 1 | 1 | |
Rock avalanche–debris flow | Volume V_{s} (m^{3}) | 0.006–0.020 | 360,000 | 1/376^{3}–1/261^{3} | 1/300^{3} |
Froude number in the water F_{s} | 0.96–3.31 | 1.52 | 0.6–2.2 | 1 |
When designing the physical model, we adopted Froude similarity and geometric similarity experiment, with a geometric similarity ratio of 1/300. We installed four identical wave gauges (P1–P4) in the water tank to record the wave height with a length of 1 m and an accuracy of ± 0.5 mm. A recording frequency of 100 Hz was used. The positions of the gauges are shown in Fig. 2. We used a digital camera to synchronize the movement of the surface waves and the movement of the rock avalanche–debris flow and installed an inclined chute with a length of 4.5 m and an inclination of 70° to simulate a landslide gully. An upper gate (H1) and a lower gate (H2) were installed on the chute. We collected data from four identical wave gauges and used this information to analyze the generation, transmission, and run-up of the landslide surge.
3.2 Experimental scheme
Experimental conditions for granular material surge tests
Tests tag no. | Experimental conditions | ||
---|---|---|---|
M_{s} (kg) | v_{s} (m/s) | h_{0} (m) | |
Cases 1–18 | 10, 20, 30 | 1.64, 3.28 | 0.1, 0.2, 0.3 |
Parameters of rock avalanche–debris flow
No. | M_{s} (kg) | ρ_{s} (× 10^{3} kg/m^{3}) | V_{s} (m^{3}) | b (m) | s_{1} (m) | s_{2} (m) |
---|---|---|---|---|---|---|
1 | 10 | 1.48 | 0.0067 | 0.4 | 0.015 | 0.023 |
2 | 20 | 1.48 | 0.0135 | 0.4 | 0.025 | 0.040 |
3 | 30 | 1.48 | 0.0201 | 0.4 | 0.025 | 0.070 |
In this table, s_{1} is the thickness of the landslide at the upper position (H1) and s_{2} is the thickness of the landslide at the lower position (H2).
4 Establishment of a formula for calculating rock avalanche–debris flow surge
4.1 Analysis of the landslide surge distribution
Due to the complex nature of the surge near the impact point, it was difficult to measure the maximum surge height near the impact pit; however, a stable surge amplitude could be measured in the propagation zone. In Fig. 5, note that the values measured at P1 exhibited a good linear correlation with theoretical values. The values measured at P2 and P4, however, were poorly correlated with the theoretical values. This can be attributed to the effects of the reflective boundary on the transmission process.
4.2 Calculation of surge height in propagation zone due to clastic landslide
4.3 Calculation of maximum surge run-up height
As shown in Fig. 6, as the initial water depth increased, the maximum landslide surge wave height decreased. Additionally, Fig. 6 shows that the maximum height of the landslide surge increased with the landslide velocity and volume. These findings are consistent with the previous studies [6, 14].
In order to further reflect the underwater movement characteristics of rock avalanche–debris flows, we take the duration of the underwater movement of the landslide as a variable.
5 Conclusions
In this work, we proposed a theoretical formula to calculate the surge height generated by a rock avalanche–debris flow near the impact pit. We carried out a series of model tests to verify the rationality of theoretical derivation. The test results show that variations in the measured value near the impact point and in the calculated result are consistent with the change of experimental variables. Due to the complexity of the wave making process, the method of calculating the surge height near the impact pit is a theoretical model, by which we can only approximately verify the rationality of theoretical derivation. In future studies, we will further refine the model and verify its rationality.
Notes
Acknowledgements
This research was supported by the National Program on Key Research Projects of China (Grant No. 2016YFC0802206) and the National Natural Science Foundation of China (Grant No. 41571004).
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