16-Civ-B4 Engineering Hydrology · May 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 16-Civ-B4 Engineering Hydrology, National Exams May 2017. Three hours, CLOSED BOOK with one two-sided candidate-prepared aid sheet and an approved Casio or Sharp calculator. Seven Problems are printed; any five constitute a complete paper and only the first five answers in the work book are marked. Each Problem is worth twenty (20) marks for a total of 100, and the page-1 Marking Scheme gives the sub-part split for all seven — Problems 1 and 7 at (6)(6)(8), Problem 2 at (10)(10), Problem 3 at (7)(5)(8), Problem 4 at (8)(6)(6), and Problems 5 and 6 at (7)(7)(6). Note 1 invites the candidate to state any assumptions made where a question is open to interpretation; this sitting needs that licence twice, and both places are flagged in the callout below. All seven Problems are worked here, because this set is a study resource rather than a timed sitting. Five of the seven are pure discussion; the numerical content sits in Problem 2(ii) and Problem 7(i), with short illustrative calculations added elsewhere so that each method is shown working on real numbers.
Reference texts. V. T. Chow, D. R. Maidment and L. W. Mays, Applied Hydrology (hydrologic cycle, unit hydrograph theory, Horton infiltration, level-pool and Muskingum routing, frequency analysis); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (areal precipitation, hydrograph analysis, conceptual watershed models); P. B. Bedient, W. C. Huber and B. E. Vieux, Hydrology and Floodplain Analysis, 5th ed. (rating curves, reservoir and river routing, urban design storms); R. S. Gupta, Hydrology and Hydraulic Systems, 4th ed. (groundwater recharge and discharge, streamflow measurement); L. W. Mays, Water Resources Engineering, 3rd ed. (Rational Method, IDF design practice). For the Canadian frame: Environment and Climate Change Canada IDF curve files, the Water Survey of Canada Hydrometric Manual (mid-section gauging to ISO 748), and the Transportation Association of Canada Drainage Manual for design-storm and runoff-coefficient practice.
Check — two source-data issues and one declared convention.
(1) Problem 7(i) cannot be solved as printed. A basin of 10 000 km² draining at 2200 m³/s sheds a runoff depth of 6937.92 mm in a year, which is 115.6 times the 60 mm of rain the question supplies. The water balance then returns a large negative evapotranspiration, which is physically impossible. The runoff depth follows from the area and the discharge alone and is not open to interpretation, so the printed precipitation is the term in error. The answer boxes the runoff depth from the printed data, demonstrates that the balance cannot close, and then adopts a declared corrected precipitation under Note 1. Two admissible repairs are carried through with numbers so the assumption is auditable.
(2) Problem 2(ii) gives the IDF relation without units on the intensity. The relation i = 7.0 − 0.2t is read here in mm/h, which is the Canadian convention for IDF work; the alternative in/h reading is carried through as a one-line sensitivity, and it produces a peak flow 25.4 times larger that no 10 ha suburban storm sewer would ever be sized for.
(3) Problems 1, 3, 4, 5 and 6 are discussion questions with no data of their own. Where a numerical illustration makes the method concrete, the input values are the solver's own representative figures and are labelled as such. Every number in those illustrations, and every number taken from the real source data.
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 deterministic hydrologic model produces exactly one output for a given input and parameter set: the same storm run twice through HEC-HMS or SWMM returns the same hydrograph to the last digit. A stochastic model treats one or more of its components as a random variable and produces a distribution of outputs — a synthetic streamflow generator, a Monte Carlo flood-frequency analysis, or a Markov chain of daily precipitation states.
Similarity 1 — both are calibrated and validated against the same observed record, and neither escapes the data. A deterministic rainfall-runoff model has its infiltration and routing parameters fitted to observed events; a stochastic model has its mean, variance, skew and serial correlation fitted to the same gauge record. Both are therefore only as good as the length and quality of that record, and both are extrapolating whenever they are used beyond it.
Similarity 2 — both must respect mass conservation and the same physical constraints. A synthetic trace that produces negative flows, or an annual runoff volume exceeding the basin's precipitation, is rejected on exactly the grounds that would reject a deterministic simulation. The physics is not suspended by making the model random; the randomness sits on top of a water balance.
Difference 1 — what the output means. The deterministic model answers what happens if this storm occurs; the stochastic model answers how often, and with what spread. A deterministic run yields a single peak flow, which must then be assigned a frequency by an argument outside the model. A stochastic run yields the frequency distribution directly, and the design value is read from it as a quantile.
Difference 2 — how they represent process detail versus variability. Deterministic models are usually process-based, resolving infiltration, overland flow and channel routing in space and time, and are correspondingly data-hungry and parameter-heavy. Stochastic models are usually statistical, resolving no physics at all but reproducing the temporal structure of the record with a handful of parameters, and they run thousands of realisations cheaply. Deterministic models can be transferred to an ungauged or a changed basin by physical reasoning; stochastic models generally cannot, because their parameters have no physical meaning outside the record that produced them.
When each is preferred. The deterministic model is preferred when the question is about a specific event and about the physical consequences of a physical change: sizing a culvert for the regional design storm, assessing the effect of a proposed subdivision on the downstream peak, real-time flood forecasting from an observed storm, or dam-break analysis. It is essential wherever the basin is about to change, since a statistical model fitted to the old regime has no standing. The stochastic model is preferred when the question is about frequency, long-term risk or persistence: reservoir yield and storage-reliability studies that need thousands of years of plausible inflows, drought sequencing, flood-frequency estimation and its confidence limits, and any economic optimisation over the distribution of outcomes rather than one outcome. In modern practice the two are combined rather than opposed — the stochastic generator supplies long synthetic input series, and the deterministic model transforms each realisation into the response of the engineered system, which is precisely the ensemble strategy described in Problem 1(iii).
Given. The method is illustrated below on a representative lake, using the solver's own values — a storage constant $K=12$ h with the reservoir weighting factor $X=0$, routed at $\Delta t=6$ h through an inflow hydrograph peaking at 190 m³/s. Find. The routing coefficients and the attenuated, lagged outflow peak.
The three key steps.
The two assumptions. First, storage is a single-valued linear function of the weighted discharge, with $K$ and $X$ constant for all magnitudes of flow. Real stage-storage and stage-discharge relations are curved, and $K$ in particular falls as discharge rises because the wave travels faster in deeper water, so the linear model is a local approximation calibrated near the flows of interest. Second, the water body is lumped and level: it is treated as a single storage element with one water-surface elevation, no lateral inflow within the element, and no backwater from downstream. That is an excellent assumption for a lake or a reservoir behind a dam, where the pool really is nearly horizontal, and a progressively worse one for a long, narrow, sloping impoundment, where a wedge of storage exists and a nonzero $X$ or a full dynamic routing is required.
| Quantity | Symbol | Value |
|---|---|---|
| Storage constant (illustrative) | K | 12 h |
| Weighting factor for level-pool storage | X | 0 |
| Routing coefficients | C₀, C₁, C₂ | 0.200, 0.200, 0.600 (sum 1.000) |
| Peak inflow | Imax | 190 m³/s at t = 18 h |
| Peak outflow | Omax | 141.5 m³/s at t = 30 h |
| Attenuation / lag | — | 25.5% / 12 h |
Factor 1 — the available storage above the operating level, expressed by the stage-storage relation. Attenuation is bought with storage: the peak is reduced by exactly the volume that the pool takes in between the moment inflow exceeds outflow and the moment they cross again. A wide, shallow lake gains a large surface area for a small rise and attenuates strongly; a narrow gorge reservoir of the same volume gains little area per metre and attenuates weakly, and a reservoir already full at the start of the flood attenuates almost nothing at all. How it enters design. It sets the flood-control allocation — the band of storage that operating rules require to be kept empty through the flood season — and, above that, it sets the surcharge storage between the full supply level and the maximum water level. That maximum water level, arrived at by routing the inflow design flood through the reservoir, is what fixes the dam crest elevation once freeboard for wind set-up and wave run-up is added. It is also what fixes the height of flood gates: gates must be tall enough to hold the full supply level and to be operable at the maximum water level, and the routing computation is the only way to establish that level.
Factor 2 — the outlet hydraulics, that is the stage-discharge relation of the spillway and low-level outlets, together with the operating rule that governs the gates. Outflow is not a free variable: it is dictated by the head on the outlets. An uncontrolled ogee spillway of crest length $L$ discharges $Q=C\,L\,H^{3/2}$, so a short crest forces a large head, a higher maximum water level and greater attenuation, while a long crest passes the flood at low head with little attenuation but a lower dam. With gates, the operator can pass flow before the crest arrives (pre-release) and so create storage, but a gate that fails to open converts the design case into an overtopping case. How it enters design. The spillway rating is chosen jointly with the crest elevation by iterating the routing computation until the maximum water level, the dam height and the cost balance. As an illustration, a spillway with $C=2.2$, crest length $L=25$ m and head $H=1.8$ m passes
$$Q=2.2\times 25\times 1.8^{3/2}=\boxed{132.8\ \text{m}^3/\text{s}}$$
so if the routed peak outflow were to exceed that value the head, and therefore the maximum water level, must rise. Canadian dam practice (the CDA Dam Safety Guidelines) then requires the inflow design flood to be selected from the dam's consequence classification, up to the probable maximum flood for high-consequence dams, and requires the routing to be repeated with the most adverse credible gate availability — typically the largest gate assumed stuck shut — because a flood-control system that depends on mechanical operation must be shown to work when part of it does not. Two further factors are worth naming even though the question asks for two: the volume and shape of the inflow hydrograph (a long-duration snowmelt flood of modest peak can fill a reservoir that easily absorbs a sharper rainfall flood of higher peak), and the antecedent reservoir level at the start of the event.
