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.
Both techniques answer the same question — what mean depth of rain fell on the basin, given a handful of point measurements — and both are area-weighted averages of the form $\bar P = \sum A_i P_i / \sum A_i$. They differ in what the weights represent and in how much the analyst is allowed to know.
Difference 1 — the weights are purely geometric in the Thiessen method and meteorological in the isohyetal method. Thiessen weighting is decided entirely by where the gauges are: perpendicular bisectors of the lines joining adjacent gauges partition the basin so that every point is assigned to its nearest gauge, and the weight of a gauge is the fraction of the basin that lies nearer to it than to any other. No judgement about the storm enters. The isohyetal method instead requires the analyst to draw contours of equal depth through the gauge values, using knowledge of the storm's structure, orographic effect and prevailing wind; the weights are then the areas of the bands between adjacent isohyets, each band being assigned the mean of the two isohyets bounding it. Consequently two analysts working the same data will always produce the same Thiessen result but may produce different isohyetal results.
Difference 2 — the Thiessen method is fixed for the network while the isohyetal method is redrawn for every storm, and only the isohyetal method can represent gradients between gauges. Thiessen weights depend only on gauge locations, so they are computed once and reused for every event until a gauge is added or removed — efficient, and well suited to routine monthly and annual summaries. But the method assumes the depth is uniform within each polygon and steps discontinuously at the boundaries, which is physically false and performs badly in mountainous terrain where depth varies strongly between gauges. The isohyetal method must be redrawn storm by storm, which is laborious, but it lets the analyst honour a known orographic gradient or a convective cell that sits between two gauges, and it is therefore the more accurate technique where the network is sparse relative to the variability of the rainfall. The arithmetic mean, the third classical technique, is a special case of Thiessen weighting with equal weights, and is acceptable only on flat basins with a dense, evenly distributed network.
A short illustration. For the four-gauge basin drawn above, with Thiessen areas of 25, 35, 20 and 20 km² and gauge depths of 42, 55, 38 and 61 mm, the weighted mean is
$$\bar P=\frac{25(42)+35(55)+20(38)+20(61)}{100}=\frac{4955}{100}=\boxed{49.55\ \text{mm}}$$
against an unweighted arithmetic mean of 49.0 mm. The difference is small here because the network is reasonably even; on a basin where one gauge commands half the area the two answers diverge sharply, and that divergence is the whole reason for weighting.
A design storm hydrograph for a chosen return period is built in three moves, and convolution is the third of them.
First the design storm itself is assembled at the required frequency. The 100-year IDF curve — in Canada, the Environment and Climate Change Canada IDF file for the nearest station — gives the 100-year depth for every duration; a temporal distribution such as the Chicago hyetograph, an SCS Type II curve or a local mass curve then arranges that depth into a sequence of blocks of the chosen computational interval $\Delta t$. Second, the losses are removed — by an infiltration model or a constant runoff coefficient — leaving a sequence of rainfall excess depths $P_1, P_2, \ldots, P_M$, one per interval.
Third, the unit hydrograph is applied to each block in turn. This is the adding and lagging procedure. The unit hydrograph $U_1, U_2, \ldots, U_N$ of duration $\Delta t$ is the response to one unit of excess; the response to a block of depth $P_m$ is therefore $P_m$ times the unit hydrograph, and because the block occurs $m-1$ intervals after the start it is lagged by $m-1$ intervals before being added to the others. Summing the lagged, scaled responses is exactly the discrete convolution
$$Q_n=\sum_{m=1}^{\min(n,M)} P_m\,U_{n-m+1}$$
with the resulting hydrograph running for $N+M-1$ intervals. The step rests entirely on the linearity and time-invariance assumptions of Problem 1(ii): proportionality licenses the scaling by $P_m$, and superposition licenses the addition of the lagged responses. Baseflow is added afterwards, since the unit hydrograph describes direct runoff only.
Illustration. With a one-hour unit hydrograph $U=[0,\ 12,\ 30,\ 18,\ 6]$ m³/s per cm and a two-block excess sequence $P=[1.5,\ 2.5]$ cm, the convolution gives ordinates $[0,\ 18,\ 75,\ 102,\ 54,\ 15]$ m³/s, so the design peak is
$$Q_{\max}=1.5(18)+2.5(30)=\boxed{102\ \text{m}^3/\text{s}}$$
occurring in the fourth hour — one interval later than the unit hydrograph's own peak, because the larger second block is lagged behind the first. Note that the frequency of the answer is inherited entirely from the rainfall: the hydrograph is called a 100-year hydrograph because the storm was a 100-year storm, and the equality of rainfall frequency and runoff frequency is itself an assumption, defensible for large events on a wet basin and weak for small events.
How the curve is generated. Discharge cannot be measured continuously; stage can. A rating curve is the empirical relation that converts the continuous stage record into a continuous discharge record. It is built by making a series of discharge measurements at the gauging section, each paired with the stage prevailing at the time, deliberately spread over as wide a range of flows as the hydrologist can catch — low flows in late summer, high flows during freshet, and any flood that can be safely gauged. Each measurement is a velocity-area computation: in the mid-section method used by the Water Survey of Canada and specified in ISO 748, the section is divided into 20 to 30 verticals, depth and mean velocity are measured in each, and the total is $Q=\sum a_i v_i$ over the verticals, with mean velocity taken at 0.6 of the depth in shallow water or averaged from readings at 0.2 and 0.8 of the depth in deeper water. The measured pairs are then fitted, conventionally with a power law
$$Q=a\,(h-h_0)^{\,b}$$
where $h_0$ is the stage of zero flow (the datum correction, usually the elevation of the control), and $a$ and $b$ are fitted by least squares on the logarithms, since taking logarithms makes the relation a straight line. For a section controlled by a broad-crested weir $b$ tends towards 1.5, and for a channel control governed by Manning's equation it tends towards about 1.7 — a fitted exponent far outside 1.4 to 2.5 is a signal that the control is compound or that the datum is wrong.
Two field gaugings on the same control, at $h = 1.20$ m giving 15.6 m³/s and $h = 2.60$ m giving 68.4 m³/s with $h_0 = 0.35$ m, fix the two constants:
$$b=\frac{\ln(68.4/15.6)}{\ln(2.25/0.85)}=1.518,\qquad a=\frac{15.6}{0.85^{1.518}}=19.97$$
and the rating then converts any recorded stage into a discharge. At a recorded stage of 2.10 m,
$$Q=19.97\,(2.10-0.35)^{1.518}=\boxed{46.7\ \text{m}^3/\text{s}}$$
Instrumentation. Stage is sensed by a float and counterweight in a stilling well, by a submerged pressure transducer, by a nitrogen bubbler, or increasingly by a non-contact radar or ultrasonic sensor mounted above the water; all are logged at 5 to 15 minute intervals and telemetered, and all are referenced to a staff gauge that is read manually on every site visit to detect drift in the sensor. Velocity for the gaugings is measured with a mechanical current meter (Price AA or pygmy), an electromagnetic meter, or an acoustic Doppler current profiler towed across the section from a bridge, cableway or boat — the situation shown in the source photograph, where the hydrographer is wading a shallow section beneath a bridge. Acoustic Doppler instruments have largely displaced mechanical meters for medium and large rivers because a complete traverse takes minutes rather than an hour, but they need care in shallow, weedy or heavily sediment-laden water. Index-velocity installations, where a side-looking acoustic meter measures velocity continuously and the rating is between index velocity and discharge, are used where a simple stage rating cannot work.
Predictability over time. This is the part of the question that matters most in practice, and the honest answer is that a rating curve is a perishable instrument. Four effects degrade it. (1) Shifting control. Scour and fill at the section, gravel movement, bar building, vegetation growth in summer and ice in winter all change the geometry, so the same stage corresponds to a different discharge; the record is corrected by shift adjustments derived from periodic check gaugings, and a station is normally regauged six to ten times a year for exactly this reason. (2) Hysteresis. On a mild-sloped river the water-surface slope is steeper on the rising limb than on the falling limb, so a passing flood wave produces more discharge at a given stage while rising than while falling — the rating becomes a loop rather than a line, and a single-valued curve can be in error by 10 to 20 per cent through a large flood. (3) Ice effects. On Canadian rivers a winter ice cover destroys the open-water rating entirely; those periods are published as estimated, derived from occasional under-ice gaugings and from the recession behaviour, and they carry much wider uncertainty than the open-water record. (4) Extrapolation. The rating is only evidence over the range actually gauged, and design work routinely needs discharges above the largest gauging ever made; extrapolating the power law, or extending it with a Manning or step-backwater computation of the section, is an unavoidable but clearly flagged approximation, and it is where the largest errors in flood-frequency work originate. The practical consequence for the engineer is that published discharge is a derived quantity, not a measurement: low flows are typically good to a few per cent, in-bank flows to 5 to 10 per cent, and extrapolated flood peaks perhaps to 25 per cent or worse, and those uncertainties should be carried into any frequency analysis built on the record.
