1 Introduction

The rational method is widely used to estimate the maximum runoff of small catchments (Schaake 1967; Guo 2001; Akan 2002; Young et al. 2009; Sabzevari 2010, 2017; Dhakal et al. 2010; Grimaldi and Petroselli 2015; Chin 2019; Baiamonte 2020; Ardekani et al. 2021; Lapides et al. 2021; Machado et al. 2022). This technique is widely used to calculate the design flood of hydraulic structures, spatially in the design of drainage systems in urban areas (Baiamonte 2020; Chin 2017; Froehlich 2016; Cleveland et al. 2011; Al-Amri et al. 2022, Młyński et al. 2020). The maximum runoff of the catchment is determined as follows (Kuichling 1889):

$${Q}_{T}=0.278\times {C}_{T\times }{I}_{T}\times A$$
(1)

where QT is the design discharge (m3s−1), CT is the design runoff coefficient (CT), A is basin area (Km2), and IT is the design rainfall intensity (mm/h). The runoff coefficient (RC) plays a crucial role in the rational formula (Eq. 1). It is defined based on soil type, land use, and slope of the basin (Longobardi et al. 2003; Sriwongsitanon and Taesombat 2011; Zhang et al. 2014), is accessible in tabular form in many hydrologic textbooks Chow (1962), and it determines the amount of excess rainfall.

Most of the available CT tables were extracted based on regional information from specific catchments and are unusable for other regions and catchments. Therefore, it is important to have a regional approach to estimate CT.

RC in the rational method is commonly adjusted by return period (Jens 1979; Bernard 1938; Dhakal et al. 2013). Based on the return period, the value of RC can be increased to calculate the CT (Ken-Bohuslav 2004). Furthermore, the RC is used to estimate plain’s water balance (Fariborzi et al. 2019; Giordano et al. 2010; Peng et al. 2019; Lu et al. 2018), to calculate the concentration time (Li and Chibber 2008), and directly in rainfall-runoff models. For example, in the Soil Conservation Service (SCS) model, excess rainfall can be calculated from different infiltration methods (e.g., Green-Ampt, Soil Conservation Service Curve Number—SCS-CN, or Horton). In SCS-CN models, infiltration rate can be estimated based on Curve Number (CN) which is correlated with RC parameters (Kim and Shin 2018; Mishra and Singh 2013). Furthermore, the infiltration process depends on the slope, topography, initial moisture, rainfall, and land cover (Dunne et al. 1991; Huang et al. 2013; Dunkerley 2012; Pishvaei et al. 2020). In addition, RC varies from one event to another because of variation in rainfall, infiltration, and soil moisture conditions, thus it is important to consider these factors in determining the RC.

Recent studies confirm the relationship between return period and runoff coefficient. (Froehlich 2016; Hotchkiss and Provaznik 1995; Titmarsh et al. 1995; Young et al. 2009; Dhakal et al. 2013; Dhakal et al. 2012). For example, Dhakal et al. (2010) found that the rational runoff coefficients increase with the return period and the rate of increase is much larger than what typically recommended in design manuals. Froehlich (2016) compared runoff coefficient adjustment factors and found that the return period adjustment factors illustrated a large difference, caused by the wide variability of parameters that influence the catchment runoff (such as precipitation, soils, and land-surface cover) within a region as vast as Texas.

However, Chin (2017) showed that Froehlich (2016) equated the incremental rainfall excess within each time interval to the incremental runoff predicted by the Natural Resources Conservation Service (NRCS 1972) Curve Number (CN) model when calculating the runoff hydrograph and this could have led to physically unrealistic results (e.g., Morel-Seytoux and Verdin 1981; Chin 2013).

Based on SCS-CN infiltration relations, equations are presented in this research to calculate CT as a function of land use (CN) and design return period.

The proposed method is compared and evaluated with the methods and standards presented in previous research. One of the important objectives of this study is to convert the standard CN tables into CT calculated and presented in tabular form for 4 different climates in Fars province, Iran.

2 Methodology

In this part, we first explain the fundamental concept of the rainfall-runoff process, then present our approach for modifying the runoff coefficient for different return periods.

3 Runoff Coefficient Concept

Runoff coefficient is the percentage of rainfall that is converted to runoff according to the following formula:

$$RC=\frac{R}{P}$$
(2)

where P and R are the rainfall and runoff depths, and RC is the runoff coefficient. In a gauged basin with available observed runoff hydrograph, the runoff volume equals the area below the hydrograph curve. The depth of runoff (R = V/A) is obtained by dividing the volume of runoff (V) by the area of basin (A). Furthermore, based on the continuity equation, the amount of rainfall (P) is equal to the sum of infiltration depth (F) and runoff depth (R). Therefore, the runoff depth can be calculated by subtracting the infiltration from the rainfall depth (R = P−F). Runoff rate changes temporally and spatially across the basin due to the spatial–temporal variation affecting factors on the infiltration process, e.g., soil permeability and soil moisture. This spatial–temporal variation affecting the parameter of infiltration is a source of uncertainty and influences the accurate estimation of the runoff coefficient.

4 Design Rainfall

The design rainfall intensity in the rational formula (Eq. 1) can be obtained by regional Intensity–Duration–Frequency (IDF) curves (Madsen et al. 2009; Soltani et al. 2017; Courty et al. 2019) as follows:

$${I}_{T}^{{t}_{d}}=\frac{a{T}^{b}}{{t}_{d}+f}$$
(3)

where td is the rainfall duration, T is the return period, a, b and f are regional coefficients. Based on this, the following equation can be applied to estimate the design rainfall in Iran (Ghahraman and Abkhezr 2004):

$${P}_{T}^{{t}_{d}}=[0.2471\mathit{ln}(T-0.6)(0.37+0.618{{t}_{d}}^{0.45}]({P}_{2}^{24}{)}^{1.14}(e{)}^{0.291}$$
(4)

where \(P_{T}^{{t_{d} }}\) is the design rainfall (mm), \(t_{d}\) is the rainfall duration (minutes), \(P_{2}^{24}\) is the 24-h maximum rainfall with a 2-year return period. Rainfall duration is usually considered equal to the time of concentration of the basin.

5 Regional Design Runoff Coefficient

To estimate the maximum runoff by rational formula, the design runoff coefficient (CT) is considered higher than the RC of the basin. Here we suggested two different approaches for calculating CT for ungauged and gauged basins.

5.1 Ungauged Basins

In ungauged basins, only rainfall data are available. Based on the maximum 24-h rainfall (maximum daily rainfall) the rainfall with different return periods (PT) will be estimated (Apollonio et al. 2018; Hajani and Rahman 2018; Merkel et al. 2017). The best-fitted probability distribution of the maximum 24-h rainfall (e.g., Normal, Lognormal, Gumbel, Pearson type III, Log Pearson type III) can be used to determine the design rainfall. Then, based on the best-fitted probability distribution, the design rainfall can be estimated as (Chow et al. 1962):

$${P}_{T}=\overline{P }+{K}_{T}S$$
(5)

where \(\overline{P }\) and S are the mean and standard deviation of maximum 24-h rainfall, and KT is the frequency factor which is calculated based on the best-fitted probability distribution (Chow et al. 1962); for example, for Gumbel distribution, it can be calculated as:

$${K}_{T}=-\frac{\sqrt{6}}{\Pi }\left[0.5772+\mathit{ln}\left(\mathit{ln}\left(\frac{T}{T-1}\right)\right)\right]$$
(6)

Equation 6 is suitable where the rainfall data are high, and if the amount of data are low, the Gumbel distribution frequency factor tables should be used. In any case, the equations of the best statistical distribution should be used by the rainfall data of the catchments. In this study, the Gumbel distribution was the best. Then the runoff depth can be estimated based on the SCS-CN method (Menberu et al. 2015):

$$R=\frac{{\left(P-0.2SR\right)}^{2}}{P+0.8SR}$$
(7)

where R and P are the runoff, and rainfall depth (mm) and SR is the maximum potential soil retention (mm), calculated as in the following:

$$SR=\left(\frac{25400}{CN}-254\right)$$
(8)

where CN is the curve number parameter that is defined based on the land use and soil type (Table 1).

Table 1 CN values for various land use and soil hydrological groups (Chow et al. 1962)
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Substituting value R (Eq. 7) in Eq. 2, we can estimate the runoff coefficient as follows:

$${C}_{T}=\frac{{\left({P}_{T}-0.2SR\right)}^{2}}{{P}_{T}\left({P}_{T}+0.8SR\right)}$$
(9)

The by substituting the calculated design rainfall (Eq. 5) in Eq. 9 the design runoff coefficient (CT) can be estimated as follows:

$$C_{T} = \frac{{\left( {\bar{P} + K_{T} s - 0.2SR} \right)^{2} }}{{\left( {\bar{P} + K_{T} s} \right)\left( {\bar{P} + K_{T} s + 0.8SR} \right)}}$$
(10)

Most RC tables provided in ASCE and WPCF (1960) are valid for return periods of less than 10 years. For higher return periods, the design rainfall intensity increases and the infiltration decreases abruptly and the runoff coefficient increases. In this situation, the design runoff coefficient should be increased for the return period of more than 10 years (Dhakal et al. 2013).

In some references (e.g., Rossmiller 1980; Chow et al. 1962), tables have been provided to increase the runoff coefficient for different return periods (e.g., Table 2), but these tables have been calibrated for a specific basin in the world and are not valid for all basins with different climates.

Table 2 Runoff coefficient for use in the rational method (Rossmiller 1980; Chow et al. 1962)
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Equation 10 shows that the design runoff coefficient is a function of the design rainfall with different return periods and the curve number as the infiltration parameter. Based on the best regional statistical distribution, you can compute design rainfall.

Equation 10 can better consider rainfall and soil infiltration conditions regionally.

5.2 Gauged Basins

In the gauged basin (basin has recorded stream discharge and flood data), the best-fitted probability distribution of the maximum of maximum 24-daily flow will be used to determine design flood for different return periods as follows:

$${Q}_{T}=\bar{Q}+{K}_{T}{S}_{Q}$$
(11)

where QT is design flood for return period T, Q and ST are mean and standard deviation of the maximum flow and KT is frequency factor which is calculated based on the best-fitted probability distribution (Chow et al. 1962); for example the KT for Gumbel distribution flow is shown in Eq. 6.

By substituting the value of QT in Eq. 1 the CT can be calculated as (Pilgrim and Cordery 1993):

$${C}_{T}=\frac{\bar{Q}+{K}_{T}{S}_{Q}}{0.278{I}_{T}\times A}$$
(12)

6 Relationship Between CT and Return Period

Bernard (1938) presented the following relation for CT:

$${C}_{T}={C}_{\mathrm{max}}\times {(\frac{T}{100})}^{a}$$
(13)

where Cmax is the value of design runoff coefficient for a return period of 100 years, T is return period and a is coefficient which ranges between 0.05 and 0.23. In this study, the information listed in Table 2 was first used to investigate the relationship between CT and T (Rossmiller 1980; Chow et al. 1962). The relationship between CT and T was defined for different land use and land cover in the city of Austin in the United States (Table 2).

This relationship for asphalt and grass cover (50–75% with slope 0–2%) is demonstrated in Fig. 1.

Fig. 1
figure 1

Relationship between design runoff coefficient (Ct) and return period (T) for a asphalt cover and b gross cover (50–75% with slope 0–2%)

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According to Table 2, for all land uses, regressions models were applied for CT and T and the best regression equation was nonlinear power. Here and based on the data in Table 2, the following equation is suggested for estimating CT is as it follows:

$${C}_{T}={C}_{5}(T{)}^{a}$$
(14)

According to Eq. 14, by knowing the 5-year runoff coefficient, which is close to the basin RC (RC in hydrologic tables), the CT can be determined for each return period.

Power a is of special importance in Eq. 14, which is related to the soil type and vegetation cover.

The values a and correlation coefficients (R) for Eq. 14 in different land uses are present in the second column of Table 2. Based on the results, the correlation coefficients for 20 different land uses were above 0.9, which is a very good result. The coefficient a was observed between 0.05 and 0.12.

Three types of slopes, 0–2%, 2–7%, and above 7%, have been considered for grass cover. The higher slope, the lower infiltration rate, and the RC increases. The higher slope, the lower coefficient a. In forest/woodlands, where the infiltration is high, the values of RC are low, and the value of a for the slope of 0–2% is equal to 0.12 and decreases with increasing slope, and for slope above 7%, this coefficient reaches 0.08. Finally, based on the results, the value of a for soils with low vegetation and permeability is 0.05, and for the higher permeability approaches to 0.12. For low slopes, a is close to 0.07, and for larger slopes, a is close to 0.12.

Since the average RC of the basin is close to the 5-year runoff coefficient, by knowing the RC of the basin, CT can be obtained for different return periods by Eq. 14. In the next part of this research, and the proposed formula will be further validated.

In order to better understand the subject matter, a practical example and methodology were presented in “Appendix A” to explain the method.

6.1 Case Studies

We calculate the CT for the large area in southern Iran (Fars province) (Fig. 2). Fars is the fourth largest province in Iran (Fig. 2a), with 122,608 km2 (Torabi Haghighi et al. 2020). The province is located in the southern part of the country, which includes 29 cities and covers 7.5% of Iranian territory. It extends between 27.02° and 31.43°N latitude and 50.42°–55.36°E longitude. The province includes different climates due to geographical configuration and position between the high Zagros Mountain in the north and west, the Sirjan desert in the north and east, and the Persian Gulf in the south (Torabi Haghighi et al. 2020). This study used the daily rainfalls of 83 rain gauge stations in Fars province (Table 3 in the supplementary material (S.M.)). Previous studies illustrated that the Gumbel distribution is the best-fitted distribution for maximum daily rainfall (Ahmadpour et al. 2017).

Fig. 2
figure 2

Study area, location of Fars province in Iran (After Samani, Jamshidi, 2017)

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Table 3 Used meteorological stations and Rainfall data (After Sabzevari et al. 2009)
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Based on the variation of daily rainfall, the Fars province is divided into four regions (Table 4 and Fig. 2.b).

Table 4 Rainfall classification of different regions of Fars province
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7 Results and Discussion

7.1 Design Runoff Coefficient CT for Fars Province

In the first region of Fars province (region 1), 83 rain gauge stations with maximum daily rainfall of less than 50 mm were analyzed. In this region, the mean and standard deviation of rainfall was 42 mm and 17.5 mm, respectively. The CT for different return periods from 5 to 1000 years and variation in CN from 30 (for highly permeable soils) to 100 (for low permeable soils) were calculated based on Eq. 10 and considering the Gumbel distribution. In this region, the CT for high permeable soil (CN below 40) is zero, and for CN = 40 only for 100 years return period, we have CT lower than 0.05. For permeable soil (i.e., CN = 70), the CT varied between 0.14 and 0.41, while for low permeable soils (i.e., CN = 90), it varied between 0.57 and 0.78 (Table 5).

Table 5 Design runoff coefficient (CT) for the region #1 to region #4 in Fars province
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Figure 3 shows the variation of design runoff coefficient (CT) for different CN values and return periods for region1 based on Table 5. As the value of CN represented the hydrological group based on land use and permeability of the soil, e.g., group A represents permeable, and group D is representative of low permeable soils (look at Table 1), we can provide the CT variation for different land use and return period (Table 6). Based on these results, the CT for meadowland use in good condition will be varied between 0 and 0.32 for 10 years return period.

Fig. 3
figure 3

Variation of design runoff coefficient (CT) for different CN values and return periods for region 1

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Table 6 Designed runoff coefficient (CT) for different land uses and different return periods in Region 1
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Furthermore, for other regions (regions 2–4), the CN- based (same as Table 5) and land-use-based (same as Table 6) tables of CT are produced and illustrated in Tables 5, 6, 7, 8, 9, 10 in SM.

Table 7 Land use-based value of Designed runoff coefficient (CT) for different land uses and different return periods in Region 2
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Table 8 Land use-based value of Designed runoff coefficient (CT) for different land uses and different return periods in Region 3
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Table 9 Land use-based value of Designed runoff coefficient (CT) for different land uses and different return periods in Region 4
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Table 10 CN-based value of α coefficients for different regions of Fars provinces
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7.2 CN-Based Value of “α” Coefficients for Different Regions

The a coefficient (in Eq. 14) was calculated from 0.05 to 0.12 for the low permeable soils to high permeable soils. Here, we will validate a coefficient for different regions based on the calculated CT in different regions in the previous part (e.g., data in Table 5 for region 1 and region 2). By using SPSS16 software and mentioned data (in Table 5), Eq.14 was re-evaluated, and the optimal nonlinear regression coefficients (a, R) were calculated (Table 10).

For region 1, the a coefficient ranged from 0.05 to 0.29 for CN variation from 90 to 60 with a significant correlation coefficient of more than 0.92 (Fig. 4a).

Fig. 4
figure 4

Relationship between CN and a for a regions 1, b region 2

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Finally following Eqs were suggested for estimating a value in different regions 1–4 of Fars province.

$$a \left(Region1\right)=9.74\times {e}^{-0.058\times CN}$$
(15)
$$a \left(Region2\right)=4.10\times {e}^{-0.051\times CN}$$
(16)
$$a \left(Region3\right)=2.98\times {e}^{-0.050\times CN}$$
(17)
$$a \left(Region2\right)=2.25\times {e}^{-0.050\times CN}$$
(18)

8 Conclusion

One method for calculating the runoff of small catchments, in particular in urban areas, is the rational method. The rational method includes a function of the catchment’s area, storm intensity, and runoff coefficient. Urban drainage system engineers use the notion of a design return period to design surface runoff disposal systems such as canals and drains.

The design runoff coefficient should be increased according to the return period of the structure design. The runoff coefficient hinges on the catchment’s slope, plant cover, and land use of the catchment. These pieces of information are usually available in the catchments.

The initial idea of this research was to formulate the design runoff coefficient on the basis of the rainfall and land use data of each catchment in the world, and to calculate it separately so that the flood estimation can be performed more accurately using the rational method.

In this study, we present a novel approach for correcting design runoff coefficients for different return periods. The regional design runoff coefficient was computed based on a statistical analysis of local maximum daily rainfall in our technical approach. The method applied for Fars province in Iran was divided into four regions based on the observed historical value of maximum daily rainfall of 83 rainfall stations. CN and land use-based runoff coefficient tables were derived for different return periods from 10 to 500 years for each region. Finally, equations were introduced where the design runoff coefficient is a function of the return period and the curve number.

The runoff coefficient for different return periods from 5 to 1000 years and variation in CN from 30 (for highly permeable soils) to 100 (for low permeable soils) were calculated based on proposed equations.

The design runoff coefficient equation was presented as \({C}_{T}={C}_{5}(T{)}^{a}\). The coefficient “a” was presented as a function of the curve number in 4 different regions in Fars province.

According to the proposed equation, it is possible to transform the standard curve number tables, which are widely used in flood prediction using the SCS method, into tables for calculating the design runoff coefficient. These tables were calculated and provided for the Fars region.

The method can be reproduced for other regions worldwide and be in service of local water authorities and consulting engineering companies to use in rainfall-runoff studies for gauged and ungauged basins. It is suggested that the results of this research and the proposed method in different basins or different climates be evaluated and validated based on observed rainfall and runoff data of the basins to further evaluate the effect of considering the return period on runoff coefficient in the rational method.