16-Civ-B4 Engineering Hydrology · May 2018

Question 1 of 7: Runoff Hydrographs, Unit Hydrographs and Conceptual Models

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

Notes on this paper

Paper format. National Examination, May 2018, 16-Civ-B4 Engineering Hydrology, three hours’ duration, closed book with one two-sided candidate-prepared aid sheet (8½″ × 11″) and one approved Casio or Sharp calculator whose model designation must be written on the first inside left-hand sheet of 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 will be marked; Note 5 weights each problem at twenty (20) points, so the examinable total is 5 × 20 = 100 points. All seven problems are solved here, because this set is a study resource rather than a timed sitting. Sub-part mark values below are the printed ones from the page-6 marking scheme, which on this sitting is internally consistent — every problem’s sub-parts sum to twenty, and the marks printed in the page margins agree with the scheme.

Reference texts. V. T. Chow, D. R. Maidment and L. W. Mays, Applied Hydrology (hydrologic cycle, unit hydrographs, conceptual models, routing, frequency analysis); L. W. Mays, Water Resources Engineering, 3rd ed. (urban and highway drainage, stormwater management, reservoir operation); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (precipitation and streamflow measurement, hydrologic modelling); C. W. Fetter, Applied Hydrogeology, 4th ed. (Darcy’s law and aquifer hydraulics); V. T. Chow, Open-Channel Hydraulics (1959) (flood-wave propagation and unsteady flow). Canadian practice references: Environment and Climate Change Canada Engineering Climate Datasets (short-duration rainfall and IDF curves) and the Water Survey of Canada HYDAT archive; the WMO Manual on Stream Gauging (WMO-No. 1044) and ISO 1100-2 (stage–discharge ratings); the Transportation Association of Canada Guide to Bridge Hydraulics and provincial highway drainage manuals; provincial stormwater management planning and design manuals (e.g. Ontario MOECC 2003, British Columbia Stormwater Planning Guidebook); and the Canadian Dam Association Dam Safety Guidelines (inflow design flood and reservoir routing).

Check — which numbers come from the paper and which are the solver’s. This sitting supplies numerical data in only three places: Problem 2(iii) (the 7 ha suburban development, its 30-minute time of concentration and the IDF relation), Problem 5(iii) (the printed IDF chart, from which an intensity must be read graphically), and Problem 6(iii) (the 50-year return period and the 10-year exposure). Those three answers are computed from the paper’s own data. Every other number below appears inside a short illustrative example whose inputs are stated in an explicit Given line as the solver’s own representative Canadian values; they exist to make a discussion answer concrete and checkable, and they are not exam data. A candidate who assumed different but reasonable values and carried them through consistently would receive the same marks.

Question 1: Runoff Hydrographs, Unit Hydrographs and Conceptual Models (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.

1(i) — Schematic of a runoff hydrograph and its three key parts (6 marks)

A storm hydrograph is the record of discharge past a gauging section through the passage of a single rainfall event. It is best drawn beneath the hyetograph that produced it, because the shape of the discharge curve is the joint product of the rainfall input and the way the drainage basin stores and delays that input. The schematic below shows a single-peaked hydrograph with the three key parts labelled.

Typical single-peak storm hydrograph beneath its hyetograph. The rainfall excess (net rain) is the part of the hyetograph above the loss rate; it is the only rainfall that becomes direct runoff. The three key parts are the rising limb, the crest segment and the recession limb, separated from the baseflow by the separation line.

Part 1 — the rising limb (concentration curve). This is the segment from the start of direct runoff to the point of inflection preceding the peak. It is the portion of the curve most strongly controlled by the storm itself: its steepness is set by the intensity of the rainfall excess, and its duration is set by how long the basin takes to bring water from its remotest point to the outlet. Physically, the rising limb is the progressive contribution of larger and larger fractions of the drainage area — at first only the areas close to the channel contribute, and as time passes the more distant time–area zones deliver their water. A basin that is small, steep, compact, densely channelled or heavily urbanised concentrates its flow quickly and produces a short, steep rising limb; a long, flat, permeable, well-vegetated basin produces a slow, gentle one. The initial part of the limb is also flattened by the initial abstractions — interception, depression storage and the high early infiltration capacity — which must be satisfied before any surface runoff appears.

Part 2 — the crest segment (peak discharge). This is the flat-topped portion containing the maximum ordinate. The peak occurs when the greatest area of the basin is contributing simultaneously, which happens when the duration of the rainfall excess is at least equal to the time of concentration. Its magnitude therefore reflects the product of the average rainfall-excess intensity over the critical duration and the contributing area, and its timing reflects the basin lag, the interval between the centroid of the net rainfall and the peak. A storm cell that moves down-basin produces a higher and later peak than the same storm moving up-basin; a storm that only covers part of a large watershed produces a lower, broader crest. The crest is the design quantity for culverts, bridge openings and sewer capacity.

Part 3 — the recession limb (falling limb, depletion curve). This is the segment from the point of inflection to the end of direct runoff. Its most important physical property is that it is independent of the rainfall: by the time the inflection point is reached the rainfall excess has ceased and the water leaving the basin is water that was in storage — surface detention, channel storage, interflow and, later, groundwater. The recession is therefore a signature of the basin, not of the storm, and it is well represented by a depletion equation of exponential form, in which the recession constant is characteristic of the basin and of the source of the storage. The inflection point itself marks the moment when overland flow to the channel effectively stops. Below the direct-runoff hydrograph lies the baseflow, the groundwater contribution, separated by the dashed line; it rises slightly during the event because bank storage and recharge are increased, and the volume between the separation line and the hydrograph is the direct runoff volume, which when divided by the basin area gives the depth of rainfall excess.

1(ii) — Three assumptions of the unit hydrograph, and two empirical corrections (7 marks)

The unit hydrograph of a given duration is the direct-runoff hydrograph produced by one unit depth (in SI practice, 1 cm or 1 mm) of rainfall excess falling in that duration. Its use as a transfer function rests on three assumptions, all of which are strained when the watershed is large.

Assumption 1 — the rainfall excess is uniform in time over the unit duration. The method treats the whole depth of net rain as having fallen at a constant intensity throughout the duration. In a large watershed the design storm is long and real storms are strongly peaked within their duration, so the effective intensity at any instant is not the average, and the true peak discharge is under-estimated.

Assumption 2 — the rainfall excess is uniform in space over the whole watershed. Every unit of area is assumed to receive the same depth of net rain at the same time. This is the assumption that fails first as area grows: convective cells are a few kilometres across, so a storm over a basin of hundreds or thousands of square kilometres covers only part of it, and the parts it covers may differ greatly in antecedent moisture, soil and land cover.

Assumption 3 — the response is linear and time-invariant. Linearity means proportionality and superposition: two units of excess produce ordinates exactly twice as large, the base time is unchanged, and the responses of successive rainfall blocks may simply be added with the appropriate lags. Time invariance means that the same unit hydrograph applies whatever the season, antecedent condition or magnitude of the event. Real basins are mildly non-linear — large floods travel faster, because velocity increases with depth, so the observed unit hydrograph of a large event is peakier and shorter than that of a small one.

Empirical correction 1 — subdivide the watershed and route. The large basin is divided into sub-basins small enough that the uniform-rainfall assumptions are defensible, each is given its own unit hydrograph and its own rainfall hyetograph from the nearest gauges or from radar, and the resulting sub-basin hydrographs are translated to the common outlet by channel routing (Muskingum or Muskingum–Cunge) and added. This is exactly what a semi-distributed model such as HEC-HMS does, and it converts an inadmissible areal-uniformity assumption into a set of admissible ones.

Empirical correction 2 — areal reduction and design-storm shaping. Point rainfall from the gauge or IDF curve is multiplied by an areal reduction factor that decreases with catchment area and increases with duration, so that the design depth applied over a large basin is the areal average rather than the point maximum. In parallel, the uniform-intensity assumption is relaxed by distributing the design depth over the storm duration with an empirical temporal pattern (SCS Type II, the Chicago or AES design storm) and convolving a short-duration unit hydrograph — obtained from the S-curve — block by block, rather than applying a single long-duration unit hydrograph. Loss rates are similarly adjusted for antecedent conditions (the SCS AMC I/II/III adjustment of the curve number).

1(iii) — Storage-based conceptual models versus synthetic unit hydrographs (7 marks)

Both families produce a unit hydrograph for a basin that has no rainfall–runoff record, but they reach it from opposite directions. The Nash and Dooge models are conceptual: they replace the basin with an idealised hydraulic system and derive the response mathematically. Snyder’s method is empirical: it correlates the observed features of the unit hydrograph with measurable basin geometry through regionally fitted coefficients. Three key differences follow.

Difference 1 — conceptual basis. Nash represents the watershed as a cascade of identical linear reservoirs, each obeying a linear storage–discharge relation. Routing an instantaneous unit input through the cascade gives an instantaneous unit hydrograph of gamma form, and Dooge generalises this to alternating linear channels (pure translation) and linear reservoirs (pure storage), so that translation and attenuation are represented separately. Snyder makes no statement about how the basin stores water at all; he observed that in the Appalachian highlands the lag, peak and widths of the unit hydrograph correlate with the main-stream length and with the length to the point opposite the centroid, and he fitted coefficients to those correlations.

Difference 2 — the meaning of the parameters and how they are obtained. The Nash parameters, the number of reservoirs and the storage constant, are abstract; they cannot be measured in the field and are normally obtained by matching the first and second moments of an observed rainfall–runoff event, which means the model is not truly ungauged unless the parameters are regionalised. Snyder’s coefficients are regional constants transferred from gauged basins in the same hydrologic region to the ungauged basin of interest, and the basin-specific inputs are lengths and areas taken straight off a map or a digital elevation model. For a genuinely ungauged Canadian basin the Snyder route (or its Canadian equivalents) is the one that can actually be executed.

Difference 3 — completeness and form of the output. The conceptual models return a continuous analytic function defined for all time, which can be differentiated, convolved with any hyetograph, or converted to any duration without further assumptions, and whose unit volume is exact by construction. Snyder’s method returns only a handful of control points — the lag, the peak discharge, the base time and the widths at 50 and 75 per cent of the peak — through which the analyst must sketch a smooth curve and then adjust it until the area under it equals one unit of runoff. The shape between the control points is therefore the analyst’s, and two engineers can obtain visibly different unit hydrographs from the same Snyder parameters.

The two are worth seeing side by side numerically. Given. For the conceptual route, a Nash cascade fitted to a basin with three reservoirs and a storage constant of 2.0 h; for the empirical route, a basin with a main-stream length of 50 km, a length to the centroid of 25 km, an area of 1000 km², and regional Snyder coefficients of 1.5 and 0.60. Find. The time to peak of the instantaneous unit hydrograph from each method, and Snyder’s peak discharge.

Nash cascade: differentiate the gamma instantaneous unit hydrograph. With n reservoirs of storage constant K the instantaneous unit hydrograph is $$u(t)=\frac{1}{K\,\Gamma(n)}\left(\frac{t}{K}\right)^{\,n-1}e^{-t/K}$$ Setting the derivative to zero gives the peak at $t=(n-1)K$, so with three reservoirs and a two-hour storage constant the response peaks at $$\boxed{t_p=(n-1)K=(3-1)(2.0)=4.0\ \text{h}}$$
Snyder: lag from basin geometry. In SI units the standard lag is $$t_p=0.75\,C_t\,(L\,L_{ca})^{0.3}$$ with the lengths in kilometres and the lag in hours. Substituting, $$t_p=0.75(1.5)(50\times25)^{0.3}=9.555\ \text{h}$$
Snyder: peak discharge of the standard unit hydrograph. The companion relation gives the peak of the unit hydrograph for one centimetre of runoff, $$Q_p=\frac{2.78\,C_p\,A}{t_p}=\frac{2.78(0.60)(1000)}{9.555}=174.6\ \text{m}^3/\text{s}$$ so that $$\boxed{t_p=9.55\ \text{h},\quad Q_p=175\ \text{m}^3/\text{s}}$$

The contrast is the point of the comparison: the Nash result is a complete curve obtained from two fitted numbers, while the Snyder result is two numbers that still need a curve drawn through them, obtained without any discharge record at all.

Topic. Runoff hydrographs, the unit hydrograph and conceptual runoff models

Key relations. Hydrograph anatomy: rising limb, crest segment, recession limb and baseflow separation. Unit hydrograph transfer function and its linearity, time-invariance and uniformity assumptions. Nash cascade instantaneous unit hydrograph $u(t)=\frac{1}{K\Gamma(n)}(t/K)^{n-1}e^{-t/K}$ with $t_p=(n-1)K$; Snyder’s synthetic unit hydrograph $t_p=0.75C_t(LL_{ca})^{0.3}$ and $Q_p=2.78C_pA/t_p$; the convolution integral $Q(t)=\int_0^t I(\tau)u(t-\tau)\,d\tau$.

Why this works. The unit hydrograph works because a drainage basin, over the range of a single ordinary flood, behaves approximately as a linear time-invariant system: doubling the input doubles the output, and the response to a sequence of inputs is the sum of the responses to each. That is what licenses convolution, and convolution is what turns a single observed storm into a general-purpose design tool. The hydrograph’s shape encodes the two things a basin does to rainfall — it delays it (translation, seen in the rising limb and the lag) and it spreads it (storage, seen in the recession). Nash makes that explicit by representing the basin as pure storage in series, Dooge by alternating storage and translation elements; Snyder skips the mechanism and correlates the answer with the geometry that controls it. The recession limb is independent of the storm precisely because by then the input has stopped and only the emptying of storage remains, which is why the recession constant is a reusable basin property.

Common pitfall. Applying a unit hydrograph to total rainfall instead of rainfall excess. The unit hydrograph converts net rain into direct runoff only; losses must be removed first and baseflow added back afterwards. A second, subtler error is to assume that a long-duration unit hydrograph derived for a large basin may be applied to a design storm of a different duration without converting it through the S-curve — the base time and the peak both change with duration.

Source. Chow, Maidment and Mays, Applied Hydrology, Ch. 7 (unit hydrographs and the convolution equation) and Ch. 8; Viessman and Lewis, Introduction to Hydrology, 5th ed., Ch. 9 (hydrograph analysis, synthetic unit hydrographs); Mays, Water Resources Engineering, 3rd ed., Ch. 8.

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