Study on the calculation method of water in�ow velocity of loose rock landslide

: Based on rock structure mechanics and water sediment dynamics, considering the additional resistance caused by the interference of loose landslide particles on the flow structure, this paper deduces the resistance calculation formula of loose rock landslide particles when entering the water. It modifies the landslide velocity formula (ASCE) recommended by the American Society of Civil Engineers. The acceleration calculation formula and velocity calculation formula of bulk rock landslide entering water are obtained, and the determination method of main parameters in the formula is given. Based on the physical model test data of a three-dimensional loose rock landslide and the example data of the Xintan Landslide in the Yangtze River, the rationality of the calculation results of this formula and the ASCE formula is compared and analyzed. It shows that the ASCE formula calculates the average velocity of landslide movement on land, and there will be a noticeable deviation from the actual situation when it is used to calculate the water entry velocity of a loose rock landslide. The formula given in this paper is more practical and can be used to calculate the velocity of loose rock landslide entering the water.

analyzed the surge characteristics caused by landslide in Kootenal River according to the investigation data, and established the relationship between landslide surge height and landslide mass velocity; Gerald collected, sorted and analyzed the common empirical calculation formulas of landslide surge height, and expounded the applicability of relevant formulas (Gerald et al., 2003); Risio et al. (2016) discussed the surge height generated during the movement of landslide along the linear reservoir bank; Evers and Hager (2016) take the attenuation rate of head wave height as the research object, specifically analyze the main factors affecting the attenuation rate of head wave height, and establish the corresponding empirical relationship; Based on Edward Noda's research, Pan (1980) proposed an approximate estimation method for initial surge and propagation wave; According to the analysis of motion mechanics, Wang and Yin (2003) applied the slice method to determine the relevant parameters of sliding slope velocity, acceleration and head wave height, and selected Xintan landslide for example analysis; Based on the analysis of a large number of hydraulic model test data, China Institute of Water Resources and Hydropower Research (2010) found that the main factors affecting the surge height of reservoir landslide are the landslide velocity and landslide volume, and put forward the calculation formula of maximum head wave height; Li et al (2016) analyzed the influence of landslide shape, volume, sliding height and reservoir water depth on surge height based on the generalized model experiment of surge tank of block landslide in near dam reservoir area; Wang et al. (2020) studied the effects of landslide width, landslide thickness and sliding surface inclination on surge height by using the landslide surge test model of the Three Gorges Reservoir.
In practice, the scale and water depth of landslide mass are easy to be measured, but the velocity of landslide mass is not suitable to be measured. Therefore, correctly calculating the landslide movement speed and mastering the variation law of its momentum in the moving process is not only an essential premise for predicting a surge but also one of the critical parameters to be used in various landslide surge theories or empirical formulas. At present, the common calculation methods of landslide velocity mainly include the method recommended by the American Society of civil engineers (Chinese society of rock mechanics and Engineering, 2021), the kinetic energy theorem method (ibid), the slice method (Pan, 1980) and time increment step method (Körner, 1976). These methods are based on taking the sliding body as a particle and analyzing the position and speed of particle movement. However, in practice, the shape of rock landslide changes continuously due to extrusion, dispersion, crushing, and abrasion in sliding. From the initial regular block to the formation of loose particles when entering the water, it is difficult to determine the position of the landslide centroid in threedimensional space. In addition, the velocity of the mass center of the whole block is constantly changing from the initial static position of the landslide to the collision with the water surface; the underwater part should also be considered when determining the three-dimensional position of the mass center, this evolution process is quite complex and challenging to measure in the test. Therefore, the empirical estimation method is mainly used to calculate sliding speed, especially the accidental mutation of landslides, which can only roughly estimate its rate.
Landslide surge is composed of the fluctuation caused by the impact of the slider on the water body and the surge height caused by the displacement of the slider from the water body. When the landslide enters the water, it will be affected by the water resistance, and the acceleration and velocity will decrease. Strictly speaking, the movement velocity of a landslide on land can not be directly used to calculate the surge formed by its water inflow impact. Based on the study of water wave dynamics, Huang et al. (2018) pointed out that for the existing landslide velocity research, the calculation results are too conservative. However, various calculation formulas consider different forms. The main reason is that when considering the movement of a landslide into the water in the past, it is generalized as sliding on a smooth inclined plane while ignoring the influence of water resistance and friction force after the landslide meets water. Therefore, the impact of wading resistance and friction coefficient should be considered in speed selection to reduce error value. Based on considering water resistance, Dai (2010) modified the method recommended by the American Society of Civil Engineers and the slice method by using the formula of particle resistance in the fluid. Then, the motion velocity of the Dayantang landslide is calculated using the modified motion equation method. Based on Pan Jiazheng's procedure, Huang (2011) comprehensively considered the influence of water resistance and friction coefficient and claimed that the underwater friction coefficient of landslide is much smaller than that onshore. Based on 30 groups of physical tests, Wang et al. (2012) discussed the variation law of plane sliding and curved sliding wading resistance on sliding block speed. The results showed that water resistance significantly impacts underwater landslide speed. In the same year, Zhang et al. (2012) deeply analyzed the influence of landslide cohesion, water pressure difference on both sides of soil strip, and water resistance on landslide speed based on the theory of Pan Jiazheng formula and Wang Yang formula; it concluded that landslide cohesion is the most significant on landslide speed. However, these studies regard the landslide mass as a single rigid block entering the water and do not consider the collision between loose landslide particles. The results can only be used to calculate the velocity of a single block landslide entering the water.
Natural rock landslide mass is mainly composed of rock structure, fissure, fault, weak interlayer, void, etc. Rock structure can be divided into block structure and plate structure. Structural planes composed of various cracks, marks, and weak interlayers can be divided into weak structural planes and complex structural planes. They are combined and arranged differently in rock mass to form different types of rock mass structures. In the process of movement and sliding, most rock landslide mass may crack under the effects of extrusion, collision, and scraping, and cut the landslide mass into blocks of different sizes and countless fragments to disperse shown in Figure 1 (Han, 2019). With the acceleration of the decline rate, the degree of dispersion will also intensify. Therefore, to determine the surge height value formed by the water inflow impact of loose rock landslide more realistically, it is necessary to consider the additional resistance to the flow structure caused by the spatial distribution of loose landslide particles and explore the calculation method of the velocity of loose rock landslide entering the water.

Derivation of calculation formula
The study of particle motion and force in fluid began with the flow field condition of prolonged flow. Stokes was the first to analyze particle motion using viscous fluid theory and first studied the linear harmonic motion of a single sphere, cylinder, and infinite plate in a viscous fluid. He deduced the resistance of the creeping viscous liquid to particles by using the N-S equation, which contains two terms related to particle acceleration and velocity, respectively. The drag term about speed is , and es R is the particle Reynolds number.
Generally, the Reynolds number used in the flow around a single sphere is: In the formula, D -sphere diameter (m);  -viscosity coefficient of water flow movement (m 2 /s); V -relative velocity of particles and water flow (m/s). Unlike the underwater movement of single block landslide particles, due to pores in the sliding block particle group of bulk landslide and the mutual collision and diffusion deformation in entering the water, complex water relative flow is generated in the sliding block particle group. If the porosity in the particle group of the loose landslide is  and the average velocity of the water flow is u , then the relative speed between the particle group and the water flow is: Therefore, the flow Reynolds number of particle group of loose landslide is: In fluid mechanics, the resistance of solid particles moving in liquid, that is, the fluid resistance, can be expressed as (Yang and Chen, 1989): In the formula, v -The movement speed of landslide mass relative to the fluid, that is, the movement speed of sliding block (m/s). According to the investigation of landslide movement process of wading examples and the physical model test of landslide surge, for bulk landslide, when the landslide moves to the water surface, it is not a single block particle, but a group of bulk particles of different sizes entering the water at the same time. The resistance coefficient of the particle group in the deformation stage is more significant than that of the single block particles. In addition to the flow resistance caused by the general relative movement, there is also additional resistance caused by the interference effect on the flow structure caused by the collision between particles. Assuming that the extra resistance caused by the crash between slider particles and the interference with the flow structure leads to the change of resistance coefficient, according to formula (4), the resistance of each sliding block particle in the particle group of bulk landslide can be written as: In the formula is the resistance coefficient, including additional resistance. The number of particles per unit slider particle group volume is: The particle number of the whole landslide particle group is: In this formula, s is the thickness of landslide mass (m).
Therefore, it can be obtained that the resistance of the sliding block particle group of the loose landslide when entering the water is: For the landslide particle group composed of spherical blocks, the resistance of water flow acting on the whole particle group is: According to formula (8) or formula (9), the commonly used calculation formula of landslide velocity recommended by the American Society of Civil Engineers (from now on referred to as "ASCE formula") can be modified to obtain the calculation formula of acceleration and speed of loose rock landslide when entering the water. ASCE formula regards the sliding body as a whole and makes the particle motion with the center of gravity. The motion speed of the sliding body is deduced according to Newton's second law and kinematics formula. The method is that the force of the sliding body along the sliding surface is equal to the difference between the sliding force and the anti-sliding force, as shown in Figure 2:

Fig. 2 Schematic diagram of formula recommended by American Society of Civil Engineers
In the formula, L -the contact surface length between the sliding body and sliding surface (m).
According to formula (9), formula (11) and formula (12) are modified to obtain the calculation formula of acceleration and velocity of loose rock landslide when entering the water as follows: 3 Determination method of main parameters When using formula (13) and formula (14), it is necessary to determine the resistance coefficient / D C , particle group's porosity, particle size D , etc.
According to formula (3), the flow Reynolds number of loose particles / e R can be calculated. Generally, when the landslide enters the water, the relative water flow velocity is tremendous. The turbulence of water flow is enhanced and / e R will be much greater than 1000. According to Stokes's theory, the inertia term caused by the acceleration of water particles is substantial. Particles will separate when moving in water flow, and the viscous force can be ignored. At this time, the resistance coefficient / D C is a constant, that is: ② Determination of porosity  : The pore volume per unit volume in the slider particle group to the volume of the whole slider particle group is called porosity. If the porosity of the slider particle group is z  and the thickness of the particle group is z s in the natural accumulation state, then the number of slider particles with size D in the slider particle group is: After the sliding block particle group enters the water, the particle group will collide and diffuse under the action of water flow resistance. When the thickness increases and the porosity changes  , the number of particles can be written as follows: . Therefore, the porosity of the particle group after the deformation of the loose landslide is: In the physical model test, the porosity under natural accumulation and the porosity of accumulation after the landslide mass enters the water can be measured directly according to the prepared loose rock landslide sliding block particles. For the actual landslide mass, the natural accumulation state and the porosity and thickness after deformation can be obtained according to the field sampling test and analysis.
③ Determination of particle size of the particle group For objects with a regular shape, such as spheres, cubes, columns, and disks, the particle size can be characterized by diameter or side length. However, for the sliding block particles of bulk landslide, the shape of each sliding block is highly complex and has unique appearance characteristics. Therefore, the concept of equivalent diameter or average particle size in sediment mechanics can be used to accurately characterize the particle size of an irregular sliding body. For single particle or loose landslide particles with a uniform particle size distribution, the particle size is generally expressed by ball equivalent diameter, triaxial particle size, projected particle size, screening average particle size, hydraulic diameter, and other methods. The equivalent particle size of the sphere refers to the fact that the landslide particles are identical to the spherical particles with the same attributes as the diameter. It is a particle feature representation method close to the characteristics of irregular particles. It is necessary to obtain the volume or surface area of the particles through high-precision measuring instruments (CT scanners). For the projected particle size of particles, the shape of landslide particles needs to be projected onto a plane. Then the projection graphics are processed to obtain the characteristics of the projection graphics, such as area, longest diameter, and so on. To screen the average particle size, thoroughly screen the rock debris particles, and then correct the pore size ratio of the adjacent screen of the rock debris of this particle size. The diameter of spherical particles with the same settling velocity as particles is generally called the hydraulic diameter of particles, which usually needs to be measured by experiments. In practice, the particle size of most bulk rocks can be counted on-site, and the triaxial length of particles, as shown in Figure 3 (Zhang, 2019), can be given. The average particle size can be calculated by the following formula. In the formula, a, b and c are the lengths of the long, medium, and short axes of rock debris particles, respectively. For the particle group of loose landslides with an uneven particle size distribution, the above representation method is not accurate enough. Generally, according to the drawn grading curve of the particle group, as shown in Figure 4 (Qian and Wan, 1983), several characterization methods such as weighted average particle size, geometric average particle size, and median particle size are used to give the average particle size of the particle group. a. weighted average particle size 100 100 In the formula, i P  represents the percentage of particles with size i D in the weight, volume, or number of the whole landslide mass.
In the formula, on the grading curve, respectively. c. median particle size On the grading curve, the particle size corresponding to % 50  P is called the median particle size, which is represented by 50 D .

Comparative calculation between this formula and ASCE formula 4.1 Comparison and calculation of test data
The physical model test of three-dimensional bulk rock landslide surge under the condition of flow dynamics in the mainstream of the Three Gorges Reservoir area is selected as a comparative calculation example (model water depth of 0.25m, flow velocity of 0.24m/s, landslide volume of 0.4m 3 , and slip surface inclination of 40 o ), as shown in Figure 5. See Table 1 for the proportioning data of the loose rock landslide model used in the test. Using the triaxial length of particles and formula (19), the average particle sizes of five small blocks can be calculated as 12.08cm, 10.26cm, 8.43cm, 6.60cm, and 4.76cm, respectively. According to the proportion percentage of each small block in Table 1 and formula (20), the average particle size of the landslide particle group can be calculated as 9.71cm. According to the test, the porosity of the natural accumulation of the landslide mass is 0.39, the porosity of the accumulation mass after the landslide mass enters the water is 0.46, the internal friction angle of the sliding surface is 18°, and the cohesion of the sliding surface is 10 kPa. The distance between the gravity center of the landslide and the water surface is 0.7m, so the height between the gravity center and the water surface is 0.45m.  According to the ASCE formula (12), the average velocity of the model landslide is 2.44m/s, which is converted to the prototype landslide mass is 20.41m/s. The water inflow velocity of the model landslide calculated by the formula (14) in this paper is 1.95m/s, which is converted to the water inflow velocity of the prototype landslide is 16.31m/s. It shows that the calculation result of the ASCE formula is larger than that of the formula given in this paper.

Comparative calculation of Xintan Landslide in the Yangtze River
Xintan landslide is located on the North Bank of the exit of Bingshu Baojian Gorge in the upper section of Xiling Gorge of the Yangtze River. It faces the Lianziya dangerous rock mass across the river and is about 26km away from the sanding dam site of the Three Gorges Project. Xintan landslide has experienced many large-scale landslides in history, a large-scale slide occurred at 3:00 on June 12, 1985, which lasted 4 minutes and 7 seconds, and the whole Xintan town was destroyed instantly. The entire terrain of the Xintan landslide is high in the north and low in the south, and the trailing edge of the landslide is steep. The plane shape is irregular quadrilateral, the rear edge elevation is about 900m, the front edge elevation is about 60m, the length is nearly 2000m, and the area is approximately 0.75×106m 2 , with an overall volume of 3×107m 3 , the relative height difference between the rear wall of the landslide. The riverbed is about 800m, the average longitudinal gradient is 23°, and the local steepness and slowness are different. The rear edge of the landslide is narrow, about 300m, the front edge is wide, about 500~1000m, and the average width is 450m, as shown in Figure 6 (Liu, 2013).

Fig. 6 Xintan landslide and site treatment
Xintan landslide belongs to accumulation landslide. The rear edge is mainly composed of Carboniferous and Permian limestone fragments. The slope section of Jiangjiapo in the middle is primarily composed of limestone fragments. The front edge consists of semi-rounded primary angular limestone fragments, intercalated with loam, calcareous nodules, and sand gravel, intercalated with stone and cemented rock. The bedrock of the sliding bed is Silurian yellow, green siltstone, shale, and sandy shale. The thickness of deposits is generally between 30~40m and 86m in local sections. According to the particle size and grading composition of landslide block stone measured by the Xintan landform investigation team of Nanjing University (see Table 2), the average particle size of Xintan landslide block stone is calculated to be 1.47m. The porosity of landslide mass is 0.45, the internal friction angle of the sliding surface is 17.5°, the friction coefficient is 0.277, and the cohesion of the sliding surface is 21.0kPa. The density of landslide mass is 2.3×10 3 kg/m 3 , and the viscous resistance coefficient is 0.18. When the landslide is unstable, the water level is about 62m, and the height from the weight center of the landslide to the water surface is about 220m. When the Xintan landslide occurs, the Yangtze River is a natural channel with a maximum water depth of about 40m, an average water depth of about 30m, a water surface width of about 380m, and an average flow velocity of about 3.0m/s. As the thickness of the deposit is generally between 30~40m, the thickness of the landslide mass is close to the water depth of the river. After the landslide is unstable, it slides into the Yangtze River with a total volume of about 3 million m 3 , blocking about half of the river course; the landslide speed is about 10~30m/s, the average rate when entering the water is about 20m/s, and the maximum head wave height is approximately 34.24m. The slope angle of the opposite bank slope is 19.71°, the wave height in front of the opposite bank slope is 16.33m, and the climbing wave on the opposite bank is about 54m; ships within 8km are overturned, which affects the range of 16km upstream and 26km downstream. According to the above relevant parameters, the average Xintan landslide mass calculated by the ASCE formula (12) is 54.34m/s, 171.7% higher than the actual average velocity of 20m/s, and 81.1% higher than the actual maximum sliding velocity of 30m/s. The water inflow velocity of Xintan landslide mass calculated by the formula (14) in this paper is 22.56m/s, which is 12.8% higher than the average velocity of actual water inflow. Thus, it is close to the actual value. Therefore, it can be shown that the calculation result of the ASCE formula is significantly larger than the reality, and the calculation result of this formula is more in line with reality.

Conclusion
(1) The commonly used landslide motion velocity method, slice method, kinetic energy theorem method, and time increment step method recommended by the American Society of Civil Engineers are based on taking the sliding body as a particle and analyzing the position and velocity of particle motion. Because its characteristic parameters neither consider the volume of the slider and the characteristics of the landslide shape, nor the change of velocity speed process of landslide movement, it can only calculate the average velocity of the landslide on the land. Therefore, there will be a noticeable deviation from the actual situation when used to calculate the surge height formed by the water inflow impact of bulk landslide.
(2) Based on rock structure mechanics and water sediment dynamics, and considering the additional resistance caused by the interference of loose landslide particles on the flow structure, this paper deduces the calculation formula of resistance when the loose rock landslide particles enter the water, and then modifies ASCE formula to obtain the acceleration and velocity calculation formula of loose rock landslide when entering the water. At the same time, the determination method of main parameters such as resistance coefficient, the porosity of particle group, and particle size of particle group in the formula has also been established.
(3) Based on the physical model test data of three-dimensional loose rock landslide surge under the condition of flow dynamics inmainstreamtream of the Three Gorges Reservoir area and the data of Xintan landslide in the Yangtze River, the ASCE formula and the formula given in this paper are used to calculate the water entry velocity of landslide. The calculation results show that the calculation results of the ASCE formula are larger than those given by the author. The calculation results of the author's formula in this paper are more in line with reality. They can be used to calculate the velocity of loose rock landslides entering the water to reasonably determine the height of landslide surge and their disaster degree.