# Difference between revisions of "Category:749 Hydrologic Analysis"

## 749.1 General

Hydrology is that branch of the applied earth sciences which deals with the occurrence and distribution of surface and ground waters. Highway drainage design is concerned with the surface water hydrology of small watersheds. In particular, the peak rate of runoff and/or the runoff hydrograph produced by flood events are of interest in analysis of highway drainage problems. Peak runoff rates are a parameter of design and their estimation is a prerequisite to hydraulic computations. Flood hydrographs are a necessary parameter of stormwater detention basin design, which can be used to reduce peak runoff rates for design of certain highway drainage structures.

In general, the design of drainage facilities may be described as follows:

• Select the design frequency
• Estimate the peak rate of runoff or runoff hydrograph resulting from the design and review floods
• Design the drainage structure to pass the design flood peak or runoff hydrograph
• Check the hydraulic capacity of the proposed system for the base flood
• Check that the final design meets the requirements of the National Flood Insurance Program (NFIP)

Criteria for determination of the design flood frequency and methods for computing peak rates of runoff and runoff hydrographs are presented in this article.

## 749.2 Design Frequency and Return Period

The frequency of occurrence or return period of an event may be defined as the average period of time between events equal to or greater than a given magnitude. The annual probability of occurrence of an event is equal to the reciprocal of the event's return period. For example, a flood with a return period of 100 years has a 1% chance of occurring in any year; whereas a flood with a return period of 25 years has a 4% chance of occurring in any year.

### 749.2.1 Design Frequency Criteria

The return period used as a criterion of design is known as the design frequency. The design frequency to be used in the design of drainage structures is a function of roadway type (major of minor), traffic volume and type of drainage facility. The design flood magnitude, design frequency and corresponding water surface elevation(s) shall be included on the project plans for all crossroad structures.

The design frequency for roadside drainage ditches should be chosen based on the function served. Ditches with a primary function of removal of water from the pavement should be based on the design frequency of the pavement drainage. Ditches with a primary function of carrying water to or from a crossroad structure should be based on the design frequency of the crossroad structure.

### 749.2.2 Base Flood and Overtopping Flood

The base flood is defined as the 100-year flood or the flood with a 1% chance of being exceeded in any given year. The overtopping flood is defined as the discharge and corresponding return period and water surface elevation at which flow occurs over the roadway.

After sizing a drainage facility using the design flood criteria, the ability of the proposed facility to pass the base (100-yr) flood shall also be evaluated, with special attention to any risks to people or property. This is done to ensure that there are no unexpected flood hazards inherent in the proposed facility for flood events exceeding the design discharge.

The overtopping flood shall be determined for all crossroad structures, and the resulting overtopping flood magnitude and approximate return period shall be included on the project plans. When the overtopping flood is greater than the 100-year flood, the return period should be noted as “>100 yr”. If the overtopping flood is less than the 100-year flood, the magnitude and water surface elevation of the 100-year flood shall also be included on the project plans.

### 749.2.3 National Flood Insurance Program Criteria

Meeting the requirements of the National Flood Insurance Program (NFIP) regulations can necessitate larger structures than those designed to meet the above design frequencies. The designer shall ensure that the final design meets all requirements of the NFIP. Federal Emergency Management Agency (FEMA) Flood Insurance Study (FIS) maps should be consulted to determine the applicable NFIP requirements. Development in a mapped regulatory floodway can cause no increase in Base Flood Elevations (BFE’s). Development in a mapped Special Flood Hazard Area where no floodways have been defined can cause no more than a 1.0 ft. increase in BFE’s.

See the following for definitions and additional information on the NFIP:

## 749.3 Design Data

In order to carry out the hydrologic analysis of a watershed, it is necessary to assemble certain data. This data includes the drainage area, the length of the hydraulically longest drainage path, the elevation of the watershed ridge, the elevation of the watershed outlet, the hydrologic soil group, the type of terrain, the land use of the watershed, and information on the extent of development in urban areas. The topographic data should be obtained from suitable maps when maps are available. Suitable maps are defined as department manuscripts and 7-1/2 or 15 minute USGS topographic maps or maps of equal quality. Only when such maps are not available should field measurements be made for the purpose of obtaining hydrologic information. The land use and terrain of the watershed should be evaluated in the field by the designer. The hydrologic soil group is either given on the soil survey performed by the Materials Division or is to be determined from county soil survey maps or from information from the soil survey and site visits.

## 749.4 Rural vs. Urban Hydrology

Small watersheds may be divided into rural and urban classifications. A rural watershed is a drainage basin whose natural response to rainfall has not been substantially altered by urban land activity. Rural watersheds may be either natural or agricultural.

An urban watershed may be defined as a drainage basin in which manmade developments in the form of impervious surfaces and/or storm drainage systems have substantially altered the basin's natural response to rainfall. Urbanization of a natural watershed progresses in one of two ways. First, the addition of impervious surfaces in the form of roads, streets, parking lots and roofs will prevent infiltration of rainfall into the covered soil surface, thus increasing the total volume and peak rate of runoff from a given rainfall volume. Second, to protect the now valuable property in a developed watershed from this increased peak and volume of runoff, storm drainage systems are installed. The installation of a storm drainage system does not increase the volume of runoff, but modifies the time distribution of runoff. Thus, when storm water drainage systems are installed, the time of concentration of the watershed is decreased. Therefore, storm water drainage systems have the effect of removing a given volume of runoff in a shorter period of time, thus further increasing the peak rate of runoff.

All hydraulic design in urban areas should consider the effect of increasing development throughout the projected life of the structure. Information on planned future development may be available from local agencies.

Two methods are presented in this article for estimating peak rates of runoff from small watersheds. The Rational Method should be used on all watersheds less than 200 acres in size. On watersheds greater than 200 acres in size, the USGS Regression Equations should be used. Watersheds that lie within Region III for the USGS Rural Regression Equations and have drainage areas between 200 to 300 acres do not fall within the limits for either set of equations. In this case, both sets of equations should be calculated, and the designer should decide which is appropriate.

## 749.5 The Rational Method

The Rational Method was developed as early as 1889, and despite its limitations is one of the most widely used methods of estimating peak flows. Several assumptions are implicit in application of the Rational Method:

• The maximum runoff rate occurs when the rainfall intensity lasts as long or longer than the time of concentration
• The frequency of the discharge is the same as that of the rainfall intensity
• The fraction of the rainfall that becomes runoff is independent of the rainfall intensity or volume

The first assumption implies that a homogeneous rainfall event is applied uniformly to the entire drainage area, and may not be valid for larger watersheds where constant rainfalls of high intensity do not occur simultaneously over the entire watershed. This assumption also provides the basis for using the watershed's time of concentration as the duration of the design storm. The second assumption again limits the size of the drainage area because for larger basins, factors other than rainfall frequency can play a large role in determining the flood frequency. Finally, the third assumption is reasonable for highly impervious areas, but less reasonable for pervious areas where the antecedent moisture condition plays a large role in determining the amount of rainfall that becomes surface runoff. For these reasons, use of the Rational Method is limited to small watersheds.

### 749.5.1 Equation

The Rational Method is expressed by the following formula.

${\displaystyle Q=k_{c}CIA\,}$ (Equation 1)
where:
Q = the peak rate of runoff, ft3/s
kc= 1.0
C = runoff coefficient, representing the ratio of direct runoff to rainfall
I = rainfall intensity of the design storm, in/hr
A = drainage area of the watershed, acres

### 749.5.2 Runoff Coefficient

The runoff coefficient represents the ratio of runoff to the total rainfall and combines the effects of several watershed characteristics such as land use, soil type, cover condition, and general terrain or watershed slope.

#### 749.5.2.1 Rural Runoff Coefficients

For rural watersheds, the soil type, land use, and terrain are used to determine runoff coefficients.

The Soil Conservation Service (SCS) has classified about 4,000 major soils found in the United States into four basic hydrologic groups as follows:

Group A (Low runoff potential) - Soils having high infiltration rates even when thoroughly wetted and consisting chiefly of deep, well-drained sands or gravels. These soils have a high rate of water transmission.
Group B - Soils having moderate infiltration rates when thoroughly wetted and consisting chiefly of moderately deep to deep, moderately well to well drained soils with moderately fine to moderately coarse textures. These soils have a moderate rate of water transmission.
Group C - Soils having slow infiltration rates when thoroughly wetted and consisting chiefly of soils with a layer that impedes downward movement of water, or soils with moderately fine to fine texture. These soils have a slow rate of water transmission.
Group D (High runoff potential) - Soils having very slow infiltration rates when thoroughly wetted and consisting chiefly of clay soils with a high swelling potential, soils with a permanent high water table, soils with a claypan or clay layer at or near the surface, and shallow soils over nearly impervious material. These soils have a very slow rate of water transmission.

The SCS has performed modern soil surveys for many counties in Missouri, classifying the soils by a soil series name. The hydrologic soil group is given in the SCS soil survey for these soils. If a modern soil survey map is not available for the county in question, the hydrologic soil group should be determined by the designer. This may be done by evaluating information available from the soil survey performed by the Materials Division and/or from a site visit and choosing the appropriate hydrologic soil group from the descriptions given above. The district soils and geology technologist may provide assistance in determining the hydrologic soil group.

For smaller projects, the soil series name and hydrologic soil group for soils along the right of way are generally given on the soil survey performed by the Construction and Materials Division. For larger projects, this soil survey does not name individual soil series, but rather gives "soil associations" along the right of way. These soil associations are groupings of soil series that have similar physical characteristics; however, the soils in an association may not have the same hydrologic soil group. Additionally, in many instances it will be necessary to determine a hydrologic soil group for soils located some distance from the right of way. In these cases, the hydrologic soil group should be determined by the designer from the SCS county soil survey map, if available, or by evaluating information available from the soil survey performed by the Construction and Materials Division and/or from a site visit.

Land use and terrain can be determined by inspection of appropriate topographic maps or by a field inspection. After the hydrologic soil group, land use, and terrain have been determined, the runoff coefficient(s) to be used for estimating the 10-year peak flow for the rural watershed can be found in the following table.

Rural Runoff Coefficients for 5 and 10-Yr Frequency Flows
Topography and Vegetation SCS Hydrologic Soil Group
A B C D
Woodland:
Flat 0.10 0.20 0.30 0.40
Rolling 0.25 0.35 0.40 0.50
Hilly 0.30 0.40 0.50 0.60
Pasture1:
Flat 0.10 0.20 0.30 0.40
Rolling 0.16 0.30 0.40 0.55
Hilly 0.22 0.32 0.42 0.60
Cultivated1:
Flat 0.30 0.40 0.50 0.60
Rolling 0.40 0.50 0.60 0.70
Hilly 0.52 0.62 0.72 0.82
1Flat 0-5% slope; Rolling 5-10%; Hilly 10-30%

#### 749.5.2.2 Urban Runoff Coefficients

For urban watersheds, the runoff coefficient is primarily a function of land use and watershed slope. The following table gives a range of "C" values for various land use types in urban areas. The value selected by the designer should reflect the watershed slope, with steeper slopes having higher "C" values.

Urban Runoff Coefficients for 5 and 10-Yr Frequency Flows
Description Runoff Coefficients
Downtown Areas 0.70 - 0.95
Neighborhood Areas 0.50 - 0.70
Residential:
Urban Single-Family 0.30 - 0.50
Urban Apartments 0.40 - 0.70
Suburban 0.25-0.40
Industrial:
Light 0.50 - 0.80
Heavy 0.60 - 0.90
Parks, Cemeteries 0.10 - 0.25
Playgrounds 0.20 - 0.40
Unimproved Areas 0.10 - 0.30
Streets and Roofs 0.70 - 0.90
Lawns:
Sandy Soil 0.10 - 0.20
Clay Soil 0.15 - 0.35
Sodded Roadway Slopes (1V:6H to 1V:1H) 0.40 - 0.60
Source: Ponce (1979)

#### 749.5.2.3 Design Frequency

The rural and urban runoff coefficients given in the tables above are appropriate for 5-year or 10-year design frequencies. Estimation of peak flows for less frequent storms requires the use of a higher runoff coefficient because infiltration and other abstractions have proportionately less effect on the amount of rainfall that becomes runoff. To obtain runoff coefficients for other frequency events, the "C" value is multiplied by a frequency correction factor. The frequency correction factor is 1.10 for the 25-year event, 1.20 for the 50-year event, and 1.25 for the 100-year event. However, the resulting runoff coefficient (original "C" multiplied by the frequency correction factor) may not be greater than 1.0.

#### 749.5.2.4 Composite Runoff Coefficients

The land use and soil type may vary from one portion of a watershed to another. In this case an average runoff coefficient, weighted by area, should be used. For example, consider a flat watershed with soils of average infiltration capacity (soil group B) and with 1/3 of the drainage area cultivated (C=0.40) and 2/3 of the drainage area in woodland cover (C=0.20). The weighted average runoff coefficient to be used to estimate the 10-year flood is.

${\displaystyle C=[(0.40)(1/3)+(0.20)(2/3)]/1.0=0.267\,}$ (Equation 2)

The frequency correction factor may be applied to the weighted average runoff coefficient to estimate lower frequency flows: in the above example, a "C" value of (1.10)(0.267)=0.294 would be used for the 25-year flood, and a "C" of (1.20)(0.267)=0.320 would be used for the 50-year flood.

### 749.5.3 Time of Concentration

In order to determine the rainfall intensity used in the Rational Method, the time of concentration of the watershed must be estimated. The time of concentration of a watershed is defined as the time required for water to travel from the most hydraulically distant point of the watershed to the watershed outlet. This is also the time required before the entire watershed begins to contribute flow to the watershed outlet. This characteristic response time of the watershed is used as the duration of the design storm and thus influences the value of rainfall intensity used in the Rational Method.

Note that the location of the most hydraulically distant point in the watershed is a function of travel time and depends on both velocity and distance. The point in the watershed used to determine time of concentration may not necessarily be the point furthest from the watershed outlet.

In general, the time of concentration of urban drainage basins will be shorter than the time of concentration of rural basins. This is true when the natural drainage channels have been altered by storm sewers, main channel straightening, paving or similar modifications.

The time of concentration of an urban watershed may consist of the summation of travel times involved in three different flow regimes; overland flow, ditch or channel flow, and storm sewer system flow. The overland flow regime consists of very shallow sheet flow over the watershed surface. The velocities in overland flow are typically much lower than in the other flow regimes. After a short distance, the sheet flow becomes concentrated in swales or ditches, which begins the open-channel flow phase. The last phase consists of the travel time through the storm sewer system, where velocities are typically greater than through natural ditches or channels. Note that the flow often returns to open-channel flow after exiting the storm sewer system. The time of concentration is then the overland flow travel time plus the storm sewer travel time plus the channel flow travel time.

The information below provide methods for estimating time of concentration for both rural and urban watersheds. Other methods such as those provided by the Soil Conservation Service (SCS) may be used as deemed necessary or appropriate by engineering judgment.

#### 749.5.3.1 Kirpich Equation

For small rural watersheds, all flow regimes may be combined into a single equation used to calculate time of concentration. The Kirpich equation is used for these watersheds:

${\displaystyle t_{c}=KL^{0.77}S^{-0.385}\,}$ (Equation 3)
where:
tc = the time of concentration (min)
K = 0.0078
L = the length of the principal watercourse from outlet to divide, ft.
S = the slope between the minimum and maximum elevation, ft./ft.

#### 749.5.3.2 Kerby-Hathaway Equation for Overland Flow

Overland flow travel time can be calculated using the Kerby-Hathaway equation:

${\displaystyle t_{o}=K(Ln)^{0.47}S^{-0.235}\,}$ (Equation 4)
where:
to = overland travel time (min)
K = 0.8262
L = the overland flow length, ft.
n = roughness coefficient
S = the overland flow slope, ft./ft.

The roughness coefficient, n, used in this equation is similar in meaning to that used in Manning's equation for open-channel flow; however, for a given type of surface, the roughness coefficient for overland flow will be considerably larger than for open-channel flow. Typical values for the roughness coefficient are given in the table below. The overland flow length should be limited to less than 500 ft., and is usually much less. After a short distance, overland flow usually begins to concentrate in swales or ditches, which begins the open channel flow phase.

Roughness Coefficients for Overland Flow
Type of Surface N
Smooth Surfaces (concrete, asphalt, gravel or bare soil) 0.011
Fallow (No Residue) 0.05
Cultivated Soils:
Residue Cover<20% 0.06
Residue Cover >20% 0.17
Grass:
Short Grass Prarie 0.15
Dense Grasses -
Bluegrass, Buffalo, Native Mixture 0.24
Bermudagrass 0.41
Range 0.13
Woods:
Light Underbrush 0.40
Dense Underbrush 0.80
Source: U.S. Army Corps of Engineers (2000)

#### 749.5.3.3 Storm Sewer Travel Time

The storm sewer travel time is the summation of travel times in each component of the sewer system between the uppermost inlet and the outlet. These times may be estimated by use of the open channel flow charts presented in Open Channels. The proper procedure is to solve for the velocity of flow in each component assuming the pipe is flowing at maximum capacity. The velocity of flow is then divided into the length of flow to obtain the travel time. In order to use this method, it is necessary to obtain detailed information on the existing storm sewer system. This information includes conduit sizes, materials, lengths and slopes. If this information is not available or if the structure being designed is not considered important enough to warrant the effort required for such a detailed analysis, the storm sewer travel time may be estimated by use of the Kirpich Equation. In this case, the travel time given by the Kirpich Equation should be multiplied by 0.20 to obtain the travel time in the storm sewer system.

#### 749.5.3.4 Channel Flow Travel Time

The travel time in the open channel flow phase can be estimated by first estimating the velocity of flow in the channel. The velocity of flow is then divided into the length of flow to obtain the travel time. For subcritical flow conditions, the velocity should be determined for bank-full conditions at the mid-point of the channel's length. If the channel is improved, the average velocity may be estimated from the flow charts or flow computations presented in Open Channels. If the channel is natural, the travel time may be estimated by open channel flow computations. Care should be taken when estimating channel flow travel time for supercritical flow conditions. In some cases, the bank-full velocity under supercritical conditions may be too high to provide a reasonable estimate of travel time and a more reasonable estimate of flow depth should be used instead.

### 749.5.4 Rainfall Intensity

The design rainfall intensity is a function of the storm duration, the design frequency and the geographic location. The storm duration is taken as the time of concentration of the watershed or five minutes, whichever is greater. Knowing the storm duration and the design frequency, the rainfall intensity may be read from the appropriate Intensity-Duration-Frequency (IDF) curve for each district.

District 1
District 2
District 3
District 4
District 5
District 6
District 7
District 8
District 9
District 10

## 749.6 USGS Regression Equations

The USGS Regression equations were developed by the United States Geological Survey Office in Rolla. Data from stream gage sites were analyzed to determine flood magnitudes with various recurrence intervals. The resulting magnitudes were then related to basin characteristics through a statistical analysis to provide the regression equations. Three sets of equations are currently available: Rural Regression Equations, Urban BDF Regression Equations and Urban Percentage of Impervious Area Regression Equations.

### 749.6.1 Rural Regression Equations

These equations were developed in 1995 by the United States Geological Survey in Rolla. Data from 278 gaged sites in Missouri were analyzed to determine flood magnitudes with recurrence intervals of 2, 5, 10, 25, 50, 100 and 500 years. The resulting magnitudes were then related to hydrologic region, drainage area and average main-channel slope by a statistical analysis to provide the regression equations.

#### 749.6.1.1 Equations

For ungaged natural floodflow sites, flood magnitudes having recurrence intervals of 2, 5, 10, 25, 50, 100 and 500 years are computed by using appropriate values of the contributing drainage basin area (A) and slope (S) in the equations shown in the Rural Regression Equations table. A computer program is available to assist in performing these calculations. The state is divided into three hydrologic regions, each with its own set of regression coefficients. The three regions are described as follows:

Region I - Central Lowlands - "Characterized by meandering stream channels in wide and flat valleys resulting in long and narrow drainage patterns with local relief generally between 50 to 150 ft. Elevations range from about 600 ft. above sea level near the Mississippi River to about 1200 ft. above sea level in the northwest parts of the region"

Region II - Ozark Plateaus - "Characterized by streams that have cut narrow valleys 200 to 500 ft. deep, resulting in sharp rugged ridges that separate streams, with local relief generally ranging from 100 to 500 ft. The drainage patterns are described as dendritic (tree shaped) with main-channel gradients steeper than elsewhere in Missouri, and karst features are locally prominent in much of the region. Elevations (generally) range from 800 to 1700 ft. above sea level."

Region III - Mississippi Alluvial Plain - "A relatively flat area of excellent farmland. Virtually all the area is drained by a series of man-made drainage ditches that slope southward at an average of about 1.5 ft./mile. Elevations range from 200 to 300 ft. above sea level with local relief seldom exceeding 30 ft."

Rural Regression Equations
Flood Freq. Flood Magnitude Std. Error of Prediction Area Limits Slope Limits
(years) ft3/s (%) (mi2) (ft/mi)
Region I
2 69.4A0.703S0.373 34.00 0.13 to 11,500 1.35 to 150
5 123A0.690S0.383 32.00 0.13 to 11,500 1.35 to 150
10 170A0.680S0.378 34.00 0.13 to 11,500 1.35 to 150
25 243A0.668S0.366 36.00 0.13 to 11,500 1.35 to 150
50 305A0.660S0.356 38.00 0.13 to 11,500 1.35 to 150
100 376A0.652S0.346 40.00 0.13 to 11,500 1.35 to 150
500 569A0.636S0.321 45.00 0.13 to 11,500 1.35 to 150
Region II
2 77.9A0.733S0.265 43.00 0.13 to 14,000 1.2 to 279
5 99.6A0.763S0.355 36.00 0.13 to 14,000 1.2 to 279
10 117A0.774S0.395 34.00 0.13 to 14,000 1.2 to 279
25 140A0.784S0.432 32.00 0.13 to 14,000 1.2 to 279
50 155A0.789S0.453 31.00 0.13 to 14,000 1.2 to 279
100 170A0.794S0.471 32.00 0.13 to 14,000 1.2 to 279
500 203A0.804S0.503 34.00 0.13 to 14,000 1.2 to 279
Region III
2 88A0.658 34.00 0.48 to 1,040 na
5 145A0.627 36.00 0.48 to 1,040 na
10 187A0.612 38.00 0.48 to 1,040 na
25 244A0.595 41.00 0.48 to 1,040 na
50 288A0.585 44.00 0.48 to 1,040 na
100 334A0.576 46.00 0.48 to 1,040 na
500 448A0.557 54.00 0.48 to 1,040 na
Source: Alexander and Wilson (1995)

#### 749.6.1.2 Drainage Area

Drainage area (A) in mi2, can be obtained by determining the area contributing surface flows to the site as outlined along the drainage divide on the best available topographic maps.

#### 749.6.1.3 Valley Slope

Valley slope (S) in ft./mile is the average slope between points 10 percent and 85 percent of the distance along the main-stream channel from the site to the basin divide. Distance is measured by setting draftsman's dividers at 0.1 mile spread and stepping along the main channel. The main channel is defined above stream junctions as the one draining the largest area. The elevation difference between the 10- and 85-percent points is divided by the distance between the points to evaluate the slope.

#### 749.6.1.4 Limitations of Equations

The USGS Rural Regression Equations may be used to estimate magnitude and frequency of floods on most Missouri streams providing the drainage area and slope are within the limits shown in the table of rural regression equations.

However, the equations are not applicable for:

• basins where manmade changes have appreciably changed the flow regimen
• the main stems of the Mississippi and Missouri Rivers
• areas near the mouth of streams draining into larger rivers where backwater effect is experienced

### 749.6.2 Urban Regression Equations

The USGS Rural Regression Equations given above are not applicable to urban watersheds where manmade changes have appreciably changed the flow regimen A set of equations were developed in 1986 by the United States Geological Survey in Rolla. Data from 37 gaged sites in both urban and rural locations in Missouri were analyzed to determine flood magnitudes with recurrence intervals of 2, 5, 10, 25, 50, and 100 years. The resulting magnitudes were then related to drainage area and urban basin characteristics to provide the regression equations.

An urban watershed may be defined as a drainage basin in which manmade developments in the form of impervious surfaces and/or storm drainage systems have substantially altered the basin's natural response to rainfall. Urbanization of a natural watershed progresses in one of two ways. First, the addition of impervious surfaces in the form of roads, streets, parking lots and roofs will prevent infiltration of rainfall into the covered soil surface, thus increasing the total volume and peak rate of runoff from a given rainfall volume. Second, to protect the now valuable property in a developed watershed from this increased peak and volume of runoff, storm drainage systems are installed. The installation of a storm drainage system does not increase the volume of runoff, but modifies the time distribution of runoff. Thus, when storm water drainage systems are installed, the time of concentration of the watershed is decreased. Therefore, storm water drainage systems have the effect of removing a given volume of runoff in a shorter period of time, thus increasing the peak rate of runoff.

A computer program is available to assist in performing these calculations.

#### 749.6.2.1 Equations

Peak discharges can be estimated at urban locations using either of the two sets of equations presented in the Peak Discharge for Urban Basins Based on Basin Development Factor table or the Peak Discharge for Urban Basins Based on Percent Impervious Area table. Both sets give peak discharge as a function of drainage area (A) and a characteristic of urbanization: either basin development factor (BDF) or percentage of impervious area (I). Choice of which set of equations to use should depend on whether it is easier to determine BDF or percentage of impervious area for a given basin. Either set of equations should provide comparable results.

All hydraulic design in urban areas should consider the effect of increasing development throughout the projected life of the structure. Information on planned future development may be available from local agencies.

#### 749.6.2.2 Drainage Area

Drainage area (A) in mi2, can be obtained by determining the area contributing surface flows to the site as outlined along the drainage divide on the best available topographic maps.

#### 749.6.2.3 Basin Development Factor

The basin development factor (BDF) is determined by dividing the drainage basin into thirds (subareas). Each subarea of the basin is then evaluated for four aspects of urbanization. For each of the four criteria, a value of either 1 (if the subarea meets the criteria) or 0 (if the subarea does not meet the criteria) is assigned. The BDF is the sum of the values for each of the four criteria and for each third of the basin. A maximum BDF of twelve results when each of the three subareas meets each of the four criteria for urbanization described below:

Channel Improvements - channel improvements such as straightening, enlarging, deepening, and clearing have been made to at least 50 percent of the main channel and principal tributaries. Channel Linings - more than 50 percent of the main channel and principal tributaries has been lined with an impervious material. (Note that the presence of the channel linings also implies the presence of channel improvements) Storm Drains or Storm Sewers - more than 50 percent of the secondary tributaries of a subarea consists of storm drains or storm sewers. Curb-and Gutter Streets - more than 50 percent of a subarea is urbanized and more than 50 percent of the streets and highways in the subarea are constructed with curbs and gutters.

The valid range for BDF is 0 to 12. See Typical Drainage Basin Shapes and the method of subdivision into thirds.

Peak discharge for Rural Basins Based on Basin Development Factor
Flood Freq. Flood Magnitude Std. Error of Estimate Area Limits Slope Limits
(years) ft3/s (%) (mi2) (ft/mi)
2 801A0.747(13-BDF)-0.400 32.90 0.25 to 40 8.7 to 120
5 1150A0.746(13-BDF)-0.318 29.40 0.65 to 100 8.7 to 120
10 1440A0.755(13-BDF)-0.300 28.40 0.25 to 40 8.7 to 120
25 1920A0.764(13-BDF)-0.307 27.30 0.25 to 40 8.7 to 120
50 2350A0.773(13-BDF)-0.319 26.50 0.25 to 40 8.7 to 120
100 2820A0.783(13-BDF)-0.330 26.40 0.25 to 40 8.7 to 120
Source: Becker (1986)

#### 749.6.2.4 Percentage of Impervious Area

The percentage of impervious area (I) is the portion of the drainage area that is nonpervious because of buildings, parking lots, streets and roads, and other impervious areas within an urban basin. The variable, I, is determined from the best available maps or aerial photos showing impervious surfaces. Field inspection to supplement the maps may be useful.

If the percentage of impervious area cannot be determined directly, a reasonable estimate may be obtained using 7-1/2 minute topographic maps and a relationship between developed area and impervious area. The drainage divide is outlined on the map, then the drainage area is divided into two subareas, open area and developed (urban) area. Open area consists of all undeveloped land, which may include scattered farmhouses and buildings, scattered single-family housing, and paved roads without significant development along the road. Developed areas include single- or multi-family housing structures, large business and office buildings, shopping centers, extensively industrialized areas, and schools. When delineating developed areas, it is important to include those areas devoted to paved parking lots around buildings. Once the amount of developed area has been determined, it can be converted into a percentage developed area (PDA) by dividing by the basin drainage area and multiplying by 100. The percentage of impervious area can then be obtained using the following equation:

${\displaystyle I=2.03PDA^{0.618}\,}$ (Equation 5)

The valid range for I is 1.0 percent to 40 percent. The values for both I and PDA are entered as percents (i.e. I = 29 for 29% impervious area and PDA = 75 for 75% developed area.)

Peak discharge for Rural Basins Based on Percent Impervious Area
Flood Freq. Flood Magnitude Std. Error of Estimate Area Limits Slope Limits
(years) ft3/s (%) (mi2) (ft/mi)
2 224A0.793(I)0.175 32.30 0.25 to 40 8.7 to 120
5 424A0.784(I)0.131 29.50 0.25 to 40 8.7 to 120
10 560A0.791(I)0.124 28.60 0.25 to 40 8.7 to 120
25 729A0.800(I)0.131 27.20 0.25 to 40 8.7 to 120
50 855A0.810(I)0.137 26.10 0.25 to 40 8.7 to 120
100 986A0.821(I)0.144 25.90 0.25 to 40 8.7 to 120
Source: Becker (1986)

#### 749.6.2.5 Limitations of Equations

The USGS Urban Regression Equations may be used to estimate magnitude and frequency of floods on most urban Missouri streams within the limits shown in the two tables above provided that the floodflows are relatively unaffected by manmade works such as dams or diversions.

## 749.7 Historical USGS Stream Gage Data

Numerous USGS recording stream gages have been maintained for many years on selected Missouri streams. For proposed structures at or near one of these gages, the gage data can be used in estimating discharge. When sufficient years of data have been collected at a stream gage, the data may be statistically analyzed to estimate discharge for the selected design flood frequency.

Stream gage data is available on the Internet at http://waterdata.usgs.gov/mo/nwis/.

Gage data is analyzed by Log-Pearson Type III regression analysis to determine the discharges associated with the relevant return periods. See Water Resources Council Bulletin #17B in References for details on this analysis method. A computer program for the analysis is available.

One statistical parameter computed in the Log Pearson analysis is the skew coefficient of the distribution of the stream gage data. Skew coefficients for the data from stream gages in Missouri are typically between -0.1 and -0.4 when sufficient years of record are available. Skew coefficients outside this range may indicate an insufficient length of record or an analysis affected by outliers in the data. In this case, other methods of determining discharges will likely provide better estimates.

Stream gage data from gages at some distance from the site on the same watershed and stream gage data from nearby hydrologically similar watersheds may also be used to estimate discharges. Discharges obtained from this type of data should be compared with discharges obtained by other methods and not given the same weight as discharges obtained from data from a stream gage at the proposed site. Better estimates of discharge using this method may be obtained by repeating the procedure for several nearby gages and averaging the results. This method should not be used when drainage areas differ by more than 50% or at sites more than 50 miles from the stream gage(s).

Transposition of discharges from one basin to another, or from one location to another within the same watershed, is accomplished using the following equation:

${\displaystyle Q_{1}=Q_{2}\left({\frac {A_{1}}{A_{2}}}\right)^{k}}$
where:
Q1 = discharge for drainage basin 1 (cfs)
A1 = drainage area for drainage basin 1 (mi2)
Q2 = discharge for drainage basin 2 (cfs)
A2 = drainage area for drainage basin 2 (mi2)
k = exponent = 0.5 to 0.7

## 749.8 NFIP Flood Insurance Study Discharges

NFIP Flood Insurance Studies (FIS) typically include estimates of 10-, 50-, 100- and 500-year discharges for streams studied by detailed methods. These discharges may be more accurate than those obtained by other methods if the FIS discharges were determined through a detailed hydrologic study, such as an HEC-1 or TR-20 hydrologic model. In some instances, the FIS discharges may have been determined using an older version of the USGS regression equations. These discharges should not be used. Careful review of the FIS report will disclose the level of detail used in the hydrologic study.

## 749.9 Flood Hydrographs

For certain design problems it may be necessary to determine flood hydrographs associated with the peak discharge for a desired frequency. A hydrograph gives flow rate as a function of time at a particular location in a watershed, usually at the watershed outlet. Two techniques are available for obtaining synthetic hydrographs for ungaged sites. The SCS Unit Hydrograph method is applicable for all watersheds up to 1000 acres, and the USGS synthetic hydrograph method is applicable for drainage areas greater than 200 acres.

### 749.9.1 SCS Unit Hydrograph

Techniques developed by the Natural Resource Conservation Service (NRCS), formerly known as the Soil Conservation Service (SCS), for calculating rates of runoff require the same basic data as the Rational Method: drainage area, a runoff factor, time of concentration, and rainfall information. The SCS method is more sophisticated in that it considers also the time distribution of rainfall, the initial rainfall losses to interception and depression storage, and an infiltration rate that decreases during the course of a storm. Additional details on the SCS methodology can be found in the SCS National Engineering Handbook, Section 4 (NEH-4) (SCS, 1985).

#### 749.9.1.1 Rainfall-Runoff Equation and Concepts

The SCS method is based on a 24-hour storm event. SCS has developed four synthetic rainfall distributions typical of storms for various geographical regions in the United States. The Type II distribution is the appropriate distribution for using the SCS method in Missouri.

The SCS rainfall-runoff equation was developed by the SCS from experimental plots for numerous soil types and vegetative cover conditions. The experimental data consisted mainly of daily rainfall totals on small watersheds, and did not include information on the time distribution of rainfall. The SCS rainfall-runoff equation is therefore used to estimate the depth of runoff resulting from a given depth of 24-hour rainfall assumed to be spatially distributed uniformly over the watershed. The equation is given by:

${\displaystyle Q={\frac {(P-I_{a})^{2}}{(P-I_{a})+S}}}$ (Equation 6)
where:
Q = accumulated direct runoff, in
P = accumulated rainfall (potential maximum runoff), in
Ia = initial abstractions including surface storage, interception and infiltration prior to runoff, in
S = potential maximum retention, in

The potential maximum retention, S, is a measure of the maximum amount of water a given watershed could retain, and is a function of land use, interception capacity, infiltration capacity, depression storage, and antecedent moisture. A relationship between Ia and S was also developed from experimental data:

${\displaystyle I_{a}=0.2S\,}$ (Equation 7)

Substituting the equation for Ia into equation 9, the SCS rainfall-runoff equation becomes:

${\displaystyle Q={\frac {(P-0.2S)^{2}}{P+0.8S}}}$ (Equation 8)

It is important to note that while both P and Q are given in units of depth, they actually represent volumes of rainfall and runoff, respectively. This is because the indicated depth of rainfall or runoff is assumed to be applied uniformly to the entire watershed area.

The SCS made additional empirical analyses to estimate the value of S for various watersheds. The principal physical characteristics affecting the relationship between rainfall and runoff are land use, land treatment, soil types and land slope. These characteristics which affect the maximum potential retention are represented by a runoff factor called the curve number (CN), which is related to S by:

${\displaystyle S={\frac {1000}{CN}}-10}$ (Equation 9)

The curve number is an index that represents the combination of hydrologic soil group, land use, hydrologic condition, and antecedent moisture condition. The table(s) below, Runoff Curve Numbers, provide curve numbers for different land uses, treatments and hydrologic conditions; separate values are given for each soil group. The SCS provides methods of adjusting the curve numbers based on varying antecedent moisture conditions; however, for design purposes, average antecedent moisture conditions are normally assumed and the values given in the table can therefore be used for design.

Runoff Curve Numbers - Urban Areas1
Cover Description Curve Numbers for Hydrologic Soil Groups
Cover Type and Hydrologic Condition Average Percent Impervious Area2 A B C D
Fully Developed Urban Areas (Vegetation Established)
Open Space (lawns, parks, golf courses, cemetaries, etc.)3
Poor Condition (grass cover<50%) 68 79 86 89
Fair Condition (grass cover 50% to 75%) 49 69 79 84
Good Condition (grass cover>75%) 39 61 74 80
Impervious Areas:
Paved Parking Lots, Roofs, Driveways, etc. (excluding right-of-way) 98 98 98 98
Paved; Curbs and Storm Drains (excluding right-of-way) 98 98 98 98
Paved; Open Ditches (including right-of-way) 83 89 92 93
Gravel (including right-of-way) 76 85 89 91
Dirt (including right-of-way) 72 82 87 89
Western Desert Urban Areas:
Natural Desert Landscaping (pervious areas only) 63 77 85 88
Artificial Desert Landscaping (impervious weed
barrier desert shrub with 25- to 50 mm sand
or gravel mulch and basin borders) 96 96 96 96
Urban Districts:
Commercial and Business 85 89 92 94 95
Industrial 72 81 88 91 93
Residential Districts by Average Lot Size:
1/8 Acre or Less (town houses) 65 77 85 90 92
1/4 Acre 38 61 75 83 87
1/3 Acre 30 57 72 81 86
1/2 Acre 25 54 70 80 85
1 Acre 20 51 68 79 84
2 Acres 12 46 65 77 82
Developing Urban Areas
Newly Graded Areas (pervious areas only, no vegetation 77 86 91 94
Idle Lands (CNs are determined using cover types similar to those in Runoff Curve Numbers - Urban Areas
1Average runoff condition, and Ia=0.2S (see equation 7)
2The average percent impervious area shown was used to develp the composite CNs. Other assumptions are as follows: impervious areas are directly connected to the drainage system, impervious areas have a CN of 98, and pervious areas are considered equivalent to open space in good hydrologic condition. If the impervious area is not connected, the SCS method has an adjustment to reduce the effect.
3CNs shown are equivalent to those of pasture. Composite CNs may be computed for other combinations of open space cover type.
Runoff Curve Numbers - Cultivated Agricultural Areas1
Cover Description Curve Numbers for Hydrologic Soil Groups
Cover Type Treatment2 Hydrologic Condition3 A B C D
Fallow Bare Soil 77 86 91 94
Crop Residue Cover (CR) Poor 76 85 90 93
Good 74 83 88 90
Row Crops Straight Row (SR) Poor 72 81 88 91
Good 67 78 85 89
SR + CR Poor 71 80 87 90
Good 64 75 82 85
Contoured (C) Poor 70 79 84 88
Good 65 75 82 86
C + CR Poor 69 78 83 87
Good 64 74 81 85
Contoured & Terraced (C&T) Poor 66 74 80 82
Good 62 71 78 81
C&T + CR Poor 65 73 79 81
Good 61 70 77 80
Small Grain SR Poor 65 76 84 88
Good 63 75 83 87
SR + CR Poor 64 75 83 86
Good 60 72 80 84
C Poor 63 74 82 85
Good 61 73 81 85
C + CR Poor 62 73 81 84
Good 60 72 80 83
C&T Poor 61 72 79 82
Good 59 70 78 81
C&T + CR Poor 60 71 78 81
Good 58 69 77 80
Close-Seeded or Broadcast Legumes or Rotation Meadow SR Poor 66 77 85 89
Good 58 72 81 85
C Poor 64 75 83 85
Good 55 69 78 83
C&T Poor 63 73 80 83
Good 51 67 76 80
1Average runoff condition, and Ia=0.2S (see equation 7)
2Crop residue cover applies only if residue is on at least 5% of the surface throughout the year.
3Hydrologic condition is based on a combination of factors that affect infiltration and runoff, including (a) density and canopy of vegetative areas, (b) amount of year-round cover, (c) amount of grass or closed-seeded legumes in rotations, (d) percent of residue cover on the land surface (good>20%) and (e) degree of roughness.

Poor: Factors impair infiltration and tend to increase runoff.

Good: Factors encourage average and better than average infiltration and tend to decrease runoff.

Runoff Curve Numbers - Other Agricultural Lands1
Cover Description Curve Numbers for Hydrologic Soil Groups
Cover Type Hydrologic Condition3 A B C D
Pasture, Grassland, or Range — Continuous Forage for Grazing Poor 68 79 86 89
Fair 49 69 79 84
Good 39 61 74 80
Meadow - Continuous Grass, Protected from Grazing and Generally Mowed for Hay 30 58 71 78
Brush - Brush-Weed-Grass Mixture with Brush the Major Element Poor 48 67 77 83
Fair 35 56 70 77
Good 430 48 65 73
Woods - Grass Combination (Orchard or Tree Farm)5 Poor 57 73 82 86
Fair 43 65 76 82
Good 32 58 72 79
Woods6 Good 45 66 77 83
Fair 36 60 73 79
Good 430 55 70 77
Farmsteads - Buildings, Lanes, Driveways and Surrounding Lots 59 74 82 86
1Average runoff condition, and Ia=0.2S (see equation 7)
2Poor: <50% ground cover or heavily grazed with no mulch

Fair: 50 to 74% ground cover

Good: >75% ground cover and lightly or only occasionally grazed

3Poor: <50% gound cover

Fair: 50 to 75% ground cover

Good: >75% ground cover

4Actual curve number is less than 30; use CN=30 for runoff computations.
5CNs shown were computed for areas with 50% grass (pasture) cover. Other combinations of conditions may be computed from CNs for woods and pasture.
6Poor: Forest litter, small trees and brush are destroyed by heavy grazing or regular burning.

Fair: Woods grazed but not burned, and some forest litter covers the soil.

Good: Woods protected from grazing, litter and brush adequately cover soil.

#### 749.9.1.2 SCS Dimensionless Unit Hydrograph

The SCS unit hydrograph was developed based on analysis of a large number of natural unit hydrographs from a wide range of drainage basin sizes and geographic locations. The SCS unit hydrograph is given in a dimensionless form and provides a standard unit hydrograph shape. Table 9-02.9 gives the ordinates of the SCS dimensionless unit hydrograph.

SCS Dimensionless Unit Hydrograph
Time Ratio Discharge Ratio Time Ratio Discharge Ratio Time Ratio Discharge Ratio Time Ratio Discharge Ratio
t/tp Q/Qp t/tp Q/Qp t/tp Q/Qp t/tp Q/Qp
0.0 0.000 0.9 0.990 1.8 0.390 3.4 0.029
0.1 0.030 1.0 1.000 1.9 0.330 3.6 0.021
0.2 0.100 1.1 0.990 2.0 0.280 3.8 0.021
0.3 0.190 1.2 0.930 2.2 0.207 4.0 0.011
0.4 0.310 1.3 0.860 2.4 0.147 4.5 0.005
0.5 0.470 1.4 0.780 2.6 0.107 5.0 0.000
0.6 0.660 1.5 0.680 2.8 0.077
0.7 0.820 1.6 0.560 3.0 0.055
0.8 0.930 1.7 0.460 3.2 0.040
Source: McCuen (1996)

Use of the SCS unit hydrograph requires calculation of the unit hydrograph peak discharge and the time to peak. The unit hydrograph peak discharge is given by:

${\displaystyle Q_{p}={\frac {K_{q}A}{t_{p}}}}$ (Equation 10)
where:
Qp = unit hydrograph peak discharge, cfs
Kq = constant, 484
A = drainage area, mi2
tp = time to peak, hrs

The time to peak is assumed to be equal to the basin lag time plus one-half the duration of rainfall. Basin lag time is estimated as 0.6 times the time of concentration, leading to the following equation for time to peak:

${\displaystyle t_{p}={\frac {t_{r}}{2}}+0.6t_{c}}$ (Equation 11)
where:
tp = time to peak, hrs
tr = duration of rainfall (unit hydrograph duration) = 0.133 tc, hrs
tc = time of concentration, hrs

#### 749.9.1.3 Application of SCS Methodology

Unit hydrograph theory depends on the principles of linearity and superposition. Given a unit hydrograph, the runoff hydrograph for a runoff depth other than unity can be obtained by multiplying the unit hydrograph ordinates by the runoff depth using the principle of linearity. The flood hydrograph for a particular storm event can be obtained by dividing the storm event into incremental periods of runoff, then applying the unit hydrograph to each incremental runoff and summing the resulting hydrographs together using the principle of superposition to obtain the total runoff hydrograph.

The unit hydrograph duration (and the corresponding duration of the period of incremental runoff used in applying the unit hydrograph method) is estimated as 0.133tc. Since the SCS Type II rainfall distribution has a 24-hour time base, application of the SCS unit hydrograph methodology to typical watersheds by hand requires calculation of runoff hydrographs for a large number of increments. This can be cumbersome and time-consuming and a computer-based implementation is recommended.

### 749.9.2 USGS Synthetic Hydrograph

A technique for generation of synthetic flood hydrographs for small basins in Missouri was developed in 1990 by the USGS in Rolla. Data from 341 floods recorded at 41 gaging stations located on small rural and urban streams in Missouri were analyzed. An average dimensionless hydrograph applicable to rural and urban basins with drainage areas from 0.25 to 0.40 mi2 was developed from this data. This procedure is applicable to flood flows that are not significantly affected by storage or diversions.

The dimensionless hydrograph developed for Missouri is given in the table below and shown graphically. The dimensionless hydrograph is given in terms of flow divided by peak flow (Qp) versus time divided by basin lag time (LT). Expansion of the dimensionless hydrograph is accomplished by multiplying each ordinate value (Q/Qp) by Qp and each abscissa value (T/LT) by LT, resulting in a flood hydrograph for the desired flood frequency.

Dimensionless Hydrograph for Small Basins in Missouri
Time Ratio Discharge Ratio Time Ratio Discharge Ratio
T/LT Q/Qp T/LT Q/Qp
0.25 0.11 1.35 0.59
0.30 0.14 1.40 0.53
0.35 0.18 1.45 0.48
0.40 0.23 1.50 0.44
0.45 0.29 1.55 0.40
0.50 0.37 1.6 0.37
0.55 0.46 1.65 0.34
0.60 0.55 1.70 0.31
0.65 0.65 1.75 0.28
0.70 0.74 1.80 0.26
0.75 0.83 1.85 0.24
0.80 0.89 1.90 0.22
0.85 0.95 1.95 0.20
0.90 0.98 2.00 0.19
0.95 1.00 2.05 0.17
1.00 0.98 2.10 0.16
1.05 0.95 2.15 0.15
1.10 0.90 2.20 0.14
1.15 0.84 2.25 0.13
1.20 0.77 2.30 0.12
1.25 0.71 2.35 0.11
1.30 0.65 2.40 0.10
Source: Becker (1990)

#### 749.9.2.1 Peak Flow Estimation

Peak flows of the desired frequency are estimated using the appropriate method given in USGS Regression Equations.

#### 749.9.2.2 Basin Lag Time

Lag time is usually defined as the time from the center of the rainfall hyetograph to the centroid of the hydrograph. A hyetograph is a plot of rainfall intensity with time. The basin lag time (LT) can be estimated using either of two equations:

English: ${\displaystyle LT=1.46A^{0.34}I^{-0.19}\,}$ (Equation 12)
Metric: ${\displaystyle LT=1.06A^{0.34}I^{-0.19}\,}$ (Equation 12)
English: ${\displaystyle LT=0.34A^{0.37}(13-BDF)^{0.52}\,}$ (Equation 13)
Metric: ${\displaystyle LT=0.24A^{0.37}(13-BDF)^{0.52}\,}$ (Equation 13)
where:
LT = basin lag time (hrs)
A = basin drainage area, mi2
I = percentage impervious area as defined in Percentage of Impervious Area
BDF = basin development factor as defined in Basin Development Factor

The choice of which equation to use should again depend on whether it is easier to determine BDF or percentage of impervious area for a given basin.

## 749.10 Detention Storage

The traditional purpose of storm drainage systems has been to collect and convey storm runoff as rapidly as possible to a suitable location where it can be discharged. As areas urbanize this type of design may result in major drainage and flooding problems downstream. Under favorable conditions, the temporary storage of some of the storm runoff can decrease downstream flows and often the cost of the downstream conveyance system. Detention storage facilities can range from small facilities contained in parking lots or other on-site facilities to large lakes and reservoirs. This article provides general procedures for detention storage analysis.

An easement must be purchased for any land, outside of the right of way, that will be flooded by water from a detention storage structure.

### 749.10.1 Data Needs

The following data will be needed to complete storage calculations.

• Inflow hydrograph for all selected design storms. The inflow hydrograph for the detention basin can be determined using the methods in Flood Hydrographs.
• Stage-storage curve for storage facility.
• Stage-discharge curve for the facility.

Using these data, the inflow hydrograph is routed through the storage facility to develop the outflow hydrograph.

#### 749.10.1.1 Stage-Storage Curve

A stage-storage curve defines the relationship between the depth of water and storage volume in a reservoir. The data for this type of curve are usually developed using a topographic map and the conic formula for irregular shaped basins, or the prismoidal formula for trapezoidal basins. The conic formula is expressed as:

${\displaystyle V_{1,2}={\frac {1}{3}}d\left(A_{1}+A_{2}+{\sqrt {A_{1}A_{2}}}\right)}$ (Equation 14)
where:
V1,2 = storage volume, ft3 (m3), between elevations 1 and 2
A1 = surface area at elevation 1, ft2
A2 = surface area at elevation 2, ft2
d = change in elevation between points 1 and 2, ft

The prismoidal formula for trapezoidal basins is expressed as:

${\displaystyle V=LWD+(L+W)ZD^{2}{\frac {4}{3}}Z^{2}D^{3}}$ (Equation 15)
where:
V = volume of trapezoidal basin, ft3
L = length of basin at base, ft
W = width of basin at base, ft
D = depth of basin, ft
Z = side slope factor, ratio of horizontal to vertical

#### 749.10.1.2 Stage-Discharge Curve

A stage-discharge curve defines the relationship between the depth of water and the discharge or outflow from a storage facility. If the detention facility has both principal and emergency spillways the stage-discharge curve should take both into account. The following equations can be used to help develop the stage-discharge curve.

Sharp-crested weir flow equations for no end contractions, two end contractions, and submerged discharge conditions are presented below, followed by equations for broad-crested weirs, v-notch weirs and orifices, or combinations of these facilities.

Sharp-Crested Weirs. A sharp-crested weir with no end contractions is illustrated in Weir Configurations. The discharge equation for this configuration is (Chow, 1959):

English: ${\displaystyle Q=\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right]LH^{1.5}}$ (Equation 16)
Metric: ${\displaystyle Q=0.55\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right]LH^{1.5}}$ (Equation 16)
where:
Q = discharge, ft3/s
Hc = height of weir crest above channel bottom, ft.
L = horizontal weir length, ft.

A sharp-crested weir with two end contractions is illustrated in Weir Configurations. The discharge equation for this configuration is (Chow, 1959):

English: ${\displaystyle Q=\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right](L-0.2H)H^{1.5}}$ (17)
Metric: ${\displaystyle Q=0.55\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right](L-0.2H)H^{1.5}}$ (Equation 17)
where: Variables are the same as for the previous equation.

A sharp-crested weir will be affected by submergence when the tailwater rises above the weir crest elevation. The result will be that the discharge over the weir will be reduced. The discharge equation for a sharp-crested submerged weir is (Brater and King, 1976):

${\displaystyle Q_{s}=Q_{f}\left[1-\left({\frac {H_{2}}{H_{1}}}\right)^{1.5}\right]^{0.385}}$ (Equation 18)
where:
Qs = submergence flow, ftA3/s
Qf = free flow, ft3/s
H1 = upstream head above crest, ft.
H2 = downstream head above crest, ft.

Broad-Crested Weirs The equation generally used for the broad-crested weir is (Brater and King, 1976):

English: ${\displaystyle Q=CLH^{1.5}\,}$ (Equation 19)
Metric: ${\displaystyle Q=0.55CLH^{1.5}\,}$ (Equation 19)
where: Q = discharge, ft3/s
L = broad-crested weir length, ft
H = head above weir crest, ft., measured at least 2.5H upstream of the weir

For weir flow over embankments with sloped sides, a C value of 3.0 should be used. For weir flow over embankments with vertical sides, a minimum C value of 2.6 should be used.

V-Notch Weirs A V-Notch weir is illustrated in Weir Configurations. The discharge through a v-notch weir can be calculated from the following equation (Brater and King, 1976):

English: ${\displaystyle Q=2.5tan\left({\frac {\theta }{2}}\right)H^{2.5}}$ (Equation 20)
Metric: ${\displaystyle Q=1.38tan\left({\frac {\theta }{2}}\right)H^{2.5}}$ (Equation 20)
where:
Q = discharge, ft3/s
q = angle of v-notch, degrees
H = head on apex of notch, ft.

Orifices Pipes smaller than 12 in. may be analyzed as a submerged orifice if H/D is greater than 1.5. For square-edged entrance conditions,

English: ${\displaystyle Q=0.6A(2gH)^{0.5}=3.78D^{2}H^{0.5}\,}$ (Equation 21)
Metric: ${\displaystyle Q=0.6A(2gH)^{0.5}=2.09D^{2}H^{0.5}\,}$ (Equation 21)
where:
Q = discharge, ft3/s
A = cross-section area of pipe, ft2/s
g = acceleration due to gravity, 32.2 ft/s2
D = diameter of pipe, ft.
H = head on pipe, from the center of pipe to the water surface, ft.

Culverts If culverts are used as outlets works, procedures presented in the Culverts should be used to develop stage-discharge data.

### 749.10.2 Routing Calculations

The following procedure is used to perform routing through a reservoir or storage facility (Storage Indication or Puhls Method of storage routing).

Routing a flood through a reservoir results in an attenuation of the peak of the inflow hydrograph and an associated change in timing of the peak. Storage of flood waters within the reservoir causes the peak outflow from the reservoir to be lower than the peak inflow, and causes the peak outflow to occur at a later time than the peak inflow. The continuity equation relates the change of storage within the detention storage basin to the inflow and outflow for the basin:

${\displaystyle I-O={\frac {\Delta S}{\Delta T}}}$ (Equation 22)

where:
I = inflow, ft3/s
O = outflow, ft3/s
DS = change in storage, ft3
DT = change in time, seconds

The Storage Indication method of reservoir routing uses a simple finite-difference form of the continuity equation. For any two points in time, the continuity equation can be written as:

${\displaystyle {\Big (}{\frac {2S_{n+1}}{\Delta T}}+O_{n+1}{\Big )}=(I_{n}+I_{n+1})+{\Big (}{\frac {2S_{n}}{\Delta T}}-O_{n}{\Big )}}$ (Equation 23)
where:
S = storage

If the values at time step n are known, the only unknowns in equation 20 are on the left-hand side.

Substituting

${\displaystyle U_{n+1}={\frac {2S_{n+1}}{\Delta T}}+O_{n+1}}$ (Equation 24)
${\displaystyle W_{n}={\frac {2S_{n}}{\Delta T}}-O_{n}}$ (Equation 25)

U is known as the Storage Indication Number. With these substitutions, equation 20 becomes:

${\displaystyle U_{n+1}=(I_{n}+I_{n+1})+W_{n}\,}$ (Equation 26)

For the first time step, Wn is calculated using the initial values of S and O, and equation 22. For subsequent time steps the following equation can be used as a shortcut.

${\displaystyle W_{n+1}=U_{n+1}-2O_{n+1}\,}$ (Equation 27)

The procedure for using the storage-indication method of reservoir routing is as follows:

• Develop an inflow hydrograph, stage-discharge curve, and stage-storage curve for the proposed storage facility.
• Select a routing time period, t, to provide at least five points on the rising limb of the inflow hydrograph.
• Use the stage-storage and stage-outflow data from Step 1 to develop a plot of U versus outflow.
• Calculate W1 using equation 22 and the initial values of S and O
• Calculate Un+1 using equation 23.
• Using Un+1 calculated in step 5 pick On+1 from the plot of U vs. outflow.
• Using Un+1 and On+1 calculate Wn+1 using equation 24
• Start over at step 5 with n = n+1. Continue repeating until inflow ceases or the outflow peak discharge has been determined.
• From the stage discharge curve, determine the stage for the peak outflow.

## 749.11 References

American Association of State Highway and Transportation Officials, 1991, Model Drainage Manual.

Alexander, T.W. and Wilson, G.L., 1995, Technique for Estimating the 2- to 500-Year Flood Discharges on Unregulated Streams in Rural Missouri, USGS Water-Resources Investigations Report 95-4231

Becker, L.D., 1986, Techniques for Estimating Flood-Peak Discharges from Urban Basins in Missouri, USGS Water-Resources Investigations Report 86-4322.

Becker, L.D., 1990, Simulation of Flood Hydrographs for Small Basins in Missouri, USGS Water-Resources Investigations Report 90-4045.

McCuen, Richard, et al., 1996, Highway Hydrology – Hydraulic Design Series No. 2, Federal Highway Administration Report No. FHWA-SA-96-067

Missouri State Highway Department, 1972, Review of Hydraulic and Drainage Design Criteria, Missouri Cooperative Highway Research Program Report 72-3.

Newton, D.W., and Herrin, J.C., 1982, Assessment of Commonly Used Methods of Estimating Flood Frequency, Transportation Research Record 896.

Ponce, V.M., 1989, Engineering Hydrology, Prentice-Hall, Inc., Englewood Cliffs, New Jersey 07632.

Soil Conservation Service, 1985, National Engineering Handbook. Section 4 – Hydrology, Washington, DC

Southard, R.E., 1986, An Alternative Basin Characteristic for Use in Estimating Impervious Area in Urban Missouri Basins, USGS Water-Resources Investigations Report 86-4362.

US Army Corps of Engineers, 2000, Hydrological Modeling System HEC-HMS Technical Reference Manual Hydrologic Engineering Center, Davis, CA