Question 2 of 7

Problem 2 (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.

(i) Conceptual runoff models (7 marks)

Conceptual (reservoir-based) runoff models represent a watershed's storage–outflow behaviour with one or more idealized "buckets" that obey a continuity equation rather than an empirical coefficient. Three important properties: (1) they are built on an explicit storage–continuity relation, $\dfrac{dS}{dt}=I-O$, typically closed with a storage law such as $S=kQ^n$ (a linear reservoir, $n=1$, is the simplest and most common case); (2) because the storage term routes and attenuates inflow, they reproduce the full shape of the outflow hydrograph — rising limb, peak and recession — not just a single peak value, which purely empirical peak-flow formulas (e.g. the Rational Method) cannot do; and (3) their parameters (storage constants, number/arrangement of reservoirs or linear channels) are physically interpretable in terms of basin lag and attenuation, so they can be transferred, with judgment, to ungauged sub-basins with similar physiographic characteristics more defensibly than a purely statistical fit.

A widely used example is the Nash cascade / unit hydrograph via linear reservoirs (a series of $n$ identical linear reservoirs routing an inflow pulse), which is used to derive a synthetic unit hydrograph for an ungauged or sparsely gauged basin from measurable basin properties (drainage area, channel length, slope), and then convolved with a design rainfall hyetograph to produce a full design flood hydrograph for reservoir or culvert routing studies — something a single Rational-Method peak value cannot support.

(ii) Stage-discharge (rating curve) approach (7 marks)

A stage-discharge (rating) curve relates the continuously and cheaply measured water-surface elevation (stage, $h$) at a gauging station to discharge ($Q$), which is expensive and slow to measure directly (current-meter or ADCP gaugings). It is derived empirically: a series of discharge measurements is made across the full range of observed stages at a stable channel cross-section (a "control" — natural riffle, weir, or artificial control structure) and a smooth curve, typically of the power form $Q = a(h-h_0)^b$ ($h_0$ = stage of zero flow), is fitted through the gauged points; once established, the rating lets the continuously recorded stage record be converted into a continuous discharge (streamflow) record without further direct gauging, until the control changes.

Two key parameters affecting the confidence level of streamflow predicted from the rating over a 25-year period: (1) stability of the control section — a natural channel control (sand/gravel bed, vegetated banks) can shift due to scour, deposition, ice effects or channel migration, causing the rating to drift over time (rating "shift"), so long-term confidence depends on how often the rating is re-verified against new gaugings and whether shifts are tracked and corrected; and (2) extrapolation beyond the gauged range — the rating is only directly verified over the range of stages actually measured, and high flows (floods) are gauged rarely and with more uncertainty (or not at all, if they exceed the largest gauged event), so the upper end of a 25-year rating is typically an extrapolation whose confidence depends on the quality of the hydraulic/geometric model used to extend it (e.g. a surveyed channel cross-section and a slope-area or step-backwater computation) rather than direct measurement.

(iii) Flood-frequency curve fitting for the 40-year and 80-year flood (6 marks)

Approach. Annual instantaneous maximum flows are treated as a sample from an extreme-value population; a theoretical frequency distribution (commonly the Gumbel Extreme Value Type I, or Log-Pearson Type III per current Canadian practice) is fitted to the sample statistics, and the fitted distribution is then used to extrapolate the flow magnitude associated with a return period beyond the 12-year observed record.

1. Compute the sample statistics of the 12 annual maxima. Mean $\bar{X}=623\ \text{m}^3/\text{s}$; sample standard deviation $s=127\ \text{m}^3/\text{s}$.
2. Fit a Gumbel (EV1) distribution using the frequency-factor method. The design flow for return period $T$ is $X_T=\bar{X}+K_T\,s$, where the Gumbel frequency factor is $$K_T = \frac{y_T-\bar{y}_n}{S_n}, \qquad y_T = -\ln\!\left[\ln\!\left(\frac{T}{T-1}\right)\right]$$ using the asymptotic reduction constants $\bar{y}_n\approx0.577$, $S_n\approx1.2825$ (large-sample values; a 12-year record would more rigorously use the finite-sample tables, but the asymptotic constants illustrate the method).
3. Evaluate at $T=40$ and $T=80$ years. $$y_{40}=3.68,\ K_{40}=2.42 \;\Rightarrow\; X_{40}=623+2.42(127)=\boxed{931\ \text{m}^3/\text{s}}$$ $$y_{80}=4.38,\ K_{80}=2.96 \;\Rightarrow\; X_{80}=623+2.96(127)=\boxed{1000\ \text{m}^3/\text{s}}$$
4. Check the fit against a plotting-position curve. Ranking the record (Weibull plotting position $T_p=(n+1)/m$, $n=12$) puts the largest observed flow (830 m³/s, 1958) at $T_p=13/1\approx13$ years; a Gumbel probability plot of $X$ vs. $-\ln[-\ln(1-1/T_p)]$ through all 12 ranked points, extended by eye or by the fitted line to $T=40$ and $80$ years, is the graphical cross-check an examiner expects alongside the frequency-factor calculation.