16-Civ-B4 Engineering Hydrology · May 2016

Question 3 of 7: Point and areal precipitation, and stream flow measurement

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. 98-Civ-B4 Engineering Hydrology, May 2016 — a three-hour closed-book examination; a candidate-prepared two-sided aid sheet and one approved Casio or Sharp calculator are permitted. The cover page states that “any five(5) questions constitute a complete paper” and that “each question is equally weighted at twenty (20) points”, so the seven printed Problems each carry 20 marks towards a 100-mark paper. The page-6 Marking Scheme confirms the sub-part split for all seven. 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 a callout below. All seven Problems are worked here, because this set is a study resource rather than a timed sitting.

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 areas, 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 1(iii) gives the IDF relation as i = 6.0 − 0.3 t_d without stating the units of i. The paper's own Problem 6(iii) figure plots rainfall on an axis labelled “Rainfall and Infiltration, mm/h” with a peak near 13, so mm/h is adopted and the alternative reading is carried through in the answer as a sensitivity.

(2) Problem 7(i) as printed cannot be satisfied: the stated area and river discharge fix the runoff depth at 5045.76 mm/a, which is 63 times the 80 mm/a of rain the question supplies, so the residual evapotranspiration comes out large and negative. The answer boxes the runoff depth, demonstrates that the balance cannot close, and then adopts a declared corrected precipitation. This is the response Note 1 asks for.

(3) Problems 3(iii), 5(iii), 6(i) and 7(iii) ask for method, not arithmetic; each is worked on a small dataset that is the solver's own representative example, clearly labelled as illustrative. Every number in those examples, and every number taken from the real source data.

Question 3: Point and areal precipitation, and stream flow measurement (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) Arithmetic average versus the isohyetal method (6 marks)

Both techniques convert a set of point gauge depths into one average depth over the watershed, but they weight the gauges differently and they use different information.

Difference 1 — weighting. The arithmetic average gives every gauge inside the watershed the same weight, so the estimate is simply $\bar{P} = \frac{1}{n}\sum_{j=1}^{n} P_j$. The isohyetal method weights by the area each depth band occupies: $\bar{P} = \frac{1}{A}\sum_{j} \bar{P}_j\,A_j$, where $\bar{P}_j$ is the mean of the two isohyets bounding band j and $A_j$ is the area of that band. Consequently a cluster of gauges in one corner of the basin biases the arithmetic average towards that corner's depth, while in the isohyetal method the same cluster only refines the position of the contours.

Difference 2 — the information used, and the judgement required. The arithmetic average uses only the gauge depths, and can be computed by anyone in a minute; it makes no use of topography, storm track or gauges just outside the boundary. The isohyetal method requires the analyst to draw lines of equal depth, and in doing so to bring in orographic knowledge, the storm's direction of travel, and depths recorded at gauges beyond the watershed boundary. It is therefore the most accurate of the standard methods in mountainous terrain — which is the normal case in British Columbia — but it is subjective and not reproducible between analysts, and it must be redrawn for every storm rather than computed once.

Figure 3.1 — Left: the arithmetic average gives every gauge equal weight. Right: the isohyetal method interpolates lines of equal depth first and then weights each band by its area, so it can honour orographic gradients that the gauges only sample.

In practice the arithmetic average is acceptable when the gauge network is dense and uniformly distributed and the terrain is flat, and the isohyetal method is preferred when either condition fails; the Thiessen polygon method sits between them, weighting by area of nearest influence without requiring contours to be drawn.

(ii) Generating real-time hydrographs from a rating curve (6 marks)

Discharge cannot be measured continuously, but stage can. A real-time hydrograph is therefore produced by measuring stage continuously and converting it to discharge through a previously established stage-discharge relation — the rating curve.

The rating curve is built by current metering: on many occasions spread over several years and over as wide a range of flows as can be caught, a gauging crew measures discharge directly (Problem 3(iii)) and records the stage at the same moment. Each pairing gives one point $(h_i, Q_i)$. A relation of the power form $$Q = a\,(h - h_0)^{b}$$ is then fitted, where $h_0$ is the stage of zero flow at the control. Plotted on logarithmic axes this becomes a straight line, which makes both the fit and the identification of a change in control straightforward; a compound channel usually needs two segments, one below and one above the flood-plain break.

In operation, a pressure transducer, float or radar sensor in the stilling well records stage at a fixed interval — five or fifteen minutes on a Water Survey of Canada station — and the logger transmits it by satellite or cellular telemetry. The rating equation resident in the station software or at the data centre converts each stage reading to a discharge, and the resulting time series is the real-time hydrograph published on the Water Survey's web service within minutes of the observation. This is the mechanism by which a flood forecast centre sees a tributary rise while the flood wave is still upstream of the town.

Three caveats keep the method honest. The rating is only valid while the control section is stable: scour, deposition, ice or weed growth shifts the relation, so the crew continues to make check gaugings and applies a shift correction or re-rates the station. Second, readings above the highest gauged flow are extrapolations, and the largest floods on record are almost always extrapolated — which is why published discharge for extreme events carries a wide uncertainty. Third, during a rapidly rising flood wave the water-surface slope is steeper on the rise than on the fall, so the true relation is a loop rather than a single curve; where that hysteresis matters, a slope-stage-discharge or index-velocity rating is used instead.

(iii) Three key components of the mid-section method (8 marks)

Given. The mid-section method computes discharge as the sum of contributions from vertical measurement sections across a channel; the illustrative gauging below is the solver's own representative dataset, used to show the three components at work.

Table 3.1 — Illustrative mid-section gauging (solver's own example data)
Vertical	Distance from left water edge, x (m)	Depth d (m)	Mean velocity v (m/s)	Panel width b (m)	Panel discharge q = v d b (m³/s)
1	1.0	0.45	0.35	1.50	0.236
2	3.0	0.95	0.62	2.00	1.178
3	5.0	1.30	0.85	2.00	2.210
4	7.0	1.55	0.95	2.00	2.945
5	9.0	1.40	0.88	2.00	2.464
6	11.0	0.85	0.55	2.00	0.935
7	13.0	0.40	0.28	1.50	0.168
Water edges at x = 0 m and x = 14 m					Σ = 10.136

Find. The three components that make the method accurate, demonstrated by computing the total discharge from the tabulated gauging.

Figure 3.2 — Mid-section layout. Each vertical stands at the centre of its own panel, whose width runs from the mid-point to the vertical on either side; the water edges close the first and last panels.

Approach. Identify the three components — sectioning of the cross-section, point-velocity measurement and vertical averaging, and summation of the panel discharges — then apply them to the tabulated data.

Component 1: sectioning the cross-section into panels. A tag line is stretched across the channel at a straight, uniform reach, and 20 to 25 verticals are located so that no single panel carries more than about 5 to 10 per cent of the total discharge (ISO 748 and the Water Survey of Canada Hydrometric Manual). Each vertical is assigned a panel width running from the mid-point to the vertical on each side: $$b_i = \frac{x_{i+1} - x_{i-1}}{2}$$ For vertical 2 this gives $b_2 = (5.0 - 1.0)/2 = 2.00$ m, and for the edge vertical 1, with the water edge at x = 0, $b_1 = (3.0 - 0)/2 = 1.50$ m. This is what distinguishes the mid-section method from the mean-section method, which instead computes each panel from the average of the two bounding verticals; mid-section needs no assumption about how depth varies between verticals, at the cost of ignoring the two small triangles at the water edges.
Component 2: measuring the point velocities and averaging them over the vertical. Depth is sounded at each vertical and a current meter (propeller, electromagnetic, or an acoustic Doppler profiler) is held at prescribed fractions of the depth. For depths above about 0.75 m the two-point method is used, averaging readings at 0.2 and 0.8 of the depth below the surface: $$\bar{v} = \tfrac{1}{2}\,(v_{0.2} + v_{0.8})$$ For shallower verticals the single-point 0.6-depth reading is taken, since $v_{0.6} \approx \bar{v}$ for a logarithmic velocity profile. Each reading is timed over at least 40 seconds so that turbulence averages out. Vertical 4, sounded at 1.55 m, would take the two-point average and returns $\bar{v} = 0.95$ m/s.
Component 3: summing the panel discharges. Each panel is treated as a rectangle of area $d_i b_i$ carrying the mean velocity of its vertical, and the total is the sum: $$Q = \sum_{i=1}^{n} \bar{v}_i\,d_i\,b_i = 0.236 + 1.178 + 2.210 + 2.945 + 2.464 + 0.935 + 0.168$$ $$\boxed{Q = 10.14\ \text{m}^3\text{/s}}$$ The same components give the cross-sectional area $A = \sum d_i b_i = 13.375$ m² and hence the section-mean velocity $\bar{V} = Q/A = 0.758$ m/s, which is the number carried to the rating curve of Problem 3(ii).
Check the panel distribution. The largest panel (vertical 4) carries $2.945/10.136 = 29$ per cent of the total, far above the 10 per cent guideline, so this seven-vertical layout is adequate only as a demonstration; a defensible field gauging of this channel would use 20 or more verticals at roughly 0.7 m spacing. Stating that limit is part of the answer: the accuracy of the method is set by the sectioning, not by the current meter.

Final results — Problem 3(iii) illustrative gauging
Quantity	Value
Number of verticals	7 (demonstration only; 20+ required in practice)
Cross-sectional area, A	13.375 m²
Total discharge, Q	10.14 m³/s
Section-mean velocity, V	0.758 m/s
Largest single panel share	29 per cent (guideline: below 10 per cent)

Topic. Areal averaging of point precipitation, stage-discharge rating, and velocity-area discharge measurement.

Key relations. Arithmetic mean $\bar{P} = \frac{1}{n}\sum P_j$; isohyetal $\bar{P} = \frac{1}{A}\sum \bar{P}_j A_j$; rating curve $Q = a(h - h_0)^b$; mid-section $Q = \sum \bar{v}_i d_i b_i$ with $b_i = (x_{i+1} - x_{i-1})/2$; two-point velocity $\bar{v} = (v_{0.2} + v_{0.8})/2$.

Why this works. All three parts are the same idea applied to different variables: an integral over an area or a cross-section is approximated by a weighted sum of point samples, and the accuracy is controlled by how the weights are assigned rather than by the precision of the individual samples. The isohyetal method beats the arithmetic average because its weights reflect the spatial structure of the storm; the mid-section method works because the panel width places each measured velocity at the centroid of the area it represents; the rating curve works because a stable channel control makes discharge a single-valued function of depth, so one variable can stand in for the other.

Common pitfall. In the mid-section method, computing the first and last panel widths as if the edge verticals had neighbours at zero distance, or averaging the depths of adjacent verticals (that is the mean-section method, and mixing the two double-counts area). In the rating curve, forgetting that discharge above the highest gauged stage is an extrapolation, and quoting it to three significant figures.

Source. Viessman and Lewis, Introduction to Hydrology, Ch. 2 (precipitation averaging) and Ch. 3 (streamflow measurement); Gupta, Hydrology and Hydraulic Systems, Ch. 4; Water Survey of Canada Hydrometric Manual and ISO 748 (velocity-area gauging); Bedient, Huber and Vieux, Ch. 2 (rating curves).

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