Comparison of Debris Flow Modeling Results with Empirical Formulas Applied to Russian Mountains Areas

Construction of debris flow protection structures is impossible without studying the processes first. Therefore, the purpose of this research was to calculate the magnitude of debris flows in three study areas. Initial information was provided by JSC Sevkavgiprovodkhoz and the Research Center “Geodinamika”. The first object of this research was the river Ardon and its tributary the Buddon, because of disastrous consequences for Mizur village of passed debris flows and floods. Modeling of unsteady water movement was carried out for estimation of potential flooding. During modeling, 5 cases of flash floods and debris flows of various probabilities from 0.5% to 1% percent were considered. Therefore, maximum floods for the cross-sections above and in the Mizur village itself were obtained. The second study area was the Chat-Bash stream, which is also situated in the north of Caucasus mountains. For this stream, the maximum discharge that could impact the mining complex at Tyrnyauz was determined. The third study area was the Krasnoselskaia river due to frequent floods in Yuzhno-Sakhalinsk. Applying three cases of various probabilities from 10% to 0.1%, the model determined maximum discharge and water level for the last cross-section above confluence into the Susuya river. Numerical experiments for all study areas with different roughness values were conducted to identify optimal ones. Comparing the model results for all study areas with empirical formulas (Golubcov V.V., Herheulidze I.I., Kkhann, Sribnyj and ASFS of EMERCOM of Russia) revealed that formulas contain only average depth slope angle and empirical coefficients and do not allow estimating flood areas and maximum characteristics of the event with a certain degree of accuracy.


Introduction
Debris flows of various densities are frequent phenomena in north Caucasus mountains [1] and in Sakhalin region [2]. Research methods for identifying the magnitude of debris flows can be divided into calculated and experimental.
The experimental method includes observation stations, where long-term monitoring of debris flows is already taking place in many countries around the world. One of the earliest instrumental observations was held by Pierson [3] in channels on the flanks of Mount St. Helens. Recently the special equipment was applied at Chalk Cliffs in the Colorado Rocky Mountains [4] [5]. In the Alpine areas, debris flow torrent and fan monitoring is carried out by several countries: Italy, France, Switzerland and Austria [6] [7]. Also, quantitative characteristics of debris flows are measured at the Spanish station in the Pyrenees [8]. The largest debris flow observation station was opened in 1961 in China in the Jiangjai River Basin in the suburb of Donchuan, Yunnan Province [9]. Moreover, special equipment was installed in Ohya landslide, central Japan [10]. Since 2002 about 19 debris flow stations have been opened in Taiwan [11]. Additionally, the experiments on reproduction of artificial debris flows in nature, which were organized by Professor Yu. B. Vinogradov, were held in [1972][1973][1974][1975] in Zailiyskiy Alatau near Alma-Ata [12].
The calculation method includes the empirical formulas and mathematical modeling. Empirical formulas give only an approximate description of the debris flow movement [13], since they only take into account the flow depth and riverbed slope. Herewith, mathematical models are used, when it is necessary to obtain more accurate data about the debris flow movement and the maximum characteristics. There are many debris flow routing models, although most of them require a specific parameter, which is difficult to obtain or to calibrate. For example, RAMMS [14] [15] and FLO-2D [16] [17] can reproduce the depositional pattern of flows on alluvial fans after being calibrated using the historical data from the torrent and the fan. Since simulation models often require calibration, data from the historical events are not available for many locations, which is a major drawback for engineering applications. In addition, rheological models are used to simulate debris flows. However, they need laboratory experiments and special equipment for proper determination of the debris flow material rheology [18]. As for the input data, the accuracy of variables has significant influence on the model results. A maximum hydrograph is the boundary condition for the most hydrodynamic models. An assessment of a maximum hydrograph can be conducted on basis of the probability curve for a stream if long-term observational datasets are available. Also, a hydrograph can be calculated by taking into account the maximum volume of a lake located in the upstream reaches  [20]. Moreover, in case of debris flows instrumental monitoring, infrasound array analysis can be used to define maximum characteristics [21].

Study Areas
The first study area was the Ardon River in North Ossetia-Alania in Russia. It flows from north and somewhat east, entering the Terek River northwest of Vladikavkaz. The length of the Ardon is about 102 km with the catchment area of 2700 km 2 [22]. This river is formed by the merger of the Mamisondon, the Nar-   years more than 10%. In that case, the residential sector will be in the flood zone [25]. Besides, bridges across the Krasnoselskaia river have an insufficient cross-section and can serve as blocks, leading to flooding of vast territories of the town. One of them (the bridge on Lenin Street) was built, most likely, without a project ( Figure 4).
In 2014, riverbanks were stabilized by stone and concrete structures in washed areas, but these protective structures proved to be short-lived and ineffective and   [25].
were destroyed in a single flood [25]. In August 1981, during the typhoon "Phyllis", the water discharge of the Krasnoselskaia river reached more than 173 m 3 /s, leading to the flooding of vast territories [26]. Calculations were carried out on the initiative of the Research Center "Geodinamika" in order to compute the maximum characteristics. of a lack of initial information for the 2-D models [27]. By applying the momentum and mass conservation laws to the mixture of a debris flow, a system of two partial differential equations is obtained, known as the Momentum equation and the Mass conservation equation. These equations can be solved using an implicit finite-difference scheme [28] [29]. A numerical scheme developed at Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the RAS was applied [30]. Differential equations of the unsteady flow in open waterways in the presence of tributary inflow have the following appearance in the model:

Methods
in (1) and (2) equations, x is the downstream coordinate, t is the time, h is the flow depth, m, V is the average velocity, m/s, Q is the water-sediment discharge, m 3 /s, ω is the cross-section area occupied by the flow, m 2 , C is the Chezy friction factor, g is the gravity acceleration, m/s 2 , R is area border ratio, m, α and β parameters depending on the shape of the cross-section and q is the tributary inflow. In the first Equation (1) As it was mentioned before calculation methods include not only modeling, but also empirical formulas. That is why various formulas for estimating velocity (v c ) and discharge of debris flows were applied. The following formulas, developed by Golubcov V.V. [31], Herheulidze I.I. [31], Kkhann [32], Sribnyj [32] and Academy of the State Fire Service Emergencies Ministry of Russia (ASFS of EMERCOM of Russia) [33] are frequently used in Russia.
Golubcov V.V. proposed a calculation formula for density flows [31]: where h is the average flow depth, m; α-average angle of slope of the mudflow bed, nondimensional.
In addition, the Kkhann formula was used to calculate debris flow velocity [32]: where h-the average flow depth, m; i-slope of the mudflow bed, nondimensional.
Accepting the assumption that volumes of debris flow's solid and liquid components are equal, the Sribnyj M.F. formula (6) changes to:  However, no theoretical description is available for this formula, so it is impossible to verify the assumptions about the reliability degree of the results. In order to obtain discharges, the values of velocity from the formulas were multiplied by cross-sectional area.

Calculation Results
Modeling was conducted to clarify the hydrological and morphometric charac-  (Table 1). These 5 cases defined maximum magnitude of debris flows and flash flood once in one hundred or even once in two hundred years. The values of these cases were obtained on basis of discharge probability curve for the Ardon and Buddon rivers [23]. The values of maximum discharges will be used for the protection territory in the valley of the Buddon River for Mizur village and Zaramagskaya hydroelectric station, which is under construction.
During the modeling, the design scheme of the Ardon River was made ( Figure 5). The most important cross-sections for the modeled flow events were the 3 rd section, at the mouth of the Buddon, and 5 th , directly near Mizur village.
No continuous observations of discharges and water levels on the Ardon or the Buddon Rivers near Mizur village were made. Therefore, short-term data provided by JSC Sevkavgiprovodkhoz were applied. The first results showed that the Froude number was overestimated. As it was mentioned above, the Froude number describes kinetic energy, meaning that the kinetic energy of the flow was exceeded. Initial discharges and water levels were specified with data from another gauge at the Ardon River. Table 2 shows the refined calculated values of the maximum discharges of the Ardon River on the cross-section before the confluence of the Buddon River, immediately after and opposite to Mizur village for the several cases.
The calculations show that during the flooding on the Ardon River, after the Buddon River flows into it, a rapid increase in water discharge occurs while a wave is forming.
Another task was to estimate the cross-sectional area of the modeled flow events at the 3 rd and 5 th cross-sections. Maximum value was obtained in 4 th case and was approximately 120 m 2 , and 116 m 2 for the 5 th cross-section, while the   flooding occurs even at a cross-sectional area of 110 m 2 [34]. Besides, for cases 1, 2 and 3, the cross-sectional area exceeds flooding area only at the 5 th cross-section. Moreover, numerical experiments were held to indicate optimal value of coefficient of roughness. In this case, optimal one was set equal to 0.08 for all cases. For the Buddon river, the calculations of debris flow velocity and discharge by different methods were conducted (Table 4). Initial information is provided in Table 3. Zaporozhchenko and A.M. Padmya [24], the scheme of the stream section was made ( Figure 6). The river section was divided into two parts, morphometry and hydrological data were anchored to the cross-sections.     [33]. The initial information for the formulas can be found in Table 5.
The results of calculations are presented in Table 6. As already mentioned, the results obtained using these formulas differed significantly. The smallest values of maximum debris flow velocity around 4.0 m/s were by the same formulas as for the Buddon (Herheulidze, Sribnyj and Golubcov formulas). The highest   Table 6. Maximum velocity and discharge of debris flow, calculated according to different methods for the Chat-Bash stream.  (Table 7) for study river was carried out according to the ultimate flow rate formula [25].
During the data preparation, the designed scheme of the river cross-sections in the city was made (Figure 8). The calculation was conducted for the section of the river near the mouth. The river section was divided into 2 parts according to changes in morphometry and presence of hydrological data. The calculation duration was 96 hours, taking into account the typhoon "Phyllis" that passed in 1981, which lasted from August 5 to 8 with the maximum discharge of 173 m 3 /s [26].      calculations is provided in Table 8. When comparing the model results with the formulas, a probability of occurrence in 100 years of 10% was used, due to the greater likelihood of occurrence ( Table 9). The highest velocities (about 13 -15 m/s) and discharges (300 -360 m 3 /s) were obtained by two empirical formulas-by Kkhann and by ASFS of EMERCOM of Russia. Calculated values by the other methods for both velocities and discharges were two times smaller. The results calculated by the model had the smallest values for both velocity (less than 1 m/s) and discharge (less than 90 m 3 /s) in comparison with empirical formulas used in this research.
The low velocities obtained by the model of unstable water movement are caused by the following: wave velocity begins to decrease if the absence of backwater occurs and the wave spreads out over nearby areas. The total width for the Krasnoselskaia river cross-section on average is 300 m, thus during the passage of the wave, spreading along the floodplain occurs. However, the formulas use only such characteristics of the channel itself as the depth and the slope, thus it is impossible to say whether an overflow will be observed or not. This model is one-dimensional, so it is possible to determine the velocity values only for the entire cross-section without specifying on the channel and the floodplain. Despite lower model values, flooding of urban infrastructure structures is observed.

Conclusions
In this research low-density debris flows and flash floods were modeled on the Ardon River. Initial information for modeling was provided by JSC Sevkavgiprovodkhoz. One of the tasks was to estimate the maximum cross-sectional area for 2 cross-sections near the Mizur village. The maximum discharges were also  Then experiments were carried out to determine optimal coefficient of roughness and it was found to be 0.075. We also compared the results of modeling with empirical formulas mentioned above. Even though the model of unsteady water movement does not take into account the size and composition of the loss material and the debris flow density, it is a linked system and gives plausible results. For more correct and reasonable calculations, it is necessary to obtain more accurate initial data.
As for the Krasnoselskaia river, maximum hydrological and morphometric characteristics of possible floods and low-density debris flows were obtained.
Three cases of various probabilities in 100 years were considered. The study revealed transformation of waveform during hazardous events. The most important hydrographs were for the 3 rd cross-section, which is located no more than