Although statistical methods, probability theory, and risk analysis are beyond the scope of this text, it is worthwhile to examine an application of such meth-ods to stormwater management system design. As an illustration, a culvert having a design return period of 50 years and an expected life of 75 years can be shown using the binomial probability distribution to have a failure probability of 0.78, or 78 percent. This probability does not reflect the likelihood of a struc-tural failure such as collapse; rather, it is the probabil-ity that the design discharge capacprobabil-ity of the culvert will be exceeded one or more times during its life-time.
The probability of failure determined for the culvert in this example seems exceedingly high. It is certainly unacceptable for a structural engineer to design facili-ties with such high structural failure rates. However, hydrologic and hydraulic engineering differs in that requiring a small failure probability can be prohibi-tively expensive. If the culvert described previously were required to have a failure probability of 5 per-cent or less, then the design storm return frequency for an expected culvert life of 75 years would need to be at least 1,500 years!
It should be clear from these examples that a trade-off must occur in the selection of the recurrence interval for which a stormwater conveyance system is to be designed. If the recurrence interval is small and the system capital costs are correspondingly low, there is
a very high probability of hydraulic failure of the sys-tem during its lifetime. Conversely, if the probability of failure must be low, the recurrence interval becomes large, and the capital costs of system con-struction are also large.
In practice, engineers often adopt design recurrence intervals specified by the regulatory and/or review agency having jurisdiction over the area in which a conveyance system will be built. In the interest of minimizing the cost of any one system (so as to per-mit resources to be allocated more or less equally among the many systems in the jurisdictional area), recurrence intervals are typically set fairly low. The recurrence interval used in the first example above is typical of real-world practice.
The high probability of operational failure, and thus overflow, typically associated with stormwater con-veyance systems means that significant attention needs to be paid to the issue of what happens when a flood does occur. The operation of the major drainage system (the often unplanned, naturally occurring sys-tem that takes over when the minor flow syssys-tem is overtaxed), which historically has not received much attention, needs some careful investigation and plan-ning. It is not enough to simply call an exceedance of a system design capacity an Act of God, especially when such Acts have such a large probability of occurring.
area is usually the most significant predictor, and is often the only one retained in the relationship. For example, the predictive formulas developed for rural basins in north-ern Alabama for the 2-, 5-, 10-, 25-, 50-, and 100-year storms are (Olin, 1984):
In these equations, QT is the peak discharge, in cfs, corresponding to the T-year recur-rence interval, and A is the drainage basin area in mi2. These northern Alabama for-mulas are limited to use for drainage basins with areas between 1 and 1,500 mi2 (259 and 388,500 ha) and have standard errors that range from 29 to 36 percent.
When applying regression-based formulas to compute peak runoff rates from a drain-age basin, the engineer must check the range of areas used in formula development [1 mi2 to 1,500 mi2 (259 and 388,500 ha) in the Alabama case] to ensure that the equation is applicable to the basin being evaluated. If the basin area is outside of this range, the predicted peak discharge for a relatively high recurrence interval can be smaller than the predicted peak discharge for a lower recurrence interval. Similarly, the predicted peak discharge for an urban area can be less than the predicted peak dis-charge for a rural area. Finally, it should be recognized that the standard errors associ-ated with regression-based formulas for rural basins are typically on the order of 30 percent (35 to 50 percent for urban basins). Therefore, significant errors may arise when using this method.
Example 5.11 – Using a Regression Equation to Compute the Peak Discharge Rate. Use the rural equations for northern Alabama to estimate the 50-year peak discharge from a drainage basin with an area of 25 mi2. The standard error for this formula is 33 percent (Olin, 1984).
For this basin, what is the standard error of the estimate in cfs?
Solution: The peak discharge is Q50 = 571(25)0.720 = 5,800 cfs
The standard error Se associated with the estimate is Se = 0.33(5,800) = 1,900 cfs
5.5 HYDROGRAPH ESTIMATION
A hydrograph represents the runoff rate (discharge) as it varies over time at a particu-lar location within a watershed. The integrated area under a hydrograph represents the volume of runoff. Base flow, if present, represents subsurface flow from groundwater that discharges into the conveyance channel. Many storm conveyance channels are dry at the beginning of a rainfall event, which equates to a base flow of zero as long as the water table does not rise into the channel during the storm. Figure 5.9 displays the various components of a surface runoff hydrograph for a case in which the base flow is zero and demonstrates that the area under the hydrograph is equal to the runoff vol-ume as initially presented in Section 5.2.
Q2 = 182A0.706 Q5 = 291A0.711 Q10 = 372A0.714 Q25 = 483A0.717 Q50 = 571A0.720 Q100 = 664A0.722
Figure 5.9
Hydrograph definition sketch
Estimation of a runoff hydrograph, as opposed to merely the peak rate of runoff, is necessary to account for the effects of storage in a drainage basin. Because a hydrograph accounts for volume and flow variations over an entire rainfall event, it is useful for analyzing complex watersheds and designing detention ponds. Of particular importance to the design engineer is assessment of the effects of storage associated with natural ponds and lakes, and with constructed stormwater detention and/or reten-tion facilities. Hydrograph estimareten-tion is also necessary in the assessment of the impacts of storm duration and/or hyetograph shape on runoff production, and in cases of two or more adjacent drainage basins (or subbasins) discharging to a common stream.
Most approaches to hydrograph estimation are based on the concept of a unit hydrograph, which is a hydrograph produced by a unit depth of runoff (usually 1 in.
or 1 cm) distributed uniformly over a basin for defined period of time. The unit hydrograph must be convolved (combined) with an effective precipitation (runoff) hyetograph to obtain the direct runoff hydrograph from a basin. Direct runoff hydrograph estimation using unit hydrograph approaches is described in detail in this section.
Several alternatives to the unit hydrograph method are used for runoff hydrograph estimation, including hydraulic methods such as the kinematic wave model. This method is a more physically-based, hydraulic approach to hydrograph estimation but requires that the surface of a drainage basin be idealized as a set of overland flow planes and channels. A brief introduction to the kinematic wave model is presented later in this section (page 162).