16-Civ-B4 Engineering Hydrology · May 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
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 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.
The technique — Muskingum hydrologic routing. Channel routing takes a known hydrograph at an upstream section and computes the hydrograph at a downstream section. The Muskingum method does this with continuity plus a storage relation, and needs no cross-section geometry. Storage in the reach is written as the sum of a prism, the storage that would exist under a steady water surface at the outflow discharge, and a wedge, the extra volume held between the sloping unsteady surface and that steady surface while the wave is rising:
$$S=K\left[X\,I+(1-X)\,O\right]$$Combining this with the discretised continuity equation over a step $\Delta t$ gives the routing equation actually used,
$$O_2=C_0I_2+C_1I_1+C_2O_1$$with the coefficients
$$C_0=\frac{-KX+0.5\Delta t}{K-KX+0.5\Delta t},\quad C_1=\frac{KX+0.5\Delta t}{K-KX+0.5\Delta t},\quad C_2=\frac{K-KX-0.5\Delta t}{K-KX+0.5\Delta t}$$which must sum to unity. The two parameters are found either by calibration, plotting computed reach storage against the weighted discharge for an observed event and choosing the $X$ that collapses the hysteresis loop to a straight line whose slope is $K$, or — for an ungauged reach — from channel properties by the Muskingum–Cunge relations $K = \Delta x/c$ and $X = \tfrac{1}{2}\left(1-q/(S_0\,c\,\Delta x)\right)$, where $c$ is the kinematic wave celerity.
Element 1 predicted — peak attenuation. The routing returns the downstream peak discharge, which is smaller than the upstream peak because part of the flood volume is taken temporarily into reach storage and released later. This is the quantity that governs the design of downstream bridges, dykes and floodplain mapping, and it cannot be obtained by simply shifting the upstream hydrograph in time.
Element 2 predicted — the travel time and timing of the peak. The routing returns when the peak arrives downstream. This is what flood forecasting and warning services issue, what determines the lead time available for evacuation or for pre-releasing a reservoir, and what allows the contributions of several tributaries to be combined correctly at a confluence — two tributary peaks that arrive together produce a far larger main-stem flood than the same two arriving hours apart.
Coupled with a stage–discharge rating, the routed hydrograph also gives the downstream stage and hence the flood extent and the duration of inundation above a threshold, which is the third element usually wanted.
Assumption 1 — reach storage is a single-valued, linear function of the weighted inflow and outflow. The whole method rests on $S=K[XI+(1-X)O]$ with $K$ and $X$ constant for the reach and for the event. This carries three things at once: that the response is linear, so that doubling the flood doubles the storage; that it is time-invariant, so that the same two parameters serve on the rising and the falling limbs and from one event to the next; and that the prism-plus-wedge picture is an adequate representation of the reach, which requires a channel of reasonably uniform geometry and slope along its length. Real channels violate this as soon as the flood leaves the banks, because the storage available on a floodplain is out of all proportion to the channel prism and the celerity changes abruptly.
Assumption 2 — there is no lateral inflow and no backwater, and the wave is not strongly dynamic. The reach is treated as a closed conduit between the two sections: no significant tributary inflow, no abstraction, no bank-storage exchange, and no downstream control — a tide, a reservoir, a constriction, or a tributary in flood — propagating a backwater effect upstream, which would make the outflow depend on something other than the reach storage. Implicit in the linear storage relation is also that the inertial terms of the full Saint-Venant equations are unimportant, so the wave is kinematic or diffusive rather than dynamic; Muskingum is not appropriate for a dam-break wave, a very flat estuarine reach or a rapidly gated release. To these must be added the numerical requirement that the time step satisfy $2KX \le \Delta t \le K$, without which the coefficients turn negative and the computed hydrograph oscillates.
The significance of the storage time constant K. $K$ has the dimension of time and is the constant of proportionality between reach storage and weighted discharge, $K = \Delta S/\Delta\left[XI+(1-X)O\right]$. Physically it is the travel time of the flood wave through the reach — the lag between the centroid of the inflow hydrograph and the centroid of the outflow hydrograph — and in the Muskingum–Cunge form it is exactly the reach length divided by the kinematic wave celerity, $K = \Delta x/c$. It is therefore the parameter that controls timing: a long, flat, heavily vegetated reach with a large $K$ delivers its peak late, a short steep one delivers it early. The division of labour between the two parameters is worth stating clearly, because it is a standard examination point: $K$ sets the translation, while $X$ sets the attenuation. At $X = 0$ there is no wedge, the reach behaves as a linear reservoir and attenuation is greatest; at $X = 0.5$ the inflow and outflow are weighted equally, the storage loop closes on a line of zero width, and the hydrograph is translated without any attenuation at all. Natural river reaches fall between about 0.1 and 0.3.
Given. A reach with $K = 12$ h and $X = 0.20$, routed with $\Delta t = 6$ h; at the start of a step the inflow rises from 60 to 140 $\text{m}^3/\text{s}$ while the outflow stands at 60 $\text{m}^3/\text{s}$. Find. The routing coefficients, the stability check, and the outflow at the end of the step.
An event-based model simulates a single storm from an assumed initial condition and stops when the direct runoff has passed. What it can and cannot represent follows almost entirely from the time scales of the processes involved: rainfall varies over minutes, overland flow responds over minutes to hours, channel routing over hours to days, interflow over hours to days, groundwater recession over weeks to months, and evapotranspiration and snowpack accumulation over days to seasons. An event model resolves the first group and must parameterise the rest.
Aspect 1 — the fast runoff-generating processes, which are INCLUDED explicitly. Rainfall excess generation (Horton or Green–Ampt infiltration, or an SCS curve number), overland flow and its concentration into the channel network (a unit hydrograph or a kinematic-wave plane), and channel translation and storage (Muskingum or Muskingum–Cunge) are all represented at a time step of minutes and a space scale of the sub-basin. These are exactly the processes whose characteristic times are comparable with the storm duration and the basin time of concentration, so they are the processes that shape the flood wave, and an event model computes them directly. This is why event models predict peak and timing well.
Aspect 2 — the slow storage processes, which are NEGLECTED and replaced by an assumed initial condition. Evapotranspiration, soil-moisture accounting, deep percolation, groundwater recharge and recession, snowpack accumulation and interception storage all evolve over weeks to seasons. Over a storm of 12 to 48 hours they change too little to be worth simulating, so an event model collapses them into two assumed numbers: an antecedent moisture condition or initial abstraction, which fixes how much of the rain infiltrates, and a constant or simply-receding baseflow added at the end. That is the central weakness of event modelling, because the answer is strongly sensitive to an initial condition that the model has no means of computing — the same storm on a dry catchment and on a saturated one produces floods that differ by a factor of several. A continuous simulation model carries the soil moisture forward through the inter-storm periods and therefore determines the antecedent condition rather than assuming it.
Aspect 3 — the spatial and temporal structure of the storm and the basin, which is NEGLECTED or represented only crudely. Event models are normally driven by a synthetic design storm — an SCS Type II or AES distribution, or a Chicago hyetograph — applied uniformly over the whole basin. Three things are given up in doing so: the movement of the storm across the basin, which shifts and sharpens or flattens the peak; the limited areal extent of a convective cell, whose few kilometres of diameter are small compared with a basin of hundreds of square kilometres, so that an areal reduction factor must be applied as a crude correction; and the spatial variability of soils and land cover within each lumped sub-basin. Where hydrologic rather than hydraulic routing is used, dynamic backwater and floodplain storage are also dropped. Underlying all of this is the equal-return-period assumption — that the 100-year rainfall produces the 100-year flood — which is the practical price of event modelling and is avoided only by continuous simulation followed by a frequency analysis of the simulated flows.
The judgement to state in an examination answer is that this bargain is usually a good one. For design of a culvert or a storm sewer, where the peak of a single event is the deliverable and a defensible return period must be attached to it, event modelling is the appropriate tool; for water-supply yield, reservoir operation, water-quality loading, or any question that depends on the sequence of wet and dry periods, it is the wrong one.
