The two-phase fluid model is applied in this study to calculate the steady velocity of a debris flow along a channel bed. By using the momentum equations of the solid and liquid phases in the debris flow together with an empirical formula to describe the interaction between two phases, the steady velocities of the solid and liquid phases are obtained theoretically. The comparison of those velocities obtained by the proposed method with the observed velocities of two real-world debris flows shows that the proposed method can estimate the velocity for a debris flow.
A debris flow is the gravity flow of soil, rock and water mixtures, which is frequently initiated by a landslide, and is a common potential hazard throughout the world. For example, the debris flow in Zhouqu (China) on 8 August 2010 killed approximately 1700 people (Wang, 2013), and the debris flow that occurred in Afghanistan on 2 May 2014 killed more than 2000 people (Ahmed and Kakar, 2014). Debris flows can often occur following bush and forest fires. They pose a significant hazard in steep, mountainous areas, and have received particular attention in China, Japan, the USA, Canada, New Zealand, the Philippines, the European Alps, Himalaya-Karakorum, Kazakhstan and Russia.
The typical characteristics of the multi-phase fluid exhibited by a debris
flow have been demonstrated by many field observations (O'Brien et al., 1993;
Hutter et al., 1996; Hutter and Schneider, 2010a, b). In Takahashi's
discussion (Takahashi, 2007), it was found that a low-viscous debris
flow with a density higher than 1400 kg m
To understand the dynamics of the debris flow, including its initiation, runout and deposition, and to help analyze these dynamics and the hazard the flow poses, it is important to ascertain the velocity of the debris flow. The reason for this being that soils or rocks, and the fluid involved in a debris flow cause the dynamics of the debris flow to become more complicated, especially the existence of interactions between the solid particles and the liquid (Pudasaini, 2012). As observed in natural debris flow, the velocities of the solid and liquid phases may deviate substantially from each other, essentially affecting flow mechanics (Prochaska et al., 2008; Pudasaini and Domnik, 2009; Pudasaini, 2011, 2012; Revellino et al., 2004; Rickenmann et al., 2006; Teufelsbauer et al., 2009; Uddin et al., 2001; Yang et al., 2011; Zhu, 1992). Pudasaini (2011) presented exact solutions for debris flow velocity for a fully two-dimensional channel flow in which the velocity field through the flow depth and also along the channel were derived analytically. Several other models have been introduced to estimate the velocity of the debris flow, such as the Fleishman formula (Fleishman, 1970) and the mean velocity formula (Takahashi, 1991; Hashimoto and Hirano, 1997; Julien and Paris, 2010; Hu et al., 2013). These models provide some rough estimations of the flow velocity and are applied to predict the risk of the debris flow. However, the assumption of one-phase flow for these models leads to large modeling errors. Few theoretical results have been obtained to estimate the solid- and liquid-phase velocities for a two-phase debris flow (Chen et al., 2004, 2006). Although some empirical formulae have been introduced to calculate the velocity of a debris flow at special locations, such as the K631 debris flow located on the Tianshan highway in Xinjiang Province of China and the Pingchuan debris flow located on the trunk highway from Xichang City to Muli County in Liangshan Yi Autonomous Prefecture, Sichuan Province, China (Chen et al., 2006). Nevertheless, there is no general formula for calculating the velocity of a debris flow.
In this study, the two-phase flow model is applied to analyze the velocity of a debris flow. To focus on the velocity of the debris flow along the channel, a simplified, one-dimensional, two-phase model is considered here, and the motion equations governing the solid and liquid phases are deduced. Following Bagnold (1954), the interaction between the solid and liquid phases is obtained and the velocities of the solid and liquid phases in a debris flow are derived theoretically. This result provides a new theoretical method for estimating the velocities of the solid and liquid phases, which would be useful for evaluating the damage caused by a debris flow, estimating its arrival time, simulating its deposition area, and predicting risk. By comparing the theoretical results for the velocity and the empirical formulae for two natural debris processes, the numerical results show that the proposed method could more accurately provide velocities of solid and liquid phases for a debris flow.
This study is arranged as follows: in Sect. 2, the formulae to calculate the velocities of a debris flow are deduced, and in Sect. 3, the numerical validation of the theoretical results is made by means of two real-world debris flows. The conclusions are presented in Sect. 4.
Two difficulties arise in the calculation of the velocity of a debris flow: firstly, the diameters of the solid phase particles have a wide range, and secondly, the interaction between the solid phase particles and liquid phase slurry is difficult to describe exactly. However, recently, by developing a general two-phase debris flow model, Pudasaini (2012) included several important physical aspects of the real two-phase debris mass flows with strong phase-interactions, including the generalized drag, virtual mass force, Newtonian, and solid particle concentration gradient enhanced non-Newtonian viscous stresses. These model equations have also been put in well structured and conservative form. Numerical simulations and possible applications of these models can be found in Pudasaini (2014), Pudasaini and Miller (2012a, b). In order to deal with solid particles with different diameters, the diameter-equivalent method (Brunelli, 1987; Chen et al., 2004), which treats all particles with different diameters as the particles with the same diameter, is applied in this study.
Configuration of the equivalent two-phase debris flow:
In order to build a simple model for a debris flow to estimate the velocities
of its solid and liquid phases, the following assumptions are made:
The downstream is set as the No mass entrainment is considered, and there is no transformation
between the solid phase particles and liquid phase slurry (Chen et al.,
2006). Three inner forces are involved in the model: the interactions among
the solid phase particles, the interactions in liquid phase slurry and the
interactions between the solid phase particles and liquid phase slurry. A debris flow is assumed to be a homogeneous flow (Major and Iverson,
1999; Kaitna et al., 2007).
With these assumptions, and following the two-phase flow theory (see,
e.g., Pudasaini, 2012 for more detail), the governing equations for a debris flow
are obtained, which are written separately for the solid and liquid phases,
denoted by subscripts “s” and “f”, respectively. The mass conservation equations
for the two phases are written as
The momentum equations for the two phases take the forms (with the buoyancy
effect considered)
For detailed model derivation, and how different types of forces and interactions can arise and should be introduced in a real two-phase mass flow model, we refer to Pudasaini (2012).
In order to estimate the velocities of a debris flow using Eqs. (
By considering the gravity and the buoyancy of solid particles, the volume
force of the solid phase is written as
For two-phase debris flow, the surface forces on a control volume can been
divided into four parts (Chen et al., 2006). The surface forces of the solid
phase
The force from the solid particles outside the control volume mainly appears in
the form of impact among all the solid particles. The mechanical effects of
impact appear as the dispersion stress,
As the liquid phase slurry in a debris flow can be regarded as a generalized
Bingham viscoplastic material (Takahashi, 2007; Chen et al., 2006), the
rheological equation of the Bingham material can reflect the internal viscous
resistance of liquid phase slurry (Chen et al., 2006), i.e.,
There are several model parameters in the proposed model including
If the effect of turbulence in the liquid slurry is not considered, then
Eq. (
Combining Eqs. (
Combining Eqs. (
Substituting Eqs. (
Substituting Eqs. (
Adding Eqs. (
Integrating from 0 to
Subtracting Eq. (
Solving this above equation yields
The velocities of the solid and liquid phases for a debris flow are then
obtained via Eqs. (
Although the model solutions (Eqs.
In this study, we developed a new formula to estimate the solid- and
liquid-phase velocities in a debris flow, which is useful for understanding
the dynamics of the debris flow. Equation (
Solid- and liquid-phase velocity variations of a debris flow from
the channel position
However, Eq. (
The velocities of the solid and liquid phases in a debris flow are then given by
Next, we provide some numerical examples to show the dynamics of a debris flow along the channel. Figure 2 shows some numerical results for the solid- and liquid-phase velocities. The figure indicates that the liquid phase is faster than the solid phase, and the ratio of the velocities for two phases is about 0.790. Such exact solutions have also been presented previously by Pudasaini (2011) for avalanche and debris flows. We note that, for such a large velocity difference, at least the drag and the virtual mass force must have been included in the model as in Pitman and Le (2005) and Pudasaini (2012). However, the model here does not consider such effects.
Solid- and liquid-phase velocities of a debris flow at the channel
position
Solid- and liquid-phase velocities of a debris flow at the channel
position
The results of velocity calculation for the K631 (G217 highway) and Pingchuan debris flows.
The solid- and liquid-phase velocities at a point 300 m in the channel are shown in Fig. 3 for the different solid volume fractions; it can be seen that the velocity of a debris flow decreases as the solid volume fraction increases. However, 10 % increase in the solid volume fraction resulted only in very slight decrease in the solid and liquid velocities.
The solid- and liquid-phase velocities at 300 m in the channel are shown in Fig. 4 for the different equivalent diameters of solid particles. Here it can be seen that, as the equivalent diameter of solid particles increases, the solid-phase velocity of a debris flow decreases very slowly whereas the liquid-phase velocity increases very slowly. However, a 10 % increase in the equivalent diameters of solid particles resulted in almost no change in the solid and liquid velocities. Such discrepancies may have emerged due to the very simplified model consideration, or some possible inconsistencies in the use of the rheological models considered here. These problems could have been avoided by using a more complete and real two-phase debris flow model (Pudasaini, 2012) which included strong phase interactions.
In order to validate the estimation of velocities presented above, two
real-world debris flows – the K631 debris flow located on the Tianshan
highway in Xinjiang Province and the Pingchuan debris flow located on the
trunk highway from Xichang City to Muli County in Liangshan Yi Autonomous
Prefecture, Sichuan Province – are considered. The velocities obtained by
observations for the two debris flows, one a viscous debris flow and the
other a thin debris flow, are 11.59 m s
A one-dimensional model for a debris flow has been introduced to estimate the velocities of the solid and liquid phases. By applying the specific form of the volume force and the surface forces for the solid and liquid phases, theoretical results are used to estimate the velocities of the solid and liquid phases. These results are found to be valid by comparing the theoretical results with the experiential data for two real-world debris flows. Furthermore, the theoretical methods can estimate the velocities of a debris flow with different solid volume fraction parameters and different equivalent diameter parameters, which makes the theoretical results more useful for analyzing the debris flow dynamics, including the associated kinetic energy and impact forces.
The authors wish to thank the editor and anonymous reviewers for their helpful and valuable comments, which greatly improved this paper. This work was supported by the Fundamental Research Funds for the Central Universities (grant no. FRF-BY-14-036) and National Natural Science Foundation of China (grant nos. 11071238 and 11471034) and the National Center for Mathematics and Interdisciplinary Sciences, CAS and the Key Lab of Random Complex Structures and Data Science, CAS. Edited by: A. Baas Reviewed by: two anonymous referees