16-Civ-B4 Engineering Hydrology · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examination, December 2017, 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.
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, stormwater management, reservoir operation); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (measurement, areal precipitation, hydrologic modelling); C. W. Fetter, Applied Hydrogeology, 4th ed. (Darcy’s law, confined and unconfined aquifers, storativity); V. T. Chow, Open-Channel Hydraulics (1959) (flood-wave propagation and unsteady flow). Canadian practice references: Environment and Climate Change Canada / Water Survey of Canada HYDAT archive and the ECCC Engineering Climate Datasets (short-duration rainfall IDF curves and the IDF_CC climate-adjustment tool); ISO 1100-2 and the WMO Manual on Stream Gauging (stage–discharge rating practice); the Transportation Association of Canada Guide to Bridge Hydraulics and provincial highway drainage manuals (culvert design); and the Canadian Dam Association Dam Safety Guidelines (inflow design flood and flood-wave routing).
Check — the illustrative data below are the solver’s own. Every problem on the December 2017 paper is a discussion question, and the paper supplies no numerical data whatsoever. Where a short calculation appears below it exists only to demonstrate the method concretely and to make the answer checkable; its input values are stated explicitly in a Given line as assumed, representative Canadian values. They are not exam data. A candidate who assumed different but reasonable values and carried them through consistently would earn full marks, and page-1 Note 1 expressly invites the candidate to state any assumptions made.
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.
A hydrologic transport model couples two balances over the same set of control volumes. The first is the water balance, which routes the flow; the second is a constituent mass balance, which is carried on the routed flow. The river is discretised into reaches, each treated as a control volume with an inflow, an outflow and a storage. Water is routed reach by reach using the continuity equation together with a storage relation — Muskingum for a channel reach, level-pool for a lake or reservoir — producing a discharge and a volume for every reach at every time step. The constituent balance is then written on the same control volume: the rate of change of constituent mass equals the mass entering, minus the mass leaving, plus or minus whatever is produced or consumed inside.
For a single well-mixed reach with a first-order decaying constituent,
$$\frac{d(VC)}{dt} = QC_{\text{in}} - QC - kVC$$in which $V$ is the reach volume, $C$ the concentration, $Q$ the discharge, $C_{\text{in}}$ the upstream concentration and $k$ the first-order rate constant. Chaining such reaches gives a river-scale model; adding lateral inflows with their own concentrations represents tributaries and outfalls; adding a settling term represents suspended solids; and coupling the oxygen and organic-matter equations gives the classical Streeter–Phelps dissolved-oxygen sag. QUAL2K, WASP and the water-quality module of HEC-RAS are all built on this structure.
Given. A well-mixed river reach of volume $V = 4.0 \times 10^5$ m³ carrying $Q = 8.0$ m³/s, receiving an upstream concentration $C_{\text{in}} = 12$ mg/L of a constituent that decays at $k = 0.25$ d−1.
Find. The steady-state concentration leaving the reach, and the percentage removed.
Approach. Set the storage derivative to zero, solve the resulting algebraic balance for $C$, and express the answer as a removal.
The structure of the answer is instructive: removal is governed entirely by the product $k\tau$, so a slow-reacting constituent in a fast river passes essentially unchanged, and lengthening the reach or impounding it — both of which raise $\tau$ — is the only hydraulic lever available. The same model run unsteadily tracks the passage of a spill through the river and gives the arrival time and peak concentration at each downstream intake, which is its most common emergency-response use.
Assumption 1 — the control volume is closed apart from the two stated fluxes. The equation as written admits exactly one inflow and one outflow, which means it assumes there is no lateral inflow along the reach, no direct precipitation on or evaporation from the water surface, no seepage to or from the bed, and no abstraction or return flow. That is often acceptable over a short reach and a short flood, where the flood wave dominates every other term by orders of magnitude. It is emphatically not acceptable over a long reach in a wet basin, over a lake in summer where evaporation is significant, or in any water-balance application over months rather than hours. When any of these terms matters it must appear explicitly, $dS/dt = I - Q + q_L - E - L_{\text{seep}}$, and quietly omitting it is the most common source of a routing model that will not close its volume.
Assumption 2 — storage is a known, single-valued function of the flow variables, so that the equation can be closed. Continuity alone is one equation in two unknowns, $S$ and $Q$. To solve it, a second relation must be supplied, and every hydrologic routing method is a different guess at that relation. Level-pool routing assumes $S = f(Q)$ alone, which is true for a reservoir whose water surface is horizontal and whose outflow is set by a fixed outlet, and false for any water body long enough that the surface tilts appreciably during a flood. Muskingum assumes $S = K[XI + (1-X)Q]$, which admits a wedge component but still assumes fixed $K$ and $X$ and therefore a fixed wave speed. Both are approximations to the momentum equation, and both fail where momentum genuinely matters — under significant backwater, in a reach with a downstream control, in a tidal estuary, or where a levee breach or bridge constriction imposes a local head loss. In those situations the hydrologic form must be abandoned in favour of the full Saint-Venant equations solved hydraulically. A subsidiary consequence of the same assumption is that the time step must be short enough to resolve the storage change — the equation is integrated numerically, and too coarse a step both loses the peak and, in Muskingum, produces the negative coefficients discussed in 4(iii).
Step 1 — prepare and quality-control the data, and split the record. Calibration cannot repair bad input. Assemble concurrent precipitation, temperature and discharge series at the model’s time step; check the precipitation for double-mass consistency between gauges so that a station move or a change of instrument does not appear as a hydrologic signal; check the discharge record for rating shifts and for ice-affected periods, which in Canada affect a large fraction of the winter record and are estimated rather than measured; and compute the annual water balance to confirm that the runoff coefficient is physically possible. Then divide the record into an independent calibration period and validation period, each containing wet years, dry years and at least one large flood.
Step 2 — reduce the free-parameter set and choose an objective function that matches the intended use. Fix everything that can be fixed from measurement or from physical reasoning, so that only the genuinely unidentifiable parameters are optimised; a discharge record can identify perhaps five or six. Then select the objective function deliberately, because it defines what “calibrated” means. Nash–Sutcliffe efficiency on untransformed discharge weights the peaks and is right for flood design; the same statistic on log-transformed discharge weights the recessions and is right for low-flow or water-supply work; percent bias constrains volume. For flood work a weighted combination of a peak-sensitive term and a volume term is normal, and the weights should be stated.
Step 3 — optimise, then validate on the withheld period and inspect the hydrographs. Search the parameter space with an algorithm robust to local optima — shuffled complex evolution is the standard choice — and then, without any further adjustment, run the calibrated set over the validation period. Judge the model on that run. Finally, look at the hydrographs and the residuals, not only at the summary statistic: a good efficiency can conceal systematic timing error, a consistently missed recession, or a snowmelt peak that arrives a week late. Errors that are structured rather than random indicate a model-structure problem that no further parameter adjustment will fix.
The Muskingum method is a hydrologic routing scheme: it uses continuity plus an assumed storage relation, and makes no attempt to solve the momentum equation. Its insight is that the storage in a river reach during a flood has two parts.
The prism is the volume beneath a water surface parallel to the channel bed, and it is a function of the outflow alone, $S_{\text{prism}} = KO$, where $K$ has units of time and is physically the travel time of the flood wave through the reach. The wedge is the additional volume held between that surface and the actual, steeper surface during the rise of the flood, $S_{\text{wedge}} = KX(I - O)$, in which the weighting factor $X$ measures how much the inflow contributes to storage. Adding them,
$$S = K\left[XI + (1-X)O\right]$$with $X$ between 0 and 0.5: at $X = 0$ the reach behaves as a level-pool reservoir with no wedge at all, and at $X = 0.5$ the wedge is symmetric and the wave translates without attenuation. Natural channels usually calibrate to $X$ between 0.2 and 0.3.
Substituting this storage relation into the continuity equation, integrating over a time step $\Delta t$ by the trapezoidal rule, and solving for the unknown outflow gives the working form
$$O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1$$ $$C_0 = \frac{\Delta t - 2KX}{2K(1-X)+\Delta t}, \qquad C_1 = \frac{\Delta t + 2KX}{2K(1-X)+\Delta t}, \qquad C_2 = \frac{2K(1-X)-\Delta t}{2K(1-X)+\Delta t}$$Given. A river reach with travel time $K = 12$ h and weighting factor $X = 0.20$, routed at $\Delta t = 6$ h. At the start of a step the inflow is $I_1 = 85$ m³/s and the outflow $O_1 = 92$ m³/s; at the end of the step the inflow is $I_2 = 210$ m³/s.
Find. The routing coefficients and the outflow at the end of the step.
Approach. Confirm that the time step lies in the stable range, evaluate the three coefficients, verify that they sum to unity, and apply the routing equation.
The inflow more than doubled over the step while the outflow rose by less than three per cent, which is the attenuation and lag the method exists to represent. Repeating the step through the whole inflow hydrograph produces the routed outflow hydrograph. The parameters are obtained by calibration on an observed pair of hydrographs: $K$ from the lag between the inflow and outflow centroids, and $X$ from the value that collapses a plot of storage against the weighted flow $XI + (1-X)O$ into a single line rather than a loop.
| Quantity | Symbol | Result |
|---|---|---|
| Hydraulic residence time of the reach | τ | 50 000 s = 0.579 d |
| Steady-state reach concentration | C | 10.5 mg/L (12.6 % removal) |
| Muskingum denominator 2K(1−X) + Δt | — | 25.2 h |
| Routing coefficient | C0 | 0.0476 |
| Routing coefficient | C1 | 0.4286 |
| Routing coefficient | C2 | 0.5238 (sum = 1.0000) |
| Routed outflow at the end of the step | O2 | 94.62 m³/s |
