A rocky granular flow is commonly formed
after the failure of rocky bank slopes. An impulse wave disaster may also be
initiated if the rocky granular flow rushes into a river with a high
velocity. Currently, the granular mass–water body coupling study is an important
trend in the field of landslide-induced impulse waves. In this paper, a full
coupling numerical model for landslide-induced impulse waves is developed
based on a non-coherent granular flow equation, i.e., the Mih equation. In this
model, the Mih equation for continuous non-coherent granular flow controls
movements of sliding mass, the two-phase flow equation regulates the interaction
between sliding mass and water, and the renormalization group (RNG)
turbulence model governs the movement of the water body. The proposed model is
validated and applied for the 2014 Tangjiaxi landslide of the Zhexi Reservoir
located in Hunan Province, China, to analyze the characteristics of both
landslide motion and its following impulse waves. On 16 July 2014, a rocky
debris flow was formed after the failure of the Tangjiaxi landslide, damming
the Tangjiaxi stream and causing an impulse wave disaster with three dead and
nine missing bodies. Based on the full coupling numerical analysis, the
granular flow impacts the water with a maximum velocity of about
22.5 m s

Impulse waves are usually generated in reservoirs, rivers, lakes, and seas as rock/soil masses impact water, resulting in huge economic losses and casualties (Wang et al., 1986; Fritz 2001; Scheffers and Kelletat, 2003; Alvarez-Cedrón et al., 2009; Silvia et al., 2011; Huang et al., 2012). This fact urges people to pay attention to landslide-induced impulse waves, which is an interdisciplinary study related to rock/soil mechanics and fluid mechanics. A large number of studies have been done on landslide-induced impulse waves, including analytical, physical, and numerical methods. The analytical solutions are derived from extensive sources, such as experimental and empirical formulae, where their application scope is limited to their sources (Kamphuis et al., 1970; Ataie-Ashtiani et al., 2008; Wieland et al., 1999; Ursell et al., 1960; Fritz et al., 2002; Huber and Hager, 1997; Heller, 2007; Yin and Wang, 2008). Due to the considered simplifications for analytical solutions, it is hard to have an overall grasp of the landslide-induced impulse wave disaster (Heller et al., 2009). The scaled physical experiment method can well reproduce or preview the dynamic process of landslide-induced impulse waves (Ball, 1970; Davidson and Whalin, 1974; Muller and Schurter, 1993). However, it requires a large amount of data, time, and money and occupies a large amount of space (Huang et al., 2014). However, the numerical analysis method can help us have a relatively comprehensive analysis of the landslide-induced impulse wave disaster; it has the advantages of being precise, economic, and reasonable, as well as having highly visible results (Heller et al., 2009). Therefore, the numerical analysis method is an efficient tool in the study of landslide-induced impulse waves (Yavari-Ramshe and Ataie-Ashtiani, 2016).

Regarding the granular mass–water body coupling system, three major numerical simulation methods have been recently applied, such as (a) single model, (b) simplified model, and (c) full coupling model (Yavari-Ramshe and Ataie-Ashtiani, 2016). Each model may apply a mesh-based (e.g., finite difference method, finite element method, FEM; finite volume method, boundary element method) or a particle-based (smoothed particle hydrodynamic, material particle method, etc.) method (Yavari-Ramshe and Ataie-Ashtiani, 2016) for numerical discretization of its model equations. In the single simulation method for a landslide-induced impulse wave, the phase of landslide movement and granular mass–water body interaction is regarded as the formation of the initial impulse wave, and generally the motion of the sliding mass is considered to the motion of a rigid block. Therefore, various kinematic formulas, such as Newton's laws of motion, are applied to calculate the motion of the sliding mass (Heller, 2009; Huang et al., 2012, 2016). Then, various empirical or experimental formulas of landslide-induced impulse waves are adopted to calculate the characteristics of the initial impulse wave caused by the landslide (Walder et al., 2003; Tappin et al., 2008; Watts et al., 2003; Ataie-Ashtiani and Malek Mohammadi, 2007). With the initial impulse wave as the initial input or boundary condition, the numerical simulation singularly aims at calculating the spread and run-up of impulse waves. Some examples of these models are TUNAMI (Fumihiko et al., 2006), MOST (Titov and Gonzalez, 1997), FUNWAVE (Joseph et al., 2003; Tappin et al., 2008), and CLAWPACK (Randall, 2006). Their accuracy and application scope largely depend on the source models for the initial impulse wave. Many scholars (Watts et al., 2003; Ataie-Ashtiani and Malek-Mohammadi, 2008; Di Risio et al., 2011; Yin et al., 2015c) have studied initial impulse wave models in a different range of application and introduced a large number of source models.

The simplified simulation for the landslide-induced impulse wave aims to
simplify landslide motion in calculation. Some landslides are simplified as
rigid bodies whose motion is mainly described with Newton's laws of motion
such as gravity, friction, and coupled water resistance (Das et al., 2009;
Basu et al., 2009; Huang et al., 2013). For example, Yin et al. (2015a)
simulated the motion of the Qianjiangping landslide as a rigid rotator and
calculated the impulse waves. Harbitz et al. (2014) simulated a rockslide
with the volume of 5

The full coupling model for landslide-induced impulse waves is a currently emerging method, which has been receiving considerable attention recently. The full coupling model can have a relatively accurate description of the motion of sliding mass, interaction with water, and consequent impulse waves. Simplified models have obvious difficulties in achieving an accurate description of the landslide motion. Accordingly, numerical models which consider the rheological behavior of the sliding mass in their calculations have been recently applied more often. The most applied continuous rheological models so far include the Coulomb model, Herschel–Bulkley model, Bagnold model, and Bingham model (Shakeri Majd and Sanders, 2014; Cremonesi et al., 2011; Yavari-Ramshe and Ataie-Ashtiani, 2016; Xing et al., 2016). Those that describe avalanche, landslide, or debris flow motions in discontinuous medium models are mainly the FEM–discrete element method model (FEM–DEM; Morris et al., 2006; Munjiza, 2004; Li et al., 2015) and DEM model (Smilauer et al., 2010; Brennen, 2005; Utili et al., 2014). For generation, propagation, and run-up of impulse waves, technologies that can finely depict large free-surface deformations, such as VOF (volume of fluid) or non-hydrostatic models (Yavari-Ramshe and Ataie-Ashtiani, 2016), are adopted. Crosta et al. (2013) used an arbitrary Lagrangian–Eulerian–FEM (ALE–FEM) approach for a 2-D–3-D simulation of landslide and impulse wave. Glimsdal et al. (2013) developed a model for submarine landslide and tsunami, where the landslide motion was simulated as a deformable viscoplastic Bingham fluid. Zhao et al. (2015) used a 3-D DEM–computational fluid dynamics (DEM–CFD) coupling method to simulate the motion of the Vajont landslide and the resulting impulse waves. By combining a landslide dynamic model and a tsunami model, Sassa (2016) presented an integrated numerical model simulating the complete evolution of a landslide-induced tsunami. This model was applied to the 1792 Unzen–Mayuyama mega-slide and tsunami disaster analysis.

In this paper, a full coupling model is developed for landslide-induced impulse waves based on a non-coherent granular flow equation. The continuous granular flow model of Mih (1999) is applied to simulate the motion process of the rocky granular flow after rockslide. Then, a two-phase flow model is adopted for granular mass–water interaction coupled calculation. Taking the Tangjiaxi rockslide and the resulting impulse wave as a case, a numerical analysis for the whole process is done to study the motion of the granular flow, its accumulation process and consequent formation, propagation, and run-up of impulse waves. Meanwhile, the validity of the full coupling model for landslide-induced impulse is checked.

Rockslides can be characterized by a rapid evolution, up to a possible transition into a rock avalanche, which can be associated with an almost instantaneous collapse and spreading (Utili et al., 2014). The failure of a rocky slope is commonly followed by a high-concentration and non-coherent rocky granular motion. A large number of non-coherent coarse solid grains as well as relatively few fine grains are densely distributed in the granular flows. They flow, deposit, or erode along their motion routes, which generally span long distances (Crosta et al., 2001). Such flowing characteristics of motion can be described through both the continuous rheological model and the discontinuous model. The discontinuous model for particle flow simulation has a natural similarity. For the discontinuous method, grains are generally simplified to be a sphere. These grains can interact with each other through well-defined microscopic contact models (Hertz, 1882; Zhang and Whiten, 1996; Johnson, 1985) and with the fluid (e.g., water or air) by empirical correlations of fluid and solid interaction models. However, the discontinuous method means a large challenge for individual researchers. That is because even for a small rockslide, the simulation will require numerous cells and huge computational resources, which is hard to process with personal computers (Utili and Crosta, 2011), whereas the model based on continuous granular flow is free from this problem.

High-concentration granular flow was studied by several researchers such as Bagnold (1954), Savage (1978), Hanes and Inman (1985), Wang and Campbell (1992), Iverson (1997), and Mih (1999). Some rheological models such as Coulomb and Voellmy consider no viscosity or shear rate in their rheological formulations (Iverson, 1997). In this study, the present continuous granular flow model is built using viscous fluid.

Landslide rheology describes landslide motions with shear stress (

Extensive work, beginning with the 1954 work of Bagnold (1954) has been
summarized and further extended to a larger range of experimental conditions
by Mih (1999). He described the shear stress of a granular flow as follows:

The equation contains fluid viscous and impact coefficients. The fluid viscous coefficient is a constant. The impact coefficient has been correlated to the properties of the solid and fluid. The equation agrees reasonably well with several sets of experiments by different investigators which cover a wide range of granular flows (Mih, 1999).

The granular flow is treated as incompressible fluid when applied with the
shear stress equation of Mih (1999). The coupling model of granular flow and
water adopts a two-phase model with two incompressible fluids having different
densities. Supposing the water has density

The renormalization group (RNG)

In particular, the RNG model is known to describe low-intensity turbulence flows and flows having strong shear regions more accurately. The RNG model selected has already been successfully used to simulate impulse waves generated by landslides (Serrano-Pacheco et al., 2009; Basu et al., 2009; Das et al., 2009; Choi et al., 2007).

A full coupling numerical analysis model for landslide-induced impulse waves is built based on coupled control equations. The model can stimulate the landslide motion of non-coherent granular flow and the generation, propagation, and run-up process of impulse waves. The Tangjiaxi landslide event in the Zhexi Reservoir, Hunan, China, is simulated as an example to analyze the whole process of the landslide motion and the impulse wave.

At 07:00 local time (LT) on 16 July, the Tangjiaxi landslide occurred on the left bank of the Tangjiaxi stream, a tributary of the Zhexi Reservoir. The impulse wave induced by the Tangjiaxi landslide destroyed the nearby residential area. The landslide is 700 m away from the mainstream of the Chanxi stream (tributary of Zi River) and 10.6 and 11.2 km away from the Tangyanguang landslide site and Zhexi Dam along the watercourse, respectively (Fig. 1). The Zhexi Dam is located in the midstream of Zi River in Anhua County, Yiyang City, Hunan Province, China, and 15 km away from the seat of Anhua County. The Zhexi hydroelectric station, which began to impound in February 1961, is a large hydroelectric station. The Tangyanguang landslide occurred on 6 March 1961. It is the first impulse wave disaster generated by landslide since the founding of the People's Republic of China. The huge wave generated by the Tangyanguang landslide overtopped Zhexi Dam and killed 64 individuals (Du, 1988). The impulse wave disaster generated by the landslide happened again in this reservoir, which drew more attention.

The location of the Tangjiaxi landslide in the Zhexi Reservoir, Hunan Province, China.

The landform of the Tangjiaxi stream valley belongs to the type of medium gorge.
The elevation of the highest mountain in this valley is 650 m, while the
bottom elevation is 140–170 m generally. The overall flow direction of
the Tangjiaxi stream is 245

The rain continued for almost half a month from late June to early July in 2014. The daily rainfall was 98.5 mm around 4 July. The Zhexi Reservoir was hit by a rainstorm on 13 July again. The rainfall reached 102.5 mm on 15 July and, more seriously, 239 mm on 16 July (Fig. 2). Rainfall increased the weight of sliding mass, formed greater underground water dynamic pressure, and decreased anti-sliding strength (Thomas, 2003; Wang et al., 2004). Persistent rainfalls and heavy rainstorm directly triggered the failure of the landslide.

Precipitation data monitored in the village of Sifang, 3.6 km from the landslide.

Photo of first slide, taken by a local villager on 16 July, 07:00 LT.

According to the description of many local survivors, the first slide occurred around 07:00 LT on 16 July. Figure 3 shows the scene of the first slide. Starting from the toe of the slope, the first slide was a shallow soil slide which destroyed one of the three houses on the sliding mass. There was a short quiet period after the first slide. At about 10:20 LT, rock blocks rolled down from the top of the slope and the global slide started. As soon as the landslide mass started to run out, rocks broke, crashed and rumbled down to the slope foot, and houses were buried quickly. The mass impacted the Tangjiaxi stream at a high speed and induced huge waves, and the still-water level was 169.5 m above sea level (a.s.l.).

As shown in Fig. 4, the morphology of the landslide scar was triangular in shape.
The crown elevation of the landslide was about 315 m, and the elevation of
the outlet was about 155 m. The height difference was 160 m. At 26 m above
the water surface, the landslide was 95 m wide, and, at 56 m above the water
surface, the landslide width reached 80 m. Much closer to the crown, the
width of the landslide was smaller. The landslide was 15 m thick on average,
with a total volume of 160 000 m

The scene of the Tangjiaxi landslide, taken on 23 July 2014, when the water level was 167 m a.s.l. The river was full of wood and debris, which were the destroyed building materials.

Geological engineering section of the Tangjiaxi landslide.

The underlying bedrock of the Tangjiaxi slope is a Nantuo Formation (

After the landslide failed, cataclasite structure rock mass disintegrated quickly. The accumulation of sliding mass was mainly composed of rock blocks of different sizes. Medium and large rock blocks were mainly in the lower-middle part, with a maximum length of rock blocks of about 2.5 m. Rock blocks in the accumulation, in a sharply angular shape with an average diameter of 30–40 cm, are overhanging stacked in the accumulation zone (Fig. 6). The few gravelly soils on the accumulation site were mainly distributed on the flanks of the landslide and at the front edge of accumulation fan. These soils were mainly derived from weathered layer and eluvial deposit of the original slope.

Accumulated blocks after the Tangjiaxi landslide failure, taken on 23 July 2014.

Part of the sliding mass was accumulated in the watercourse and some stayed
on the slope. The landslide dam raised the river bed and halted part of
the upstream water to form a small landslide lake. The landslide dam was high
downstream and low upstream, with a bulge in the middle. Two terraces were
formed on the vertical section. The dip angle of the deposits on the terrace
was about 33

Profile photo of the Tangjiaxi landslide, taken on 23 July 2014, when the water level was 167 m a.s.l.

The plot of run-up of the impulse wave generated by the Tangjiaxi landslide. The photos describe the scenarios of houses and trees damaged where marked by

Witnesses reported that it took only several seconds for the landslide to
slide into the water and form the landslide dam. Calculated at 10 s for
the sliding duration time, the landslide barycenter is about 70 m above
the still-water surface, and the sliding distance is about 120 m. It is roughly estimated
that the biggest impact speed is about 24 m s

Despite the 10 m depth of the watercourse in the landslide zone, the limited water gained a large amount of energy from the high-speed sliding and formed huge impulse waves. As shown in the field survey, the maximum run-up was 22.7 m and occurred in the opposite bank of the landslide; the upstream maximum run-up was 19.5 m and occurred in a gully about 100 m upstream. At the downstream, with the increase in distance from the source of impulse waves, the run-up decayed. The maximum run-up at the river mouth, where the Tangjiaxi stream flowed into the Chanxi stream, was 1.8 m (Fig. 8). As the Tangjiaxi stream flowed into the Chanxi stream nearly vertically, the water surface suddenly became very wide, the impulse waves decayed rapidly, and no sign of impulse waves was seen on either bank of the Chanxi stream.

The computational domain which is considered to simulate the Tangjiaxi
landslide-induced impulse wave by the full coupling numerical model covers
the landforms of the valley where the Tangjiaxi landslide occurred. The domain is
792 m long and 684 m wide including the valley source of the Tangjiaxi stream
at the tail of the Zhexi Reservoir, with the lowest elevation at 140.0 m and the
maximum mountain elevation at 740.2 m (Fig. 9). The digital elevation model
of the Tangjiaxi sliding mass is plotted based on the drilling survey and the
topographic maps before and after the landslide, with a volume of about
158 000 m

Main parameters for Mih equation calculation.

The water surface elevation in the model is 169.5 m a.s.l., and the still-water surface is the initial condition.

Numerical model for the Tangjiaxi landslide-induced impulse waves. The red points refer to the velocity monitoring points of the sliding mass motion, and the blue ones refer to the process monitoring points for water level.

In this simulation, the following aspects of the Tangjiaxi landslide event are analyzed: the motion process of the sliding mass and the process of impulse wave and the model's validity, which was also checked through comparison with the field survey results.

The model analysis starts with the movement of the sliding mass. The
depth-averaged velocity curves at different elevation points of the sliding
mass show that the time to reach the maximum velocity varies for
different parts of the landslide. Most of the landslide parts reached the
maximum velocity before impacting the opposite valley at the sixth second. The
maximum sliding velocity of the area at the rear edge (V0) was about 16.6 m s

Depth-averaged velocity process plot of monitoring points in the sliding mass. See Fig. 9 for positions of VO–V1.

Observed from the landslide configuration at different times, the motion of
the sliding granular flow on land is generally within the scope of the
sliding mass. After

Instantaneous state of the Tangjiaxi landslide and river surface at

Changes of plane shape after the Tangjiaxi landslide failure.

The depth profile of section A-A

A-A

Depth process plot of monitoring points in the sliding mass.

Transient condition of river water and the vector diagram of mass. The arrows indicate the direction of movement and the colors indicate the magnitudes shown in legend.

The motion results of the Tangjiaxi landslide simulated by the granular flow model show no significant differences from those seen in the field survey, basically reflecting the real motion process and characteristics of the landslide. A huge impulse wave was induced in the stream due to the motion of granular flow.

After the sliding mass occupied the watercourse, it pushed and supported the
river water to move outwards and upwards in an arc shape (Fig. 13 and I in
Fig. 15), similar to the forming of the impulse wave induced by the Qianjiangping
landslide. At

As can be seen in Fig. 2, the Tangjiaxi valley is narrow. Therefore, it is hard to distinguish the generation, propagation, and run-up phases of the impulse wave. Accordingly, this event was not a typical landslide-induced impulse wave. As can be observed in the water level lines of various points in the Tangjiaxi river surface in Fig. 16, there was only one large peak for the impulse waves, close to the landslide impact area (H3 in Fig. 16). Since the upstream of the landslide was quickly dammed after the impulse waves arrived, water arriving upstream failed to flow smoothly and therefore formed a temporary upsurge upstream (Wang et al., 1986). The maximum upsurge in the upstream was up to 172.5 m (H2 in Fig. 16), and the upstream water level remained at about 171.6 m at 30 s. After a relatively large impulse wave, wave amplitude fluctuation in the landslide downstream watercourse attenuated (H4 in Fig. 16).

Hydro-process line of various points in watercourse. See Fig. 9 for locations of H1–H5.

During the generation of this atypical landslide-induced impulse wave, it was
hard to determine the maximum height of the first wave in the watercourse.
The maximum propagating height of the wave in the peripheral watercourse of
the landslide zone was about 8.0 m, located at the downstream of the
landslide. The maximum run-up of the landslide was calculated to be 21.8 m
at the opposite bank of the landslide; the run-up of this point in the field
survey was 22.7 m. The slope at the opposite bank of the landslide was
directly impacted by the impulse wave, with a relatively higher run-up. Overall, the run-up was higher in the area where the landslide slid into
water and gradually decreased in the periphery with the increase in distance.
Table 2 shows the run-up at the bank surveyed in the field and the corresponding
calculated values. The correlation coefficient (

The calculated and measured run-up values at different points.

The equations of Baglad and Mih were obtained from the experiments of sphere grains, and there is non-coherence among the grains. Although some parameters are taken by back analysis in this case, the dynamic capacity of sphere grains is bigger than grains with other shape, which makes the energy transferred to water higher. Meanwhile, as rock mass begins to disintegrate in the actual situation as it slides into water, the coherence between rocks deserves consideration in the dynamic process. Therefore, the run-up values simulated are larger than investigations in general. Consideration of coherence and shape of grain is a main modification direction for this granular flow coupling model, which might improve its realism for a wider range of applications.

In this paper, a full coupling numerical model for landslide-induced impulse wave was developed. The non-coherent granular flow model of Mih (1999) was used to simulate the dynamic characteristics of the Tangjiaxi rockslide, and the two-phase flow model and RNG model were used to simulate the impulse waves while the granular flow impacted water.

The Tangjiaxi rocky granular flow slid into the watercourse and then moved to the
upstream and the downstream, forming a fan shape, and depositing a
landslide dam in the valley, damming the watercourse. The sliding mass
impacted water at the maximum velocity of 22.5 m s

The landslide dam configuration and impulse wave run-up calculated were well fit with the actual survey results. Therefore, the coupling model based on non-coherent Mih granular flow performed well in the whole-process analysis of the Tangjiaxi landslide-induced impulse wave. The framework of this coupling numerical model deserves more attention and further improvement.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

This work was supported by the National Natural Science Foundation of China (project ID: 41372321) and National Science and Technology Support (ID: 2012BAK10B01). Additionally, the authors would like to thank Xie from the Tangjiaxi village, who provided us with his photos and other useful information to us. Edited by: K.-T. Chang Reviewed by: two anonymous referees