Reconstruction of debris flow in the Gerkhozhan-Su river valley based on the chain modeling

. The paper presents the reconstruction of the most catastrophic debris flows in the Gerkhozhan-Su river valley (North Caucasus, Russia) based on a chain of mathematical models. Transport-shift model was applied for 6 sections of debris flow formation or intense material increment, while for the rest the FLO-2D hydrodynamic model was used. A number of numerical experiments were carried out in the transport-shift model for a section of a debris flow origination site with a change in the parameters of a potential debris flow-forming soils, such as initial moisture content and density. According to the simulation results, with a rise in moisture content, the maximum debris flow discharge can increase by about 2.8 times. Modeling of debris flow in the Gerkhozhan-Su river valley was conducted for the case of low density debris flow. Based on the results, debris flow wave hydrographs were obtained at 12 sections. The results of modeling are in general agreement with the commonly accepted reconstructed pattern of the debris flow in 2000 and values obtained by videomaterials.


Introduction
Debris flows are one of the most frequent mass movement processes. Their high velocity, impact forces, long runout, combined with poor temporal predictability, cause debris flows to be one of the most hazardous processes [1]. Numerical simulation is one of the ways to estimate debris flows characteristics [2]. FLO-2D and RAMMS are one of the most popular hydrodynamic models for debris flows assessment [3][4][5]. Also, discrete models such as RocFall can be used [6]. These models are based on representation of debris flows as a cluster of material points or solid bodies. Previously, in the chosen valley debris flow of 2000year modeling was conducted with the use of discrete DEBRIS model [7]. A one-dimensional two-phase continuous Bozhinsky-Nazarov model based on the equations of motion and continuity was also used [8]. However, these works do not present the dynamics of debris flow discharge from the moment of its formation to the top of the alluvial fan.
The aim of this paper was to apply model chain to reconstruct debris flows of 2000 year in the Gerkhozhan-Su river valley. Transport-shift model, developed by Yu.B. Vinogradov [9], was used for debris flow origination site and sections of intense material increment. For the areas between these sections hydrodynamic FLO-2D model [10] was applied. In this paper transport-shift model was for the first time adapted for the sections of intense solid material * Corresponding author: viktoriiakurovskaia@gmail.com increment in the river course, whereas earlier it was used only for debris flow origination site [11][12].

Study area
The Gerkhozhan-Su river watershed is located on the northern macroslope of the Central Caucasus on the territory of the Kabardino-Balkarian Republic, Russia. The Gerkhozhan-Su River is the right tributary of the Baksan River and flows into it 124 km from the mouth. The length of the Gerkhozhan-Su River is 12 km, the basin area is equal to 76.3 km 2 [13]. The average slope of the course is 7°. The average and maximum watershed elevations are 2760 and 4149 m, respectively. The Gerkhozhan-Su river itself begins after the confluence of the Kayaarty-Su and Saakashil-Su tributaries at an altitude of 1665 m.
The basin of Gerkhozhan-Su river is among the North Caucasian regions with highest debris flow hazard [14]. The debris flows in 18-25 July 2000 were the most catastrophic [15].  According to estimates [18], debris flows volume was approximately 2.1 million m 3 . In the river mouth an extensive debris flow cone about 2 km long and 0.7 km 2 in area was formed [16]. As a result, the Baksan River was blocked with a subsequent formation of dammed lake 2 km long and 0.6 km wide within the city of Tyrnyauz [19]. The area of the lake was 0.55 km 2 , and the maximum depth was 12 m. According to [19], there were 42 buildings and a number of small structures in the flood zone.

Materials and methods
In this study, chain of mathematical models was used to estimate debris flows characteristics. Previously this chain of mathematical models was implied to assess the effect of the present position of the Buzulgan rockslide on the conditions for the passage of possible debris flows in the Gerkhozhan-Su river valley [20]. The results showed the effectiveness of these models. A distinctive feature of the debris flows in 2000 was the catastrophic incision of the river course. In total, 6 sections of material increment were identified in the river course: 1 -the main debris flow origination site, 2 -the area between the sandrs, 3 -the Western Canyon, 4 -the Upper Gorge, 5 -the Lower Gorge, 6 -the Buzulgan landslide and the lower valley cutting in the area of the destroyed dam and 6 sections of solid material accumulation ( fig.1). For the sections of material increment transport-shift model was applied, since this model allows taking into account the increment of solid material during flow formation [9]. Hydrodynamic model FLO-2D [10] was applied for sections of material accumulation. The simulated flow discharge was subsequently used as an input data for the transport-shift model and vice versa.

Transport -shift model
The transport-shift model is a one-dimensional and is intended for calculating high-density flows in the origination site.
where l is the distance by the thalweg of the debris flow source, m; l0 is the distance to the current reach, m; G is solid material discharge, m 3 /s; G0 is the initial value of variable G for a certain section and the result of calculation for the preceding one (for the first upstream section, G0 = 0), m 3 /s; α is the slope of the thalweg of debris flow source, deg; Q is water discharge, m 3 /s; θml is the volume ratio of water to solid matter at the yield (rigidity) point of the mixture of water and debris flowforming soils; ζ is the volume ratio of water to solid matter in the potential debris flow massif, dimensionless; g is the acceleration of gravity, m/s 2 ; ρ0 is water density kg/m 3 ; ρ is the density of debris flowforming soils , kg/m 3 ; А is proportionality factor, m/s 2 kg. The simulation results include a debris flow wave hydrograph, the flow density and velocity.
Obtained hydrograph was applied in FLO-2D model for the next section. Then the simulated hygrograph was again used in the transport-shift model. The amount of https://doi.org/10.1051/e3sconf/202341505007 , 05007 (2023) E3S Web of Conferences 415 DFHM8 solid material discharge (Go) for the previous section, where FLO-2D model was applied, was identified according to this formula [9]: = + (1 + ζ) * (2) Where Qc is debris flow discharge from FLO-2D model, m 3 /s; Q is water discharge from FLO-2D model, m 3 /s; ζ is the volume ratio of water to solid matter in the potential debris flow massif, dimensionless. In this research, the transport-shift model equations were implemented into FLOVI program in Python.

FLO-2D model
FLO-2D is two-dimensional hydrodynamic model developed by J. O'Brien [10]. It is based on solving Saint-Venan equations, which are widely used in studies of water flows [21]. The motion of debris flow in FLO-2D model is simulated under the assumption that debris flows move as Bingham (viscoplastic) liquid [22]. Input hydrograph from transport-shift model was divided into water and sediment according to following formula [10]: = − ( * ) (3) Where Qc is debris flow discharge from transport-shift model, m 3 /s, Q is water discharge, m 3 /s ; Cv is sediment concentration, dimensionless.

Initial data
The data on the relief for the modeling were taken from a 1:25000 map reflecting the situation in the valley before catastrophic debris flows in 2000. The mean slope in the sections of intense material increment varies from 18° for the origination site to 6 ° for the last section. Data from field trips, that were carried out immediately after the disaster, were used during input hydrograph construction [16]. The maximum water discharge was set as 4 m 3 /s. Also, a number of numerical experiments were carried out in the transport-shear model with a change in the parameters of the debris flow-forming soils. The debris flow volume concentration in the FLO-2D model was taken as 25%. Due to the fact that for this basin there are no measured values of rheological parameters required for modeling in FLO-2D, such as viscosity and yield stress, we chose the parameters of Aspen Natural Soil, proposed by the authors [22].

Results and discussion
Initial data for transport-shift model includes values of a potential debris flow-forming soils, such as initial moisture content (ζ) and density (ρ). Due to the fact that these data are not available, numerical experiments were carried out for debris flow origination site (table 1).
According to the simulation results initial moisture content of a potential debris flow-forming soils has the greatest influence on the maximum value of debris flow discharge. Maximum discharges for absolutely dry massif and a massif filled with water differ by almost 2.8 times. An increase in the density of a potential debris flow-forming soils by 600 kg/m 3 leads to growth in discharges by 4 m 3 /s. The 3rd case was chosen for modeling in the downstream valley, as it was the smallest one. The simulated flow density was 1492 kg/m 3 . The average flow velocity in the source was about 9.7 m/s. For the sections lying downstream, the same case of parameters was used. There is a sharp increase in discharges already for the 7 section and is approximately 231 m 3 /s. Further, for the 9 and 11 sections an increase in maximum discharges by about 1.8 times is observed.
At the top of the alluvial fan the maximum discharge reaches 1007 m 3 /s (12 line). In [8], maximum discharges were calculated using the flow cross-section in the contuit and flow velocity from the videomaterial; the obtained discharges were 1356 m 3 /s at the mean velocity of 11.3 m/s, 1764 m 3 /s at maximal velocity of 14.7 m/s and 936 m 3 /s at minimum velocity of 7.8 m/s. The relative error between discharge obtained for the minimum velocity and the simulated maximum at the top of the alluvial fan will be 8%. Difference between simulated maximum discharge and value estimated at the maximum velocity will be 757 m 3 /s, the average velocity will be 349 m 3 /s/. However, discharges obtained from video material were determined for the most powerful waves observed on July 19th. Thus, the simulated discharge value could be observed from 18 to 25 July. On the whole, the obtained results correspond to the generally accepted reconstructed flow pattern [16].

Conclusions
The study assessed the possibility of chain modeling to calculate discharges of catastrophic debris flows in the Gerkhozhan-Su river valley in 2000. The river course was divided into 12 sections depending on the observed processes. Transport-shift model was applied for debris flow origination site and sections of intense material increment, FLO-2D was used for the accumulation ones.
Several numerical experiments with various parameters of a potential debris flow-forming soils were conducted in the transport-shift model. According to the simulation results initial moisture content plays the most important role in the maximum discharge formation. In this work modeling was conducted for the case of absolutely dry and low density a potential debris flowforming soils.
In general, the simulation results are in agreement with the estimations of debris flows in 2000. Thus, the possibility of applying the transport-shift model for sections of intense material increment is confirmed. In the future, the authors plan to model high-density debris flows using various rheological parameters of the debris flow block in the FLO-2D model.