# Developing a Rational Method Hydrograph Model for the Urban Desert Southwest USA

#### Introduction

## Conceptual Outcomes

- Students demonstrate understanding of the physical causes of flood frequency and intensity, especially the roles of land use and climate.

- Students identify the roles and responsibilities of U.S. Federal hydrology and flood management organizations.

- Students demonstrate increased perception of the value of geoscience and hydrology education and information.

- Students demonstrate understanding of the utility of mathematical geoscience models, especially for prediction and risk management.

## Practical Outcomes

- MS Excel and spreadsheet-based calculation and visualization of a simple model

- Use of published charts, figures, and tables to obtain design rainfall intensities and frequencies for an urban area

- Manual calibration of a simple model to produce results that match observations for an example

- Prediction and estimation of the effects of varying a key model parameter on model results

## Time Required

## Computing/Data Inputs

- Value: Average watershed slope, feet per mile, S

- Value: Length of longest flow path, miles, L

- Value: Area of basin, acres, A

- Value: Channel maximum flow capacity at outlet, cubic feet per second, Qmax

- Description: Qualitative description of watershed land cover

- Value: Design Storm duration, hours, ts

- Value: Design Storm total rainfall depth, inches, ds

- MS Excel 2010 Spreadsheet: 1432241251 (Excel 2007 (.xlsx) 564kB May21 15)

## Computing/Data Outputs

- MS Excel 2010 Spreadsheet: Rational Method model for current conditions
- Value: current time of concentration, hours, Tc
- Value: current peak runoff flow rate, cubic feet per second, Qpeak
- Value: current total storm runoff volume, cubic feet, Vs

- MS Excel 2010 Spreadsheet: Rational Method model for fully urbanized conditions
- Value: fully urbanized time of concentration, hours, Tc
- Value: fully urbanized peak runoff flow rate, cubic feet per second, Qpeak
- Value: fully urbanized total storm runoff volume, cubic feet, Vs

## Hardware/Software Required

- You will need a computer with an internet browser and Microsoft Excel 2010 or equivalent.

## Supporting References and Files

- USGS Flood Resources: http://water.usgs.gov/floods/resources/
- USGS WaterWatch: http://waterwatch.usgs.gov/?id=wwdp2

- USGS NWIS peak flow data: http://nwis.waterdata.usgs.gov/usa/nwis/peak

## Instructions

**1.**Open the module spreadsheet file "Model_v1.xlsx" in Excel. Here is what you will find in there:

The "TITLE" page gives metadata for the model.

The "MODEL" page contains the Rational Method model; you will need to enter data into the YELLOW and BLUE boxes to parameterize the model. Do not change the red boxes because these are computed results of the model.

The model automatically displays the results of its simulation in the figure titled "Rational Method Triangle Hydrograph". The vertical axis is the volumetric water flow rate Q in cubic feet per second, and the horizontal axis is the time t after the start of rainfall in hour.

- The blue line is the stormwater runoff hydrograph; this is the flow rate of rainwater runoff that is flowing out of the watershed at its outlet at any given point in time.

- The red line is the channel capacity; this is the maximum capacity of the stream channel to convey water before a flood occurs.

- The green line is the rainfall hyetograph; this is the rate at which rainwater is entering the watershed at any given point in time.

- The black diamond is the observed runoff peak for a storm; this is for comparing the model hydrograph with actual observations of the timing and magnitude of the runoff peak. We will not use this feature in the current exercise.

The default values in the model spreadsheet are for a "2-year 24-hour" design storm for the East Maricopa Floodway; this is a storm of 24 hour duration that has a 50% chance of being exceeded in intensity in any given year (e.g. a "2-year" storm frequency; note that this does not mean the storm recurs every two years, but rather that this depth has a 50% chance per year of being exceeded).

The Rational Method's simplest implementation, used for urban watersheds of relatively small size, assumes a triangular shape of hydrograph that is defined by a linear rise in flow rate Q cfs starting from the point in time when rainfall begins t = 0 (hrs) until the peak flow rate Qpeak = C*i*A (cfs) is reached when the full watershed area is contributing runoff at the time of concentration at Tc hrs. Then, the flow rate linearly decreases until it reaches zero at t = th*Tc (hrs), where th is the length of the hydrograph recession curve, generally assumed to be roughly 1.67. A is the watershed's total area draining to the channel outlet, in acres, as defined above. C is a dimensionless runoff coefficient which gives the effective fraction of the land surface that is contributing runoff; it is a function of the land cover of the watershed, and is usually obtained from a table based on a description of the watershed's land use. i is the adjusted average excess rainfall intensity (in/hr), a rainfall rate measured as the rainfall which is not intercepted or evaporated or infiltrated and therefore contributes to runoff. i is computed by dividing the total storm rainfall depth ds (in) by the storm duration ts (hrs) and then multiplying by the dimensionless depth area reduction factor D to obtain an average rainfall rate over the entire watershed during the entire storm. This assumes negligible interception, evaporation, or infiltration, which is an appropriate assumption for small urban watersheds during brief storms, especially in desert regions with relatively low vegetation cover and relatively impervious soils. D = 1 for watersheds under one square mile in area, but decreases for larger watersheds because a storm will not rain at its full intensity over larger watershed areas. Tc is a function of the watershed's land cover properties and geometry; a specific method of estimation is assumed in this model.

**2.**On the "MODEL" tab of the spreadsheet, enter S, L, A, and Qmax into the "Watershed Properties" yellow boxes. The model now knows the geometry of your chosen watershed.

**3.**On the "C Chart" tab of the spreadsheet, or using an equivalent chart containing appropriate data for your location (this chart is for Maricopa County Arizona), look up a mean runoff coefficient based on the typical land uses in your watershed and a "2 year" storm frequency. Enter it in the model's blue box for C.

4. On the "D Chart" tab of the spreadsheet, or using an equivalent chart containing appropriate data for your location (this chart is for Maricopa County Arizona), look up the depth area reduction factor based on your watershed's area. Enter it in the model's blue box for D.

**5.**On the "m and b Chart" tab of the spreadsheet, or using an equivalent chart containing appropriate data for your location (this chart is for Maricopa County Arizona), look up the coefficients m and b based on the typical land uses in your watershed. Enter these in the model's blue box for m and b.

**6.**On the "ds Chart" tab of the spreadsheet, or using an equivalent chart containing appropriate data for your location (this chart is for a location at the center of the East Maricopa Floodway), look up the total storm rainfall depth ds for a 50% annual probability of exceedance. This means that in any given year, based on historical precipitation data that is generally kept by agencies such as the National Oceanographic and Atmospheric Administration (NOAA) National Weather Service (NWS), there is a 50% chance that a depth of rainfall greater than ds will occur during at least one 24-hour period. This is often called a "2-year" storm frequency because of the math (1 / 50% = 2). Enter the value you find in the model's yellow box for ds. In the yellow box for ts, enter 24 hours; this is typically the duration of a design storm used in the USA, because this length of rainfall tends to have a moderate rainfall intensity and to saturate the watershed's soil and retention capacity and create an increased volume of runoff that is more likely to create floods as compared with shorter rainfall events.

**7.** For the procedure explained above and using the example of the East Maricopa Floodway, for a 24 hr storm of depth 1.7 in, the model should give the following results: Tc = 58 hrs, Qpeak = 1110 cfs, and Qrain = 4070 cfs. You may use these numbers to check that your model is working.

**8.**Repeat the above steps 3-8 for several different 24-hr rainfall events reflecting different storm rainfall annual exceedance probabilities for storm events, and in the provided table in the "Results Tables" tab record the values Tc, Qpeak, Qrain, Vs, and Vrain. For each event, compare Qpeak with Qmax; if Qpeak > Qmax, then the channel draining the watershed will flood. For the first storm event that causes a flood, make a copy of the hydrograph figure to record your results. The spreadsheet tab "EMF EXAMPLE 100 YR DESIGN STORM" gives an example of the parameters and results that would be used for a 100 year storm event. When you are finished, save your model spreadsheet as "RationalModelCurrent.xslx". Note the smallest rainfall event that causes a flood.

**9.**Make a change in the watershed's land cover type to simulate significantly increased urbanization of the watershed. You will need to change C, m, and b in the model. Now repeat step 9 above for the urbanized watershed. When you are finished, save your model spreadsheet as "RationalModelUrbanized.xslx". Note the smallest rainfall event that causes a flood.

## Additional Activities and Variants

- Combine this exercise with GIS and mapping activities to derive watershed properties for your own local watershed.

- Utilize the resources of NOAA to obtain rainfall statistics for your local watershed.

- Utilize USGS stream gage data and USGS or NOAA rainfall data from the internet to obtain your own rainfall/runoff observations for a large rainfall event in your watershed, and use the data to calibrate a Rational Method model to match your local watershed's conditions exactly.

- For lower level courses where this exercise is being completed in isolation from other activities that teach students how to obtain watershed properties, it is sufficient to use the default model design for the East Maricopa Floodway.

## Related Steps

- Calibrating a Rational Method Hydrograph Model for the Urban Desert Southwest USA