Question 5 of 7

Problem 5 (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 models of runoff (7 marks)

Three important properties of a conceptual runoff model: (1) lumped or semi-lumped storage representation — the catchment is represented as a small number of interconnected "storage" elements (interception, soil moisture, groundwater) with simplified routing between them, rather than resolving the full spatial detail of the watershed; (2) parameters are calibrated, not purely physically measured — each storage's capacity and routing coefficients are typically fitted to reproduce an observed rainfall–runoff/streamflow record rather than derived solely from field survey, so the model must be calibrated (and ideally validated on an independent period) before it is trusted for design; and (3) continuity is enforced at every storage and at every time step, so the model is mass-conservative even though it is a simplification of the true, spatially-distributed hydrologic process.

Two important simplifying assumptions: (1) spatial uniformity within each storage/sub-catchment — rainfall, soil properties and antecedent moisture are assumed uniform over each modelled unit, avoiding the need for fully distributed (grid-cell) input data; and (2) time-invariant (or simply parameterized) routing — the storage-outflow relationships (e.g. linear reservoir routing) are assumed to keep the same functional form storm after storm, avoiding the need to re-derive routing behaviour for every individual event.

(ii) Stage-discharge approach to streamflow prediction (7 marks)

Two important ways the stage-discharge (rating-curve) approach is used to predict streamflow: (1) converting a continuously recorded water-level record into a continuous discharge record — once a rating curve $Q=f(\text{stage})$ has been established from a set of concurrent stage/discharge gaugings, an inexpensive continuous stage recorder can be used to generate a full discharge hydrograph without having to gauge flow directly at every time step; and (2) extrapolating the rating curve to estimate flows for stages beyond the gauged range — e.g. extending the curve (using the channel's known geometry and slope, or a hydraulic model) to estimate the discharge of an extreme flood stage that was observed but never directly gauged, which is central to building a long enough peak-flow record for flood-frequency analysis.

Two key parameters that affect the confidence of the predicted streamflow, which must be considered to reduce liability: (1) channel/section stability (rating shift) — if the control section scours, deposits, or grows vegetation, the stage-discharge relationship itself shifts over time, so a rating curve must be periodically re-verified with new gaugings or the predicted flows silently drift from the truth; and (2) extrapolation beyond the gauged range — confidence drops sharply for stages well above or below the range that was actually gauged (especially above bankfull, where the channel's effective cross-section and roughness both change abruptly at the floodplain), so any rating-curve-based flow reported outside the gauged range should be flagged with a wider confidence band, not presented with the same precision as an interpolated value.

(iii) Fitting the 12-year annual flood series to estimate the 100-year flood (6 marks)

The general method is flood-frequency analysis: the series of $n$ annual instantaneous peak flows is treated as a sample from an underlying extreme-value probability distribution, and a theoretical distribution is fitted to the sample so that discharges corresponding to rare, unobserved return periods (like 100 years, roughly 8× longer than the 12-year record) can be estimated by extrapolation rather than by waiting for the event to actually occur. The standard steps are: (1) compute the sample mean $\bar{Q}$ and standard deviation $s$ of the annual maxima; (2) select and fit a probability distribution commonly used for flood peaks — the Gumbel (Extreme Value Type I) or Log-Pearson Type III distribution are the two most common in Canadian practice; (3) for the target return period $T$, compute the corresponding frequency factor $K_T$ from the fitted distribution (from tables or the distribution's reduced-variate formula, with a finite-sample correction for small $n$); and (4) estimate the design flood as $Q_T=\bar{Q}+K_T\,s$.

Illustrative Gumbel fit (not required by the question, shown to demonstrate the method). For this 12-year record, $\bar{Q}=636.7\ \text{m}^3/\text{s}$ and $s=131.3\ \text{m}^3/\text{s}$. Using the Gumbel reduced variate $y_T=-\ln[\ln(T/(T-1))]$ with the finite-sample correction for $n=12$ ($\overline{y}_n=0.5035$, $S_n=0.9833$): $$y_{100}=-\ln\!\left[\ln\!\left(\frac{100}{99}\right)\right]=4.600\qquad K_{100}=\frac{y_{100}-\overline{y}_n}{S_n}=\frac{4.600-0.5035}{0.9833}=4.166$$ $$Q_{100}=\bar{Q}+K_{100}\,s=636.7+4.166(131.3)=\boxed{1184\ \text{m}^3/\text{s}}$$ A 12-year record is short for a 100-year estimate (the extrapolation is roughly 8 record-lengths beyond the data), so this number carries wide confidence bounds and would normally be cross-checked against a second distribution (e.g. Log-Pearson III) and, where available, regional flood-frequency data from nearby gauged rivers.