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.
Both methods answer the same question — what average depth fell over the basin, given depths at a handful of gauges — and both are area-weighted averages of the form $\bar{P} = \sum A_j P_j / \sum A_j$. They differ in how the weights are obtained and in what they assume about the rainfall field.
Difference 1 — how the weights are obtained: geometry versus interpretation. The Thiessen method connects adjacent gauges, draws the perpendicular bisector of each connector, and assigns to every gauge the polygon of points closer to it than to any other. The weight $A_j/A$ is a pure function of gauge positions, so the calculation is objective, reproducible by anyone, and easily automated in GIS — and the same weights can be reused for every storm as long as the gauge network does not change. Isohyetal analysis instead requires the analyst to draw contours of equal depth through the gauge data, and the weight for each isohyetal band is the band’s area; the mean of the two bounding isohyets is used as that band’s depth. The contours must be redrawn for every storm, and two competent analysts will draw them slightly differently, so the method is subjective and slower.
Difference 2 — what each assumes about the rainfall field, and hence accuracy. Thiessen assumes rainfall is uniform within each polygon and changes discontinuously at polygon boundaries, and it carries no information beyond the gauge values themselves. It is therefore the right default for flat terrain with a reasonable gauge density, and it is superior to a simple arithmetic mean whenever gauges are unevenly spaced. Isohyetal analysis assumes a smooth, continuous rainfall field and — crucially — lets the analyst incorporate outside knowledge: orographic effect in mountainous terrain, radar imagery, storm-cell tracks, and elevation–precipitation relationships. In mountainous British Columbia, where valley-bottom gauges systematically under-represent the depth falling at elevation, an isohyetal analysis guided by elevation gives a materially better areal mean than any purely geometric scheme. Its price is subjectivity, and with too few gauges it can be misleading precision.
For three gauges reading 62, 48 and 75 mm with Thiessen areas of 45, 30 and 25 km2 in a 100 km2 basin:
$$\bar{P} = \frac{45(62) + 30(48) + 25(75)}{100} = \frac{2790 + 1440 + 1875}{100} = \boxed{61.05\ \text{mm}}$$
against an arithmetic mean of 61.67 mm — the weighting has pulled the estimate toward the gauge that represents the largest share of the basin.
Discharge cannot be measured continuously, but water level can. A stream gauge therefore records stage $G$ — the water-surface elevation above a fixed local datum — continuously and cheaply with a float-and-counterweight in a stilling well, a pressure transducer, or a non-contact radar sensor. The rating curve is the calibrated relationship that converts that record to discharge, so that a 15-minute stage series becomes a 15-minute discharge series. It is almost always fitted as a power law:
$$Q = a\,(G - G_0)^{b}$$
where $G_0$ is the stage of zero flow (the elevation of the lowest point of the control section), and $a$ and $b$ are fitted constants; $b$ typically falls between about 1.5 and 2.5, reflecting the geometry of the control. Plotted on log–log axes the relation is a straight line, which is how the fit is usually made and checked.
How the rating is developed. The hydrometric technician makes repeated discharge measurements at the section over a range of flows, ideally including at least one high-flow gauging. In the velocity–area method (ISO 748 / WMO practice, and the Water Survey of Canada standard), the cross-section is divided into 20–30 verticals; at each vertical the depth is sounded and the velocity is measured with a current meter or ADCP at 0.6 of the depth (or averaged at 0.2 and 0.8 for deeper verticals), and the panel discharge is $q_i = v_i a_i$. Summing the panels gives $Q = \sum v_i a_i$ for that occasion, paired with the stage recorded at the same time. Each measurement is one point on the rating; several dozen points spanning low to high flow define the curve, which is fitted by least squares in log space and then applied to the continuous stage record.
For a rating $Q = 12.5\,(G - 0.35)^{2.15}$ and a recorded stage of 3.20 m:
$$Q = 12.5\,(3.20 - 0.35)^{2.15} = 12.5\,(2.85)^{2.15} = \boxed{118.8\ \text{m}^3/\text{s}}$$
Two cautions belong with every rating. It is only valid while the control — the riffle, weir or channel constriction that fixes the stage–discharge relation — is stable; scour, deposition, ice or heavy weed growth shifts the curve, so ratings are re-checked and periodically re-issued, and winter records on Canadian rivers are estimated separately under ice-affected conditions. And extrapolation beyond the highest gauging — unavoidable for flood flows, which are rarely measured — is the largest single source of error in a flood record; it is constrained using the slope–area method, a step-backwater model, or Manning’s equation at the section.
The hydrograph is produced by applying the rating curve to the continuous stage record. The sensor logs stage at, say, 15-minute intervals; each stage value is passed through $Q = a(G-G_0)^b$ (with the shift applicable to that period), producing a discharge at the same interval; plotting discharge against time gives the stream discharge hydrograph. In practice, the “other related watershed information” that must be combined with the rating is substantial: the applicable rating shift or ice-affected estimate for the period, the concurrent precipitation record used to check that the response is consistent with the storm, a baseflow separation to distinguish the storm response, upstream regulation or diversion records, and the drainage area, which converts discharge to a runoff depth or to a unit-area yield for comparison with other basins. Quality assurance — comparing computed volumes with rainfall, checking mass balance against upstream gauges — is part of generating a defensible hydrograph, not an optional extra.
Use for flood prediction. The hydrograph supports flood work in three distinct ways. Real-time forecasting: the rising limb at an upstream gauge is a direct forecast for a downstream community, because a flood wave takes a repeatable travel time to move between sections; the rate of rise and the observed peak upstream, routed downstream (Problem 5(i)), give both the expected level and the hours available to act. Provincial river-forecast centres in Canada run exactly this on the HYDAT real-time network, comparing the hydrograph against pre-set warning and evacuation trigger levels, and the crossing times give the operational lead time. Design-flood estimation: the annual maximum peak is extracted from the hydrograph for every year of record, and the resulting series is fitted with a frequency distribution to give the 100-year or 200-year flood used to size a spillway or map a floodplain (Problem 6). Volume and duration: reservoir routing, detention-pond sizing and dyke overtopping depend not on the peak alone but on the volume above a threshold and the time spent there, both of which are areas and intervals read directly off the hydrograph. The same record also supports the low-flow end — the 7Q10 statistic used for effluent-dilution permitting comes from the recession.
