A new design method enhances multipurpose uses and allows more economical storage sizing for urban stormwater detention facilities.

By Yosif A. Ibrahim

This article proposes a new storage routing method for the design of small stormwater management facilities in Fairfax County, VA. The method assumes that the temporal variation of rainfall intensities for different storm events follows a gamma distribution function and that a gamma distribution function can aid in the release from the stormwater management facility. The proposed procedure provides an effective design tool to control the impact of urbanization.

The general practice in Fairfax County is to use the Storage Indication or Modified Pulse Method for the routing and design of stormwater management facilities. The allowable release from the pond is limited to the predeveloped level. The inflow to the pond is estimated based on the Rational Method for a relatively smaller drainage area. Although the method proves to be simple, effective, and easier to perform, in some situations a reverse design procedure might be required to solve problems related to inadequacy of outfall. This article presents a reservoir sizing method that allows the designer the option of predefining the release from the pond and sizing the detention facility accordingly. The article includes a full description of the proposed method and its application to three design cases.

Description of the Proposed Method

Basic Equations. Routing inflow hydrograph through a reservoir element can be made with the aid of the continuity equation. In a conservation form, this equation can be written as follows (Equation 1):

where Q₁(t) is the inflow into the pond in m³/sec. (cfs), Q₂(t) is the outflow in m³/sec. (cfs), and S(t) is the storage in m³ (ft.³). Following the Rational Method, the inflow hydrograph can be estimated from Equation 2:

where A is the total contributed drainage area in km² (ac.), C is the runoff coefficient, and I(t) is the incremental unit hydrograph intensity in mm/hr. (in./hr.), which is a function of time of concentration and the frequency of the storm event.

Assuming that the temporal variation of these intensities follows a gamma distribution function, the inflow hydrograph can be formulated as Equation 3:

where n₁ is a shape parameter, K₁ is a scale parameter, and G is the gamma function, which has the formula (Equation 4):

Following the Nash (1959) unit hydrograph concept, it can be further assumed that the outflow from a reservoir is given by Equation 5:

where C_f is a factor that has a value between 0 and 1 and is used to specify the percentage in peak reduction, and n₁ and K₁ are shape and scale parameters that control the outflow hydrograph. Substituting Equations 3 and 5 into Equation 1 and integrating results in Equation 6:

The integral term between brackets in Equation 6 is known as the incomplete gamma function, which can easily be determined with standard algorithms.

Introducing the Concept of Storage Response Function

The unit storage response function is defined as the amount of storage required in m³ (ac.-ft.) per km² (ac.) of impervious drainage area that has a total rainfall depth of 1 mm (in.) falling during a period of two hours. (Peak intensity distribution is based on one given in Fairfax County's Public Facilities Manual [2001].) For a given time of concentration and a specific storm event, the storage response function will determine the required capacity for a certain percentage of peak reduction. Rewriting Equation 6 in terms of the storage response function results in Equation 7:

where I_p is the peak rainfall intensity in mm/hr. (in./hr.), c_s is a conversion factor having a unit of a km²-sec./m² (ac.-sec./ft.²), and S_u(t) is the dimensionless unit storage response function given by Equation 8:

Table 1 shows values of I_p and c_s for different storm sizes and times of concentration.

Click here for Table 1. Tabulation of Peak Intensity and Conversion Factor for Different Storm Events

Figure 1 is a plot of a typical storage response function.

Methodology

Estimation of Model Parameters. The proposed model contains five parameters: n₁, K₁, n₂, K_2, and C_f. The first two parameters, n₁ and K_1, are shape and scale parameters used to describe the temporal variation of rainfall intensities in a form of gamma distribution function. Parameters n₂ and K₂ provide the required lag and attenuation to the inflow hydrograph, and parameter C_f determines the percentage of peak reduction needed. A value of C_f equal to 0 means that there is no release from the pond, and a value of C_f equal to unity indicates no attenuation in peak. The optimum value of n₂ and K₂ depends largely on the specific design case. However, a value of n₂ equal to 2 and K₂ equal to 5 seems to be typical. Parameters n₁ and K₁ need to be calibrated once for each storm event and time of concentration.

The calibration procedure followed here is a manual iterative searching method that seeks to match the outcome of Equation 3 with values of incremental unit hydrograph shown in Public Facilities Manual (PFM). Four criteria or objective functions were selected to judge the goodness of fit: minimizing the variance, or sum of square of errors; matching the peak; plotting the time to peak; and conserving the volume under the hydrograph. The results of optimization are shown in Table 2 and plotted in Figures 2 through 4.

As shown in the table, the gamma distribution function fits exactly the peak and time to peak of the incremental unit hydrograph rainfall intensities without any significant loss or error in volume.

Design Procedure

Once the optimum values of n₁ and K₁ have been determined, the design procedure can be summarized as follows:

1. Determine the maximum allowable release from the pond and find the corresponding C_f factor. The C_f is calculated as the ratio of the maximum allowable outflow to the peak inflow. The maximum allowable outflow is determined on the basis of limiting the releases to the predeveloped level and/or satisfying certain downstream outfall protection measures.

2. Find the corresponding model parameters n₂ and K₂ that should match the peak and time to peak with the aid of the following equations (Equation 9):

3. Determine the unit storage response function from Equation 8.

4. Find the corresponding I_P and c_s for the specific storm event from Table 1 and calculate the required storage from Equation 7.

Applying the Model to Real Design Cases

To test the accuracy of the model in reproducing the same outflow hydrograph and storage size, the model was applied to three design cases in Fairfax County. Table 3 provides a brief description of these stormwater management facilities.

The results of model applications were shown in Figures 5 through 8 and summarized in Table 4. The results show that the efficiency of the model in estimating the required storage and reproducing the same outflow hydrographs is reasonably accurate.

Click here to view Table 4. - Results of Model Application to Three Design Cases

Advantages of the Proposed Method

The major advantage of the proposed reservoir sizing procedure lies in its capability of allowing the maximum release from the pond to be specified as a direct input. This might provide a more economical storage sizing procedure and enhance the multipurpose functionality of the pond. The proposed method requires less computational effort. Most of the parameters can be coded as standard constant values for a certain time of concentration and frequency of storm event. The incomplete gamma function has a standard solution algorithm, which can be found in most spreadsheet functions. The method seems to be theoretically sound, and the concept of Gamma Function Unit Hydrograph Pulse response function is well documented in the literature as a diffusion analogy type of catchment model (Dooge, 1973; Rosso, 1984).

Summary and Recommendations

The results of this study highlight the possibility of employing a new hydrologic storage routing method that can be used for the design of small stormwater management facilities in Fairfax County. The Rational Method was used to estimate the peak runoff into the detention facility. A gamma distribution function was fitted to describe the rainfall distribution for a set of storm events, and the outflow, or the release from the pond, was approximated with the aid of a nonlinear gamma distribution function. A mathematical model for the resulting unit storage response function was derived and applied to a set of three design cases in Fairfax County. The results are promising, and the scope of a reverse sizing procedure seems to be justified.

Various extensions of this method are recommended, particularly in the following:

• Further refinement might be needed on the optimization and calibration procedure utilized to estimate the two model parameters n₁ and K₁.
• The Rational Method cannot be used for drainage areas greater than 20 ac.; research on incorporating the Soil Conservation Service hydrology in estimating the peak runoff might expand the applicability of the proposed routing method to larger drainage areas.
• The proposed method can be tested and applied for other local jurisdictions.

References

Dooge, J.C.I. "Linear Theory of Hydrologic System." USDA Technical Bulletin No. 1468. Washington, DC. 1973

Fairfax County Department of Public Works and Environmental Services. Public Facilities Manual, pp 6-24. 2001.

Nash, J.E. "Synthetic Determination of Unit Hydrograph Parameters." Journal of Geophysical Research, Vol. 64, No. 1. 1959.

Rosso, R. "Nash Model Relation to Horton Order Ratios." Water Resources Research, Vol. 20, No. 7. 1984.

Guest author Yosif A. Ibrahim, Ph.D., is a senior engineer with the Department of Public Works and Environmental Services in Fairfax, VA, specializing in site development.

SW - November/December 2002

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