Unit Hydrograph S Curve Calculator
Calculate unit hydrograph curve with our free science calculator. Uses standard scientific formulas with unit conversions and explanations.
Formula
Tp = D/2 + tL; qp = 2.08 * A / Tp; S-curve equilibrium = A / (D * 3600)
Where Tp is time to peak, D is unit duration, tL is lag time, qp is peak flow, A is watershed area, and the S-curve approaches equilibrium discharge equal to area times unit rainfall rate.
Worked Examples
Example 1: SCS Synthetic UH Derivation
Problem: Watershed area 50 km2, unit duration 2 hr, lag time 3 hr. Derive the SCS triangular UH.
Solution: Tp = 2/2 + 3 = 4 hr\nTr = 1.67 x 4 = 6.68 hr\nTb = 4 + 6.68 = 10.68 hr\nqp = 2.08 x 50 / 4 = 26.0 m3/s
Result: Tp = 4.00 hr | Qp = 26.0 m3/s | Tb = 10.68 hr
Example 2: Duration Conversion via S-Curve
Problem: Convert a 2-hr UH to a 4-hr UH. Original peak 25 m3/s, lag 3 hr, area 50 km2.
Solution: Peak ratio = D/Dt = 2/4 = 0.5\nAdjusted peak = 25 x 0.5 = 12.5 m3/s (approx)\nNew rise time = 4/2 + 3 = 5 hr
Result: Adjusted Peak: 12.5 m3/s | Rise Time: 5.0 hr | Ratio: 0.500
Frequently Asked Questions
What is a unit hydrograph in hydrology?
A unit hydrograph (UH) is the direct runoff hydrograph resulting from one unit of effective rainfall (typically 1 cm or 1 inch) applied uniformly over a watershed for a specified duration. It is a fundamental tool in flood hydrology that allows engineers to predict runoff hydrographs for any rainfall event by superposition and scaling. The concept assumes linearity (runoff is proportional to rainfall excess) and time invariance (the response shape remains constant). Sherman introduced the concept in 1932.
What is an S-curve in unit hydrograph theory?
An S-curve is the cumulative runoff response to a continuous uniform rainfall of unit intensity, constructed by lagging and summing successive unit hydrographs at intervals equal to the unit duration D. The S-curve rises from zero and asymptotically approaches an equilibrium discharge equal to the watershed area times the unit rainfall rate. The S-curve method is essential for converting a unit hydrograph from one duration to another, which is needed when the available UH duration does not match the desired computational time step.
How do you use S-curves to change unit hydrograph duration?
To derive a new-duration UH from an existing one, first construct the S-curve by successively lagging and summing the original UH at intervals of its duration D. Then create a second S-curve lagged by the desired new duration Dt. The difference between the two S-curves, divided by the ratio Dt/D, gives the new-duration unit hydrograph. This method preserves the volume of the unit hydrograph while adjusting its temporal distribution. It works for both increasing and decreasing the duration.
What is the SCS triangular unit hydrograph?
The SCS triangular unit hydrograph is a simplified synthetic UH approximated by a triangle with time to peak Tp = D/2 + tL (where D is duration and tL is lag time), and recession time Tr = 1.67 * Tp. The peak flow rate is qp = 2.08 * A / Tp where A is area in km2 and Tp in hours. This gives a base time Tb = 2.67 * Tp. The triangle has the same volume as 1 cm of runoff over the watershed area. It is the most widely used synthetic UH for ungauged watersheds.
What is lag time and how does it relate to the unit hydrograph?
Lag time (tL) is the time interval between the centroid of effective rainfall and the peak of the direct runoff hydrograph. In the SCS method, lag time is estimated as tL = L^0.8 * (S+1)^0.7 / (1140 * Y^0.5) where L is hydraulic length, S is potential retention from CN, and Y is average slope percent. The time to peak of the UH is Tp = D/2 + tL. Lag time is approximately 0.6 times the time of concentration for most watersheds.
What assumptions underlie unit hydrograph theory?
Unit hydrograph theory assumes linearity (the principle of proportionality and superposition applies), time invariance (the watershed response shape does not change with time or antecedent conditions), and uniform spatial distribution of rainfall excess. In reality, these assumptions are only approximately met since infiltration is nonlinear, watershed response changes with soil moisture, and rainfall is rarely uniform. Despite these limitations, UH methods remain practical and widely used for engineering design.