Peak Flow Reduced by Upstream Storage (Estimate)

Before spending the effort on a full storage routing, it helps to know whether the storage upstream of a culvert will do much. This formula gives that quick answer. It takes the unrouted peak flow $Q_p$, the volume $s$ that can be stored upstream of the crossing up to the allowable headwater, and the time to peak $t_p$ of the inflow hydrograph, and it returns the reduced peak flow $Q_r$ that the culvert would have to pass. The subtracted term, a volume over a time, is a flow.

The Federal Highway Administration's culvert manual (HDS 5) prints the rule as $Q_r = Q_p - s/(80\,t_p)$ for cubic feet per second, cubic feet and minutes; the 80 belongs to those units and carries the 60 seconds in a minute. With the time in seconds, as this page takes it, the constant is 80/60 = 4/3. The manual does not say where its 80 comes from, but one reading is ours: written as $Q_p(1 - s/V)$ with $V = (80/60)\,Q_p t_p$, the 80 matches the volume of a triangular hydrograph whose base is 2.67 times the time to peak (60 × 2.67 / 2 = 80.1), so the rule takes off the peak the share of the runoff that the storage can hold back.

The manual's own example is the check. With 220 ft³/s of peak flow, 261,360 ft³ of storage and a time to peak of 75 min, the reduced peak is 176 ft³/s (176.44 before rounding), which the manual calls a 20 percent reduction (19.8 by the numbers). The same culvert in SI, with 6.23 m³/s, 7,400 m³ and 75 min, gives 4.9967 m³/s. Run the other way, the storage that takes 220 ft³/s down to 176 ft³/s over 75 min is 80 × 75 × 44 = 264,000 ft³.

This is an estimate for deciding whether to route, not the routing. In the same example the full storage routing of the hydrograph gives a peak outflow of 150 ft³/s, lower than the 176 ft³/s the shortcut returns. The manual notes that the storage needed for a significant cut in the peak is typically quite large, and that leaving storage out only makes the barrel somewhat larger, which is a margin of safety. If the estimate shows a worthwhile cut, the full level-pool routing is the next step.