16-Civ-B4 Engineering Hydrology · May 2016

Question 1 of 7: Hydrologic cycle processes, groundwater flow and surface runoff

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 1: Hydrologic cycle processes, groundwater flow and surface runoff (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) Temperature, precipitation and streamflow (7 marks)

The three quantities are the energy input, the water input and the water output of a drainage basin, and each one conditions the other two. Temperature governs the state in which water arrives and the rate at which it leaves as vapour. Below roughly 0 °C precipitation arrives as snow and is placed in seasonal storage instead of running off, so the basin's response is deferred by weeks or months; above that threshold the same precipitation depth produces runoff within hours. Temperature also sets the saturation vapour pressure of the air, and through it the evaporative demand, so a warm basin returns a larger share of its precipitation to the atmosphere and yields less streamflow from the same rainfall. Precipitation is the only significant input of water to a land basin; its depth fixes the volume available, and its intensity, duration and areal distribution fix how much of that volume can infiltrate and how much must run off. Streamflow is the integrated output — the water the basin could neither store nor evaporate — and because it is measured continuously at a single point it is the most reliable observation we have of what the other two processes did.

The interrelationships close on themselves. Precipitation that infiltrates recharges soil moisture and groundwater, which sustains streamflow through the dry season as baseflow; that same soil moisture supplies transpiration, which is temperature-driven and which returns water to the atmosphere where it becomes precipitation somewhere downwind. A rise in temperature therefore reduces streamflow twice over, once by increasing evapotranspiration and once by converting snow storage into earlier, flashier runoff. This is why a hydrologic assessment in British Columbia cannot treat a snowmelt-dominated interior basin and a rain-dominated coastal basin with the same model even when their annual precipitation is identical.

Figure 1.1 — Temperature sets the phase of the input and the evaporative demand; precipitation supplies the water; streamflow is what the basin can neither store nor evaporate. The dashed return through evaporation and transpiration closes the cycle.

Each component matters to the cycle in a different way: precipitation is the mass input without which nothing else occurs, temperature is the energy control that decides the pathway the mass takes, and streamflow is both the output and the measurement that lets the whole budget be checked.

(ii) Recharge areas, percolation and discharge areas (7 marks)

A recharge area is that part of the land surface from which water moves downward into the saturated zone; a discharge area is where groundwater returns to the surface, most often as baseflow in a stream, but also as springs, seeps and wetlands. Recharge areas matter because they are the only entry point to the aquifer: the yield an aquifer can sustain, and the quality of the water it delivers, are both determined at the recharge area rather than at the well.

The sequence is infiltration, then percolation, then flow. Infiltration is the entry of water across the ground surface, and its rate is limited by the capacity of the surface soil, which falls during a storm as the pores fill. Water held in the unsaturated (vadose) zone above the water table then percolates downward under gravity, at a rate set by the unsaturated hydraulic conductivity, until it reaches the water table — the surface on which pore-water pressure equals atmospheric pressure. Arrival at the water table is what constitutes recharge; water that stops short of it and is later transpired never becomes groundwater at all. Below the water table, flow follows the hydraulic gradient from the high water-table elevations under the recharge area towards the low elevations at the discharge area, so the water table is both the upper boundary of the saturated zone and the map of the driving head.

Figure 1.2 — Precipitation on the upland recharge area infiltrates, percolates through the vadose zone to the water table, and then flows down the hydraulic gradient to the stream, which is the discharge area. The water table is a subdued replica of the topography.

Three consequences follow, and they are the reason recharge areas are mapped and protected. First, the water table under a recharge area is a subdued replica of the topography, so recharge areas are usually the topographic highs and discharge areas the valley bottoms; a stream that gains along its length is being fed by the aquifer, and one that loses is recharging it. Second, because travel times from recharge area to discharge area are measured in years to millennia, a contaminant introduced at the recharge area is effectively irretrievable — which is why British Columbia's Water Sustainability Act and municipal well-protection plans regulate land use over recharge areas rather than only at wellheads. Third, paving a recharge area removes the recharge without removing the demand: baseflow falls, the water table declines, and the same storm that once recharged the aquifer now arrives at the stream within the hour as a flood peak.

(iii) Peak runoff by the Rational Method (6 marks)

Given. A new suburban development is to be drained by a storm sewer, and the sub-watershed's response time and local IDF relation are supplied:

Given data — Problem 1(iii)
Quantity	Symbol	Value
Drainage area	A	4 ha
Time of concentration	t_c	50 min = 0.8333 h
Local IDF relation (t_d in hours)	i	6.0 − 0.3 t_d
Land use	—	suburban residential development

Find. The peak surface runoff rate to be carried by the storm sewer connection, with the assumptions behind each adopted coefficient stated.

Figure 1.3 — The supplied IDF relation is linear and decreasing, so intensity is largest at short durations while contributing area is largest at t_d ≥ t_c. The product peaks exactly at t_d = t_c.

Approach. Adopt the design duration equal to the time of concentration, read the intensity from the IDF relation, select a runoff coefficient for suburban land use, and apply the metric Rational formula.

Justify the design duration. The Rational Method assumes the whole catchment contributes simultaneously, which first happens at $t_d = t_c$. For $t_d < t_c$ only part of the area contributes; for $t_d > t_c$ the full area contributes but the intensity is lower. Because the given IDF relation decreases monotonically with duration, the product $i \cdot A$ is largest at the critical duration $$t_d = t_c = 50\ \text{min} = 0.8333\ \text{h}.$$
Read the design intensity from the IDF relation. Substituting the critical duration into the supplied linear approximation, $$i = 6.0 - 0.3\,t_d = 6.0 - 0.3(0.8333) = 6.0 - 0.25 = \boxed{5.75\ \text{mm/h}}$$ The relation is only physically meaningful while $i > 0$, that is for $t_d < 6.0/0.3 = 20$ h, and it should not be extrapolated to the very short durations that govern small inlet catchments.
Select the runoff coefficient. Canadian practice (TAC Drainage Manual, and the runoff-coefficient tables reproduced in Mays) gives C = 0.35 to 0.50 for suburban residential development on flat-to-moderate slopes with 30 to 45 per cent imperviousness. Adopt $$C = 0.40$$ as the mid-range value for a newly built suburban subdivision with grassed boulevards and a piped minor system.
Apply the Rational formula in metric units. With intensity in mm/h and area in hectares the constant is 1/360: $$Q_p = \frac{C\,i\,A}{360} = \frac{0.40 \times 5.75 \times 4}{360} = \boxed{0.0256\ \text{m}^3\text{/s}\ (25.6\ \text{L/s})}$$ As a check on magnitude, the 50-minute storm delivers $5.75\ \text{mm/h} \times 0.8333\ \text{h} = 4.79$ mm over 4 ha, or 191.7 m³ of rain, of which 40 per cent appears as the 76.7 m³ that the peak flow of 25.6 L/s would carry over the same period — internally consistent, as the Rational Method requires.
Bracket the answer against the coefficient and the units. Holding the intensity fixed, C = 0.35 gives 22.4 L/s and C = 0.50 gives 31.9 L/s, so the design flow is 23 to 32 L/s; a 250 mm connection at 1 per cent grade carries far more than this, and the governing constraint at the connection will be the municipal system's capacity, not the site's runoff.

Check — the units of i are not stated in the question. If the relation were intended in inches per hour, then i = 5.75 in/h = 146.05 mm/h and the peak flow becomes 0.649 m³/s — twenty-five times larger and a materially different pipe. The mm/h reading is adopted because the paper's own Problem 6(iii) figure plots rainfall intensity on a “mm/h” axis peaking near 13 mm/h, because these are metric national examinations, and because an intensity of 146 mm/h sustained for 50 minutes exceeds any Canadian 100-year IDF value. The assumption is stated here as Note 1 on the cover page requires.

Final results — Problem 1(iii)
Quantity	Value
Design duration (= t_c)	0.8333 h (50 min)
Design rainfall intensity, i	5.75 mm/h
Adopted runoff coefficient, C	0.40 (suburban residential)
Peak surface runoff, Q_p	0.0256 m³/s = 25.6 L/s
Range for C = 0.35 to 0.50	22.4 to 31.9 L/s

Topic. The hydrologic cycle as a coupled mass-and-energy system, recharge and discharge areas, and the Rational Method for small urban catchments.

Key relations. Rational Method $Q_p = C\,i\,A/360$ (Q in m³/s, i in mm/h, A in ha), equivalently $Q_p = 0.00278\,C\,i\,A$; critical duration $t_d = t_c$; Darcy's law for the groundwater limb $Q = K A\,(dh/dl)$; the water balance $P - R - G - E - T = \Delta S$.

Why this works. The Rational Method is a steady-state argument: if rainfall of constant intensity falls for at least as long as the catchment's travel time, the outflow reaches equilibrium with the rainfall rate, reduced by the fraction C that is not abstracted. That is why the critical duration is the time of concentration and not the storm duration, and why C is a volumetric fraction rather than a physical loss rate. The method is defensible for areas up to roughly 100 ha with a single dominant land use; beyond that, spatial variation in intensity and land use makes a unit-hydrograph or continuous-simulation approach necessary. The recharge-area argument rests on the same conservation logic applied to the subsurface: water that reaches the water table must eventually leave at a discharge area, so protecting the entry point protects the exit.

Common pitfall. Two recur. Students set t_d to the storm duration rather than t_c, which understates the intensity and therefore the peak; and they mix units, using the American Q = CiA (cfs, in/h, acres) constant of unity with metric data, which is wrong by a factor of 360. A third, subtler error is to apply the Rational Method to compute a runoff volume — it estimates a peak rate only.

Source. Chow, Maidment and Mays, Applied Hydrology, Ch. 1 (the cycle) and Ch. 15 (design storms); Mays, Water Resources Engineering, §15.3 (Rational Method); Gupta, Hydrology and Hydraulic Systems, Ch. 9 (recharge and discharge areas); TAC Drainage Manual, runoff-coefficient tables.

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