Stormwater Runoff and Detention: Method

Rational Method, Curve Number, Unit Hydrograph, Detention Routing

Scope

The tool answers the usual question of a site drainage report: how much does development raise the peak runoff of a small watershed, and how large a detention basin, with what outlet, brings the peak back to the pre-development value. It computes the peak three ways, the Rational method, the NRCS (SCS) graphical peak discharge method of Technical Release 55, and a runoff hydrograph from the NRCS dimensionless unit hydrograph, each for the condition before and after development; it then estimates the storage by the TR-55 chapter 6 curve and by the modified rational method and routes the developed hydrograph through the basin by the level-pool (storage-indication) method. Every relation comes from a public U.S. government document: TR-55 (1986), the National Engineering Handbook Part 630 chapters 15 and 16, FHWA Hydraulic Engineering Circular No. 22 (HEC-22) and Hydraulic Design Series No. 2 (HDS-2).

Units

The source relations are empirical in U.S. customary units and the tool evaluates them there. Rainfall and runoff depths are plain numbers in inches (millimetres in SI), areas in acres (hectares), intensities in in/h (mm/h), times of concentration in minutes; lengths, elevations, storage volumes and the allowable release go through the unit dictionary. Plain numbers are not converted when the unit system is switched: re-enter them in the other system.

Curve number and runoff depth

Each subarea takes its curve number from TR-55 Table 2-2 (a: urban areas, b: cultivated land, c: other agricultural land, d: arid rangeland) by cover and hydrologic soil group, or an entered value. The composite CN is the area-weighted mean, used as the whole number as TR-55 worksheet 2 does. For other impervious percentages than the table assumes, TR-55 figures 2-3 and 2-4 give CN_c = CN_p + (P_imp/100)(98 − CN_p) for connected impervious area and CN_c = CN_p + (P_imp/100)(98 − CN_p)(1 − 0.5 R) for unconnected impervious area under 30 percent (R the unconnected share); enter the result in the CN column.

The runoff depth follows TR-55 equations 2-1 to 2-4: S = 1000/CN − 10, I_a = λ S with λ = 0.2, and Q = (P − I_a)² / (P − I_a + S) when P exceeds I_a. TR-55 warns that the equation is less accurate below 0.5 in of runoff and that another procedure is needed when the weighted CN is under 40; the curve numbers of Table 2-2 belong to λ = 0.2, so another ratio needs curve numbers derived for it.

Time of concentration

The TR-55 velocity method adds the travel times of consecutive segments (eq. 3-2). Sheet flow, at most 300 ft, uses Manning's kinematic solution T_t = 0.007 (n L)^0.8 / (P₂^0.5 s^0.4) hours (eq. 3-3) with the roughness of Table 3-1 and the 2-year 24-hour rainfall. Shallow concentrated flow moves at V = 16.1345 s^0.5 ft/s unpaved or 20.3282 s^0.5 paved (figure 3-1, Appendix F), and channel flow at the Manning velocity V = 1.49 r^2/3 s^1/2/n (eq. 3-4); T_t = L/(3600 V). The Kirpich formula t_c = 0.0078 L^0.77 S^−0.385 (minutes, feet) and the FAA overland-flow formula t = 1.8 (1.1 − C) L^0.5/S^1/3 (minutes, feet, S in percent) are offered for comparison. The National Engineering Handbook prints the Kirpich constant as 0.007; the tool keeps Kirpich's 0.0078.

Rational method

Q = C_f C i A (HEC-22 eq. 3-1; divided by 360 with mm/h and hectares), C the area-weighted coefficient (eq. 3-2) and i the intensity of the design return period at a duration equal to the time of concentration. The intensity comes from the fit i = a/(t + b)^c or from a pasted duration-intensity table read along straight lines on log-log paper; the values belong to the site and are taken by the analyst from NOAA Atlas 14. The frequency factor C_f (1.1 at 25 years, 1.2 at 50, 1.25 at 100) is listed by HEC-22 with the remark that FHWA does not endorse it; C_f C is limited to 1. HEC-22 limits the method to about 200 acres. In U.S. units the formula drops the factor 1.008 between acre-inches per hour and cubic feet per second, while the SI form with 360 is exact, so the same case entered in the two systems differs by 0.8 percent.

Graphical peak discharge

q_p = q_u A_m Q F_p (TR-55 eq. 4-1) with A_m in square miles and Q in inches. The unit peak discharge of Exhibit 4 is the Appendix F equation log q_u = C₀ + C₁ log T_c + C₂ (log T_c)² with the coefficients of Table F-1 for the rainfall type and the ratio I_a/P; between two tabulated ratios q_u is interpolated linearly, and outside the tabulated range the limiting curve is used, as the release directs. T_c is held between 0.1 and 10 hours. The pond and swamp factor F_p of Table 4-2 is read along straight lines between its five points (the release rounds to the nearest). The method applies to one homogeneous watershed.

Unit hydrograph and design storm

The NRCS unit hydrograph for a burst of duration Δt peaks at T_p = Δt/2 + 0.6 T_c with q_p = 484 A_m Q / T_p (NEH Part 630 eqs. 16A-7, 16A-6, and lag = 0.6 T_c from chapter 15); its shape is the dimensionless unit hydrograph of Table 16-1. For another peak rate factor the one-parameter curve q/q_p = (x e^1−x)^m, x = t/T_p (eq. 16-1) is used, with m found from PRF = 645.33 m^m+1/(e^m Γ(m+1)), which reproduces the handbook's Table 16-5. The computation interval is a whole number of minutes, at most six and not above 0.133 T_c (eq. 16A-13); the ordinates are scaled by a factor close to one so that the unit hydrograph holds exactly one unit of runoff.

The 24-hour design storm is the cumulative NRCS distribution of the chosen type times the rainfall depth. At each interval the cumulative rainfall is turned into cumulative runoff by the curve-number equation; the increments of runoff are convolved with the unit hydrograph (the procedure of NEH Part 630 example 16-1). The built-in distributions are the NRCS tables of types I, IA, II and III at 0.1-hour steps (241 cumulative fractions each), the rainfall table files of the NRCS programs WinTR-20 and WinTR-55; the cumulative rainfall is taken by straight lines between the tabulated points. With these tables the hydrograph peak of a homogeneous watershed (curve number 75, times of concentration from 0.1 to 2 hours) lies within about 10 % of the graphical peak discharge for I_a/P from 0.10 to 0.30 in all four types; on the limiting curve I_a/P = 0.50, where little of the rain runs off, the two differ by up to 16 % (types I, II, III) and 26 % (type IA). TR-55 gives no figure for the accuracy of its graphical method; its curves were fitted to output of the 1983 TR-20 program, whose rainfall tables were coarser (type II at quarter hours, the others at half hours). A NOAA Atlas 14 temporal distribution or a measured storm can be pasted instead.

Storage estimates

TR-55 chapter 6 relates the ratio of storage to runoff volume to the ratio of peak outflow to peak inflow: V_s/V_r = C₀ + C₁ (q_o/q_i) + C₂ (q_o/q_i)² + C₃ (q_o/q_i)³ (figure 6-1, Table F-2: 0.660, −1.76, 1.96, −0.730 for types I and IA; 0.682, −1.43, 1.64, −0.804 for types II and III), with both peaks from the graphical method. The release states that the curve is biased to the safe side and may overestimate the storage by up to 25 percent.

The modified rational method considers storms longer than the time of concentration. A storm of duration t_d gives a trapezoidal inflow of peak Q(t_d) = C_f C i(t_d) A and volume Q t_d; the release is taken as a triangle that reaches Q_o at t_d and ends with the inflow at t_d + t_c, so V_s(t_d) = Q t_d − Q_o (t_d + t_c)/2, and the largest value over all durations is the estimate. This release shape is the tool's stated convention; local manuals vary it. The simpler rational-hydrograph estimate of HDS-2 (eq. 8.4), 60 (Q_a − Q_b) t_c, is listed beside it.

Outlet rating and level-pool routing

The basin is described by a table of stage against storage (straight lines between rows) and by its outlet devices. An orifice discharges Q = C_o A (2 g H)^0.5 with H measured to the centre of the opening (HEC-22 eq. 8-17); while the water surface is below the crown, the tool uses the wetted circular segment with the head to its centroid, which joins the submerged formula without a step. A weir or spillway discharges Q = C_d (2/3)(2 g)^0.5 L H^1.5; the dimensionless C_d = 0.62 is the sharp-crested 3.33 ft^0.5/s (1.84 in SI) of HEC-22 eq. 8-19, and a broad crest (eq. 8-22) runs from 0.49 to 0.58. A riser passes the smaller of weir flow over its rim, with L = π D, and orifice flow through its throat (HEC-22 example 8-6). The devices act together and their flows add.

Routing solves the continuity equation over each interval, (I₁ + I₂)/2 Δt + S₁ − O₁ Δt/2 = S₂ + O₂ Δt/2 (HEC-22 eqs. 8-29 and 8-30). The right side is a known function of stage, so the new stage is found on the rating; because storage and outflow are both taken linear between rating points, the solution is exact on each segment and the mass balance closes to rounding. Where the outlet would empty a storage increment within one interval (the storage-indication curve crossing the equal-value line, HEC-22 step 4) the interval is subdivided. After the storm the pool is drawn down, and the drawdown time is counted from the peak stage until the storage has fallen to 1 percent of its peak.

Limits

One homogeneous watershed and one basin at its outlet; no channel routing, no tailwater on the outlet, no infiltration or evaporation, and the outlet conduit is assumed not to control. The TR-55 tabular method for several subareas with reach travel times is not included. The emergency spillway coefficient is an entered constant (HEC-22 figure 8-18 gives it as a function of head). The design rainfall values are the analyst's, from NOAA Atlas 14 or the local authority.