Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9578928, 9 pages

https://doi.org/10.1155/2017/9578928

## Superelevation Calculation of Debris Flow Climbing Ascending Slopes

^{1}School of Civil Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China^{2}Key Laboratory of Mountain Surface Process and Hazards, CAS, Chengdu 610041, China^{3}Institute of Mountain Hazards and Environment, CAS, Chengdu 610041, China

Correspondence should be addressed to Yong You

Received 28 November 2016; Revised 18 February 2017; Accepted 12 March 2017; Published 30 March 2017

Academic Editor: Sandro Longo

Copyright © 2017 HaiXin Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present a new method for calculating the superelevation of debris flow when it encounters obstacles in the process of flowing. Our calculation method is based on the Bingham Model for debris flow determination and considers the vertical difference of debris flow velocity and characteristic parameters of debris flow on a hypothetical basis. Moreover, we conducted an indoor flume experiment to verify the accuracy and reasonability of our calculation method. The experimental results showed that our method is able to accurately calculate the superelevation of debris flow with a root-mean-square error (16%). Furthermore, we provide an in-depth example of how our calculation method can be employed. Ultimately, we conclusively prove that our calculation method can be used for the superelevation calculation of debris flow climbing ascending slopes. Finally, we provide more exact parameters for debris flow protection engineering.

#### 1. Introduction

Debris flow causes superelevation when it encounters obstacles in the flow process, or when channels suddenly become narrow. This is the result of the kinetic energy of the debris flow being converted to potential energy in the space of a single instant [1]. Extremely high superelevation of debris flow can be disastrous and cause casualties, in part because people do not pay sufficient attention to the risks posed by debris flow [2]. For instance, debris flow in Qiongshan Gully in the upper Dadu River basin climbed a slope of more than 15 meters and smashed a building, killing 51 people in 2003 [3].

The accurate calculation of maximum superelevation of debris flow is crucial for designing debris flow prevention projects [1, 4, 5]. For example, when designing debris dams, builders must factor in how debris flow will impact the dam, such as the occurrence of superelevation on the wing wall. Superelevation on the wing wall is disadvantageous to the stability of the dam body because the debris flow could potentially bypass the wing wall and flush out on the sides of the dam, causing corrosion between the back of the dam and nearby gullies. Thus, the accurate calculation of maximum superelevation is paramount.

Additionally, design parameter calculations for drainage canals and protection embankments also require the exact calculation of debris flow superelevation in terms of debris flow encountering obstacles. If the calculation of superelevation is too low, the debris flow can bypass protection engineering projects and damage farmlands, villages, towns, and roads.

Previous studies have proposed calculation methods that use the mean velocity of debris flow and the law of conservation of energy for determining the maximum superelevation of debris flow climbing ascending slopes [1]. These superelevation results were typically used to back-calculate the debris flow velocity of large-scale flume experiments [6–9]. Mancarella and Hungr [10] used a dynamic model based on shallow-flow assumptions to study debris flow predictions for rapid landslides and avalanches against protective dikes and walls placed perpendicular to the path of the debris flow. Choic et al. [11] used flume experiments and the discrete element method (DEM) to study the interaction between debris flow and the baffles, a kind of structural countermeasure positioned along the flow path, which can change flow depth and cause superelevation. Iverson et al. [12] have argued that debris flow run-up is influenced by both smooth momentum fluxes and abrupt momentum jumps and that heights vary systematically with slope angle, effective basal friction, and the flow Froude number. To be successful, run-up predictions must account for the unsteadiness of incoming flows.

On one hand, the results of the above calculation methods have been flawed and inaccurate. For instance, calculations based on the mean velocity of debris flow had large errors, which implies that there are implicit dangers in current prevention and control engineering methods. On the other hand, alternative and potentially more accurate calculation methods are too complex to be applied in practical engineering. As such, we propose a new method to calculate the superelevation of debris flow climbing ascending slopes. The accuracy and applicability of our method were verified by the results of 9 indoor flume experiments. We conclude by providing more exact parameters for debris flow protection engineering.

#### 2. Theoretical Equation Based on the Bingham Model

We assume that the debris flow impinging on an obstacle is 2-dimensional (2D), based on Bingham Model and unaffected by the dynamics of run-up on the obstacle.

By comparing the differences between measured data and calculated data [1], we found that most superelevation calculations for debris flow were smaller than the measured value. The main reasons for this occurrence are as follows:

() The former calculation methods were deduced by energy methods:where is the superelevation of debris flow, is the function of the slope, is the mean velocity of the debris flow, and is the gravitational acceleration.

Equation (1) regards debris flow as a moving whole which treats a debris flow as a highly idealized frictionless point mass and calculates the superelevation of debris flow climbing ascending slopes as the height of the debris flow center of gravity (as a whole, ), whereas the actual superelevation of debris flow climbing ascending slopes is the edge height of the mud depth from the surface of the debris flow , (as shown in Figure 1).