The Development of a Hydrological Method for Computing Extreme Hydrographs in Engineering Dam Projects

Abstract

Engineering dam projects benefit society, including hydropower, water supply, agriculture, and flood control. During the planning stage, it is crucial to calculate extreme hydrographs associated with different return periods for spillways and diversion structures (such as tunnels, conduits, temporary diversions, multiple-stage diversions, and cofferdams). In many countries, spillways have return periods ranging from 1000 to 10,000 years, while diversion structures are designed with shorter return periods. This study introduces a hydrological method based on data from large rivers which can be used to compute extreme hydrographs for different return periods in engineering dam projects. The proposed model relies solely on frequency analysis data of peak flow, base flow, and water volume for various return periods, along with recorded maximum hydrographs, to compute design hydrographs associated with different return periods. The proposed method is applied to the El Quimbo Hydropower Plant in Colombia, which has a drainage area of 6832 km². The results demonstrate that this method effectively captures peak flows and evaluates hydrograph volumes and base flows associated with different return periods, as a Root Mean Square Error of 11.9% of the maximum volume for various return periods was achieved during the validation stage of the proposed model. A comprehensive comparison with the rainfall–runoff method is also provided to evaluate the relative magnitudes of the various variables analysed, ensuring a thorough and reliable assessment of the proposed method.

1. Introduction

Dam engineering projects are crucial for water supply, power generation, irrigation, and flood control, among other needs [1]. An essential component in the design of these projects is the hydrological study conducted to define design hydrographs used for sizing diversion and spillway structures [2,3].

Numerous dam failures and incidents are documented, with the International Commission on Large Dams (ICOLD) having exerted significant efforts to compile the underlying causes of such occurrences [4]. According to these studies, spanning until 1975, it can be inferred that out of approximately 15,800 dams constructed (excluding China), failure occurred in roughly 1% of cases. Among all failure incidents, overtopping (water overflow over the crest) emerges as one of the foremost causes globally, with percentages ranging from 30 to 41%, according to various publications [4,5,6].

Table 1. Examples of dam failures due to overtopping.

Dam	Location	Watershed (km2)	Observed Peak Flow (m3/s)	Failure Year	Meteorological Conditions	References
Gangneung	South Korea	258.7	3781	2002	Around 900 mm of rainfall dropped in one day.	[7,8]
Lake Ha!Ha!	Canada	610	910	1996	A small low-pressure system generated a rainfall event for 58 h.	[9]
Noppikoski	Sweden	520	600	1985	The design flow was determined as the maximum observed flow with a safety factor equivalent to approximately a return period of 1000 years.	[4]
Tous	Spain	17,820	15,000	1982	Intensities around 500 mm dropped in one day.	[5,10]
Machhu II	India	1930	14,000	1979	Continuous rainfall over three consecutive days.	[4]
Sella Zerbino	Italy	141	2500	1935	A severe rainfall event was presented.	[4]

presents some examples of dam failures.

Various committees have adopted differing stances on diversion structure failures during dam construction. Consequently, compiling a dataset of such failures proved unattainable [4].

Careful consideration should be made of the return period of the peak discharge of the inflow hydrograph used for spillway design. The ICOLD conducted two surveys to establish a framework for estimating return periods in spillways [6,11]. The initial survey revealed that return periods for spillway design could range from 1000 to 10,000 years in countries such as France, Belgium, Germany, Portugal, Mexico, Canada, and Austria. The subsequent survey highlighted that in the United States, the criterion of the Probable Maximum Flood was utilised for spillway sizing.

Generally, design hydrographs employed for the hydraulic design of diversion structures encompass various return periods. Meanwhile, for spillways, designs adhere to country-specific regulations, where the design parameters fluctuate concerning floods associated with different return periods up to the Probable Maximum Flood [12].

The inflow hydrographs associated with different return periods are utilised to design hydraulic structures considering the reservoir routing method [13]. Various methods have been developed for calculating design hydrographs [14], all of which are compromised by the available information. Considering the available information, the following techniques have been developed for estimating design hydrographs: methods based on hydrological records [15], only used when this information is reliable and adequately available, and strategies based on rainfall–runoff relationships [16,17,18] employed when enough rainfall information is available. The understanding and suitable assessment of using appropriate hydrological methods are crucial in dam engineering projects [19,20].

Techniques to assess uncertainty in hydrological models have been used by Mishra (2009) [21], ensuring reliability in the conducted designs. In this sense, the Bayesian and Generalized Likelihood Uncertainty Estimation (GLUE) and Bayesian total error analysis (BATEA) approaches can be used for computing uncertainty in the main parameters of rainfall–runoff processes in watersheds [22,23]. The main guidelines for the hydrological design of diversion and spillway structures in dams recommend evaluating uncertainty, although none specify a formal procedure for this type of analysis.

This study presents the development of a proposed model based on hydrological records. The proposed method utilises the annual maximum peak flow, base flow, and water volume series to compute design hydrographs associated with different return periods and information from the main recorded hydrographs. The utilisation of the proposed model allows for time savings during the design stage of dam engineering projects. It facilitates the calibration of key hydrological variables for maximum floods associated with different return periods. This study covers several topics to estimate design hydrographs for diversion structures and spillways in dams, such as governing equations and the required dataset for making calculations [24] for an analysed case study. This study aims to enable engineers to refine their calculations, thereby enhancing the reliability of estimated hydrographs. Engineers can utilise this research to compute extreme hydrographs for various return periods, thereby facilitating the analysis of hydrological risks based on the findings of this study. Implementing rainfall–runoff models in large watersheds is complex, and the proposed model seeks to address this challenge by offering a novel tool.

For these analyses, the El Quimbo Hydropower Plant is chosen as the case study, one of Colombia’s most important energy generation hydropower plants. The hydropower plant was inaugurated in 2015. It features a dam with a height of 151 m, a reservoir covering an area of 8250 hectares, an average multi-year flow of 235 m³/s, and an installed capacity of 400 megawatts. The dam is situated on the Magdalena River, within a drainage area of 6832 square kilometres. The approximate length of the river up to the dam site is around 180 km, with an average slope of 0.42% [25].

2. Case Study

The case study pertains to the El Quimbo Hydropower Plant’s dam site in Colombia. This project was selected based on the implementation of good practices during the design stage, corroborated by the hydropower plant’s current performance. The spillway is equipped with four (4) radial gates measuring 14.25 × 18.0 m, enabling a maximum operating level of 720 m a.s.l., with a crest at an elevation of 702 m a.s.l.

The dam site is positioned at coordinates 2°27′ N and 75°33′ W. The spillway was engineered to withstand the Probable Maximum Flood. The El Quimbo Hydropower Project commenced in 2008, with construction beginning in 2010 and concluding in 2015.

The nearest hydrological station to the dam site is Puente Balseadero, situated in the Magdalena River (code 2104701), overseen by the IDEAM, the public entity for managing hydro-meteorological information in Colombia. This station encompasses a drainage area of 5584 km². Data were compiled from the record period from 1972 to 2014, encompassing parameters such as daily flow, maximum annual flows, hourly water levels, maximum annual water levels, and discharge-stage measurements. Furthermore, data from 46 rainfall stations were employed, covering records of both daily and 10 min interval precipitation. Figure 1 presents the location of the case study.

The total drainage area until the dam site is 6832 km².

Table 2. Drainage area of each sub-basin.

ID	Catchment	Drainage Area (km2)
1	Upper sub-basin—Magdalena River	1527
2	Guarapa	807
3	Negra	292
4	Bordones	706
5	Timaná	209
6	Hígado	401
7	Suaza	1575
8	Seca	75
9	Lower sub-basin—Magdalena River	1238
Total	Location of dam site	6832

shows the nine sub-basins of the Magdalena River with their corresponding drainage areas.

Table 3. Used rainfall stations.

Rainfall Station	Station Code	Rainfall Station	Station Code
Altamira	2102002	Agrado	2104001
Guadalupe	2103005	Gigante N	2106007
Antena TV	2104002	San José	2101005
El Hatillo	2105014	Insfopal	2101011
Acevedo	2103008	Pita La	2106004
El Carmen	2113006	Laguna La	2101004
Ins. El Belén	2101017	Tesalia N2	2105029
San Adolfo	2103006	Bajo Frutal	2101013
Palestina	2101010	La Candela	2101014
Hornitos	2101025	Villa Fátima	2101016
La argentina	2105006	El Tabor	2101018
Hda. Meremberg	2105019	Alto del	2101019
Es Agr La Plata	2105502	Montecristo	2101021
Paez Paicol	2105015	Morelia	2101022
Yaguara	2108003	La Jagua	2103009
San Vicente	2105016	Oporapa	2104003
Sta Rosa	2108007	Pt Balseadero	2104004
Buenavista Hda	2108012	Tarqui	2104005
Totumo Hda	2108013	Tres esquinas	2104006
Armena La	2108009	Escalereta La	2104007
Mediania	2101006	Belalcazar	2105007
Garzón	2106008	Valencia	4401503
Sta Leticia	2105027

presents the code number and the location of the used rainfall stations.

The results of the primary critical meteorological variables are presented in

Table 4. Summary of annual average meteorological variables.

Variable	Range	Units
Temperature	15.8–24.3	°C
Relative humidity	76.5–84.6	%
Evaporation	668.5–1338.2	mm
Precipitation	1049–2202	mm

.

The project site lies within the Neiva sub-basin of the Upper Magdalena Valley, which divides Colombia’s Central and Eastern Ranges. Its geological history dates back to the late Paleozoic and early Mesozoic eras. Through geological data, the project area delineates distinct hydrogeological units: Porous Aquifers, Fractured Aquifers, Semi-confining Layers, and Confining Layers.

3. Materials and Methods

This section presents the proposed method for calculating design hydrographs employed in this study. It can be applied to watersheds with reliable hydrological records.

The methodology used in this study presents three steps: (i) case study definition, where a watershed with available hydro-meteorological data must be obtained, as shown in Section 2; (ii) the determination of design hydrographs through the proposed model (based on hydrological records) and the traditional rainfall–runoff models, where a frequency analysis for different return periods is performed; and (iii) a comparison and discussion of the results. Figure 2 presents the methodology used in this study.

3.1. Determination of Design Hydrographs

3.1.1. Proposed Method

This section presents the proposed method for computing hydrographs associated with different return periods. It is based on hydrological records of mean daily water flow, annual maximum flow, and hydrograph shapes (hourly water flow).

The proposed method has the following assumptions:

• Hydrological records are enough for simulating hydrographs associated with various return periods.
• The processes of the hydrological cycle for an analysed watershed are represented considering records of a hydrological station.
• A hydrograph can be divided into runoff and base volumes, where each one can be mathematically modelled.
• The peak flow is computed based on a frequency analysis associated with different return periods from the annual maximum flow.
• The base flow is computed based on the mean monthly flow since water level variations produced in aquifers occur slowly.

The proposed method is based on a mass balance considering a hydrograph, as shown in Figure 3.

The total volume ($V$) of a hydrograph for a return period can be expressed as follows:

$$V=V_{R-O}+V_{B-F}$$

(1)

where $V_{R-O}$ = the runoff volume, and $V_{B-F}$ = the volume produced by the base flow.

Considering low variations in water levels in aquifers, a constant base flow ($Q_b$) can be used to simulate the volume $V_{B-F}$, as follows:

$$V=\sum _{t=0}^T\left(Q_p-Q_b\right)q∆t+\sum _{t=0}^T{Q}_b∆t$$

(2)

Simplifying and organising terms, we obtain the following:

$$V=\sum _{t=0}^T{Q}_b∆t+Q_p\sum _{t=0}^{T}q∆t-\sum _{t=0}^T{Q}_{b}q∆t$$

(3)

where $Q_b$ = the base flow, $Q_p$ = the peak flow, $∆t$ = the time step, $T$ = the duration of a hydrograph, and $q$ = a dimensionless hydrograph that depends on a recorded hydrograph.

The dimensionless hydrograph ($q$) relates water flows over time ($Q_t$) with regard to peak ($Q_p$) and base ($Q_b$) flows, as presented in Figure 4. It can be computed by considering registered maximum hydrographs in a hydrological station. The peak time ($t_p$) is related to the maximum peak flow.

Considering that $Q_p$ and $Q_b$ are constant, then

$$V=T{Q}_b+Q_p{\sum }_{t=0}^{T}q∆t-Q_b{\sum }_{t=0}^{T}q∆t.$$

(4)

Finally, the total volume of a hydrograph can be modelled as follows:

$$V=T{Q}_b+(Q_p-Q_b)\sum _{t=0}^{T}q∆t$$

(5)

3.1.2. Rainfall–Runoff Models

These models can be developed across any catchment, and spatial and temporal variability plays a crucial role in hydrological modelling; consequently, they can be categorised into lumped, semi-distributed, and distributed models [26]. Lumped and semi-distributed models simulate the catchment’s behaviour by considering factors such as the average rainfall, a unit hydrograph, and the morphometric characteristics of each sub-catchment. The software HEC-HMS 4.10 was used to make calculations [16,27]. These models were applied to compare hydrographs obtained for different return periods with those obtained for the proposed model. Figure 5 presents various conditions for modelling hydrographs using rainfall–runoff methods. The SCS and Snyder unit hydrographs were employed to assess hydrographs for different return periods, in combination with two antecedent runoff conditions (ARCs). These include ARC—Type II, which applies to 5-day antecedent precipitation ranging from 36 to 53 mm, and ARC—Type III, for values exceeding 53 mm.

3.2. Frequency Analysis

The frequency analysis must be applied considering stationary and non-stationary methods. Stationarity methods consider a stationary distribution (exceedance probability is constant over time) and the independence of extreme events. The expected value $E\left(x\right)$ (average waiting time) in practice is defined as the return period:

$$T=E\left(X\right)={\displaystyle \frac{1}{p}}$$

(6)

where $T$ = the return period and $p$ = the exceedance probability.

If exceedance probability changes over time, then the expected value is given by Reference [28].

To perform the frequency analysis, the general expression of the Generalized Extreme Value distribution is as follows [28,29,30]:

$$F\left(z,\underset{\_}{{\theta }_t}\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left\{-{\left[1+\epsilon \left({\displaystyle \frac{z-{\mu }_t}{\sigma }}\right)\right]}^{-1/\epsilon }\right\}$$

(7)

where $\underset{\_}{{\theta }_t}=\{{\mu }_t,\sigma ,\epsilon \}$ represents the parameters.

The parameter ${\mu }_t$ varies over time (${\mu }_{t }$ = ${\beta }_0+{\beta }_{1}t$), and $\sigma$ and $\epsilon$ remain constant. If ${\mu }_t$ remains constant, then the stationarity condition is achieved (${\mu }_t$ = $\mu$). If $\epsilon \to 0$, then the Gumbel distribution is obtained; when $\epsilon >0$ occurs, the Frechet distribution is derived (Type II), and when $\epsilon <0$, the Weibull distribution is reached.

The Pearson Type III distribution under stationary conditions is yielded by the following:

$$F\left(z\right)={\displaystyle \frac{1}{\alpha \mathsf{\Gamma }\left(k\right)}}\underset{0}{\overset{\infty }{\int }}{\left({\displaystyle \frac{z-\beta }{\alpha }}\right)}^{k-1}{e}^{-{\displaystyle \frac{z-\beta }{\alpha }}}dz$$

(8)

where $\mathsf{\Gamma }$ = the gamma function, $\beta$ = the location parameter, $\alpha$ = the scale parameter, and $k$ = the shape parameter.

A two-sided confidence interval was employed in this study as follows [31,32]:

$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y} \left\{L_{P,c}\left(X\right)\le X_P^{*}\le U_{P,c}\left(X\right)\right\}=2c-1$$

(9)

where $X_P^{\mathrm{*}}$ = the population logarithmic variable that has an exceedance probability $P$, $c$ = the confidence level, $U_{P,c}\left(X\right)$ = the upper confidence limit, and $L_{P,c}\left(X\right)$ = the lower confidence limit.

3.3. Comparison of Hydrological Methods

This section compares the proposed model and the rainfall–runoff methods, considering data available in a specific watershed.

Table 5. Advantages and disadvantages of hydrological methods.

Hydrological Method	Advantages	Disadvantages
Rainfall–runoff models	They may be founded upon physical and empirical formulations to represent genuine hydrological responses accurately. They are capable of identifying all processes occurring within an entire catchment area, considering spatial and temporal variability. They are applicable in locations with ungauged catchments.	The calibration process proves particularly complex in expansive catchments due to the requirement for numerous parameters. Developing nations have not yet produced unit hydrographs for most of their catchments or established suitable formulations for calculating the concentration time.
Proposed model (based on hydrological records)	The generation of design flows for various return periods can be established for any diversion structures. It does not require the analysis of rainfall processes across catchments. Determining design hydrographs proves more straightforward when compared to rainfall–runoff models. It only necessitates data gathered from hydrometric monitoring stations for application.	There are no methodologies for computing the Probable Maximum Flood (PMF); hence, it is not applicable in countries where this event is crucial for computing design hydrographs in dams. Design hydrographs can only be computed in locations with recorded data from hydrometric monitoring stations; therefore, intermediate points along rivers rely on hydrograph transposition. There is a scarcity of information regarding these methodologies.

presents the advantages and disadvantages of applying both models.

4. Results

4.1. Proposed Model

The results of the proposed model are presented in this section. Figure 6 presents the peak flow, base flow, and 48 h maximum volume associated with different return periods. Peak flows ($Q_p$) are computed based on maximum daily water flow series of the hydrological station Puente Balseadero—Magdalena River (code 2104701). The transposition method was employed to compute peak flows at the dam site using the following expression:

$$Q_{p,dam}=Q_{p,PB}{\left({\displaystyle \frac{A_{dam}}{A_{PB}}}\right)}^{0.6}$$

(10)

where $Q_{p,dam}$ = the peak flows at the dam site, $Q_{p,PB}$ = the peak flows at the hydrological station Puente Balseadero, $A_{dam}$ = the drainage area until the dam site, and $A_{PB}$ = the drainage area until the hydrological station Puente Balseadero. A coefficient of 0.6 was used for the analysis [25]. Peak flows at the hydrological station were computed based on the rating curve interpreted in the design phase of the project [25]. The transposition for daily flows was conducted using a relationship of drainage areas.

The analysis was carried out using data from 1972 to 2014. The hydrological station was suspended for the filling of the reservoir. A frequency analysis was performed using observed data from the dam site, employing the Gumbel, Generalized Extreme Value, and Pearson Type III distributions. According to the results, the Generalized Extreme Value (GEV) distribution exhibited the best fit using the L-moments (LM) method, as shown in Section 5. Peak flows ranged from 1946.4 to 6341.3 m³/s for return periods of 5 and 10,000 years, respectively (Figure 6a). Figure 6a also displays the confidence limits for the selected hydrological distribution, calculated at a 95% confidence level.

The time duration ($T$) of hydrographs was defined considering a mean duration of 48 h, which was computed based on recorded hydrographs of May 1985, June 1988, July 1933, June 1944, and April 1999. The GEV using the ML method was employed for the annual maximum 48 h volume series frequency analysis, obtaining values from 211.4 to 446.1 hm³ for return periods varying from 5 to 10,000 years, respectively (Figure 6b).

The base flow analysis used the dam site’s average monthly water flow series. In this sense, the wet month (July) was employed for frequency analysis for different return periods, as shown in Figure 6c, obtaining values from 490.2 to 1144.6 m³/s for return periods ranging from 5 to 10,000 years, respectively [33]. The best fit was obtained using the GEV distribution with the ML method. The frequency analysis was conducted using the HEC-SSP 2.3 software developed by the U.S. Army Corps of Engineers.

Appendix A shows the dataset of the annual maximum series of peak flow ($Q_p$), 48 h volume ($V$), and base flow ($Q_b$). The Supplementary Materials contain the dataset of the base flow and 48 h maximum volume series at the dam site.

Considering the extreme values of peak and base flows presented in Figure 6, the design hydrographs associated with different return periods were computed using the following expression:

$$Q_t=q\left(Q_p-Q_b\right)+Q_b$$

(11)

The average dimensionless hydrograph ($q$) was computed based on recorded hydrographs of May 1985, June 1988, June 1993, June 1994, and April 1999. A peak time of 23 h provides good accuracy since its value is suitable for representing the recorded volume of hydrographs (see Figure 6). The nonlinear GRG was used as a numerical resolution method. Figure 7 presents the results of the proposed model. Figure 7a shows the different recorded dimensionless hydrographs, and Figure 7b presents the calculation of design hydrographs associated with return periods varying from 5 to 10,000 years. The generated hydrograph can represent the peak and base flows, recorded volume, and shape of the registered hydrographs.

The complexity of rainfall–runoff models is significant, especially for large watersheds, such as in the case study. In this context, the proposed model tends to be more precise as it accounts for all interactions upstream of the analysed watershed. This analysis estimated that the proposed model is more suitable for calculating floods associated with different return periods in this case study.

Figure 8 compares hydrograph volumes for various return periods for the proposed model. During the calibration process, a Root Mean Square Error ($RMSE$) was computed as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left({\displaystyle \frac{V_{computed}-V_{registered}}{V_{registered}}}\right)}^2}$$

(12)

where $V_{computed}$ = the computed volume using the proposed model, $V_{registered}$ = the registered volume for 48 h (Figure 6b), and $N$ = the total number of analysed hydrographs associated with various return periods.

An $RMSE$ of 11.9% was obtained, confirming the selected method’s adequacy. The obtained $RMSE$ is low considering the order of magnitude of the recorded and modelled volumes. The orange line in Figure 8 represents a perfect adjustment.

Engineers and designers should select return periods depending on the purpose of diversion structures. In this study, return periods from 5 to 10,000 years were calculated using the proposed model. However, the proposed model cannot compute the Probable Maximum Flood, which can only be established using the rainfall–runoff models.

4.2. Computation of Rainfall–Runoff Models

Hydrographs for Return Periods from 5 to 10,000 Years

Applying the rainfall–runoff models requires determining the drainage area, time of concentration, basin lag, ARC type II and III, and river routing parameters, presented in

Table 6. Parameters for application of rainfall–runoff models.

Catchment	Antecedent Runoff Condition (ARC)
	ARCII	ARCIII
Upper sub-basin—Magdalena River	78.2	90.0
Guarapa	75.6	88.6
Negra	75.7	89.0
Bordones	75.0	88.0
Timaná	78.4	90.5
Hígado	74.5	88.0
Suaza	75.5	88.5
Seca	79.4	91.0
Lower sub-basin—Magdalena River	75.0	88.0
At dam site	76.0	88.8

. The values of antecedent runoff conditions were considered based on the analysis conducted in the project [25].

Afterwards, considering the rainfall stations within the case study watershed, the the maximum daily precipitation associated with return periods ranging from 5 to 10,000 years was determined. Figure 9 illustrates the isohyetal maps of the daily maximum precipitation for various return periods within the case study area.

Table 7. Average daily maximum precipitation for each sub-basin.

Catchment	Rp (Year)
	5	10	20	50	100	200	1000	2000	10,000
Upper sub-basin—Magdalena River	81.3	96.5	111.0	129.9	143.4	158.0	190.3	204.0	236.5
Guarapa	72.2	83.1	93.5	107.0	117.1	127.3	150.7	160.6	184.1
Negra	78.2	93.2	107.4	126.0	139.2	153.7	185.5	199.0	231.0
Bordones	81.0	93.8	106.2	122.4	133.8	146.2	173.7	185.5	213.1
Timaná	78.1	88.2	98.0	110.7	120.0	129.3	151.0	160.6	182.4
Hígado	86.4	101.5	115.8	134.4	148.3	162.2	194.4	208.2	240.5
Suaza	79.8	92.5	105.4	121.5	133.4	145.2	173.0	185.4	212.7
Seca	74.3	86.1	97.6	112.4	123.4	134.4	159.8	170.8	196.2
Lower sub-basin—Magdalena River	96.2	108.8	121.0	137.0	148.8	160.6	188.0	199.8	227.0
Average	82.5	95.7	108.5	125.1	137.1	149.6	178.0	190.2	218.6

delineates the average maximum daily rainfall for each sub-basin, which was input for the semi-distributed model. Furthermore, the lumped model was executed using the average maximum daily rainfall across the watershed. The upper sub-basin and the Hígado watersheds exhibit the highest values of maximum daily precipitation, registering at 81.3 and 86.4 mm, respectively, for a return period of 5 years, while values of 236.5 and 240.5 mm are reached for a return period of 10,000 years.

Utilising the maximum daily precipitation values corresponding to various return periods for each sub-basin and incorporating the SCS and Snyder unit hydrograph methods alongside antecedent runoff conditions (types II and III), the maximum flow rates linked to return periods ranging from 5 to 10,000 years were calculated, as depicted in

Table 8. Maximum flow rates for different return periods considering semi-distributed model.

Unit Hydrograph	Rp (Years)
	10,000	2000	1000	200	100	50	20	10	5
ARCII
SCS	3592.9	2791.9	2467.6	1776.1	1501.9	1256.5	954.8	629.6	580.6
Snyder	2473.0	1972.1	1767.6	1325.2	1148.2	987.0	786.6	576.3	527.3
ARCIII
SCS	6160.0	5054.6	4588.4	3535.8	3089.9	2672.6	2122.4	1385.8	1336.8
Snyder	3916.4	3244.2	2960.1	2319.1	2047.5	1791.0	1452.9	1013.6	964.6

for the semi-distributed model.

Table 9. Maximum flow rates for different return periods considering lumped model.

Unit Hydrograph	Rp (Years)
	10,000	2000	1000	200	100	50	20	10	5
ARCII
SCS	3411.0	2715.3	2429.1	1803.8	1550.9	1321.6	1029.1	827.2	644.3
Snyder	2362.1	1918.4	1735.7	1332.5	1167.3	1016.1	821.7	685.7	559.1
ARCIII
SCS	5286.2	4401.1	4025.4	3171.0	2805.1	2461.7	1999.4	1658.1	1323.7
Snyder	3429.8	2877.7	2643.8	2109.6	1880.9	1665.9	1375.2	1160.5	948.0

shows the results of the simulations for the lumped model. The semi-distributed model yields higher flow rate values for return periods ranging from 1000 to 10,000 years compared to the lumped model. Conversely, the lumped model produces higher values for return periods ranging from 5 to 200 years. The Kirpich equation was used to calculate rainfall–runoff models. Moreover, opting for the SCS unit hydrograph results in higher peak flow rate values compared to the Snyder unit hydrograph.

5. Discussion

XLSTAT 2020 software was used to analyse the stationarity, homogeneity, and trend of the original dataset of the annual maximum peak flow, base flow, and 48 h volume series (see Appendix A), employing the Dickey–Fuller (trend criterion), Phillips–Perron (trend criterion), KPSS (trend criterion), Pettitt, and Mann–Kendall tests. The three series analysed ($Q_p$, $Q_b$, and $V$) are stationary and homogeneous, and show no trend, as indicated in

Table 10. Test of stationarity, homogeneity, and trend of original dataset.

Variable	Stationarity			Homogeneity	Trend
	Dickey–Fuller	Phillips–Perron	KPSS	Pettitt	Mann–Kendall
$Q_p$ (m3/s)	Stationarity	Stationarity	Stationarity	Homogenity	No Trend
$Q_b$ (m3/s)	Stationarity	Stationarity	Stationarity	Homogenity	No Trend
$V$ (hm3)	Stationarity	Stationarity	Stationarity	Homogenity	No Trend

.

In addition, the selection of the hydrological distribution was analysed for calculating the extreme peak flows, 48 h volume, and base flow. The proposed model depends on the chosen hydrological distribution. The dataset covers the period from 1972 to 2014 at the dam site.

Table 11. The goodness-of-fit summary statistics.

Distribution (Fitting Method)		Test
	Kolmogorov–Smirnov	Chi-Square	Anderson–Darling
$\mathrm{Peak} \mathrm{Flow} (Q_p$)
Generalized Extreme Value (LM)	0.069	2.581	0.199 *
Pearson III (LM)	0.081	3.698	0.203
Gumbel (LM)	0.087	6.302	0.270
Generalized Extreme Value (PM)	0.078	4.814	0.217
Pearson III (PM)	0.079	3.698	0.284
Gumbel (PM)	0.091	7.047	0.326
Generalized Extreme Value (MLE)	0.075	6.302	0.192
Gumbel (MLE)	0.076	6.302	0.218
Pearson III (MLE)	0.092	8.535	0.320
$\mathrm{Base} \mathrm{flow} (Q_b$)
Generalized Extreme Value (LM)	0.081	5.558	0.228 *
Gumbel (LM)	0.086	5.558	0.233
Pearson III (LM)	Not applicable
Gumbel (PM)	0.084	5.558	0.228
Generalized Extreme Value (PM)	0.090	5.558	0.244
Pearson III (PM)	0.093	5.558	0.292
Gumbel (MLE)	0.084	5.558	0.227
Generalized Extreme Value (MLE)	0.085	5.558	0.229
Pearson III (MLE)	0.096	5.558	0.283
$48 \mathrm{h} \mathrm{maximum} \mathrm{volume} (V$)
Generalized Extreme Value (LM)	0.063	5.558 *	0.241 *
Pearson III (LM)	0.066	5.558	0.234
Gumbel (LM)	0.069	5.558	0.269
Pearson III (PM)	0.065	5.186	0.241
Generalized Extreme Value (PM)	0.066	5.558	0.245
Gumbel (PM)	0.082	7.791	0.336
Generalized Extreme Value (MLE)	0.075	7.791	0.287
Gumbel (MLE)	0.076	7.791	0.296
Pearson III (MLE)	0.310	25.279	4.847

Notes: The grey cells represent the best fit using a hydrological distribution. * The Generalized Extreme Value (LM) obtained the second best fit using this test.

displays the results for the mean peak flow values associated with different return periods using the Gumbel, GEV, and Pearson III hydrological distributions. Additionally, the methods of L-moments (LMs), product moments (PMs), and maximum likelihood estimation (MLE) were applied to fit the parameters of the hydrological distributions. The calculations were carried out using the HEC-SSP 2.3.

The Generalized Extreme Value distribution, employing the L-moments method, was utilised for calculating design hydrographs for various return periods within the proposed model for the following reasons:

• The computed extreme peak flow, 48 h volume, and base flow series, using the GEV distribution for return periods ranging from 5 to 10,000 years, had a range of reasonable values.
• The GEV distribution using the ML method provided the best fit using the Kolmogorov–Smirnov test for all analysed series. In addition, the Chi-square test provided the best fit for the peak and base flow series, while the 48 h volume series obtained a good agreement. By comparing this with the Anderson–Darling test, the selected distribution reached the second best fit.

Selecting a hydrological distribution is crucial for appropriately applying the proposed model. Although the Gumbel distribution has been widely used in several publications for analysing extreme flows [28,34,35], it does not always accurately capture the skewness and kurtosis coefficients of observed maximum annual peak flow data [36]. A comprehensive analysis of various hydrological distributions (GEV, Gumbel, Pearson III, among others) must be conducted using a range of statistical hypothesis tests [37]. Furthermore, anthropogenic and climatic changes should also be considered by incorporating parameters in hydrological distributions that can vary over time [28].

This section presents a comparative analysis of the outcomes of applying hydrological methods.

Table 12. Comparison of peak flow rates ($Q_p$) for different hydrological methods.

Rp (Years)	Peak Flow (m3/s)
	Hydrological Method (Generalized Extreme Value (LM))	Lumped Method				Semi-Distributed Method
		SCS		Snyder		SCS		Snyder
		ARC—II	ARC—III	ARC—II	ARC—III	ARC—II	ARC—III	ARC—II	ARC—III
10,000	6341	3593	6160	2473	3916	3411	5286	2362	3430
2000	5121	2792	5055	1972	3244	2715	4401	1918	2878
200	3687	1776	3536	1325	2319	1804	3171	1333	2110
100	3316	1502	3090	1148	2048	1551	2805	1167	1881
50	2969	1257	2673	987	1791	1322	2462	1016	1666
20	2542	955	2122	787	1453	1029	1999	822	1375
10	2239	630	1386	576	1014	827	1658	686	1161
5	1946	581	1337	527	965	644	1324	559	948

shows the peak flows corresponding to various return periods. The highlighted values represent those greater than those obtained using the hydrological distribution (GEV-ML).

Based on the results of Table 12, the following results can be drawn: (i) the hydrological method estimates a higher peak flow compared to the rainfall–runoff method; and (ii) using the SCS unit hydrograph results in higher peak flows than using the Snyder unit hydrograph.

Understanding the variations in design hydrographs resulting from different hydrological methods is essential for selecting the method. Figure 10 illustrates the shapes of the hydrographs, with the y-axis representing the relationship between water flow rates and peak flow (dimensionless hydrograph, $q$), and the x-axis denoting time.

The hydrological method (proposed model) obtained the lowest dimensionless hydrographs. The peak time computed by the proposed model was 23 h. A sensitivity analysis was conducted for the peak time, considering values ranging from 19 to 27 h, as shown in Figure 11. The results indicate that the proposed model is a suitable tool, as it can reproduce annual maximum peak and base flows for various return periods, with an $RMSE$ that does not change significantly when this parameter is varied.

6. Conclusions

This study provides an analysis that can be applied to compute design hydrographs linked to various return periods, thereby facilitating the calculation of diversion structures and spillways in dam engineering projects. A hydrological model was developed in this study, which offers a comprehensive understanding of the potential applications of such projects. The following conclusions can be drawn:

• The proposed model is based on hydrological records and can be used to compute design hydrographs associated with different return periods. It requires only the frequency analysis of the annual maximum series of peak flow, base flow, and water volume for various return periods and registered hydrographs.
• The model was validated by comparing the computed and observed hydrograph volumes, resulting in a Root Mean Square Error ($RMSE$) of 11.9%. This is significant as it demonstrates the method’s robustness when applied to this case study.
• The model can compute design hydrographs for various return periods, specifically for spillways and diversion structures in dam engineering projects.
• The proposed model is an innovative tool that enables faster computation of design hydrographs compared to traditional rainfall–runoff models.

The proposed model cannot be used to calculate spillways when local regulations establish that the Probable Maximum Flood must be computed.