07-Str-B11 · May 2013
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Examinations — May 2013 — 07-Str-B11 Hydraulic Engineering. Three-hour, CLOSED-BOOK examination; one 8.5 × 11 in aid sheet (both sides) and one approved Casio or Sharp calculator are permitted. The paper prints six questions of 20 marks each and instructs the candidate to complete any five; where a question has more than one part, the parts carry equal marks. Candidates are urged to state any interpretive assumptions with their answer. All six questions are worked below, because this set is intended as a study resource.
Reference texts: Mays, L.W., Water Resources Engineering (3rd ed., Wiley) — Hazen-Williams pipe hydraulics, pipe networks, pump-system curves and quasi-steady reservoir routing; Chow, V.T., Open-Channel Hydraulics (McGraw-Hill, 1959) — uniform flow, Manning's equation, compound and divided channel sections, and the specific-energy and momentum treatment of channel obstructions; Chin, D.A., Water-Resources Engineering (3rd ed., Pearson) — North-American design practice for transmission mains, minimum service pressures and pump selection; Henderson, F.M., Open Channel Flow (Macmillan) — surges, hydraulic jumps and the unsteady response of a channel to a sudden blockage.
Notation and conventions used throughout. The paper supplies the SI Hazen-Williams form \(Q = 0.278\,C\,D^{2.63}S^{0.54}\) with \(Q\) in m3/s, \(D\) in metres and \(S = h_f/L\); it is used exactly as printed. Local losses and velocity head are neglected, as instructed in Note 6, so the hydraulic grade line (HGL) and the energy grade line coincide and "pressure head at a node" means \(p/\gamma = \mathrm{HGL} - z\). Water properties are \(\rho = 1000\) kg/m3 and \(\nu = 1.31\times10^{-6}\) m2/s.
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 sudden and complete blockage of a river is the open-channel analogue of instantaneous valve closure in a pipeline, and the response divides cleanly into a positive surge propagating upstream and a negative surge propagating downstream. Before the failure the reach is in approximately uniform flow at normal depth \(y_n\) with mean velocity \(V_n\) and Froude number well below unity, typical of a gravel-bed river. Within seconds of the slide the discharge at the blockage face drops to zero, and the two disturbances that carry that information away from the face set the conditions on either side.
Continuity. The one-dimensional continuity equation for unsteady open-channel flow, \(\partial A/\partial t + \partial Q/\partial x = 0\), states that any imbalance between the discharge entering and leaving a reach must appear as a change of stored volume. Upstream of the blockage the catchment continues to deliver the pre-failure discharge \(Q_0\) while the outflow is zero, so the entire inflow goes into storage and the water surface rises. The rise is not uniform: it begins at the face and advances upstream as a distinct front, behind which the water is nearly stagnant and above which the surface is essentially horizontal. The impoundment therefore grows both in depth and in length, and the rate at which the pool level rises is simply \(\mathrm{d}h/\mathrm{d}t = Q_0/A_s\), where \(A_s\) is the surface area of the pool at that instant. Because the pool area grows rapidly at first as the front runs upstream over a wide floodplain and then more slowly, the initial rise is fast and progressively decelerates. Downstream of the blockage the reverse holds: inflow has fallen to zero while the reach continues to drain, so storage is released, the depth falls and the discharge decays. Only tributaries, groundwater discharge, and whatever seeps through the porous gravel and rock of the slide mass keep any water moving at all.
Momentum. Both fronts are surges, and a surge is a hydraulic jump translating along the channel; the correct tool is therefore the momentum equation applied to a control volume moving with the front, not the energy equation. For the upstream-advancing positive surge of celerity \(c\) entering flow of depth \(y_1\) and velocity \(V_1\), superimposing a velocity \(c\) on the whole field renders the front stationary and gives the moving-jump relation
$$\left(V_1 + c\right)^2 = \frac{g\,y_2}{2\,y_1}\left(y_1 + y_2\right)$$where \(y_2\) is the depth immediately behind the front. Because the blockage is complete, the water behind the front is brought to rest, and the additional continuity condition across the front, \((V_1 + c)y_1 = (V_2 + c)y_2\) with \(V_2 = 0\), closes the pair of equations for \(y_2\) and \(c\). The result is a steep-fronted wave that travels faster than the natural wave celerity \(\sqrt{g y_1}\) of the undisturbed flow, which is exactly why it forms an abrupt front rather than spreading out. The blockage itself must resist the resulting thrust: applying the momentum equation to a control volume enclosing the slide mass, the debris dam carries the full hydrostatic force of the growing pool plus the rate of destruction of the incoming momentum flux, \(F = \gamma \bar{h} A + \rho Q_0 V_n\), a load that increases as the square of the impounded depth and which is what ultimately determines whether the natural dam survives or fails. Downstream the negative surge is quite different in character: a depletion wave steepens nowhere and instead flattens as it runs, because its deeper parts travel faster than its shallower parts, so the downstream reach experiences a gradual drawdown rather than a sharp front.
Energy. Energy is not conserved across either front, and the asymmetry between them is instructive. The upstream positive surge is a moving hydraulic jump and dissipates energy at the classical rate
$$\Delta E = \frac{\left(y_2 - y_1\right)^3}{4\,y_1\,y_2}$$per unit weight, converting the kinetic energy of the arrested flow into turbulence and heat in the roller at the front. Behind the front the velocity head is essentially zero, so the specific energy is almost entirely potential and the energy grade line collapses onto the water surface; the pool becomes a horizontal-surfaced reservoir joined to the undisturbed river upstream by an M1 backwater profile that lengthens as the pool deepens. Downstream, the negative surge dissipates very little energy, since it is a gradual and continuous transition rather than a jump; the specific energy there simply declines with the falling depth as the reach drains toward a residual chain of pools and riffles. If the blockage is even slightly permeable, as a gravel and rock slide mass certainly is, a small throughflow will emerge from its downstream face at high velocity under the full head of the pool, producing a locally supercritical jet, a scour hole and a small hydraulic jump immediately below the toe. This seepage is the seed of the most dangerous mode of behaviour, because internal erosion of the fines can enlarge the flow path and lead to a rapid breach.
Two further consequences deserve to be stated explicitly, since they follow directly from the same three principles. First, the assumption of a completely impermeable blockage is an idealisation: real landslide dams pass some flow, and the working question for an engineer arriving on site is whether that seepage is stable or progressive. Second, the impoundment will eventually either overtop the crest of the slide mass or breach it, and because the crest is uncompacted debris rather than engineered fill, overtopping normally leads to rapid erosion and an outburst flood whose peak discharge can greatly exceed the natural flood of record. In Canadian practice a landslide dam of this kind is treated as an unregulated dam under provincial dam-safety legislation, and the immediate engineering response is to survey the pool, estimate the impounded volume and the rate of rise, model the breach outflow, warn the downstream valley, and consider controlled lowering of the crest by notching so that the impoundment is released gradually instead of catastrophically.
The storm sewer outfall lies immediately upstream of the blockage, which is the worst possible position: it is the first structure to be submerged by the rising impoundment, and the pool level climbs past it within minutes rather than hours. The impacts follow from the fact that the outfall's tailwater, which the storm system was designed to discharge against, has been replaced by a rising and eventually deep reservoir.
Loss of outfall capacity and surcharging of the storm system. A storm outfall is normally designed to discharge freely, or at most with the tailwater below the pipe obvert. Once the pool submerges the outfall the discharge becomes a submerged-orifice or full-pipe condition governed by the difference between the sewer hydraulic grade line and the pool level, and as the pool rises that difference collapses toward zero. The hydraulic grade line in the trunk sewer is forced upward by exactly the amount the tailwater rises, and this backwater propagates upstream through the network. Manholes that previously ran part-full begin to surcharge, gully and catch-basin inlets cease to drain, and if the blockage coincides with or is followed by any rainfall the town's minor drainage system will fail from the outfall backwards. The consequences are surface ponding at low points, flooded intersections, and basement flooding through foundation drains and any storm connections in the affected service area.
Backflow of river water into the town's drainage system. If the outfall has no flap gate, duckbill valve or other backflow preventer — and many older municipal outfalls do not — the rising pool will drive water back up the storm sewer. River water carrying suspended sediment, organic debris and slide-derived fines will be forced into the pipe network, depositing in the flatter reaches and reducing capacity long after the emergency has passed. Where the storm and sanitary systems share any cross-connection, or where a combined sewer exists, this backflow becomes a public health issue as well as a hydraulic one, and it can also flood the town through the same manholes and inlets that would otherwise have drained it. The single most effective immediate mitigation is therefore to confirm the condition of any existing backflow prevention at the outfall and to plug or gate the outfall if none exists, accepting that the storm system must then be pumped.
Structural and geotechnical damage to the outfall itself. Submergence imposes external hydrostatic pressure on a structure designed for free discharge, and a buoyant uplift on the outfall pipe and headwall if the pipe is empty or gated. Saturation of the bank behind the headwall raises pore pressures and reduces effective stress in the embankment, which can cause the headwall to rotate or the outfall apron to be undermined; the erosion protection at the outfall is designed for the velocities of a discharging jet, not for wave action on a reservoir shoreline. The greatest risk to the structure, however, comes later: if the debris dam breaches, the pool drains rapidly, and the resulting rapid drawdown leaves the saturated bank with high pore pressures and no supporting water load, a classic slope-stability failure case that frequently takes out outfall structures and the bank they are founded in.
Inundation, environmental and service impacts on the town. The impoundment does not stop at the outfall. As it grows upstream it will inundate low-lying land, roads, riverside trails and any services in the floodplain, and depending on the town's elevation relative to the crest of the slide it may threaten property directly. Standing water in the pool becomes stratified and oxygen-depleted, sediment settles out of the arrested flow, and fish passage is entirely blocked, all of which engage the federal Fisheries Act and Department of Fisheries and Oceans authorisation requirements for any works undertaken in the channel. If the town also draws water from the river, its intake is affected by the same stagnation and turbidity. Finally, the downstream reach is left with almost no flow, which strands fish, dewaters any downstream intakes and destroys the assimilative capacity that the town's outfall and any wastewater discharges rely upon.
Recommended immediate actions. Establish continuous monitoring of the pool level and the rate of rise so the time to overtopping can be estimated; survey the slide mass and impoundment to model the breach hydrograph; notify the provincial emergency management authority, the downstream communities and the dam-safety regulator; isolate the storm outfall with a temporary gate or plug and provide emergency pumping of the storm system; restrict access to both the impoundment shoreline and the downstream reach; and plan for controlled notching of the crest under professional supervision so the pool is lowered at a rate the downstream channel can accept. Each of these follows directly from the continuity, momentum and energy reasoning in part (a): the pool grows because continuity demands it, the dam may fail because momentum loads it beyond the strength of uncompacted debris, and the breach would be violent because the energy stored in the impoundment is released over minutes rather than days.
