# Hydrological Methodology Evolution for Runoff Estimations at Ungauged Sites

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

- (1)
- The rate of rainfall;
- (2)
- The kind and condition of the soil;
- (3)
- The character and inclination of the surface;
- (4)
- The condition and inclination of the bed of the stream;
- (5)
- The shape of the area to be drained, and the position of the branches of the stream;
- (6)
- The form of the mouth and the inclination of the bed of the culvert; and
- (7)
- Whether it is permissible to back water up above the culvert, thereby causing it to discharge under a head.

## 2. Early Methodologies

- (1)
- Observing the existing openings on the same stream;
- (2)
- Measuring, preferably at time of high water, a cross-section of the stream at some narrow place; and
- (3)
- Determining the height of high water as indicated by drift and the evidence of the inhabitants of the neighborhood. With these data and a careful consideration of the various matters, it is possible to determine the proper area of the waterway with a reasonable degree of accuracy.

#### 2.1. Flood Magnitude and Economics

#### 2.2. Dun’s Table

#### 2.3. Myers Formula

^{2}), D is the drainage area (acres), and C is a coefficient recommended to be 1.0 as a minimum for flat lands, 1.6 for hilly compact ground, 4.0 as a minimum for mountainous and rocky lands, and higher values in exceptional cases.

^{3}/s. The application of this expression requires an expert view of the water structure location.

#### 2.4. Talbot Formula

^{2}), D is the drainage area (acres), and C is a coefficient. Talbot offered the following guidance for selection of the coefficient, C.

- (i)
- For rolling agricultural country, subject to floods at the time of melting snow, and with the length of valley three or four times the width, one third is the proper value for C;
- (ii)
- In districts not affected by snow, and where the length of the valley is several times the width, one-fifth or one-sixth or even less may be used; and
- (iii)
- C should be increased for steep side slopes, especially if the upper part of the valley has a much greater fall than the channel at the culvert.

_{25}= 4.985 C(A)

^{0.5}

_{50}= 1.2Q

_{25}

_{100}= 1.17 Q

_{50}

_{25}, Q

_{50,}and Q

_{100}are runoff discharges (m

^{3}/s) for return periods of 25 years, 50 years and 100 years, respectively. Parameter C is for summation of C1, C2 and C3 which depends on terrain condition, slope, width, and length of drainage area (see Table 1) and A is drainage area in hectares.

- (a)
- The same straight line is used to derive the 25 year return period discharge as Q
_{25}= Q_{basis}S, where S is the slope of wadi main channel. This implies that as long as the slopes and the wadi drainage areas are the same, the discharge value can be obtained, irrespective of location anywhere in the world. However, this cannot hold true for any location. It has already been stated that there should be some difference in the discharge between arid regions and humid regions [22]; - (b)
- The 25 year return period discharge is converted to discharges for other periods (50 year, 100 year, etc.) using the same constant multiplications. These constants can be derived at least from rainfall records in a particular area according to some suitable theoretical probability distribution functions (PDF), such as extreme value distribution functions. These constants cannot be adopted by taking the same value for use in different areas of the regions;
- (c)
- The Modified Talbot formulation does not take into consideration the rainfall intensity/design of the storm based on return period;
- (d)
- Runoff coefficient (C) has maximum value of 0.9 when taking into consideration bedrock nature, channel slope, and some topographic factors. Coefficient factors, C1, C2 and C3 are not mathematical expressions and thus cannot be adapted as constants for the whole catchment. The reason for this is that any catchment may have mixed land use which varies according to topography and mountainous areas, vegetation cover, low lands, urban areas, rural areas, and residential areas. The Talbot approach has a limited choice of land-use types and this makes its application questionable, particularly in arid regions with mountainous settings; and
- (e)
- The Modified Talbot formulation does not take into consideration the percentage of imperviousness, soil infiltration parameters (i.e., initial and constant losses) and transformation parameters (i.e., time of concentration and storage coefficient) within the catchment. On the other hand, the time of concentration is a very important parameter in hydrograph analysis and is estimated generally by empirical formulations based on watershed major channel length and slope values [23].

- (i)
- It takes into consideration the area of drainage basin and then, according to this size, slope and shape empirical formulations are given with subjective specification. Basically, it yields a 25 year return period discharge, and then subsequent proposed factors are used for its increment to other return periods of 50 years, 100 years, etc. (see Equations (4)–(6)).
- (ii)
- Shape factor is another basic variable, but it is dependent on the areal width and area of the basin, hence it does not take into consideration the length of the major stream, which contributes to flood discharge more than any other drainage parameter.
- (iii)
- It does not take into consideration rainfall intensity. However, there are other methodologies that do not take into account rainfall intensity, but they are more recent (e.g., the Snyder [23] method, which has logical foundations). However, it is recommended in modern times that rainfall intensity is taken into consideration as the major factor in flood calculations.
- (iv)
- C, as the runoff coefficient, has been considered compositely as maximum values equal to 0.9. It is decided in a subjective manner by taking into consideration the bedrock nature, channel slope, and some topographic factors. In fact, nowadays, there are detailed tables for the determination of C depending on surface soil types, and for hydrological soil classification taking into consideration vegetation cover, land use, geology, topography, etc., in Chow [15], both of which are not considered in the Talbot approach,
- (v)
- It is not clear how the Discharge (Q)-Area (A) straight line is obtained on double logarithmic paper. Accordingly, the basic equation Q
_{25}= Q_{basic}S.F takes only into consideration the area. Inclusion of S.F. is not clear, although it is mentioned that it is dependent on the slope factor for the drainage area. - (vi)
- W is a factor that reflects the elongation ratio of the catchment, but not the drainage density and main channel slope, which are important in discharge calculations.
- (vii)
- There is no reason why n is taken as 3/4 (see Equation (3)); in fact, there should be some empirical basis for such an exponent. In the Saudi Geological Survey (SGS) reports, this figure is based on factual data, and the exponent appears as 0.522.
- (viii)
- C1, C2 and C3 are based on expert views and not mathematical expressions. They cannot be taken as constants for the whole catchment, because in any catchment there may be a mixture of mountainous areas, vegetation cover and low lands. Perhaps the best figure that can be used is the weighted average, but such an approach is not available in the Talbot method.
- (ix)
- The factors for Q
_{50}= 1.2Q_{25}and Q_{100}= 1.4Q_{25}can be considered frequency factors, but they should be based on a certain probability distribution function (PDF) such as Gumbel, Pearson Type-II, log-Pearson, or extreme value distributions with validation. These factors cannot be adopted as constant in the different areas of a region. Additionally, they should also be dependent on the rainfall PDF. - (x)
- The width, W, of the wadi is taken as constant, for a given discharge, as W = 4.84√Q; this should vary along the main channel of the wadi [22].

#### 2.5. Burkli-Ziegler Formula

#### 2.6. Flood Frequency

^{2}.

^{2}. The first regional flood–frequency relationships for small streams in Kansas were published by the USGS in 1966 [26]. These relationships for drainage areas under 70 mi

^{2}were labeled “preliminary” because they were based on only 8 years of peak flow data (1957–1964) for 95 stations. The report presented statewide regression equations for discharges with recurrence intervals of 1.2 years, 2.33 years (the mean annual flood for a Gumbel [27] probability distribution), 5-years and 10-years. The three inputs to these equations are drainage area, average channel slope, and the average number of wet days per year. The standard error of estimation for the 10 year equation, expressed in percent, is from +100% to −49%. A more comprehensive USGS study of Kansas flood frequency, published by the Kansas Water Resources Board in 1975 [28] provided statewide equations for flood discharges with recurrence intervals from 2 years to 100 years for unregulated rural streams with drainage areas from 0.4 to 10,000 mi

^{2}. The two inputs to these equations are the drainage area and the 2 year, 24 h rainfall, which are obtained from a map. The standard errors range from +50% to −31% for the 5-year equations, and from +74% to −42% for the 100 year equations. The 1975 USGS equations were the first regional flood frequency equations to be widely used to compute design flows for highway culverts and bridges in Kansas. The USGS published updated flood-frequency equations for Kansas in 1987 Clement [29] and 2000 Rasmussen and Perry [30]. The 1987 equations have four inputs: drainage area, 2 year 24 h rainfall, average channel slope and generalized soil permeability. The 2 year 24 h rainfall and the generalized soil permeability are obtained from maps. Standard errors range from +35% to −26% for the 10 year equations, and from +46% to −31% for the 100 year equations.

^{2}and another set for drainage areas under 30 mi

^{2}. The equations for drainage areas over 30 mi

^{2}have four inputs: drainage area, mean annual precipitation, channel slope and generalized soil permeability. Standard errors range from +35% to −26% for the 10-year equations, and from +42% to −30% for the 100 year equations. The equations for drainage areas under 30 square miles have only two inputs: drainage area and mean annual precipitation. These equations for small watersheds have large standard errors that vary from +52% to −34% for the 10-year and from +71% to −41% for the 100-year equations.

#### 2.7. Critics of Early Methods

^{3}/s is insufficient, a 3 m

^{3}/s design discharge may be considered and this will mean an increase of 225%. The real question is which one to adapt in practical applications.

## 3. Recent Methodologies

#### 3.1. Recurrence Intervals for Design

#### 3.2. Rainfall Frequency

#### 3.3. Flood Frequency Analysis

## 4. Practical Estimation Methodologies

#### 4.1. Rational (RM) Method Versions

_{X}/C

_{Y}, from Table 2, leading to Table 3.

- (i)
- The runoff coefficients are equal, which is hardly the case in practical applications due to the differences in geomorphology, land use, geologic lithology and vegetation cover, etc.;
- (ii)
- Equal rainfall intensities generate equal discharges per area in both basins; and
- (iii)
- Since the RM is valid only for small drainage basins [16], the validity of Equation (11) has areal limitation.

#### 4.2. Mean Runoff Versions

_{i}; Q

_{1}, Q

_{2}, ⋯, Q

_{n}) at a site:

_{i}is the arithmetic average, σ

_{i}is the standard deviation, ρ

_{i}is the first order serial correlation coefficient, and finally ε

_{i}is the standard normal PDF with zero mean and unit standard deviation. It is mentioned by Farmer and Vogel [45] that such an approach is used in flood frequency analysis. This statement implies that the serial correlation coefficient is equal to zero, which reduces Equation (16) into the following form.

#### 4.3. Mean Runoff and Standard Deviation Versions

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Values of C1, C2 and C3 [21].

C1 | 0.30 | Mountainous area |

0.20 | Semi-mountainous | |

0.10 | Low land | |

C2 | 0.50 | S > 15% |

0.40 | 10% < S < 15% | |

0.30 | 5% < S < 10% | |

0.25 | 2% < S < 5% | |

0.20 | 1% < S < 2% | |

0.15 | 0.5% < S < 1% | |

0.10 | S < 0.5% | |

C3 | 0.30 | W = L |

0.20 | W = 0.4 L | |

0.10 | W = 0.2 L |

**Table 2.**Runoff coefficient for lawns [16].

Land Use | C |
---|---|

Sandy soil flat, (2%) | 0.05–0.10 |

Sandy soil avg., (2–7%) | 0.10–0.15 |

Sandy soil steep, (7%) | 0.15–0.20 |

Heavy soil flat, (2%) | 0.13–0.17 |

Heavy soil avg., (2–7%) | 0.18–0.22 |

Heavy soil steep, (7%) | 0.25–0.35 |

Ungauged | X | ||||||
---|---|---|---|---|---|---|---|

Gauged | Sandy Soil Flat, (2%) | Sandy Soil avg. (2–7%) | Sandy Soil Steep, (7%) | Heavy Soil Flat, (2%) | Heavy Soil avg., (2–7%) | Heavy Soil Steep, (7%) | |

Y | Sandy soil flat, (2%) | 1 | 0.5–0.67 (0.59) | 0.33–0.5 (0.43) | 0.77–0.59 (0.69) | 0.28–0.57 (0.43) | 0.20–0.29 (0.25) |

Sandy soil avg., (2–7%) | 1 | 0.67–0.75 (0.71) | 0.77–0.88 (0.83) | 0.56–0.68 (0.62) | 0.40–0.43 (0.42) | ||

Sandy soil steep, (7%) | 1 | 1.15–1.18 (1.17) | 083–0.91 (0.87) | 0.60–0.57 (0.59) | |||

Heavy soil flat, (2%) | 1 | 0.72–0.77 (0.75) | 0.53–0.49 (0.51) | ||||

Heavy soil avg., (2–7%) | 1 | 0.72–0.63 (0.68) | |||||

Heavy soil steep, (7%) | 1 |

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**MDPI and ACS Style**

Şen, Z.
Hydrological Methodology Evolution for Runoff Estimations at Ungauged Sites. *Water* **2023**, *15*, 702.
https://doi.org/10.3390/w15040702

**AMA Style**

Şen Z.
Hydrological Methodology Evolution for Runoff Estimations at Ungauged Sites. *Water*. 2023; 15(4):702.
https://doi.org/10.3390/w15040702

**Chicago/Turabian Style**

Şen, Zekâi.
2023. "Hydrological Methodology Evolution for Runoff Estimations at Ungauged Sites" *Water* 15, no. 4: 702.
https://doi.org/10.3390/w15040702