Design Flood Calculation Model for Extra-Small Watersheds in Ungauged Basin

Abstract

Designing floods in ungauged watersheds with limited data is a significant challenge in water conservancy projects. To address this, the method of calculating the design flood peak and flood volume using the weighted average method was proposed, which is based on the instantaneous unit hydrograph method and the inference formula method, combined with the characteristics of heavy rainfall floods in ungauged watersheds. The calculation results are analyzed in terms of reasonableness through the distribution pattern of the flood peak modulus under different frequencies of the constructed reservoirs, the relative error analysis, and the HEC-RAS model. Based on the one-day flood process of the adjacent basin, the calculation of deducing the design flood process using the hydrological comparison method was proposed. Taking the “Stormwater Runoff Chart” as the data source, the runoff generation, and concentration model was established with the design flood of Baludi Reservoir in the Gelangram River basin of Menglian, Yunnan Province as the research object. A comparative study of the results of the design floods calculated by different methods was carried out. The results show that the new method can well describe the rainstorm process. The method has better performance in the application to the design flood calculation of ungauged basins due to its consideration of the influence of subsurface conditions. The method not only reduces the construction cost but also improves the safety of the reservoir through a better-fitted design flood calculation.

1. Introduction

China’s mountain and hilly areas, with a humid climate, are rich in rain and flood resources, but the spatial and temporal distribution of precipitation is uneven. And the water conservancy infrastructure is weak due to a lack of effective control of stormwater, and flash floods triggered by localized torrential rains occur from time to time. This has resulted in a large number of casualties and socio-economic losses downstream of the small watershed areas [1]. Reservoir construction is an important way to prevent regional floods and protect the lives and property of downstream residents. Designing floods in ungauged watersheds with limited data is a significant challenge in water conservancy projects.

In areas without data, there are generally two approaches for estimating design floods: using streamflow data or rainfall data [2,3,4,5,6,7,8,9]. For example, Zhou [10] investigated the applicability of various design flood calculation methods in small and medium-sized catchments without observed data. The results indicated that the instantaneous unit hydrograph method is more suitable for estimating large-scale flood events in large catchments with high design standards, while the rational formula method is generally applicable to catchments smaller than 300 km². However, the lower boundary of its applicability is difficult to define. And the accuracy of the contour maps in the “Regional Handbook” plays a critical role in this method’s reliability. Xiong [11] dividing the river basin into distinct flood zones based on rainfall distribution and runoff characteristics enhanced the accuracy of the flood prediction formula, making it more aligned with the actual hydrological conditions of the region. Tan [12] develops a regional comprehensive formula for calculating the design peak discharge, taking into account multiple factors, such as watershed area, rainfall amount, and topographic conditions. The result shows that the mean and coefficient of variation (Cv) errors were reduced by about 52% and 8%. Ma [13] applied several methods (Time Series Decomposition and Synthesis Method, Hilbert-Huang Transform Method, Mixed Distribution Model Method, and Series Reconstruction Method) to design flood calculation of flood series in restoration, reconstruction, and mixed states. Wu [14] found that hydrological models can accurately characterize the variation in infiltration capacity of different soil types during rainfall events and the design flood results were similar to those from the Inference Formula Method, with relative errors in flood peak and volume not exceeding 30%. In recent years, several studies have addressed flood design in ungauged basins, contributing to the growing body of knowledge on this subject [15,16,17]. For example, Wu [18] uses the Hierarchical Bayesian Method for flood frequency analysis in ungauged regions, which shows that the method significantly improved the accuracy of flood design value estimation for ungauged regions compared to traditional methods. Moreover, machine learning models have been employed to estimate design floods, demonstrating the potential of data-driven techniques for enhancing flood prediction accuracy in ungauged watersheds [19]. Kan [20] proposed a coupled machine learning model that integrates Artificial Neural Networks (ANN) and the k-Nearest Neighbors (k-NN) method for watershed flood estimation. The results demonstrated that the model exhibits good accuracy and reliability, with promising potential for practical applications. Furthermore, studies have demonstrated that parameter optimization and uncertainty estimation techniques can significantly enhance regional water balance and flood prediction models in data-limited basins [21]. These studies collectively underscore the importance of integrating various methodologies to improve flood estimation and management in ungauged basins, but they often focus on larger or more general watersheds, leaving a gap for more refined approaches in smaller, extra-small ungauged basins. However, due to the influence of mountainous and hilly terrain in China, many small-scale water conservation and hydropower projects, such as irrigation water conservancy projects, soil and water conservation projects, and urban flood control projects, are mostly located in extra-small watersheds with a control basin area of less than 50 km². Therefore, there is a critical need to improve the flood calculation methods to better address the challenges posed by extra-small ungauged watersheds.

In this paper, with the upstream of the Gelangramu River basin (Baludi Reservoir Control Basin) in Menglian County, Yunnan Province as the study area, the method of calculating the design flood peak and flood volume using the weighted average method is proposed, which is based on the instantaneous unit hydrograph method and the inference formula method. And the reasonableness of the result is demonstrated by the distribution pattern of flood peak modulus of constructed reservoirs under different frequencies. Considering the subsurface factor, the design flood process is obtained by amplifying the same-frequency amplification method with the hydrological station of the adjacent basin as the reference station. To further assess the reliability of the proposed method, relative error analysis and the HEC-RAS model are conducted, which demonstrates the accuracy and consistency of the design flood estimates. Through comparative analysis with traditional design flood calculation methods, such as the instantaneous unit hydrograph method and the inference formula method, we explore the optimal method for estimating the design flood in the uninformative extra-small watershed. This is of certain reference significance for the planning and designing of engineering projects in uninformative areas of extra-small watersheds in the whole country.

2. Generalization of the Study Area and Data Sources

2.1. Basin and Project Overview

The Gelangramu River is located in the northwest of Menglian County, Yunnan Province, is a primary tributary on the left bank of the Nanka River, and belongs to the Salween River system. The mainstream is 15.5 km long, with a watershed area of 32.5 km². The Gelangramu River basin belongs to the subtropical humid monsoon climate zone; the annual mean precipitation reached 1372.0 mm, the annual mean temperature reached 19.6 °C, the annual mean evaporation was as high as 1250 mm; precipitation is mostly concentrated in May–October, accounting for 88.6% of the annual precipitation; and temperature, rainfall, evaporation, and other characteristics of the values change with altitude (the three-dimensional climate is more obvious).

The Gelangramu is a typical mountainous and hilly area with a scarcity of river hydrological stations and insufficient hydrological observation data and belongs to an ungauged region. According to the national survey on natural disaster risks and mountain torrent ditch field survey data statistics, the city has 126 mountain torrent ditches, 8 mudslide ditches, 126 landslides, and flood disasters that occur year after year [22,23]. The flooding process has a short duration, the flood volume is relatively concentrated, the flow rate rises and falls steeply, the basin flood forecast period is short, flood flow is large, and flood control and scheduling difficulty is high.

Baludi Reservoir is located in the upstream section of the Gelangramu River, and the dam site is located at 22°22′2″ N latitude and 99°21′50″ E longitude (Figure 1). The total design capacity of the reservoir is 2.693 million m³, the design flood level is 1460.52 m, and the reservoir scale for a small (Ⅰ) type reservoir project is the key small reservoir in Menglian County’s “14th Five-Year Water Security Plan” (2021–2025). The project is mainly for flood control, taking into account irrigation, water supply, and other tasks.

2.2. Data Sources

(1) Topographic Data: Through the DEM topographic data (5 m) provided by the Yunnan Provincial Hydrological Bureau, the characteristic data such as the control basin area, river length, and average specific drop of Baludi Reservoir were obtained through quantitative calculation, as shown in

Table 1. Characterization data for the Baludi Reservoir.

Reservoirs	Watershed Area/km2	River Length/m	Average Gradient/‰	Average Elevation/m
Paludi Reservoir	7.21	3.94	58	1563

.

(2) Precipitation Data: The measured daily-scale runoff, precipitation, evaporation, and hourly flood element data from 1959 to 2022 at Menglian Hydrological Station were selected and provided by the Pu’er Branch of the Yunnan Provincial Hydrology and Water Resources Bureau.

(3) Land Use Data: The land use data is sourced from the ’Results of the Three Zones and Three Lines in Menglian County’.

(4) Other Data: The hydrological data of Yunnan Province, such as Comprehensive Point and Area Reduction Coefficients (α) for Rainstorm Zones in Yunnan Province, the table of the integrated rainfall patterns of one-day rainstorm zones in Yunnan Province, the table of the Nash instantaneous unit hydrograph S(t) curves, the table of the descriptions of the zones of runoff generation and concentration in Yunnan Province, the map of the zones of runoff generation and concentration, the map of the average value of the storm, and the map of the storm zones, etc., are all from the “Practical Handbook of Storm Flood Calculations in Yunnan Province”.

3. Research Methodology

To effectively address the challenges of calculating the design flood for ungauged, extra-small watersheds, this study adopts a structured and integrative methodological system. Within this context, two primary methods are implemented: Design Flood Calculations and D4esign Flood Process Line Extrapolation.

The method begins by generalizing the rainfall-runoff relationship through the inference formula method, which conceptualizes the hydrological system as a semi-empirical lumped model. To address the inherent variability of rainfall and runoff distribution, the instantaneous unit hydrograph method is applied, facilitating the modeling of dynamic flood concentration processes. The weighted average approach integrates these two methods, yielding a refined estimate of design flood characteristics.

Subsequently, the design flood process line is derived using a hydrological comparison method. This process involves scaling historical flood data from a representative basin to align with the hydrological and geomorphological parameters of the study basin. The resulting flood process line encapsulates the temporal and spatial characteristics of the watershed, providing a robust basis for designing flood applications.

3.1. Design Flood Calculations

Because of its simple structure, the traditional inference formula method is commonly used in small and medium-sized basins where the number of hydrological monitoring stations is insufficient and the data on storm floods are not sufficiently representative, but the accuracy is limited and it is difficult to generalize the rainfall and runoff processes in complex situations [14]. The Nash instantaneous unit hydrograph is used to generalize the rainfall-runoff process in complex situations. The Nash instantaneous unit hydrograph uses a specific mathematical structure to simulate the watershed confluence, which is applicable and widely used, but there are problems such as fixed flood confluence parameters remaining unchanged, and the Nash instantaneous unit hydrograph flood process being short and fat [24]. Based on the problems of the above methods, the flood flow calculation method is now improved from the consideration of engineering safety and practical situations. The improved design flood calculation method is a weighted average of the results of the above inference formula method and instantaneous unit hydrograph method. The specific formulas are as follows:

$$Q_p=a{Q}_{p1}+b{Q}_{p2}$$

(1)

where: $Q_p$—flood flow with frequency p, m³/s; $Q_{p1}$—flood flow with frequency p by the inference formula method, m³/s; $Q_{p2}$—flood flow with frequency p by the instantaneous unit hydrograph method, m³/s; and $a,b$—weighting factor.

3.1.1. Inference Formula Method

The inference formula method is one of the earliest methods of inferring the flood flow based on rainfall data, which generalizes the spatial and temporal distributions of various elements with the concept of surface averaging, and belongs to the semi-inferential, semi-empirical lumped conceptual model [25]. In this analysis and calculation of the heavy rainfall flood in the Gelangramu River basin, the calculation formula applicable to extra small basins was adopted considering the characteristics of the basin [26]. In the calculation of the runoff generation, the focus is on the design of the rainstorm time-range distribution characteristics; in terms of flow concentration, this is divided into the full confluence and partial confluence of two cases, so that the water supply calculation results are closer to the actual. For China, the inference formula is generally considered to be applicable to the calculation of design flood flow for small and medium-sized watersheds (watershed area less than 500 km²). Hongmin Lin [27] proposed that the inference formula method is applicable to the design flood calculation of small watersheds with a basin area of more than 10 km² and less than 100 km², which further narrows down the theoretical scope of application of the inference formula method. The flood peak flow calculation formula is as follows:

$$Q_{p1}=0.278F{h}_t/t$$

(2)

$$\tau =0.278\theta /\left(m{Q_{p1}}^{1/4}\right)$$

(3)

where: F—watershed area, km²; $h_t$—maximum net rainfall at calendar time t; $\tau$—watershed confluence calendar time, h; $\theta$—geographic parameter, and $\theta =L/J^{1/3}$; L—the longest distance along the main river from the outlet section to the watershed, km; J—the average gradient of the process L; and m—the-confluence parameter.

3.1.2. Instantaneous Unit Hydrograph Method

The instantaneous unit hydrograph method is one of the commonly used approaches for estimating design floods, especially in regions where limited hydrological data are available. The method assumes that the response of a watershed to a unit rainfall event can be represented as an instantaneous unit hydrograph, which is then scaled based on the actual rainfall intensity and duration. It is applicable to watersheds with a catchment area of 200 to 1000 km², and is generally not used for design flood calculations in small watersheds where flood data are missing [28].

The instantaneous unit hydrograph is the flow process line formed at the outlet section of the watershed by a uniform distribution of rainfall on the watershed with an infinitesimal rainfall duration and infinite rainfall intensity, but totaling one unit of net rainfall. Its mathematical expression is as follows:

$$Q_{p2}={\displaystyle \underset{0}{\overset{t}{\int }}{U}_{\left(t\right)}}I\left(t-\tau \right)dt$$

(4)

$$U_{\left(t\right)}=1/k{\Gamma }_{\left(n\right)}{\left(t/k\right)}^{n-1}{e}^{t/k}$$

(5)

where: ${\Gamma }_{\left(n\right)}$—gamma function of order n; n—number of linear reservoirs equivalent to the basin’s storage capacity; and k—Regulation and storage coefficient, a parameter equivalent to the basin’s time of confluence.

In order to apply the instantaneous unit hydrograph to areas without hydrological data, indirectly derive the elements or parameters of the unit hydrograph based on watershed characteristics and rainfall characteristics, and estimate the design flood, it is necessary to establish a synthesis formula between the elements of the unit hydrograph or the instantaneous unit parameters of the watershed with hydrological data and the characteristics of the watershed [29].

3.2. Design Flood Rationalization Analysis Methodology

a. The Relative Error Method.

The relative error is a key metric used to assess the performance and accuracy of flood flow estimation methods. It quantifies the deviation between the estimated flood volumes and the reference flood volumes, providing an indication of how closely the estimated values align with observed data.

To provide a more detailed comparison and evaluate the reliability of the results, we calculated the relative error for each method by comparing the estimated flood volumes to the reference flood volumes. The formula for the relative error is expressed as follows:

$$\epsilon =\left|Q_{estimated}-\left.Q_{reference}\right|\right./Q_{reference}$$

(6)

where: $Q_{estimated}$ represents the flood volume calculated by each method; $Q_{reference}$ is the reference flood volume.

b. The Flood Peak Modulus Method.

The Flood Peak Modulus (FPM) is defined as the ratio of the peak flood discharge to the corresponding basin area during a specific flood event, which is a critical approach used to validate the rationality of flood peak design in hydrological studies, particularly for ungauged basins and regions where limited data is available. This methodology relies on the relationship between the flood peak modulus and various hydrological parameters to assess the accuracy of design flood estimations. This parameter is crucial for understanding the magnitude of peak flood flows and their relationship with the basin characteristics. Mathematically, it is expressed as follows:

$$FPM=Q_{peak}/A$$

(7)

where: $Q_{peak}$ is the peak flood discharge (m³/s), A is the watershed area (km²).

c. HEC-RAS Model introduction.

The HEC-RAS model, developed by the Hydrologic Engineering Center, is a River Analysis System designed for one-dimensional steady flow, one-dimensional or two-dimensional unsteady flow hydraulic calculations [30], and sediment transport modeling. It is capable of numerically simulating the two-dimensional evolution of floods and modeling the impact of hydraulic structures such as bridges, culverts, and weirs on river flow [31].

This study uses the HEC-RAS two-dimensional hydrodynamic model to analyze and calculate the watershed of the Baludi Reservoir, in order to determine the flow rates under different design flood standards. The governing equations of the model are the continuity equation and the momentum equation, which are expressed as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial x_i}}=0$$

(8)

$${\displaystyle \frac{\partial u_i}{\partial t}}+u_j{\displaystyle \frac{\partial u_i}{\partial x_i}}=f_i-{\displaystyle \frac{1}{P}}{\displaystyle \frac{\partial P}{\partial x_i}}+\lambda {\displaystyle \frac{{\partial }^2{u}_i}{\partial x_j\partial x_i}}$$

(9)

where: $\rho$ is the fluid density (kg/m³); t is the time(s); $u_i$ and $u_j$ are the flow velocities at cross-sections $i$ and $j$ (m/s); $f_i$ is the body force (m/s²); $P$ is the pressure (N/m²); $\lambda$ is the kinematic viscosity of the fluid ((N·s)/m²); and $x_i$ and $x_j$ represent the distances between sections $i$ and $j$ and their subsequent sections, respectively (m).

3.3. Design Flood Process Line Extrapolation

The hydrological comparison method is to modify and transfer the measured hydrological characteristic values of similar watersheds in hydrological similarity areas to watersheds without hydrological data. When there is a lack of measured data in the designed basin, the measured hydrological data in the upstream and downstream or in the hydrological similar area can be selected as the reference station, and the designed basin can be estimated by correction. The accuracy of the hydrological comparison method depends on the degree of similarity between the designed basin and the reference basin, especially the degree of proximity of the subsurface of the basin [32]. Therefore, the prerequisite for applying the hydrological analogy method is the assumption that the hydrological processes of the watersheds exhibit consistent trends of variation at the given time scale. The formula is as follows:

$$Q_{design}={\left({\displaystyle \frac{F_{\mathrm{de}sign}}{F_{reference}}}\right)}^n{Q}_{reference}$$

(10)

where: $Q_{design}$ is the design watershed flow flood volume, (m³/s); $Q_{reference}$ is the reference watershed flood volume, (m³/s); $F_{\mathrm{de}sign}$ is the designed watershed area, (km²); $F_{reference}$ is the reference watershed area, (km²); n is the proportionality factor, generally taken as 2/3.

In this study, according to the measured data of the hydrological station in the adjacent basin for many years, the flooding process in the basin is mostly single-peak type. The flood rise lasts for a short time, from the rise to the peak of about 2 h, with a peak lag time of 1~2 h. The water retreats after the peak and lasts for a long time; a flood process generally lasts 24 h, with the process being sharp and thin, presenting the characteristics of mountainous rivers. For this reason, the design of the flood process selection time is T = 24 h, and the time period is Δt = 1 h.

4. Results and Analysis

4.1. Calculation Results of Flood Flow Estimation Methods

a. Inference formula method. The watershed outlet cross-section flooding process actually reflects the result of the interaction between the stormwater and the subsurface that lands on the watershed. Since the watershed area of the extra small watershed is small, the inhomogeneity of the rainstorm over the watershed area is not considered, so only the changes in the characteristics of the subsurface need to be paid special attention. In this study, the relevant parameters are checked according to the relevant local hydrological manuals, and the flow concentration parameter—m is selected with reference to the comprehensive table for the classification of flood parameters m in small watersheds [33]. The characteristics of the subsurface in the watershed where the Baludi Reservoir is located basically belong to type II, and the forest cover is high and rainfall is abundant. The flow concentration parameter is m = 0.39, which is brought into Equations (2) and (3) to obtain the design flood peak flow.

b. Instantaneous unit hydrograph method. Baludi Reservoir is located in the upper reaches of the Gelangramu River, with a small watershed area and similar meteorological and climatic conditions. Due to the lack of measured heavy rainfall series data, the reservoir is located in the 9th heavy rainfall sub-district according to the location of the reservoir watershed center, checking the “Atlas of Stormwater Statistical Parameters of Yunnan Province” to get the maximum 1 h, 6 h, and 24 h mean rainstorm values and Cv values. The maximum rainfall mean values and Cv values for each frequency in each time period are shown in

Table 2. Parameters of rainfall at each time period of the Baludi Reservoir.

	1 h	6 h	24 h
Mean value/mm	40	70	89
Cv	0.35	0.36	0.35
Cs/Cv	3.5	3.5	3.5

. The Baludi Reservoir is located in the 7th runoff generation zoning, and the runoff generation parameters are Wm = 200 mm, Wt = 180 mm, fc = 3.0 mm, and ΔR = 6 mm. The reservoir is located in the 6th runoff concentration zoning, and the flow concentration parameters are Cm = 0.60, and Cn = 0.70. According to the selected maximum rainfall parameters of 1 h, 6 h, and 24 h, the design rainfall process is derived (taking into account the correction for the point-plane discount factor corrections), and the design net rainfall process is derived according to the loss parameters by deducting the losses, and then the design peak flow is obtained by considering the characteristic values such as the river length, area, and specific drop of the design section through a certain confluence calculation.

c. Weighted average method. According to the similarity theory, the adjacent basins with similar slopes, land use, soil, and other subsurface characteristics are selected. The weighting coefficients are determined through the relationship between the design flood and historical flood information of the adjacent basins, and the design flood flow obtained by using the inference formula method and the instantaneous unit hydrograph method is brought into Equation (1). Particularly, the weighting factor α = 0.61, β = 0.39.

d. HEC-RAS Model. In this study, the Isohyet Method for rainfall was applied to calculate the design rainfall for the study area for return periods of 5, 10, 20, 30, 100, 300, and 500 years. It is assumed that rainfall and flood events share the same frequency. Based on the “Yunnan Province Rainfall and Flood Atlas”, typical rainfall events were selected as the basis for the temporal distribution of the rainfall. Temporal distribution of the rainfall was then carried out for the study area. The HEC-RAS 6.6 software was used to create a grid division for the study area, with grid quality being adjusted and verified. Land use type data was incorporated to overlay the existing surface data within the HEC-RAS software. The rainfall module of HEC-RAS was employed to distribute the design rainfall temporally, while boundary conditions were set accordingly. The simulation area is entirely within the riverbed and is relatively small for which a composite roughness value was used. The riverbed in the simulated reach consists of pebbles, gravel, and sandy soil. The bed slope is relatively uniform, though the riverbed surface is uneven. Based on previous engineering experience and existing design results from the treated sections, a composite roughness value of 0.038 was selected. After running the model, the design flood results for the study area were obtained.

In summary, the design flood flow and the one-day flood volume deduced by the above method are shown in Figure 2 and Figure 3.

4.2. Design Flood Rationalization Analysis

a. The Relative Error Method Rationalization Analysis.

To assess the performance and accuracy of the three flood flow estimation methods, Figure 2 and Figure 3 present the calculated flood volumes for each method. To facilitate a more detailed comparison and evaluate the reliability of these results, we tested the reasonableness of the results of the four methods on the basis of the design flood flow of the adjacent basins. The relative error results are shown in Figure 4.

Among the four methods, the weighted average method demonstrates superior accuracy and stability in estimating flood volumes. As shown in Figure 4, this method consistently achieves the smallest relative errors across all design flood frequencies, with values ranging between 1.25% and 6.14%. This indicates that the results generated by the weighted average method are much closer to the reference flood volumes when compared to the other methods. In contrast, the Instantaneous Unit Hydrograph Method and the Inference Formula Method exhibit significant fluctuations in relative error as the design flood frequency changes, leading to less stable performance. The primary strength of the weighted average method lies in its consistent performance, as it minimizes fluctuations in relative error across different probabilities. This stability underscores the method’s robustness and reliability in handling flood volume estimations under diverse conditions. Furthermore, the integration of catchment characteristics, rainfall data, and subsurface conditions into the flood design process significantly enhances the method’s accuracy. This improvement is particularly valuable in regions with sparse hydrological data, where traditional methods may struggle to provide reliable results. In conclusion, the weighted average method stands out for its accuracy, stability, and adaptability, making it a highly effective tool for flood volume estimation in ungauged or data-scarce basins. It offers a scientifically sound and practical solution for improving the precision of flood design processes.

b. HEC-RAS Model Validation.

To assess the reasonableness and accuracy of the flood flow estimates, the HEC-RAS model was used to validate the results obtained from the Weighed Average Method. As seen in Figure 4, the weighted average method produced flood volume estimates that were reasonably close to those generated by HEC-RAS, with relative errors ranging from 1.25% to 6.14% across different flood frequencies. These relatively small errors indicate that the weighted average method provides a highly accurate approximation of flood volumes. The close alignment between the HEC-RAS model and weighted average method results further suggests that the empirical method is reliable and capable of providing useful estimates, particularly in the context of limited data availability, such as in ungauged or poorly monitored basins.

However, it was observed that the relative error between the HEC-RAS model and the weighted average method increased as the flood frequency rose because the accuracy of the HEC-RAS model is influenced by the correct selection of parameters such as Manning’s roughness coefficient, soil type, and DEM resolution. These parameters have a significant effect on the flow characteristics and can cause deviations in predicted flood volumes and peak discharge. For smaller events, the precision of these parameters becomes more critical, as minor changes in roughness or soil texture can cause significant variations in runoff and flood behavior. In contrast, for larger flood events, the overall magnitude of the event overshadows the impact of small-scale parameter variations, reducing the relative error.

c. The Flood Peak Modulus Rationalization Analysis.

The Baludi Reservoir is close to the LaFu (medium-sized) Reservoir (F = 25.8 km²) and the DongMi (medium-sized) Reservoir (F = 87.8 km²), belongs to the right bank of the Lancangjiang River, and has close statistical parameters of the maximum 24 h point rainfall, and the main weather systems that form the heavy rainfalls are also basically the same. Therefore, the design flood results of the two reservoirs, Lafu and Dongmi, are used as the reference basis for the reasonableness analysis to further discuss the reasonableness of the design flood results of the Baludi Reservoir.

When compared to the flood peak modulus for each frequency of the reviewed LaFu Reservoir and DongMi Reservoir, the flood peak modulus of the flood in Barruthi Reservoir decreases with increasing area, which is consistent with general flooding patterns [34]. And the flood peak modulus of the design flood of each frequency of the Baludi Reservoir has a tendency to decrease gradually with the increase of frequency, which is in line with the general hydrological law, indicating that there is coordination between the results of the peak volume of each frequency. The rationality analysis of the results is shown in Figure 5.

4.3. Calculation Results of the Design Flood Process Line

The Menglian station in the adjacent watershed of Baludi Reservoir has a more complete flood process, and the conditions of the subsurface are similar. From the catchment area, flood magnitude, and degree of flood concentration for consideration, the Menglian Hydrological Station was selected as a representative station for the typical flood of the dam site of the Baludi Reservoir. After analyzing and selecting the Menglian Hydrological Station, the measured flood process of 25 July 2012 was considered typical, respectively, with the design flood peak flow of the Baludi Reservoir dam site and the flood volume of one day as a control. The design flood process line of the reservoir dam site was deduced by scaling at the same frequency. The results are shown in Figure 6.

4.4. Discussion

From the calculation results of the three methods obtained in Section 4.1 above, it is known that the results of the inference formula method are significantly larger and the results of the instantaneous unit hydrograph method are significantly smaller. This is due to the fundamental assumptions and sensitivities of these approaches. The Inference Formula Method tends to overestimate design floods due to its reliance on simplified assumptions of uniform rainfall and runoff across the basin. This method assumes a constant rainfall intensity and uniform subsurface conditions, which do not adequately account for the spatial and temporal variability of hydrological processes in complex basins. Furthermore, the method’s high sensitivity to the runoff coefficient parameter amplifies the potential for overestimation, especially in basins with significant land-use variability or altered subsurface conditions. When runoff coefficients are derived from outdated hydrological manuals, the resulting flood predictions may be significantly larger due to changes in catchment characteristics over time. In contrast, the Instantaneous Unit Hydrograph Method often is significantly smaller, primarily because it simplifies the dynamic interaction between rainfall and catchment response. This method assumes a linear relationship between net rainfall and runoff, which may not hold true for basins with heterogeneous topography or complex runoff pathways.

These contrasting biases underscore the need for methodological refinements or the integration of multiple approaches to achieve more reliable flood predictions. The use of a weighted average method that integrates the inference formula method and the instantaneous unit hydrograph method yielded design flood results that balance accuracy and reliability. By calibrating weighting factors ($\alpha$ = 0.61, $\beta$ = 0.39) using hydrological data from adjacent watersheds, the results effectively addressed the challenges posed by limited observational data in the extra-small watershed. The method reduces errors and improves the robustness of flood estimation. For instance, in the case of maximum flow estimation, the weighted average method produced results with an error margin of only 10%, while traditional methods yielded errors exceeding 20%. Similarly, the relative error in flood volume estimates was lower with the weighted average method (less than 15%), compared to discrepancies of up to 30% observed in traditional methods. These improvements can be attributed to the model’s ability to incorporate subsurface conditions and regional rainfall patterns. Traditional methods often fail to account for these factors, leading to less accurate flood predictions. Additionally, the weighted average method provided a more realistic representation of the temporal distribution of flood events, which is particularly important in small watersheds where rainfall intensity can vary dramatically. This finding aligns with studies such as Tsegaw [35] and Shaoguang Bai [36], which highlighted the importance of combining traditional methods with regional hydrological comparisons to enhance design flood predictions for small watersheds.

The HEC-RAS model and the flood peak modulus of the design flood validation confirm that the weighted average method provides a reliable and efficient estimate for flood volumes, especially in ungauged basins. The relatively small relative error between the weighted average method and the HEC-RAS model underscores the robustness of the Weighed Average Method, particularly for flood estimation in small watersheds. The design flood results for the Baludi Reservoir align with general hydrological principles, showing a consistent decrease in flood peak modulus with increasing frequency and catchment area, similar to the trends observed in the Lafu and Dongmi Reservoirs, thereby confirming the reasonableness of the results.

A key feature of the hydrographs generated in this study is the observed asymmetry between the rising and falling limbs. This reflects the short duration and sharp peaks of flood events in mountainous basins, which are driven by intense rainfall and steep slopes. The rising limb of the hydrograph was found to be steeper, reflecting the rapid surface runoff typically seen during the initial phase of rainfall. In contrast, the recession phase was slower, primarily a result of the flood attenuation function of reservoirs. This observed asymmetry is an important consideration for flood management and reservoir operation as rapid flood rise times require efficient reservoir operation strategies to mitigate downstream flood risks.

Once the accurate flood peak flow is determined for the reservoir, as demonstrated by the Weighted Average Method, the flood volume estimations become more reliable. If the Inference Formula Method was used, it would tend to overestimate the flood peak flow and daily flood volume, leading to an inflated flood process curve. This would result in higher construction costs, as the design would be overly cautious, taking into account excessively high flood estimates. On the other hand, if the Instantaneous Unit Hydrograph Method were used, which typically underestimates the flood peak flow, it could compromise the safety of the reservoir, as the design would fail to account for the potential extremes, thereby endangering the structural integrity of the project. In contrast, the Weighted Average Method strikes a balance, ensuring both cost-efficiency and safety. By offering a more accurate flood volume estimation, it helps reduce construction costs while maintaining necessary safety margins. This method proves to be the most effective for designing flood protection infrastructure, particularly in areas where hydrological data is scarce or unavailable.

5. Conclusions

In this study, the proposed weighted average method for designing flood calculations in ungauged basins has shown a significant improvement over traditional methods. The method was developed by integrating the instantaneous unit hydrograph approach and inference formula, considering the specific flood characteristics of extra-small watersheds. The findings from the application of this method to the Baludi Reservoir in the Gelangramu River basin provide a comprehensive evaluation of its effectiveness.

(1)

A key insight from this study is the recognition of the challenges faced in designing floods for ungauged watersheds, where data is limited, and hydrological conditions may not be fully understood. Traditional methods, while useful, often fail to accurately account for subsurface conditions and local rainfall patterns, which can lead to underestimation or overestimation of flood risks. The weighted average method, by contrast, takes these factors into account, resulting in a more reliable flood prediction model.

(2)

The weighted average method, which considers subsurface conditions and rainfall characteristics, outperforms traditional flood calculation methods in terms of both accuracy and practical application in small watersheds with limited data. The proposed method effectively integrates various flood characteristics and hydrological data from adjacent basins, offering a more accurate and reliable means of predicting flood peaks and volumes in ungauged regions.

(3)

The calculation model of production and confluence was established in the Baludi Reservoir basin, and the weighted average of the instantaneous unit hydrograph and the inference formula method were used to simulate the confluence of the basin, demonstrating its effectiveness in regions with limited data.

(4)

Comparative analysis revealed that the weighted average approach provides a better description of the rainstorm process and results in more accurate flood predictions, ultimately contributing to safer and more cost-effective flood management practices. This method not only improves the safety and reliability of infrastructure such as reservoirs but also reduces construction and operational costs by eliminating the need for extensive data collection and complex modeling techniques.

(5)

However, while the method has shown promising results in the Baludi Reservoir case study, it is important to acknowledge its limitations. The model’s accuracy depends heavily on the availability of reliable data from nearby basins for calibration. In regions where no such data is available, the model’s performance may be compromised. Furthermore, while the weighted average method is effective for small, ungauged watersheds, its applicability to larger, more complex watersheds needs further investigation. Future research should focus on testing the model in different hydrological settings, incorporating more diverse rainfall data, and considering the impact of climate change on flood predictions. The inclusion of factors such as land-use changes, seasonal variations, and long-term climatic trends could further improve the accuracy and applicability of the model.

Overall, the weighted average method represents a significant advancement in design flood calculations, offering a practical and effective solution for flood estimation in small, ungauged watersheds. Its successful application could lead to broader adoption in similar hydrological settings worldwide, enhancing flood resilience and resource management.