1. Introduction

a)

Flow depth mapping

b)

Flow velocity mapping

c)

Flood hazard classification based on flood intensity values

d)

Time of arrival mapping

e)

Area of inundation

2. Materials and Methods

2.1. Area of interest

The Kulekhni reservoir dam is the oldest rockfill dam in Nepal located at Dhorsing, Makwanpur [26]. The impounded water of the reservoir has been used to generate electricity by three hydropower stations, Kulekhani I (60MW) commissioned in 1982, Kulekhani II (32MW) commissioned in 1986, and Kulekhani III (14MW) commissioned in 2019 owned by Nepal Electricity Authority[26]. The designed Kulekhani reservoir capacity can withhold Probable Maximum Flood (PMF) up to 2540 $m^3$/s considering a watershed area of 126$k{m}^2$[27]. The zonal rock-fill dam consists of an inclined clay core, coarse and fine filter materials, and boulder rip rap for slope protection. The dam was constructed with an overall upstream slope of 1:2.35 and a downstream slope of 1:1.8. The clay core zone has been excavated until the bedrock, and provided with the grout curtain to prevent flow through seepage [27]. The 10m crest-width dam has a central axis cross-sectional bottom length of 97m and a top length of 397m with a height of 114m. The meandering river of 16km reach length from the downstream of the dam to the confluence of the Bagmati River has a steep overall slope of 1:42 and a narrow valley with a vertical side slope between $45-{60}^o$ [27].

Figure 1. Kulekhani reservoir inclined clay core rockfill dam [28].

Figure 1. Kulekhani reservoir inclined clay core rockfill dam [28].

The reservoir volume elevation-area curve is an essential parameter for dam break that determines the change in volume with respect to change in elevation. The curve accurately represents the hydrodynamics of the dam breach to determine the dam release flow rate, often used in dam break simulation to calibrate and validate real case failure results as illustrated by Alcrudo, F. [29].

Figure 2. Kulekhani reservoir elevation volume-area curve [27].

Figure 2. Kulekhani reservoir elevation volume-area curve [27].

The reservoir holds a volume of 85.285 million $m^3$ with a total dam height of 114m consisting of 397m crest length and 10m width. The reservoir is spread up to 7km with a surface area of 2.2$k{m}^2$ [27].

2.2. Breach model

The dam breach modeling process is the mathematical and physical representation of the dam’s failure to release significant flow from the reservoir. The selection of an appropriate dam breach model is often considered a challenging task due to uncertainties of dam failure and resulting downstream peak, which is directly affected by the dam breach parameter[30]. Several validated regression equations have been established to determine the dam breach parameter i.e. average breach width, side slope, volume of erosion, and time of breach based on the dam failure cases. HECRAS implements some well-known regression models such as Froehlich (1995), Froehlich (2008), MacDonald and Langridge-Monopolis (1984), Von Thun and Gillette (1990), and Xu and Zhang (2009) [31]. The breach parameters were estimated, Figure 3 considering the breaching depth up to 75m (1459 m.a.s.l) with a reservoir volume of 85.285 million $m^3$.

Figure 3. Dam breach parameters through regression equations.

Figure 3. Dam breach parameters through regression equations.

The regression models using Froehlich (1995), Froehlich (2008), and MacDonald and Langridge-Monopolis (1984) were tested using the over-topping scenario excluding Von Thun and Gillette (1990), and Xu and Zhang (2009) as they were providing higher breach bottom width than the existing bottom width geometry of the dam section.

Figure 4. Dam breach section parameters.

Figure 4. Dam breach section parameters.

Table 1. Regression model equations for breach parameter [31].

Table 1. Regression model equations for breach parameter [31].

Regression equation	Average breach width (${\mathit{B}}_{\mathit{a}\mathit{v}\mathit{e}}$)	Breach formation time (${\mathit{t}}_{\mathit{f}}$)
Froehlich (1995)	$B_{ave}=0.1803K_0{V}_w^{0.32}{h}_b^{0.19}$	$t_f=0.00254V_w^{0.53}{h}_b^{-0.90}$
Froehlich (2008)	$B_{ave}=0.27K_0{V}_w^{0.32}{h}_b^{0.04}$	$t_f=63.2\sqrt{\frac{V_w}{g{h}_b^2}}$
MacDonald and Langridge-Monopolis (1984)	$V_{\mathrm{eroded}}=0.00348{\left(V_{\mathrm{out}}\times h_w\right)}^{0.852}$$W_b=\frac{V_{\mathrm{eroded}}-h_b^2\left(C{Z}_b+h_b{Z}_b{Z}_3/3\right)}{h_b\left(C+h_b{Z}_3/2\right)}$	$t_f=0.0179{\left(V_{\mathrm{eroded}}\right)}^{0.364}$

Figure 5. Hydrograph through regression equations, downstream of the dam at Section-0, Figure 12

Figure 5. Hydrograph through regression equations, downstream of the dam at Section-0, Figure 12

The peak discharge of 61389 $m^3/s$ using Froehlich (2008), 49066 $m^3/s$ implementing Froehlich (1995), and 17594 $m^3/s$ with MacDonald and Langridge (1984) were evaluated downstream of the dam. The time of breaching and bottom width breach progression highly dominate the downstream flow peaks as obtained. Although breaching time and breach width are uncertain, Table 2 shows the breaching time for most of the observed real cases earth fill dams are above 2.51h for large dams. Therefore, MacDonald and Langridge (1984) regression equation was implemented to obtain the breach parameters.

Sixteen dam breaks observed downstream peaks were tested against the hydrograph method suggested by Zenz, G. (2015) [20] for estimation of the breach downstream peak considering reservoir volume and time of the breach to validate the model results.

$$Q_p=\frac{2V_w}{t_f(2\alpha +\beta -\alpha \beta )}$$

(1)

• Where,
• Height of rectangle $=\left(Q_p\right)$
• Height of triangular shape $=\beta \times Q_p$
• Width of rectangular shape $=\alpha \times t_f$
• Width of triangular shape $=t_f-\left(\propto \times t_f\right)/2$

Figure 6. Hydrograph method, Zenz, G. (2015) [20] .

Figure 6. Hydrograph method, Zenz, G. (2015) [20] .

Table 2. Sixteen dam break cases with observed and estimated peak using hydrograph method [32,33].

Table 2. Sixteen dam break cases with observed and estimated peak using hydrograph method [32,33].

Name	Height (m)	Reservoir volume (Million m3)	Duration of failure (h)	Observed peak (m3/s)	Estimated peak (m3/s)
Teton Dam (1976)	94	356	4	46817	44949
Baldwin Hills Dam (1963)	70.7	1.1	4.5	141	123
Euclides Da Cunha Dam (1977)	53.07	13.56	7.25	1004	945
Mammoth Dam (1943)	21.3	13.56	3	2521	2283
Hatchtown Dam (1914)	19.8	14.8	4	1973	1869
Whitewater Brook Upper Dam (1972)	18.8	0.51	3	70	86
Horse Creek Earth Dam (1914)	16.7	20.96	3	3885	3529
Puddingstone Dam (1926)	15.24	0.61	3	283	103
Wheatland Reservoir No. 1 Dam (1969)	13.7	11.55	1.5	4285	3889
Lake Latonka Dam (1966)	13.1	1.58	3	294	266
Frenchman Dam (1952)	12.1	8.6	3	1603	1448
Frankfurt Dam (1975)	9.75	0.35	2.5	79	71
Elk City Dam (1963)	9.14	0.9	0.83	607	545
Kelly Barnes Lake Dam (1977)	7.92	1.62	1.5	549	545
Hemet Dam (1927)	6.1	28.22	3	1602	4751
Goose Creek Dam (1916)	6.1	68.79	3	3880	11581

Figure 7. Regression plot between observed and estimated peak values.

Figure 7. Regression plot between observed and estimated peak values.

The regression model shows a good correlation (R² =0.96) between observed and estimated values from the hydrograph method. The empirical relation was used to evaluate and compare the simulated results considering 2.51h of breach time with 85.285 million $m^3$ volume.

Table 3. Simulated and estimated peak values downstream of the dam at Section-0, Figure 12.

Table 3. Simulated and estimated peak values downstream of the dam at Section-0, Figure 12.

Simulated peak ($m^3$/s)	Estimated peak ($m^3$/s)
17594	17160

The model produces more or less similar peak flow with an error of 2.5%, which is negligible for the flood event of 17594 $m^3$/s.

2.3. Data acquisition

Alaska Satellite Facility (ASF) [34], an available 12.5m resolution data set of Digital Elevation Model (DEM) was considered for defining the terrain up to the confluence of Bagmati River, 16km downstream from the dam site. In order to accurately capture the hydrodynamic behavior in the river channel, the central line and banks were assigned to refine and construct an aligned mesh along the direction of flow using HECRAS. The refining of mesh helps to precisely evaluate the hydraulic parameters like velocity gradient and water surface elevation. Esri Land Cover [35] map was used to define varying Manning’s roughness. The expected river channel flow was drawn to represent the channel roughness precisely due to the poor resolution of the land cover map. The roughness values were assigned based on the Ven Te Chow,1959 [36].

Table 4. Model defined Manning’s roughness

Table 4. Model defined Manning’s roughness

Manning’s roughness
Open water	0.035
Forest	0.14
Crops	0.05
Bare ground	0.04
River	0.037
No data	0.06

Figure 8. Roughness mapping

Figure 8. Roughness mapping

Figure 9. Model mesh

Figure 9. Model mesh

2.4. Governing equations

HECRAS implements a hyperbolic partial differential equation of the Shallow Water Equation, derived from the depth average integration of the Navier-Stokes equation assuming the horizontal length scale larger than the vertical length scale [37]. The shallow Water Equation assumes a vertical pressure gradient nearly hydrostatic, with a vertical velocity much smaller than the horizontal velocity [37]. HECRAS uses the Finite Volume Method to solve the Shallow Water Equations by discretizing the computational domain to the grid cells and integrating the conservation equations with different implicit and explicit time integration schemes [31]. The Shallow Water Equation is widely used for solving rapidly varying unsteady flow in open channels problems such as dam break scenarios, change in channel geometry with contraction and expansion, and steep mountain terrain meandering rivers[38].

2.4.1. Mass Conservation Equation

The continuity equation is based on the principle of conservation of mass, which defines the mass of flow into a defined control volume equal to the mass of outflow without an additional external source/sink [31].

$$\frac{\partial H}{\partial t}+\frac{\partial \left(hu\right)}{\partial x}+\frac{\partial \left(hv\right)}{\partial y}+q=0$$

(2)

• Where,
• H = Water surface elevation
• u and v = Velocity in x direction and y direction
• q = Flow domain source/sink
• x and y = Spatial coordinate
• t = Time

2.4.2. Full Dynamic Momentum Equation

The Full Dynamic Momentum Equation is based on the principle of conservation of momentum, which accounts for the forces acting on the fluid including pressure, gravity, friction, and external force to estimate the flow velocity and water surface elevation at a particular domain[31].

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}-f_{c}v=-g\frac{\partial z_s}{\partial x}+\frac{1}{h}\frac{\partial }{\partial x}\left(v_{t,xx}h\frac{\partial u}{\partial x}\right)+\frac{1}{h}\frac{\partial }{\partial y}\left(v_{t,yy}h\frac{\partial u}{\partial y}\right)-\frac{{\tau }_{b,x}}{\rho R}+\frac{{\tau }_{s,x}}{\rho h}-\frac{1}{\rho }\frac{\partial p_a}{\partial x}$$

(3)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+f_{c}u=-g\frac{\partial z_s}{\partial y}+\frac{1}{h}\frac{\partial }{\partial x}\left(v_{t,xx}h\frac{\partial v}{\partial x}\right)+\frac{1}{h}\frac{\partial }{\partial y}\left(v_{t,yy}h\frac{\partial v}{\partial y}\right)-\frac{{\tau }_{b,y}}{\rho R}+\frac{{\tau }_{s,y}}{\rho h}-\frac{1}{\rho }\frac{\partial p_a}{\partial y}$$

(4)

• where,
• $\frac{\partial v}{\partial t}$ represents the local acceleration term
• $u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}$ represents convective acceleration term
• $f_{c}u$ represents coriolis effects term , $f_c=2\omega sin\left(\varphi \right),\varphi =27.5^o$ for Kulekhani River
• $g\frac{\partial z_s}{\partial y}$ represents hydrostatic pressure term
• $\frac{1}{h}\frac{\partial }{\partial x}\left(v_{t,xx}h\frac{\partial v}{\partial x}\right)+\frac{1}{h}\frac{\partial }{\partial y}\left(v_{t,yy}h\frac{\partial v}{\partial y}\right)$ represents conservative turbulence viscosity term
• $\frac{{\tau }_{b,y}}{\rho R}$ represents bed shear stress term
• $\frac{{\tau }_{s,y}}{\rho h}$ represents surface wind stress term
• $\frac{1}{\rho }\frac{\partial p_a}{\partial y}$ represents the atmospheric pressure term

2.4.3. Diffusion Wave Equation

The Diffusion Wave Equation neglects the local acceleration, convective acceleration, turbulence viscosity, and Coriolis effects terms. Complex physical flow behavior such as sudden changes in channel geometry defining expansion and contraction losses, meandering effects defining bend losses, and re-circulation defining eddies losses are not captured with the simplified momentum equation, resulting in false flow velocity and water depth[31].

$$c_{f}u=-g\frac{\partial z_s}{\partial x}-\frac{1}{\rho }\frac{\partial p_a}{\partial x}+\frac{{\tau }_{s,x}}{\rho h}$$

(5)

$$c_{f}v=-g\frac{\partial z_s}{\partial y}-\frac{1}{\rho }\frac{\partial p_a}{\partial y}+\frac{{\tau }_{s,y}}{\rho h}$$

(6)

• Where,
• $c_{f}v$ represents the bed friction term
• $g\frac{\partial z_s}{\partial y}$ represents hydrostatic pressure term
• $\frac{1}{\rho }\frac{\partial p_a}{\partial y}$ represents the atmospheric pressure term
• $\frac{{\tau }_{s,y}}{\rho h}$ represents surface wind stress term

2.4.4. Mesh sensitivity analysis

The mesh sensitivity analysis was carried out considering fixed time steps of $\Delta t=0.2s$ with varying spatial steps of $\Delta x=30m$, $\Delta x=20m$, $\Delta x=10m$, $\Delta x=5m$, and $\Delta x=4m$ to evaluated hydrograph at cross-section II. The spatial step of $\Delta x=5m$ was selected for the model run as the result starts to converge with the decreasing spatial steps below $\Delta x=5m$.

Figure 10. Mesh sensitivity for different spatial steps ($\Delta x$), considering section II Figure 12 hydrograph

Figure 10. Mesh sensitivity for different spatial steps ($\Delta x$), considering section II Figure 12 hydrograph

$\Delta x$ (m)	Peak flow ($m^3$/s)
30	17455.25
20	17319.44
10	17256.97
5	17086.11
4	17085.94

2.4.5. Stability criteria

The courant criterion was satisfied using the fixed time step method to ensure the model’s stability and robustness considering $\Delta x=5m$, $\Delta t=0.2s$.

$$\mathrm{Courant}\mathrm{Number}\left(\mathrm{CFL}\right)=\frac{\Delta t·V}{\Delta x}\le 1$$

(7)

• Where,
• $\Delta t$ = Time step
• $\Delta x$ = Spatial step or computational grid cell length
• V = Flow velocity

3. Result and Discussion

3.1. Model compression

The 2D Diffusion and Full Dynamic Wave Equations were modeled to compare and contrast the results regarding water depth, flow velocity, inundation extent, and peak hydrograph.

Figure 11. Flow behavior in meandering of the river [39]

Figure 11. Flow behavior in meandering of the river [39]

The Full Dynamic Wave Equation captures the physical flow phenomena like the formation of eddies, higher flow velocity at the outer bend adapting centripetal force, and flow depth variation along the cross-section as expected by increasing the flow extension. The secondary flow by meandering is not captured, as it does not account for vertical flow velocity variation [31,37]. The obtained flow results illustrated that there is a higher possibility of deposition of transported sediment and debris at the inner bend of the river resulting point bar, changing a river morphology during flood events. Whereas, the Diffusion Wave Equation does not capture these effects as local acceleration, convective acceleration, and Coriolis effects terms are neglected in the momentum equation, which produces average horizontal water surface elevation, and unrealistic higher central flow velocity resulting in less water depth and less flow extension.

Figure 12. Flow hydrograph from different sections of the Kulekhani River

Figure 12. Flow hydrograph from different sections of the Kulekhani River

Figure 13. Section-I flow hydrograph

Figure 13. Section-I flow hydrograph

Figure 14. Section-II flow hydrograph

Figure 14. Section-II flow hydrograph

Figure 15. Section-III flow hydrograph

Figure 15. Section-III flow hydrograph

Figure 16. Section-IV flow hydrograph

Figure 16. Section-IV flow hydrograph

Figure 17. Diffusion Wave Equation hydrograph

Figure 17. Diffusion Wave Equation hydrograph

Figure 18. Full Dynamic Wave Equation hydrograph

Figure 18. Full Dynamic Wave Equation hydrograph

The flow hydrograph at four different river cross sections was evaluated over 16km of river reach downstream of the dam. The flood hydrograph for Diffusion and Dynamic Wave clearly indicated less attenuation in the flow peak due to a steep slope, narrow valley, and less floodplain river. The river reach from the dam to the Bagmati River confluence has an overall slope of 1:42, a side slope of $45-{60}^o$ vertical, and a bottom width ranging from 50 to 250m. The results have also shown that there is a time lag in the flow peaks using both models. The Diffusion Wave Equation does not account for the inertial and acceleration effects, resulting in higher flow velocity by increasing the peak. Whereas the Full Dynamic Wave Equation incorporates friction and energy dissipation more accurately, which decreases the flow velocity leading to lower peak flows. However, sometimes numerical dissipation while solving a Full Dynamic Equation might lead to smoothing effects on the peak of the hydrograph by reducing the magnitude[40].

Table 5. Peak flow attenuation and time lag

Table 5. Peak flow attenuation and time lag

Section	Diffusion peak ($m^3/s$)	Time of Peak (h)	Dynamic peak ($m^3/s$)	Time of Peak (h)
Section-I	17588.76	1.73	17582.21	1.75
Section-II	17522.24	1.78	17086.11	1.90
Section-III	17469.44	1.87	16832.48	2.12
Section-IV	17354.52	1.92	16683.82	2.20

3.2. Flood hazard assessment

The flood hazard assessment was carried out by mapping water depth, flow velocity, flood intensity classified by the American Society of Civil Engineers (ASCE) [41], and arrival time to evaluate the flood extension on the affected settlements.

Figure 19. Possible vulnerable areas with the identified town, 2785 05D Bhimphedi [42]

Figure 19. Possible vulnerable areas with the identified town, 2785 05D Bhimphedi [42]

Twelve major downstream towns are expected to be flooded due to a dam breach of the Kulekhnai Reservoir dam till the confluence of the Bagmati River as shown in Figure 19. The existing total number of living households in the town was estimated based on the provided number of households on topo map, 2785 05D Bhimphedi [42] and open street map.

Table 6. Expected total number of households along the river to be flooded Figure 19, [42]

Table 6. Expected total number of households along the river to be flooded Figure 19, [42]

Town	Households
Thulchaur	13
Ranche,Nagmar,Khanikhet, and Debaltar	71
Lambagr and Sulikot	48
Tallogau, and Tinghare	10
Sano Tar	32
Simletar	20
Kuntar	18

3.2.1. Flow depth mapping

Figure 20. Possible vulnerable areas with flow depth mapping

Figure 20. Possible vulnerable areas with flow depth mapping

3.2.2. Flow velocity mapping

Figure 21. Possible vulnerable areas with flow velocity mapping

Figure 21. Possible vulnerable areas with flow velocity mapping

3.2.3. Flood intensity mapping (depth x velocity ($m^2$/s) )

Figure 22. Possible vulnerable areas with flood intensity mapping

Figure 22. Possible vulnerable areas with flood intensity mapping

3.2.4. Time of arrival

Figure 23. Possible vulnerable areas with arrival time mapping

Figure 23. Possible vulnerable areas with arrival time mapping

The obtained flow depth mapping has shown that all twelve towns will be affected by both models due to the dam breach.

Table 7. Flooded households along the river

Table 7. Flooded households along the river

Town	$\begin{array}{c}\mathrm{Expected}\\ \mathrm{to}\mathrm{be}\mathrm{flooded}\end{array}$	$\begin{array}{c}\mathrm{Flooded}\\ (\mathrm{Dynamic}\\ \mathrm{Wave})\end{array}$	$\begin{array}{c}\mathrm{Flooded}\\ (\mathrm{Diffusion}\\ \mathrm{Wave})\end{array}$
Thulchaur	13	7	4
$\begin{array}{c}\mathrm{Ranche},\mathrm{Nagmar},\\ \mathrm{Khanikhet},\mathrm{and}\\ \mathrm{Debaltar}\end{array}$	71	47	21
$\begin{array}{c}\mathrm{Lambagr}\mathrm{and}\\ \mathrm{Sulikot}\end{array}$	48	15	11
$\begin{array}{c}\mathrm{Tallogau},\mathrm{and}\\ \mathrm{Tinghare}\end{array}$	10	8	5
Sano Tar	32	14	12
Simletar	20	16	10
Kuntar	18	13	9

Table 8. Total inundation area till the confluence of Bigmati River

Table 8. Total inundation area till the confluence of Bigmati River

Area of inundation ($k{m}^2$)
Dynamic Wave	Diffusion Wave
2.92	2.25

Comparatively, most households are affected using a Full Dynamic Wave rather than a Diffusion Wave Equation which is reasonable, the dynamic wave considers the effect of inertial and convective acceleration losses resulting in less flow velocity by increasing the water depth. The increased water depth increases the flood extension, which can be clearly visualized through flood flow depth mapping, Figure 20. The calculated total extension of the flow area, Table 8 till the Bagmati River confluence has also shown that the flow is likely to be extended using a Full Dynamic Wave Equation. Both the model predicts that most of the households of Ranche, Nagmar, Khanikhet, and Debalta are vulnerable due to dam breach flooding, Figure 20 and Table 7.

The intensity of flood flow due to the dam breach was classified as per the American Society of Civil Engineers (ASCE) flood intensity classification guideline.

Table 9. Flood hazard classification, (ASCE).

Table 9. Flood hazard classification, (ASCE).

Flood Hazard Classification	Flood intensity (depth x velocity, ${\mathit{m}}^2$/s)
Low- Medium	< 2.1
High	2.1- 3
Very High	> 3

The flood intensity map, Figure 22 clearly indicates that flood hazard is "Very High" vulnerable to almost all of the towns with flood intensity above $3m^2/s$ as expected, due to the steep and narrow vertical valley with less floodplain till the confluence of the Bagmati River. The result has shown only some flow area in the towns of Ranche, Nagmar, Khanikhet, and Debaltar lies in the "Low-Medium" and "High" regions.

The flood arrival time mappings Figure 23 have shown, that the flood quickly reaches the confluence of Bagmati River considering Diffusion Wave while Full Dynamic Wave slows down the time of flood as a result of less flow velocity considering losses. The difference in time of peak to reach the confluence between Diffusion and Full Dynamic wave has been found to be 0.28h almost 17 minutes, Table 5.

3.2.5. Flood mitigation measures

Several flood mitigation hydraulic measures such as Levees, Dikes, and vertical Flood walls are essential to prevent the town from flooding [41].

Figure 24. Construction of Levee / Flood wall for the flood protection

Figure 24. Construction of Levee / Flood wall for the flood protection

In order to protect the most affected towns of Ranche, Nagmar, Khanikhet, and Debalta with a higher number of households, the construction of a Levee / Flood wall along the river has been suggested.

Table 10. Earth fill levee geometry

Table 10. Earth fill levee geometry

Height	23m
Length	375m
Crest width	8m
Side slope	1:1.5

The river diversion works along the river help to prevent the towns from flooding, Figure 24. This implies the necessity of a Levee / Floodwall along the Kulekhani River in case of a dam breach to prevent towns from flooding. Most of the households could be protected after the construction of the Levee, excluding 11 houses that lie below the maximum inundation elevation of 1269.6 m.a.s.l along the river. Therefore, proper river flood management along the Kulekhani River is needed to prevent the devastating effects of floods on human settlements in flood-prone areas through restricted zones.

4. Conclusion and Recommendation

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