16-Civ-B4 Engineering Hydrology · December 2013

Question 1 of 7: Precipitation, Groundwater and Surface Runoff in a Rural Watershed

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

Notes on this paper

Paper format. National Examinations, December 2013 — 98-Civ-B4 Engineering Hydrology. Three hours, closed book, one candidate-prepared two-sided 8½″ × 11″ aid sheet, and one approved Casio or Sharp calculator whose model must be declared. 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 workbook are marked. Each problem carries twenty (20) points, so the examinable total is 5 × 20 = 100 points. The page-6 marking scheme breaks each problem into its sub-parts. 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, recharge). 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); the ISO 1100 / WMO Manual on Stream Gauging series as adopted by the Water Survey of Canada; the Canadian Dam Association Dam Safety Guidelines (inflow design flood and dam-break consequence classification); and the Transportation Association of Canada Guide to Bridge Hydraulics, 2nd ed.

Check — conventions used throughout this paper. A hydrologic year is taken as 365 days = 31 536 000 s unless a question says otherwise. Water density is 1000 kg/m³, gravitational acceleration is 9.81 m/s², and the latent heat of vaporisation of water is 2.45 MJ/kg at 20 °C. Several sub-parts ask for an explanation with an example rather than for the solution of stated data; in those cases a realistic Canadian data set is declared at the point of use and every number arising from it. Where the printed data are internally inconsistent — and Question 6(i) is such a case — the inconsistency is demonstrated arithmetically, the governing conservation requirement is stated, and the corrected reading actually used is declared, as page-1 Note 1 invites (“the candidate is urged to submit… a clear statement of any assumptions made”).

Question 1: Precipitation, Groundwater and Surface Runoff in a Rural Watershed (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) Three relationships among precipitation, groundwater and surface runoff (10 marks)

Within a large rural watershed the arriving precipitation depth is not a runoff depth: it is a supply that the land surface partitions. What decides the partition is the soil and the landscape — the infiltration capacity of the surface horizon, the storage available beneath it, and the slope and roughness that control how long water is given to infiltrate. The three relationships below are the ones an engineer actually uses, and they are ordered the way water encounters them.

Figure 1.1 — A rural watershed in cross-section. The same rainfall produces almost no overland flow on the deep permeable till of the forested upper slopes and a large overland-flow fraction on the clay flat and the thin soil over bedrock; the infiltrated water reappears at the stream days to months later as baseflow.

Relationship 1 — precipitation intensity against infiltration capacity fixes how much rainfall becomes runoff (the Hortonian relation). Overland flow is generated only while the rainfall intensity i exceeds the instantaneous infiltration capacity f, and that capacity decays as the surface horizon wets up. Horton’s expression captures both facts,

$$f(t) = f_c + (f_0 - f_c)\,e^{-kt}, \qquad q_{\text{overland}} = \max(i - f(t),\,0)$$

where $f_0$ is the dry-soil capacity, $f_c$ the saturated (final) capacity and k a soil decay constant. The soil type enters entirely through those three parameters: a forested sandy till may hold $f_c \approx 15$ mm/h and shed almost nothing under a 10 mm/h rain, whereas a compacted clay pasture with $f_c \approx 2$ mm/h sheds most of the same storm. This is why two neighbouring sub-catchments of equal area and equal rainfall can deliver peaks that differ by an order of magnitude.

Relationship 2 — the antecedent-moisture / storage relation, expressed operationally by the SCS curve number. Over a whole storm the split between retention and runoff is controlled not by intensity but by how much storage the soil profile still has. The SCS–CN method writes that remaining storage as S and gives the storm runoff depth directly:

$$S = \frac{25400}{CN} - 254 \ \text{(mm)}, \qquad Q = \frac{(P - 0.2S)^2}{P + 0.8S} \quad \text{for } P > 0.2S$$

Example. Take a mixed rural watershed of pasture and row crop on a group-C soil, $CN = 75$, receiving a storm of $P = 60$ mm. Then $S = 25400/75 - 254 = 84.67$ mm, the initial abstraction is $I_a = 0.2S = 16.93$ mm, and

$$\boxed{Q = \frac{(60 - 16.93)^2}{(60 - 16.93) + 84.67} = \frac{1855.0}{127.73} = 14.52\ \text{mm}}$$

so only about a quarter of the storm leaves as direct runoff and the remaining 45 mm is retained in the profile, available to evapotranspiration and to recharge. Raise the curve number to 90 for the same landscape after it has been drained and cropped, and the runoff depth more than doubles — the curve number is the soil-and-landscape descriptor.

Relationship 3 — the infiltrated water returns to the stream as baseflow, and Darcy’s law fixes the rate. Recharge that reaches the water table moves toward the channel under the water-table gradient:

$$Q_g = K\,b\,w\,\frac{dH}{dL}$$

with K the hydraulic conductivity, b the saturated thickness, w the width of the flow section and $dH/dL$ the water-table slope. For a valley-bottom aquifer with $K = 2$ m/d, $b = 20$ m, $w = 4000$ m and a gradient of 0.003, $Q_g = 2 \times 20 \times 4000 \times 0.003 = 480$ m³/d. The number is small beside a storm peak, but it is the flow that is still there in February and in a August drought, and it is what sustains fish habitat and rural wells. The engineering consequence is that the three quantities are not independent: any landscape change that raises the runoff coefficient (tile drainage, land clearing, pavement) simultaneously raises the flood peak and lowers the recharge that produces baseflow, so the stream becomes both flashier and drier.

(ii) Three hydrological issues over the 100-year design life of a hydroelectric dam (10 marks)

Issue 1 — the inflow design flood and the safe passage of extreme events. The spillway of a hydroelectric dam is not sized on the flow that generates power; it is sized on the flood that must pass without overtopping the embankment. Canadian Dam Association practice ties the inflow design flood to the dam’s consequence classification, running from a 100-year event for a very low consequence structure up to the probable maximum flood for a very high consequence one. Over a hundred years the exceedance risk of any given design event is substantial: for a $T = 1000$-year spillway design flood the probability of at least one exceedance in 100 years is $1 - (1 - 1/1000)^{100} = 9.5\%$. The hydrology work is therefore a flood-frequency analysis on the regional record, a PMP/PMF study where the classification demands it, and a freeboard allowance for wind setup and wave run-up on the reservoir.

Issue 2 — reservoir sedimentation and the progressive loss of live storage. A reservoir is a sediment trap, and the storage it loses is never recovered cheaply. If the contributing basin of 5000 km² yields 200 t/km² per year and the deposited material has a bulk density of 1.3 t/m³, the annual trapped volume is

$$V_s = \frac{200 \times 5000}{1.3} = 7.692 \times 10^{5}\ \text{m}^3/\text{a}$$

which over the 100-year life accumulates to $7.692 \times 10^{7}$ m³, or 3.85 percent of a 2.0 × 10⁹ m³ reservoir. That headline figure is reassuring and misleading: the deposition is concentrated as a delta at the head of the reservoir and as fines against the dam, so the intake and the low-level outlet silt long before the gross figure matters. The design response is a sediment-yield study, a sluicing or flushing capability at the low-level outlet, an intake set above the projected deposition profile, and a downstream sediment-continuity assessment, because a starved river erodes its bed and its banks.

Issue 3 — non-stationarity of the hydrologic record over the design life. Frequency analysis assumes the annual maxima are drawn from one unchanging distribution. Over a century that assumption fails on two counts: land use in the basin changes, and the climate changes. Warming shifts a snowmelt-dominated Canadian basin toward earlier and smaller freshets with more rain-on-snow winter peaks, which alters both the energy revenue and the seasonal position of the design flood. The engineering response is to design for a range of inflow scenarios rather than one, to keep the operating rule curve adaptable, to use the ECCC IDF_CC-style climate-adjusted rainfall inputs where short-duration rainfall governs, and to build a monitoring programme (basin precipitation, snow surveys, inflow gauging) whose data are re-analysed on a stated cycle so the flood estimates can be revised while the structure is still young.

Topic. Rainfall partitioning at the land surface, groundwater–streamflow coupling, and the long-horizon hydrologic risks of storage projects.

Key relations. Horton infiltration $f(t) = f_c + (f_0-f_c)e^{-kt}$; SCS runoff $Q = (P-0.2S)^2/(P+0.8S)$ with $S = 25400/CN - 254$ mm; Darcy $Q_g = KbW(dH/dL)$; risk of exceedance in n years $= 1-(1-1/T)^n$.

Why this works. Every one of these relations is a statement about a rate meeting a capacity. Horton compares a supply rate with an instantaneous capacity and so predicts the flash response; the curve number compares an event depth with a remaining storage and so predicts the event volume; Darcy compares a hydraulic gradient with a conductivity and so predicts the slow return of that stored water to the channel. Soil and landscape enter all three through a single physical property — the pore space available and how readily water moves through it — which is why one field observation (soil group, depth to bedrock, land cover) constrains flood peak, storm volume and dry-weather flow at once.

Common pitfall. Treating the runoff coefficient or curve number as a fixed property of the catchment. It is a property of the catchment in a given antecedent condition; the same watershed can run at CN 65 in July and CN 88 on frozen ground in March, and a design that ignores the seasonal state under-predicts the spring peak. A second pitfall is applying the SCS equation when $P \le 0.2S$, where it returns a meaningless value — the runoff is simply zero.

Source. Chow, Maidment & Mays, Applied Hydrology, Ch. 4–5 (infiltration and abstractions); Fetter, Applied Hydrogeology, Ch. 4 (Darcy’s law); Canadian Dam Association, Dam Safety Guidelines, §6 (inflow design flood).

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