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.
The hydrologic cycle is a closed mass balance driven by solar energy and gravity: solar radiation lifts water from ocean and land surfaces into the atmosphere, and gravity returns it. An engineer cares about the cycle because every design quantity — a design flood, a firm yield, a well capacity, a culvert size — is a statement about one of the transfers between its stores. The four processes below are the ones that carry the largest fluxes and that a Canadian hydrologic study must quantify.
Precipitation (P). Precipitation is the only input to a land basin, so every other flux is bounded by it. It arrives as rain, snow, freezing rain or hail, and in Canada the snow fraction matters as much as the total: a winter’s precipitation is held in the snowpack for months and then released over a few weeks of melt, which is why the annual maximum flow on most Canadian rivers is a spring freshet rather than a rainstorm event. Precipitation is measured at points by tipping-bucket and weighing gauges and mapped over an area by radar; the engineer converts point depths to an areal depth before using them (Problem 3(i)).
Evaporation and transpiration (E, T). Evaporation from open water, soil and intercepted storage, together with transpiration through plant stomata, returns roughly 60–70 % of continental precipitation to the atmosphere. The two are lumped as evapotranspiration because they are driven by the same variables — available energy, vapour-pressure deficit, wind and moisture supply — and are almost impossible to separate in the field. ET is the term that makes a water balance close (Problem 7(i)) and the term that decides how much of a rainfall is actually available for runoff or recharge; it is usually obtained as a residual, from a Penman–Monteith energy balance, or from pan data with a pan coefficient.
Infiltration and groundwater flow (G). Infiltration is the entry of water through the soil surface; what infiltrates beyond the root zone becomes recharge and joins the saturated groundwater system, where it moves under Darcy’s law at speeds of metres per year to metres per day. Groundwater is important twice over: it is the store that sustains streamflow between storms (baseflow, and hence the low-flow reliability of a water supply), and it is the pathway by which surface contamination reaches wells. The infiltration rate also controls the partition at the surface — rainfall in excess of the infiltration capacity becomes runoff, so the same storm on a clay till and on a sand outwash produces entirely different hydrographs.
Surface runoff (R). Runoff is the water that reaches a channel over or just beneath the surface and is routed to the outlet. It is the flux the engineer measures most directly (a gauged discharge record) and the one that design is usually about: culverts, bridges, spillways, floodplain mapping and reservoir yield are all statements about the runoff time series. Runoff couples to the other three processes — it is what precipitation leaves after ET and infiltration have taken their share — which is why a change to the land surface (urbanisation, forest harvest, wildfire) alters flood peaks without altering the rainfall at all.
The distinction is a matter of what lies above the saturated zone. An unconfined (water-table) aquifer has its upper surface at atmospheric pressure, with a permeable unsaturated zone above it that connects it directly to the ground surface. A confined (artesian) aquifer is bounded above by an aquitard — clay, till, shale — and is under pressure greater than atmospheric, so water in a well rises above the top of the unit to the potentiometric surface, and may flow at ground level.
Source waters. An unconfined aquifer is recharged over its whole footprint by infiltration of local, recent precipitation, so its water is young — often months to a few years old — and its water level responds visibly to seasonal wet and dry cycles. A confined aquifer is recharged only where the permeable unit outcrops or subcrops beneath permeable cover, which may be tens of kilometres updip; its water is old (decades to millennia), chemically evolved through longer rock contact (higher dissolved solids, often higher iron, hardness or fluoride), and its potentiometric level responds to pumping and regional loading rather than to last week’s rain. Because recharge is remote, a confined aquifer’s sustainable yield is fixed by that distant recharge area and cannot be increased by local conservation.
Vulnerability to advective transport. Advection — transport of a dissolved contaminant at the average linear (seepage) velocity of the water — is the dominant transport mechanism in both units, but the access differs. In an unconfined aquifer a surface spill infiltrates directly into the saturated zone, so the aquifer is highly vulnerable; the only attenuation is the travel time and the sorption and degradation that occur in the unsaturated zone. The seepage velocity follows from Darcy’s law divided by effective porosity:
$$v = \frac{K\,i}{n_e}$$
where $K$ is hydraulic conductivity, $i = \mathrm{d}h/\mathrm{d}L$ the hydraulic gradient and $n_e$ the effective porosity. For a typical sand aquifer with $K = 25\ \text{m/d}$, $i = 0.004$ and $n_e = 0.22$,
$$v = \frac{25 \times 0.004}{0.22} = 0.455\ \text{m/d} \qquad\Rightarrow\qquad t = \frac{500\ \text{m}}{0.455\ \text{m/d}} = \boxed{1100\ \text{d} \approx 3.0\ \text{a}}$$
to reach a receptor 500 m down-gradient — short enough that a spill today is a wellhead problem within one election cycle. A confined aquifer at the same depth is protected by the aquitard: vertical movement across a clay layer with $K$ of order $10^{-9}$–$10^{-11}$ m/s takes decades to millennia, so the confined unit is far less vulnerable to a local surface release. Its vulnerability instead concentrates at three places — the outcrop recharge area, where it behaves as an unconfined aquifer; any breach of the aquitard by an improperly sealed or abandoned borehole, which short-circuits the protection entirely; and downward leakage induced by heavy pumping, which can reverse the natural upward gradient. Canadian wellhead-protection practice reflects this: capture-zone delineation and time-of-travel zones are computed from the advective velocity above, and decommissioning of abandoned wells is a regulated activity precisely because a single unsealed hole defeats a hundred metres of natural protection.
The figure printed with the question is the standard single-peaked storm hydrograph — discharge on the ordinate against time on the abscissa, with a steep rising limb, a crest, and a long recession. It is the basin’s transfer function made visible: the rainfall that fell over the basin is redistributed in time by the travel paths water takes to the outlet, and the shape of that redistribution is what lets an engineer predict discharge.
Prediction proceeds in four steps. First, separate baseflow. The measured discharge is the sum of direct runoff (the storm response) and baseflow (drainage from groundwater storage). A straight-line or constant-slope separation from the start of the rise to the inflection point on the recession splits the two; only the direct runoff scales with the storm.
Second, establish the basin’s response. The area under the direct-runoff hydrograph divided by the basin area is the depth of effective (runoff-producing) rainfall, and dividing the whole direct-runoff hydrograph by that depth yields the unit hydrograph — the response to one unit of effective rainfall in a fixed duration. Once derived from one or more observed storms, it characterises the basin.
Third, convolve. For a new storm, subtract losses to get an effective-rainfall hyetograph, then superimpose scaled and lagged copies of the unit hydrograph:
$$Q_n = \sum_{m=1}^{n} P_m\,U_{n-m+1} + Q_{\text{base}}$$
where $P_m$ is the effective rainfall in interval $m$ and $U_k$ the $k$-th unit-hydrograph ordinate. Adding the separated baseflow back gives the predicted river discharge as a full time series, not just a peak.
Fourth, use the timing. The lag $t_p$ between the rainfall centroid and the peak is roughly constant for a basin, so once the rain has fallen the hydrograph gives a real forecast lead time — hours on a small urban catchment, days on a large one. That lead time is what makes flood warning, reservoir pre-release and evacuation possible. The recession limb, which decays approximately as $Q_t = Q_0 K^{t}$, extends the prediction into the drawdown period and supports low-flow and water-supply forecasting.
