Topic 7 of 10 — free theory
Hydrology & Runoff
Rational method, SCS curve number, unit hydrographs, and runoff fundamentals.
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How much runoff? Two methods, two scales
The exam tests two runoff methods. The Rational method gives a peak flow rate for small watersheds; the SCS curve number method gives a runoff depth for a design storm. Know which output each one produces.
Rational method — peak flow
Qp = C · i · A
| Qp | peak runoff rate (cfs when i is in in/hr and A in acres — the units work out) |
| C | runoff coefficient (0–1); use an area-weighted composite for mixed land use |
| i | rainfall intensity for a duration equal to the time of concentration, tc |
| A | drainage area |
Composite C = Σ(CjAj) / ΣAj
| tc | time of concentration — travel time from the hydraulically most distant point |
Rule of thumb: the Rational method suits small, mostly uniform watersheds (commonly under ~200 acres). The intensity i comes from an IDF curve at duration tc.
SCS curve number — runoff depth
The SCS (now NRCS) method converts a storm's rainfall depth into direct runoff depth from the watershed's soil and cover, encoded in a curve number CN:
S = 1000/CN − 10 (inches) Ia = 0.2S
| CN | curve number, 0–100 (higher = more runoff); from TR-55 tables by soil group and cover |
| S | potential maximum retention after runoff begins |
| Ia | initial abstraction — interception, depression storage, early infiltration |
Q = (P − Ia)²(P − Ia) + S for P > Ia; Q = 0 otherwise
| Q | direct runoff depth (inches) |
| P | storm rainfall depth (inches) |
With Ia = 0.2S this simplifies to Q = (P − 0.2S)²/(P + 0.8S). If P ≤ 0.2S, every drop is abstracted and Q = 0 — a favourite trick question.
Unit hydrographs — distributing runoff in time
A unit hydrograph (UH) is the watershed's runoff response to 1 inch of direct runoff in a given duration. Real storms are handled by convolution: scale and lag the UH ordinates. The SCS triangular UH is the exam staple:
tp = ΔD/2 + 0.6 tc qp = 484 A / tp
| tp | time to peak (hr) |
| qp | peak of the unit hydrograph (cfs) |
| ΔD | unit storm duration (hr) |
| A | area in square miles |
The 484 constant assumes the standard SCS dimensionless hydrograph shape (37.5% of volume under the rising limb).
PE depth: frequency, detention, and level-pool routing
A return period is a probability statement, not a schedule. For an annual-maximum event with return period T, the annual exceedance probability is p = 1/T. The chance of at least one exceedance in n years is:
P(≥1) = 1 − (1 − 1/T)n
Detention design routes an inflow hydrograph through storage. Combining I − O = dS/dt with trapezoidal integration gives the Modified Puls equation:
2S2Δt + O2 = I1 + I2 + 2S1Δt − O1
At each time step, the right side is known. Use the reservoir's storage–outflow table to find the matching stage, S2, and O2. Repeat through the hydrograph; the routed peak should be delayed and usually reduced.
PE trap: detention controls release rate but usually drains after the event; retention includes permanent storage. Read the outlet geometry and stage-storage relation before assuming either behaviour.
Worked example Rational peak flow and SCS runoff depth
Given:
- Part A (Rational): 12-acre commercial site, composite C = 0.55, tc = 25 min, IDF intensity i = 3.2 in/hr.
- Part B (SCS): separate 40-acre watershed, CN = 78, design storm P = 5.0 in.
Solution:
- Part A: Qp = C·i·A = 0.55 × 3.2 × 12 = 21.1 cfs. Units check: (in/hr)×(acres) converts directly to cfs with C dimensionless.
- Part B: S = 1000/78 − 10 = 2.821 in; Ia = 0.2 × 2.821 = 0.564 in. Since P = 5.0 > 0.564, runoff occurs.
- Q = (5.0 − 0.564)² / (5.0 − 0.564 + 2.821) = 19.68 / 7.26 = 2.71 in of direct runoff.
Answer: Peak flow ≈ 21.1 cfs (Rational); runoff depth ≈ 2.71 in (SCS).