Rainfall Intensity, Depth and Runoff Hydrographs for a Frozen Watershed: 22-Agric-B7 Principles of Hydrology, May 2014, Q1

Question 1: Rainfall Intensity, Depth and Runoff Hydrographs for a Frozen Watershed (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Watershed area $A=5\ \text{km}^2$, completely frozen with little/no snow cover (so essentially all rainfall becomes direct runoff — no infiltration losses to speak of). Time of concentration $t_c=60$ min. IDF curve (read from the attached figure) passes through $(30\ \text{min},\,100\ \text{mm/hr})$, $(60\ \text{min},\,50\ \text{mm/hr})$, $(120\ \text{min},\,30\ \text{mm/hr})$, $(240\ \text{min},\,15\ \text{mm/hr})$.

Find. (a) The rainfall intensity and total depth for storms of duration $t_c/2$, $t_c$, and $2t_c$. (b) A qualitative sketch comparing the three resulting runoff hydrographs.

Figure 1a — the paper's own IDF curve, with the three storm durations asked for ($t_c/2$, $t_c$, $2t_c$) marked at their read intensities.

Approach. Read the intensity directly off the given IDF curve at each of the three specified durations (they coincide with the curve's own labelled points), then multiply intensity by duration to get each storm's total rainfall depth.

Storm durations relative to $t_c$. $$t_1=\tfrac{t_c}{2}=30\ \text{min}\qquad t_2=t_c=60\ \text{min}\qquad t_3=2t_c=120\ \text{min}$$
Intensities from the IDF curve and resulting depths $P=i\times t$. Reading each duration straight off the curve: $$i_1=100\ \text{mm/hr}\ \Rightarrow\ P_1=100\left(\tfrac{30}{60}\right)=\boxed{50\ \text{mm}}$$ $$i_2=50\ \text{mm/hr}\ \Rightarrow\ P_2=50\left(\tfrac{60}{60}\right)=\boxed{50\ \text{mm}}$$ $$i_3=30\ \text{mm/hr}\ \Rightarrow\ P_3=30\left(\tfrac{120}{60}\right)=\boxed{60\ \text{mm}}$$ Discussion: intensity keeps falling as duration doubles, but not fast enough to offset the longer duration — the half-$t_c$ and full-$t_c$ storms happen to deliver the same 50 mm depth here, while the $2t_c$ storm delivers slightly more total rain (60 mm) despite its much lower intensity.

(b) Comparative hydrograph sketch (no calculation required). Because the frozen, snow-free ground produces essentially 100% direct runoff, each storm's hydrograph shape is governed purely by how its duration compares with $t_c=60$ min. The 30-min storm ($t

Figure 1b — qualitative comparison: storm A (30 min) is short and sharp with the lowest peak; storm B (60 min) reaches the full equilibrium peak, the highest of the three; storm C (120 min) reaches the same equilibrium peak but holds it flat for the extra duration, giving the longest base and greatest volume.

Storm duration	Intensity	Depth
$t_c/2=30$ min	100 mm/hr	50 mm
$t_c=60$ min	50 mm/hr	50 mm
$2t_c=120$ min	30 mm/hr	60 mm

Concept & Theory

Topic. IDF curves, time of concentration, and the critical-duration effect on runoff hydrograph shape.

Key relations. Rainfall depth $P=i\,t$; a storm reaches watershed equilibrium (peak) discharge once $t\ge t_c$, since only then has runoff from every part of the watershed had time to reach the outlet simultaneously.

Why this works. An IDF curve encodes the universal inverse relationship between rainfall intensity and duration (intense bursts are short-lived; sustained rain is gentler). Time of concentration is the travel time from the hydraulically most remote point of the watershed to its outlet; a storm shorter than $t_c$ can never engage the full watershed area at once, capping its peak below the equilibrium value, while any storm $t\ge t_c$ reaches that same equilibrium peak — extra duration beyond $t_c$ only sustains the peak (adds volume), it does not raise it further.

Common pitfall. Assuming the longest storm always gives the highest peak — it gives the greatest volume and the longest-duration peak, but the peak magnitude itself is capped once $t=t_c$; conflating "biggest hydrograph" with "highest peak" is a common design error.

Source. Chow, Maidment & Mays, Applied Hydrology, Ch. 6 & 14 (IDF curves; time of concentration and design storms).

Question 1 of 6: Rainfall Intensity, Depth and Runoff Hydrographs for a Frozen Watershed

Question 1: Rainfall Intensity, Depth and Runoff Hydrographs for a Frozen Watershed (20 marks)