16-Civ-B4 Engineering Hydrology · May 2018

Question 2 of 7: Hydrologic Cycle, Groundwater Flow and Surface Runoff

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 2: Hydrologic Cycle, 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.

2(i) — Three ways watershed characteristics govern surface runoff (7 marks)

Precipitation is the input to a watershed, but the hydrograph that leaves it is the product of the watershed itself. The schematic below shows the processes of the hydrologic cycle that stand between the two; the three groups of characteristics discussed afterwards act on different links in that chain.

The main processes of the hydrologic cycle. Water reaches the basin as precipitation, is partitioned at the surface between interception, infiltration and overland flow, and returns to the channel by three routes of very different speed (overland flow, interflow, baseflow) before evaporation and transpiration return it to the atmosphere.

1. Size, shape and drainage density — they set the volume and the timing. The drainage area fixes the volume of water available from a given depth of rainfall, and it is the multiplier in every peak-flow formula. Shape and drainage density fix how that volume is delivered: a compact, fan-shaped basin brings the contributions of its sub-areas to the outlet at nearly the same time and produces a sharp peak, whereas a long, narrow basin of the same area staggers them and produces a lower, broader hydrograph. A dense channel network shortens the length of slow overland flow and lengthens the fast channel path, so it reduces the time of concentration and raises the peak — which is precisely why urban storm sewer systems, which are artificial drainage networks of very high density, increase peak flows even before any change in imperviousness is counted.

2. Slope, relief and channel gradient — they set the velocity. The time of concentration is a travel time, and travel time is length divided by velocity. Steep overland slopes and steep channel gradients raise velocities, shorten the time of concentration, and therefore both raise the peak (because the critical rainfall duration is shorter and short-duration intensities are higher) and reduce the opportunity time for infiltration. Relief also controls orographic enhancement of precipitation, the elevation distribution of snowpack and hence the timing of the freshet, and the aspect-driven variation in melt and evapotranspiration — all first-order controls in mountainous British Columbian and Albertan basins.

3. Soils, geology and land cover — they set the partition between runoff and storage. Infiltration capacity is a soil and land-cover property: a coarse, deep, well-vegetated soil may absorb the whole of an ordinary storm and yield no surface runoff at all, while a compacted clay, a frozen or already saturated soil, or an impervious surface converts nearly all of it to overland flow. This is the characteristic that the runoff coefficient of the rational method and the SCS curve number both encode. Land cover also controls interception storage and evapotranspiration, and the underlying geology controls how much water is stored in the saturated zone and therefore how large and how sustained the baseflow is. Urbanisation acts on this group and on the first group simultaneously, which is why it changes runoff so sharply: the same storm now produces more direct runoff, delivered faster.

2(ii) — The terms of Darcy’s law and their consistent dimensions (8 marks)

Darcy’s law states that the volumetric flow of groundwater through a porous medium is proportional to the cross-sectional area and to the hydraulic gradient. Written in the form given on the paper for flow through a slice of aquifer,

$$Q = K\,Z\,W\,\frac{\Delta H}{\Delta L}$$

Term	Meaning	Dimensions	Typical SI units
$Q$	Volumetric discharge of groundwater through the section — the quantity of water crossing the whole face per unit time	$\mathrm{L}^3\mathrm{T}^{-1}$	$\text{m}^3/\text{s}$ or $\text{m}^3/\text{d}$
$K$	Hydraulic conductivity of the aquifer material — the ease with which this fluid moves through this medium; it combines an intrinsic permeability of the solid with the density and viscosity of the water	$\mathrm{L}\,\mathrm{T}^{-1}$	m/s or m/d
$Z$	Saturated thickness of the aquifer normal to the flow — the full thickness between confining beds for a confined aquifer, or the depth of water above the aquifer base for an unconfined one	$\mathrm{L}$	m
$W$	Width of the flow section measured horizontally and perpendicular to the direction of flow	$\mathrm{L}$	m
$\Delta H$	Difference in hydraulic (piezometric) head between the two sections — elevation head plus pressure head, i.e. the difference in water level in two piezometers, not a difference in ground elevation	$\mathrm{L}$	m
$\Delta L$	Distance between those two sections measured along the flow path	$\mathrm{L}$	m
$\Delta H/\Delta L$	Hydraulic gradient $i$ — the slope of the potentiometric surface, and the driving force for the flow	dimensionless	m/m

Two groupings inside the equation are worth naming, because they are how the terms are usually reported in practice. The product $Z\,W$ is simply the cross-sectional area $A$ of saturated aquifer through which the flow passes, so the equation reduces to the familiar $Q = KAi$. The product $K\,Z$ is the transmissivity $T$, with dimensions $\mathrm{L}^2\mathrm{T}^{-1}$ (typically $\text{m}^2/\text{d}$), which is what a pumping test actually measures; in that form $Q = T\,W\,(\Delta H/\Delta L)$. The dimensional check on the whole equation is immediate:

$$[Q]=\left(\frac{\mathrm{L}}{\mathrm{T}}\right)(\mathrm{L})(\mathrm{L})\left(\frac{\mathrm{L}}{\mathrm{L}}\right)=\frac{\mathrm{L}^3}{\mathrm{T}}$$

A short numerical illustration fixes the units. Given. A confined sand aquifer with $K = 30$ m/d, saturated thickness $Z = 12$ m, section width $W = 500$ m, and a head drop of $\Delta H = 3.0$ m over a flow path of $\Delta L = 1500$ m; effective porosity 0.25. Find. The discharge through the section and the true velocity of a water particle. The gradient is $i = 3.0/1500 = 0.0020$, so

$$Q=(30)(12)(500)(0.0020)=360\ \text{m}^3/\text{d}=4.17\times10^{-3}\ \text{m}^3/\text{s}$$

The quantity $q = Ki = (30)(0.0020) = 0.060$ m/d is the Darcy velocity or specific discharge. It is a flux per unit of total cross-sectional area, not a particle speed, because water can only travel through the pores; dividing by the effective porosity gives the seepage velocity that a tracer would show, $v = q/n_e = 0.060/0.25 = 0.24$ m/d. Confusing the two by a factor of four is the classic contaminant-transport error. Finally, the law is valid only for laminar flow, in practice for a Reynolds number based on grain size below about unity, so it fails in fractured rock, in karst, and immediately adjacent to a heavily pumped well.

2(iii) — Peak runoff from the 7 ha suburban development by the rational method (5 marks)

Given.

Contributing area of the new development	$A = 7$ ha
Time of concentration	$t_c = 30$ min $= 0.50$ h
Local IDF relation ($t_d$ in hours, $i$ in mm/h)	$i = 6 - 0.3\,t_d$
Land use	suburban residential development
Runoff coefficient assumed for that land use	$C = 0.40$
Outlet	trunk storm sewer to a local stream

Find. The peak surface runoff from the development, expressed in $\text{m}^3/\text{d}$.

Approach. Take the critical storm duration equal to the time of concentration, read the corresponding intensity from the given IDF relation, and apply the rational formula in its consistent SI form before converting the peak rate to cubic metres per day.

Select the design storm duration. The rational method assumes a steady storm of uniform intensity lasting at least as long as the time of concentration, so that the whole catchment contributes at the moment of peak. Because intensity falls as duration rises, the largest peak is obtained by taking the shortest duration that still allows full contribution, that is $t_d = t_c$: $$t_d=t_c=30\ \text{min}=0.50\ \text{h}$$
Read the design intensity from the IDF relation. Substituting the design duration in hours into the relation supplied with the question, $$i = 6 - 0.3\,t_d = 6 - 0.3(0.50) = 6 - 0.15 = 5.85\ \text{mm/h}$$ This corresponds to a storm depth over the time of concentration of only $(5.85)(0.50) = 2.925$ mm, which is a very small event; see the note below on the form of this IDF relation.
Choose the runoff coefficient. The question asks for a typical value for suburban development. Standard tables give 0.25 to 0.40 for suburban residential areas, the upper end applying where lot coverage, roads and driveways are extensive and the soils are poorly drained. A serviced suburban development discharging to a trunk sewer is at the upper end, so C = 0.40 is adopted, and the sensitivity to that choice is reported below.
Apply the rational formula in consistent SI units. With $i$ in mm/h and $A$ in hectares the conversion factor is $1/360$, since $1\ \text{mm/h}\times1\ \text{ha}=10^{-3}\times10^{4}/3600=1/360\ \text{m}^3/\text{s}$. Hence $$Q=\frac{C\,i\,A}{360}=\frac{(0.40)(5.85)(7)}{360}=0.04550\ \text{m}^3/\text{s}=45.5\ \text{L/s}$$
Convert the peak rate to cubic metres per day. The question asks for the answer in $\text{m}^3/\text{d}$, so the instantaneous peak is expressed as the volume that would pass in a day if it were sustained: $$Q=(0.04550)(86\,400)=\boxed{Q_p \approx 3931\ \text{m}^3/\text{d}}$$

Assumptions stated. (a) The storm duration equals the time of concentration and the intensity is uniform in time and over the whole 7 ha. (b) The runoff coefficient is constant at 0.40, with no frequency-adjustment factor and no allowance for antecedent wetness. (c) The development drains entirely to the one trunk sewer, with no on-site detention, no infiltration facilities and no storage routing — the rational method gives the peak inflow to the sewer, not a routed outflow. (d) The peak is converted to a daily volume purely arithmetically; it is a rate expressed in daily units, not a real 24-hour runoff volume, which would be very much smaller. (e) The rational method is appropriate here because the catchment is small and largely impervious, well within the usual limit of about 80 ha.

Quantity	Symbol	Value
Design storm duration	$t_d$	30 min (0.50 h)
Design rainfall intensity	$i$	5.85 mm/h
Adopted runoff coefficient	$C$	0.40 (suburban)
Peak runoff	$Q_p$	0.0455 $\text{m}^3/\text{s}$ = 45.5 L/s
Peak runoff in the units asked for	$Q_p$	3931 $\text{m}^3/\text{d}$
Sensitivity to C (0.35 to 0.45)	$Q_p$	3440 to 4423 $\text{m}^3/\text{d}$

Check — two features of the supplied IDF relation. First, its units are not stated in the question beyond “$i$ is the intensity in mm/hr”, and that reading is adopted; the paper’s own IDF chart in Problem 5 is plotted in mm/h, which confirms the convention used throughout this sitting. Second, the relation is linear in duration, whereas a real IDF curve is a hyperbolic or power decay of the form $i = a/(t+b)^c$; a linear relation gives an implausibly low intensity at short durations (6 mm/h as the duration tends to zero) and reaches zero intensity at a duration of 20 hours. It is solved exactly as printed, but the resulting peak of about 46 L/s should be recognised as far below what a real 7 ha suburban catchment would deliver in a design storm — at a realistic 5-year, 30-minute Canadian intensity near 50 mm/h the same calculation would give roughly 390 L/s. The method, not the magnitude, is what the question is testing.

Topic. Watershed controls on runoff, Darcy’s law, and the rational method

Key relations. Darcy’s law $Q=KZW(\Delta H/\Delta L)=KAi$, transmissivity $T=KZ$, Darcy velocity $q=Ki$ and seepage velocity $v=q/n_e$. Rational method $Q=CiA$, in SI $Q(\text{m}^3/\text{s})=C\,i(\text{mm/h})\,A(\text{ha})/360$, with the critical duration $t_d=t_c$.

Why this works. Darcy’s law is an empirical statement that in a porous medium the resistance to flow is linear in velocity, which is what laminar flow through small pores implies; the hydraulic gradient rather than the pressure gradient appears because both elevation and pressure energy drive the flow and only their sum matters. The rational method is the steady-state limit of the same conservation idea applied to a surface catchment: if a storm of constant intensity lasts long enough for every part of the catchment to be contributing simultaneously, the outflow reaches a steady rate equal to the rainfall rate times the area times the fraction that runs off. The critical duration is $t_c$ because any shorter storm leaves part of the catchment not yet contributing, and any longer one has a lower intensity.

Common pitfall. In Darcy’s law, taking $\Delta H$ as a difference in ground surface elevation or in water-table depth below surface rather than in hydraulic head; and quoting the Darcy velocity as the speed of the water. In the rational method, using a storm duration longer than $t_c$ (which under-estimates the peak) and using $Q=CiA$ with $i$ in mm/h and $A$ in hectares without the 1/360 factor, which is wrong by more than two orders of magnitude.

Source. Fetter, Applied Hydrogeology, 4th ed., Ch. 3 (Darcy’s law, hydraulic conductivity, transmissivity); Chow, Maidment and Mays, Applied Hydrology, Ch. 1 and Ch. 15 (design storms and the rational method); Mays, Water Resources Engineering, 3rd ed., Ch. 15; Transportation Association of Canada, Guide to Bridge Hydraulics (runoff coefficients in Canadian practice).

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