Return Period and Flood Water Surface Elevation: 18-Env-B2 Water Resources, May 2016, Q4

Problem 4: Return Period and Flood Water Surface Elevation (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.

(a) Return Period and the 100-Year Flood

The return period (or recurrence interval), $T$, is the average time interval, in years, between occurrences of a flood equal to or greater than a given magnitude, computed as the reciprocal of the flood's annual exceedance probability: $T=1/P$. It is a statistical average derived from a flood-frequency analysis of the historical annual maximum flow series, not a literal, evenly spaced schedule of events.

The 1-in-100-year flood is therefore the flow magnitude that has a $P=1/100=1\%$ probability of being equalled or exceeded in any given year — it is entirely possible for two 100-year floods to occur in successive years, or for none to occur in 150 years, and the term does not mean "once every 100 years exactly." Because the annual probability is independent from year to year, the probability of experiencing at least one 100-year flood over a 30-year design/development life is $1-(1-0.01)^{30}\approx 26\%$, which is why the 100-year flood elevation (not a smaller, more frequent event) is the conventional regulatory benchmark for setting floodplain development limits.

(b) 100-Year Flood Water Surface Elevation

Given. A trapezoidal river channel section carrying the 100-year design flow:

Given data
Quantity	Symbol	Value
100-year design flow	$Q$	129 m³/s
Bottom width	$b$	12 m
Side slope (H:V)	$z$	2.5
Manning's roughness	$n$	0.026
River bed slope	$S$	0.01 (1%)
Bed elevation	$z_{bed}$	239.00 m

Find. The normal (uniform-flow) depth $y$ that conveys $Q=129$ m³/s, and the resulting 100-year flood water surface elevation.

Check: uniform (normal-depth) flow is assumed — the question gives only the local channel geometry, roughness, and bed slope, with no downstream control or backwater data, so normal depth via Manning's equation is the appropriate first-order design elevation.

Approach. Write the trapezoidal section's area, wetted perimeter, and hydraulic radius as functions of depth $y$, solve Manning's equation for the $y$ that conveys $Q$, then add that depth to the given bed elevation.

Trapezoidal section geometry.$$A(y)=by+zy^2, \qquad P(y)=b+2y\sqrt{1+z^2}, \qquad R(y)=\frac{A(y)}{P(y)}.$$
Solve Manning's equation for normal depth. $Q=\dfrac1n A R^{2/3}S^{1/2}$ has no closed form in $y$ for a trapezoid, so it is solved by iteration (bisection) on $y$ until the computed $Q$ matches 129 m³/s; this converges to$$y=\boxed{1.72\ \text{m}}.$$
Section properties at normal depth. $A=(12)(1.715)+(2.5)(1.715)^2=27.94$ m², $P=12+2(1.715)\sqrt{1+2.5^2}=21.24$ m, $R=A/P=1.315$ m, giving a check velocity$$V=\frac{Q}{A}=\frac{129}{27.94}=\boxed{4.62\ \text{m/s}}.$$
Flood water surface elevation. Adding the normal depth to the given bed elevation:$$\text{WSE}=239.00+1.72=\boxed{240.72\ \text{m}}.$$The 100-year flood water surface elevation, 240.72 m, sets the minimum floor/development elevation for structures in the downtown floodplain (typically with an added freeboard allowance per local floodplain policy).

Trapezoidal river cross-section at normal (100-year) flow: bed width 12 m, side slopes 2.5H:1V, normal depth 1.72 m above the 239.00 m bed elevation.

Quantity	Value
Normal depth, $y$	1.72 m
Flow area, $A$	27.94 m²
Hydraulic radius, $R$	1.315 m
Check velocity, $V$	4.62 m/s
100-yr flood WSE	240.72 m

Concept & Theory

Topic. Return period and flood-frequency terminology; normal depth in a trapezoidal open channel via Manning's equation.

Key relations. $T=1/P$ (return period); $P(\ge 1\ \text{event in}\ n\ \text{yr})=1-(1-1/T)^n$; $A=by+zy^2$, $P=b+2y\sqrt{1+z^2}$, $Q=\tfrac1n AR^{2/3}S^{1/2}$.

Why this works. A flood-frequency return period is a probabilistic statement about any single year, not a deterministic clock, which is why regulators standardize on a fixed annual-exceedance-probability event (the 1% / 100-year flood) as the floodplain-mapping benchmark; Manning's equation then converts that design discharge into a physical water-surface elevation once the channel geometry, roughness, and slope are known.

Common pitfall. Interpreting "100-year flood" as "happens once every 100 years, so we're safe until then" instead of a 1% chance in any given year; using the side-slope ratio backwards (2.5:1 means $z=2.5$ horizontal per unit vertical, not $z=1/2.5$).

Source. Linsley, Kohler & Paulhus, Hydrology for Engineers, ch. 6 (flood-frequency analysis, return period); Chow, Open-Channel Hydraulics, ch. 3 & 7 (Manning's equation, trapezoidal channels, floodplain design).

Question 4 of 6: Return Period and Flood Water Surface Elevation

Problem 4: Return Period and Flood Water Surface Elevation (20 marks)

(a) Return Period and the 100-Year Flood

(b) 100-Year Flood Water Surface Elevation