16-Civ-B4 Engineering Hydrology · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examination, May 2015, 98-Civ-B4 Engineering Hydrology, three hours’ duration, closed book with one two-sided candidate-prepared aid sheet (8½″ × 11″) and an approved Casio or Sharp calculator whose model must be declared in the work book. Seven problems are printed; page-1 Note 4 states that any five (5) questions constitute a complete paper and that only the first five answers appearing in the work book are marked. Each problem is weighted at twenty (20) points, so the examinable total is 5 × 20 = 100 points. The page-6 marking scheme gives the sub-part split for every problem. All seven problems are solved here, because this set is a study resource rather than a timed sitting; the sub-part mark values shown below are the printed ones.
Reference texts. V. T. Chow, D. R. Maidment and L. W. Mays, Applied Hydrology (hydrologic cycle, unit hydrographs, routing, frequency analysis, infiltration); L. W. Mays, Water Resources Engineering, 3rd ed. (design application, rainfall–runoff, reservoir operation); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (measurement, areal precipitation, energy budget); V. T. Chow, Open-Channel Hydraulics (1959) (flood-wave propagation, dam-break and gradually varied unsteady flow); C. W. Fetter, Applied Hydrogeology, 4th ed. (Darcy’s law, hydraulic conductivity, advective transport). Canadian practice references: Environment and Climate Change Canada / Water Survey of Canada HYDAT archive and the ECCC Engineering Climate Datasets (IDF curves and the IDF_CC climate-adjustment tool); ISO 1100 / WMO Manual on Stream Gauging (stage–discharge rating practice); and the Canadian Dam Association Dam Safety Guidelines (inflow design flood, dam-break inundation mapping).
Check — Problem 7(i) source data are internally inconsistent. As printed, the 10 000 km2 basin receives 50 mm of rain in a year while its river carries 200 m3/s, which is a runoff depth of 630.72 mm — about 12.6 times the stated rainfall. No basin can discharge more water than it receives, so the printed precipitation is in error (almost certainly a lost order of magnitude), not the discharge. Problem 7(i) below boxes the runoff depth first, demonstrates that the balance cannot close, and then adopts a declared corrected annual precipitation of 1000 mm/a to complete the estimate. Every number is flagged where the correction is used.
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.
Channel routing needs two relations: continuity, which conserves water, and a momentum or resistance relation, which says how fast the water moves for a given depth and slope. Manning’s equation supplies the second in its simplest useful form:
$$V = \frac{1}{n} R^{2/3} S^{1/2}, \qquad Q = VA = \frac{1}{n} A R^{2/3} S^{1/2}, \qquad R = \frac{A}{P}$$
where $n$ is Manning’s roughness, $R$ the hydraulic radius, $A$ the flow area, $P$ the wetted perimeter and $S$ the friction slope (taken as the bed slope $S_0$ in the kinematic approximation). Paired with the continuity equation in its distributed form,
$$\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = q_L$$
the two close the problem. The combination gives the kinematic wave model, in which the wave travels at the celerity
$$c_k = \frac{\mathrm{d}Q}{\mathrm{d}A} \approx \frac{5}{3}V \quad \text{(wide rectangular channel, Manning)}$$
and is translated downstream with little attenuation. In routing practice this appears in two guises. As the storage relation: Manning applied at successive sections gives the stage–discharge and stage–area relations from which reach storage is computed, letting a Muskingum or storage-indication routing proceed. And as the parameter estimator: in Muskingum–Cunge, $K$ and $X$ are not fitted to a record but computed from the channel itself,
$$K = \frac{\Delta x}{c_k}, \qquad X = \frac{1}{2}\left(1 - \frac{Q}{B\,S_0\,c_k\,\Delta x}\right)$$
which is what makes routing possible on an ungauged reach.
Worked illustration. For a trapezoidal channel with bed width $b = 20$ m, side slopes 2H:1V, normal depth $y = 2.5$ m, $n = 0.035$ and $S_0 = 0.0008$:
$$A = (b + zy)y = (20 + 5)(2.5) = 62.5\ \text{m}^2, \qquad P = b + 2y\sqrt{1+z^2} = 20 + 2(2.5)\sqrt{5} = 31.18\ \text{m}$$
so $R = 62.5/31.18 = 2.004$ m and
$$V = \frac{1}{0.035}(2.004)^{2/3}(0.0008)^{1/2} = 1.285\ \text{m/s}, \qquad Q = VA = \boxed{80.3\ \text{m}^3/\text{s}}$$
The kinematic celerity is $c_k \approx \tfrac{5}{3}(1.285) = 2.14$ m/s — the flood wave outruns the water — and the Froude number, with hydraulic depth $D = A/T = 62.5/30 = 2.083$ m, is $Fr = 1.285/\sqrt{9.81(2.083)} = 0.284$, comfortably subcritical, which confirms that downstream conditions can influence this reach.
Consideration 1 — the roughness coefficient dominates the answer and is the least certain input. Discharge is inversely proportional to $n$, so a value taken as 0.030 rather than 0.040 changes the computed flow by a third. Manning’s $n$ is not a constant for a reach: it varies with stage (a weedy or treed floodplain is far rougher than the main channel), with season (summer vegetation, winter ice cover, which also adds a second wetted perimeter), and with bed form. Where a high-water mark and a surveyed profile exist, $n$ should be back-calculated from that observed event rather than taken from a table; otherwise a range should be carried through and the sensitivity reported.
Consideration 2 — check that the kinematic assumption is actually valid for the reach. Using $S_f \approx S_0$ throws away the pressure and inertia terms of the full Saint-Venant momentum equation, which is defensible only on steep, well-defined channels where the wave translates with little attenuation. On mild slopes, in reaches with backwater from a downstream lake, tidal boundary, bridge or dam, in reaches with large floodplain storage, or where the wave is strongly attenuated, the kinematic model is wrong — it cannot attenuate a peak at all — and a diffusive (Muskingum–Cunge) or full dynamic model is needed. A quick screening criterion is the kinematic wave number; when in doubt, the diffusive form costs little and is far more robust.
Consideration 3 — cross-section geometry, floodplain engagement and the choice of $\Delta x$ and $\Delta t$. Manning is applied to a representative section, but real reaches vary; sections must be surveyed closely enough to capture contractions, expansions and the point at which flow leaves the bank. Once flow spills onto the floodplain, $A$, $P$ and $n$ change abruptly and the section must be treated as compound with separately weighted conveyances, otherwise the computed velocity is badly overestimated. Numerically, $\Delta x$ and $\Delta t$ must satisfy the Courant condition $c_k\,\Delta t/\Delta x \approx 1$ for accuracy, and the routing interval must resolve the rising limb. Steady uniform flow is also assumed at each section, so the method should not be pushed through hydraulic jumps, sharp bends or structures — those are handled with local energy relations, not Manning.
Only the method is asked for. The problem is a dam-break routing problem: convert a stored volume and a breach description into a downstream flood wave, route the wave to the protection site, extract the maximum water-surface elevation there, and add freeboard. The method below is the sequence a dam-safety consultant would follow under the Canadian Dam Association guidelines.
Step 1 — define the failure scenario and the breach. Establish the reservoir elevation–storage curve and the water level at failure (usually the full-supply level, or the level during the inflow design flood for a “flood-induced” failure). Then parameterise the breach, because the outflow depends far more on the breach than on anything else: final bottom width, side slopes, and formation time, taken from regression relations (Froehlich, MacDonald–Langridge-Monopolis) appropriate to the dam type. Both a sunny-day (piping) failure and a flood-induced overtopping failure should be run, since the second superimposes the breach outflow on an already-high river.
Step 2 — compute the breach outflow hydrograph. Treat the breach as a broad-crested weir whose crest falls as the breach develops, coupled to the reservoir drawdown by continuity:
$$Q_b = C\,L\,H^{3/2}, \qquad \frac{\mathrm{d}S}{\mathrm{d}t} = I - Q_b$$
integrated with the elevation–storage curve. The result is the peak outflow and the outflow hydrograph — typically a very sharp, short-duration wave whose volume is bounded by the reservoir storage.
Step 3 — obtain a first estimate with the analytical (Ritter) dam-break solution. For an instantaneous removal of a vertical barrier on a dry, horizontal, frictionless bed, the solution to the Saint-Venant equations gives the wave celerity, front speed and depth profile:
$$c_0 = \sqrt{g h_0}, \qquad u_{\text{front}} = 2c_0, \qquad h(x,t) = \frac{1}{9g}\left(2c_0 - \frac{x}{t}\right)^2, \qquad h_{\text{dam}} = \tfrac{4}{9}h_0$$
For $h_0 = 25$ m: $c_0 = \sqrt{9.81 \times 25} = 15.66$ m/s, the front advances at $u = 31.32$ m/s, and the depth held at the dam site is $\tfrac{4}{9}(25) = 11.11$ m. At a site 5 km downstream the front arrives at $t = 5000/31.32 = 160$ s, and at $t = 600$ s the depth there is
$$h = \frac{1}{9(9.81)}\left(31.32 - \frac{5000}{600}\right)^2 = \boxed{5.99\ \text{m}}$$
This is a screening number only — it neglects friction, valley storage and channel shape, all of which reduce the real depth — but it brackets the problem, gives the order of magnitude of the arrival time, and provides an independent check on the numerical model that follows.
Step 4 — route the breach hydrograph numerically to the site. Use a one-dimensional unsteady (full dynamic Saint-Venant) model — HEC-RAS unsteady flow or equivalent, or a 2D model where the valley is wide or the flow leaves the channel — built on surveyed and LiDAR-derived cross-sections, with Manning’s $n$ assigned by land cover and bridges, culverts and constrictions represented explicitly. Because the wave is steep, the model must handle mixed flow regimes and wetting of a dry bed. The output is a water-surface profile and a maximum stage at every section, including the protected site.
Step 5 — set the wall height and confirm it. Take the maximum computed water-surface elevation at the wall alignment, add superelevation on bends and any wave or debris allowance, then add freeboard for model uncertainty — typically 1–1.5 m for a dam-break application, more where the consequence classification is high. For the screening case above, $5.99 + 1.5 = 7.5$ m of wall above the local bed. Because the breach parameters are the dominant uncertainty, the whole chain must be re-run for a sensitivity envelope (breach width, formation time, roughness) and the wall set on the upper bound rather than the central estimate. Finally, check the wall as a structure — hydrostatic and hydrodynamic load, debris and impact, overtopping stability, foundation seepage and scour at the toe — and confirm that the design does not simply displace the flood onto a neighbouring property. The dam-break inundation mapping, warning system and emergency preparedness plan required by the CDA guidelines are produced from the same routing run.
