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 or minor), traffic volume and type of drainage facility.

749.2.1.1 Crossroad Structures

The design flood magnitude, design frequencies and corresponding water surface elevation(s) shall be included on the project plans for all crossroad structures.

749.2.1.2 Roadside Ditches and Culverts

The design frequency for roadside drainage ditches should be selected 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.

Culverts not considered crossroad structures should be designed to the same frequency as the ditch.

749.2.1.3 Pavement Drainage

See EPG 640.1.2.1 Design Frequency.

749.2.1.4 Median Drainage

See EPG 640.2.3 Design Discharge.

749.2.2 Base 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.

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.

749.2.3 Overtopping Flood

The overtopping flood is defined as the discharge and corresponding return period and water surface elevation at which flow occurs over the roadway. The overtopping flood shall be determined for all crossroad structures. The resulting overtopping flood discharge 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. For reporting overtopping for bridges, refer to EPG 750.3.2.4.5 Overtopping Discharge and Frequency.

749.2.4 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.

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

• Floodplain Management and the Regulatory Floodway for roadway construction
• National Flood Insurance Program (NFIP) for structures

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.

Three methods are presented in this article for estimating peak rates of runoff from small watersheds. The Rational Method may be used on all watersheds smaller than 200 acres. On watersheds within the area limits shown in the table below, United States Geological Survey (USGS) Regression Equations may be used. The Natural Resource Conservation Service (NRCS) Unit Hydrograph may also be used for drainage areas larger than 6.40 acres.

For more information on watershed area limits for the above methods see Summary of Small Watershed Peak Runoff Calculation Method Drainage Area Limits table.

749.4.1 Rural Watersheds

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.

The Rational Method may be used on rural watersheds smaller than 200 acres. On watersheds larger than 200 acres, the USGS Regression Equations should be used. The 2014 USGS Rural Regression Equations may be used for Watersheds that lie in Region I and Region II if the drainage area is greater than 70 acres and 110 acres respectively. Watersheds that lie within Region III for the 2014 USGS Rural Regression Equations and have drainage areas between 200 to 1350 acres do not fall within the limits for either set of equations. In this case one of the following approaches may be used:

a. Use an alternate method such as Flood Hydrographs;

b. for drainage areas greater than 300 acres use the 1995 USGS Rural Regression Equations;

c. for drainage areas between 200 and 300 acres calculate the peak rate of runoff using the Rational Method and the 1995 USGS Rural Regression Equations, and use the one that is most appropriate.

The NRCS Unit Hydrograph may also be used for drainage areas larger than 6.40 acres.

For more information on watershed area limits for the above methods see Summary of Small Watershed Peak Runoff Calculation Method Drainage Area Limits table.

749.4.2 Urban Watersheds

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.

The Rational Method should be used on urban watersheds smaller than 200 acres. On watersheds larger than 200 acres, the USGS Urban Regression Equations should be used. The NRCS Unit Hydrograph may also be used for drainage areas larger than 6.40 acres.

For more information on watershed area limits for the above methods see Summary of Small Watershed Peak Runoff Calculation Method Drainage Area Limits table.

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.

${\displaystyle Q=k_{c}CIA\,}$

where:

Q = the peak rate of runoff, ft³/s

k_c= 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 Natural Resource Conservation Service (NRCS) 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 NRCS 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 NRCS 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 Construction and 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 NRCS 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.

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.

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:

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

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 a watershed will consist of the summation of travel times from one or more of the following flow regimes:

• overland flow,
• ditch or open channel flow,
• and storm sewer system flow.

Overland 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. Overland flow will become concentrated after a short distance and transform into one of the other flow regimes.

Open Channel Flow. The open channel flow regime consists of:

• flow that has concentrated into swales or ditches, or
• an outlet discharge from a storm sewer system
• If open channel flow is a combination of the two, both contributors should be evaluated and engineering judgment used to determine the appropriate time of concentration.

Storm Sewer System Flow. Velocities are typically greater in sewer systems than in natural ditches or channels. The storm sewer system flow regime consists of:

• Overland flow entering drain inlets, or
• open channel flow entering drain inlets.
• If sewer system flow is a combination of the two, both contributors should be evaluated and engineering judgment used to determine the appropriate time of concentration.

749.5.3.1 Rural Watersheds

For rural drainage basins the Kerby-Kirpich Method described below is a recommended method of determining time of concentration. Other methods such as those provided by the Natural Resource Conservation Service (NRCS) may be used as deemed necessary or appropriate by engineering judgment. See Velocity Method in Chapter 2 of FHWA publication Hydraulic Design Series No. 2 (HDS-2) for details on the NRCS method.

749.5.3.2 Urban Watersheds

For urban drainage basins the Kerby-Hathaway Equation for Overland Flow plus the appropriate Channel or Storm Sewer Travel Time as described below is a recommended method of determining time of concentration. Other methods such as those provided by the Natural Resource Conservation Service (NRCS) may be used as deemed necessary or appropriate by engineering judgment. See Velocity Method in Chapter 2 of FHWA publication Hydraulic Design Series No. 2 (HDS-2) for details on the NRCS method.

749.5.3.3 Kerby-Kirpich Method

For rural drainage basins the Kerby- Hathaway and Kirpich methods can be used in conjunction to estimate the time of concentration.

T_c = t_o + t_c

where:

T_c = the total time of concentration (min)

t_o = the overland flow time of concentration from the Kerby-Hathaway equation (min)

t_c = the channel flow time of concentration from the Kirpich equation (min)

749.5.3.3.1 Low (Flat) Slopes

In watersheds with low topographical relief, time of concentration calculations will often result in unreasonably large values. Cleveland, et al. 2012 recommends the following adjustment to the slope in the Kirpich and Kerby-Hathaway equations to provide a more realistic estimate of the time of concentration:

• For slopes flatter than 0.002 ft/ft (0.2%) the adjusted slope should be increased by 0.0005 (S_adj = S + 0.0005).
• For slopes between 0.002 ft/ft (0.2%) and 0.003 ft/ft (0.3%) engineering judgment should be used to determine if the adjustment to the slope should be made.

749.5.3.3.2 Kirpich Equation

Open channel flow travel time for streams with natural beds, or that mimic natural beds, can be calculated using the Kirpich Equations If flow regimes are combined the upstream end of the flow length (L) should extent to the basin divide.

${\displaystyle t_{c}=KL^{0.77}S^{-0.385}\,}$

where:

t_c = the time of concentration (min)

K = 0.0078

L = the flow length of the principal watercourse, ft.

S = the slope between the minimum and maximum elevation, ft./ft.

749.5.3.3.3 Kerby-Hathaway Equation for Overland Flow

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

${\displaystyle t_{o}=K(Ln)^{0.47}S^{-0.235}\,}$

where:

t_o = 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.;it is usually much less and engineering judgment should be used for lengths of more than 200 ft. After a short distance, overland flow usually changes flow regimes and begins to concentrate in swales or ditches, or into a storm sewer system.

749.5.3.4 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.

749.5.3.5 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.

Northwest District

Northeast District

Kansas City District

Central District

St. Louis District

Southwest District

Southeast District

749.6 USGS Regression Equations

The USGS Regression equations were developed by the United States Geological Survey. 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: 2014 and 1995 Rural Regression Equations, and Urban Regression Equations.

Stream flows for ungaged sites on streams that have a stream gage(s) may be determined using the area weighted method shown in EPG 749.7, Historical USGS Stream Gage Data.

749.6.1 Rural Regression Equations

For ungaged natural flood flow sites, flood magnitudes having recurrence intervals of 2, 5, 10, 25, 50, 100, 200 (2014 only) and 500 years are computed by using appropriate values of the contributing drainage basin area (A) and basin shape (B) (2014) or slope (S) (1995) in the equations shown in the Rural Regression Equations tables. 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 with local relief generally between 50 to 150 feet. Elevations range from about 600 ft. above sea level near the Mississippi River to about 1200 ft. above sea level in the northwestern part of the state.

Region II - Ozark Plateaus - - Characterized by streams that have narrow valleys 200 to 500 ft. deep, with local relief generally ranging from 100 to 500 feet. Main channel gradients generally are 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 II also includes the geological formation known as Crowley’s Ridge which rises 250 to 550 ft. above the Mississippi Alluvial Plain and reflects hydrologic and hydraulic characteristics similar to the Ozark Plateau.

Region III - Mississippi Alluvial Plain - This is a relatively flat area and is predominately agricultural cropland. Virtually all the area is drained by a series of man-made drainage ditches with average slopes of about 1.5 ft./mile. Elevations range from 200 to 300 ft. above sea level with local relief seldom exceeding 30 feet.

749.6.1.1 2014 Rural Regression Equations

Engineering judgment should be exercised when using these equations to ensure that the subject drainage area is similar in characteristics to the USGS region being used. Portions of eastern Butler and Ripley counties are especially sensitive as some streams will be consistent with slopes for Region 2 and others with Region 3.

Valid data ranges for each of the regional equations are listed in the Valid Data Ranges table below. The Supplemental Data Ranges table may be used to help determine which regional equation to use, but should not be used to determine the validity of the equations.

2014 Rural Regression Equations
Flood Freq. (years)	Flood Magnitude ft3/s	Std. Error of Prediction (%)
Region I
2	(102.594)*(A0.618)*(BS-0.282)	38
5	(102.861)*(A0.593)*(BS-0.266)	31
10	(102.990)*(A0.580)*(BS-0.258)	29
25	(103.120)*(A0.568)*(BS-0.248)	29
50	(103.199)*(A0.560)*(BS-0.242)	29
100	(103.266)*(A0.554)*(BS-0.236)	30
200	(103.324)*(A0.550)*(BS-0.231)	31
500	(103.391)*(A0.544)*(BS-0.226)	33
Region II
2	(102.493)*(A0.686)*(BS-0.222)	44
5	(102.801)*(A0.679)*(BS-0.251)	32
10	(102.955)*(A0.676)*(BS-0.268)	28
25	(103.113)*(A0.673)*(BS-0.287)	24
50	(103.205)*(A0.671)*(BS-0.296)	24
100	(103.282)*(A0.669)*(BS-0.302)	24
200	(103.349)*(A0.668)*(BS-0.307)	25
500	(103.422)*(A0.667)*(BS-0.311)	27
Region III
2	(101.933)*(A0.665)	30
5	(102.026)*(A0.681)	26
10	(102.070)*(A0.689)	26
25	(102.113)*(A0.698)	26
50	(102.139)*(A0.703)	27
100	(102.162)*(A0.708)	27
200	(102.182)*(A0.713)	28
500	(102.204)*(A0.718)	29
Source: Southard and Veilleux (2014)

749.6.1.1.1 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 or computer program.

749.6.1.1.2 Basin Shape

The Basin Shape (BS) is dimensionless and is a statistical method used to determine the magnitude and arrival time of a peak discharge. Basins with elongated shapes will have longer duration hydrographs and lower peak discharges than a circular basin that will have shorter duration hydrographs and higher peak discharges. The value used for the Basin Shape (BS) is calculated by dividing the square of the longest flow path or stream length by the Drainage Area (A). Stream length can be determined using the best available topographic maps or computer program.

749.6.1.1.3 Limitations of Equations

The USGS Rural Regression Equations may be used to estimate magnitude and frequency of floods on most Missouri streams. Caution should be used when using these equations when the drainage area and basin shape are outside of the limits of the Valid Data Ranges shown in EPG 749.6.1.1 2014 Rural Regression Equations.

However, the equations are generally not applicable for:

a) basins where manmade changes have appreciably changed the flow regimen such as:

• Regulation of discharge (i.e. dams);
• Channelization;
• Diversion;
• Urbanization

b) the main stems of the Mississippi and Missouri Rivers;

c) basins with greater than 5% impervious area;

d) areas near the mouth of streams draining into larger rivers where backwater effect is experienced may require special consideration, see EPG 749.11 Stream Confluences.

749.6.1.2 1995 Rural Regression Equations

Valid data ranges for each of the regional equations are listed in the Valid Data Ranges table below:

749.6.1.2.1 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 or computer program.

749.6.1.2.2 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. 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.

749.6.1.2.3 Limitations of Equations

The USGS Rural Regression Equations may be used to estimate magnitude and frequency of floods on most Missouri streams Caution should be used when using these equations when the drainage area and slope are outside of the limits of the Valid Data Ranges shown in EPG 749.6.1.2 1995 Rural Regression Equations.

However, the equations are generally not applicable for:

a) basins where manmade changes have appreciably changed the flow regimen such as:

• Regulation of discharge (i.e. dams);
• Channelization;
• Diversion;
• Urbanization

b) the main stems of the Mississippi and Missouri Rivers;

c) basins with greater than 5% impervious area.

d) areas near the mouth of streams draining into larger rivers where backwater effect is experienced may require special consideration, see EPG 749.11 Stream Confluences.

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 2010 by the United States Geological Survey. Data from 35 gaged sites in urban locations in and adjacent to 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 drainage area and urban basin characteristics to provide the regression equations. A computer program is available to assist in performing these calculations.

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.

749.6.2.1 Equations

Peak discharges can be estimated at urban locations using the equations presented in the Urban Regression Equations table, below. The Equations give peak discharge as a function of drainage area (A) and percentage of impervious area (I).

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.

The range of valid data for USGS Urban Regression Equations are:

a) Drainage Area – 0.28 mi² to 189 mi²

b) Impervious Area - 2.3% to 46%

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 or computer program.

749.6.2.3 Percentage of Impervious Area

The percentage of impervious area (I) is the portion of the drainage area that is impervious 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, satellite imagery or aerial photos showing impervious surfaces. Field inspection to supplement the imagery may be useful.

749.6.2.4 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 noted in EPG 749.6.2.1 Equations 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/.

749.7.1 Analysis

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.

749.7.2 Ungaged Sites Near Stream Gage

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 Area-Weighting equation:

${\displaystyle Q_{1}=Q_{2}\left({\frac {A_{1}}{A_{2}}}\right)^{k}}$

where:

Q1 = discharge for ungaged drainage basin (cfs)

A1 = drainage area for ungaged drainage basin (mi²)

Q2 = discharge for gaged drainage basin (cfs)

A2 = drainage area for gaged drainage basin (mi²)

k = exponent = see table

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, HEC-HMS or TR-20 hydrologic model. In some instances, the FIS discharges may have been determined using an older version of the USGS regression equations. Engineering judgment should be exercised to determine if the FIS discharges determined from older USGS regression equations should be used for the hydraulic design. When a No-Rise Certificate is required, and the FIS discharges are not used for hydraulic design, the 100-year FIS discharge should be used to verify the no-rise condition based on the Base Flood elevations in the FIS. 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.

749.9.1 NRCS 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 NRCS 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. The NRCS Unit Hydrograph method is applicable for all watersheds between 6.4 and 16,000 acres. Additional details on the NRCS methodology can be found in the National Engineering Handbook, (NEH), Part 630, Chapter 10 and Chapter 16(USDA/NRCS).

749.9.1.1 Rainfall-Runoff Equation and Concepts

The NRCS method is based on a 24-hour storm event. NRCS 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 NRCS method in Missouri.

The NRCS rainfall-runoff equation was developed by the NRCS 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 NRCS 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:

${\displaystyle Q={\frac {(P-I_{a})^{2}}{(P-I_{a})+S}}}$

where:

Q = accumulated direct runoff, in

P = accumulated rainfall (potential maximum runoff), in

I_a = 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:

${\displaystyle I_{a}=0.2S\,}$

Substituting the equation for I_a into the equation above, the NRCS rainfall-runoff equation becomes:

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

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 NRCS 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:

${\displaystyle S={\frac {1000}{CN}}-10}$

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 NRCS 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.

749.9.1.2 NRCS Dimensionless Unit Hydrograph

The NRCS 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 NRCS unit hydrograph is given in a dimensionless form and provides a standard unit hydrograph shape.

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

${\displaystyle q_{p}={\frac {484AQ}{T_{p}}}}$

where:

q_p = unit hydrograph peak discharge, cfs

A = drainage area, mi²

Q = runoff in inches

T_p = 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:

${\displaystyle {\frac {\Delta D}{2}}+0.6T_{c}=T_{p}}$

where:

t_p = time to peak, hrs

ΔD = duration of rainfall (unit hydrograph duration) = 0.133 t_c, hrs

t_c = time of concentration, hrs

749.9.1.3 Application of NRCS 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 0.133t_c. Since the NRCS Type II rainfall distribution has a 24-hour time base, application of the NRCS 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.

749.10 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 1)

where:

V_1,2 = 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 2)

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 3)

Metric: ${\displaystyle Q=0.55\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right]LH^{1.5}}$ (Equation 4)

where:

Q = discharge, ft³/s

H = head above weir crest excluding velocity head, ft.

H_c = 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: ${\displaystyle Q=\left[3.27+0.4\left({\frac {H}{H_{c}}}\right)\right](L-0.2H)H^{1.5}}$ (Equation 5)

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 6)

where:

Q_s = submergence flow, ftA³/s

Q_f = 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 7)

where: Q = discharge, ft³/s

C = broad-crested weir coefficient (see Broad-Crested Weir Coefficients)

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 8)

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 9)

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 10)

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 11)

where:

S = storage

If the values at time step n are known, the only unknowns in equation 8 are on the left-hand side.

Substituting

${\displaystyle U_{n+1}={\frac {2S_{n+1}}{\Delta T}}+O_{n+1}}$ (Equation 12)

${\displaystyle W_{n}={\frac {2S_{n}}{\Delta T}}-O_{n}}$ (Equation 13)

U is known as the Storage Indication Number. With these substitutions, equation 8 becomes:

${\displaystyle U_{n+1}=(I_{n}+I_{n+1})+W_{n}\,}$ (Equation 14)

For the first time step, W_n is calculated using the initial values of S and O, and equation 13. For subsequent time steps the following equation can be used as a shortcut.

${\displaystyle W_{n+1}=U_{n+1}-2O_{n+1}\,}$ (Equation 15)

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

1) Develop an inflow hydrograph, stage-discharge curve, and stage-storage curve for the proposed storage facility.

2) Select a routing time period, t, to provide at least five points on the rising limb of the inflow hydrograph.

3) Use the stage-storage and stage-outflow data from Step 1 to develop a plot of U versus outflow.

4) Calculate W₁ using equation 22 and the initial values of S and O

5) Calculate U_n+1 using equation 23.

6) Using U_n+1 calculated in step 5 pick O_n+1 from the plot of U vs. outflow.

7) Using U_n+1 and O_n+1 calculate W_n+1 using equation 24

8) Start over at step 5 with n = n+1. Continue repeating until inflow ceases or the outflow peak discharge has been determined.

9) From the stage discharge curve, determine the stage for the peak outflow.

749.11 Stream Confluences

Water surface elevations at stream crossings near the confluence with another stream are often affected by backwater from the other stream. Several methods may be used to determine the water surface elevations at such crossings, such as:

• Coincident Peaks (i.e., 100-yr event on both streams)
• Exclusive Peaks (i.e., 100-yr event on one stream, no flow from other stream)
• Flow Combinations (i.e., 86-yr Event on Main Reach, 62-yr Event on Tributary)

749.11.1 Coincident Peaks

When both streams reach their peak flow, for a given return period, at the confluence of the streams at or near the same time the peak flows can be considered to happen at the same time. The water surface elevation of each stream is effected by the other. This method may provide overly conservative water surface elevations if applied to confluent streams that do not have similar peak times.

FEMA, Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix C: Guidelines for Riverine Flood Analysis and Mapping gives the following guidance on when coincident peaks are appropriate to use:

749.11.2 Exclusive Peaks

Hydraulic analysis is conducted individually for each stream, for a given return period, without inclusion of any flow from the other stream, as each stream is assumed to reach its peak flow at different times. Analysis of the crossed stream will yield water surface elevations for the headwater flow of the crossed stream. If a structure is included in this model the effects of the structure on the water surface elevations will be included in the model. Analysis of the confluent stream will yield water surface elevations for the headwater flow of the confluent stream which can be used to determine the backwater surface elevation on the crossed stream. This backwater surface elevation does not include any effects from the flow of the crossed stream or any structures on the crossed stream. Since there would likely be some flow from the crossed stream this method may provide un-conservative water surface elevations in some cases.

This method is commonly used in hydraulic modeling for FEMA Flood Insurance Studies.

749.11.3 Flow Combinations

Similar to Coincident Peaks this method accounts for effects of one streams water surface elevation on the other, but uses flows from different return periods for each stream. Using a combination of flows can more closely simulate actual flooding conditions; the difficulty is in selecting a flow combination that will mimic those conditions. A comparison of hydrographs of both watersheds to determine the maximum peak flow at a given return period, or the probability based procedure shown below may be used to estimate flow combinations.

For streams that are within the influence reach of the confluent streams backwater the procedure provided in NCHRP 199, Estimating Joint Probabilities of Design Coincident Flows at Stream Confluences, produces a probability based estimate for combinations of return periods for confluent streams that will produce the maximum water surface elevation and/or velocity for a design flood event. An excerpt below from NCHRP 199 provides information on which cases this procedure is applicable to (Note: For Case A this procedure may still be useful to determine flow downstream of the junction based on flow combinations).

749.12 References

AASHTO, 2014 AASHTO Drainage Manual, American Association of State Highway and Transportation Officials.

Southard, R.E. and Veilleux, A.G.2014, Methods of Estimating Annual Exceedance-Probability Discharges and largest Recorded Floods for Unregulated Streams in Rural Missouri, USGS Scientific Investigations Report 2014-5165.0000

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

Southard, R.E., 2010, Estimation of the Magnitude and Frequency of Floods in Urban Basins in Missouri, USGS Scientific Investigations Report 2010-5073.

McCuen, R.H., et al., 2002 Highway Hydrology – Hydraulic Design Series No. 2 (HDS-2), Federal Highway Administration, Publication No. FHWA-NHI-02-001

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

United States Department of Agriculture: National Resource Conservation Service, National Engineering Handbook. NEH part 630 – Hydrology & Hydraulics

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

Kilgore, R.T., et. al., 2013, Estimating Joint Probabilities of Design Coincident Flows at Stream Confluences, NCHRP 199.

Roussel, M. C., et al, 2005. Time-Parameter Estimation for Applicable Texas Watersheds, U.S. Geological Survey, Texas Water Science Center, Research Report No. 0-4696-2

Cleveland, T. G., et al, 2012, Guidance for Estimation of Time of Concentration in Texas for Low-Slope Conditions, Texas Tech University, Lubbock Texas, Report No. 0-6382-P1