Question 5 of 7: Channel and river routing, flood-wave behaviour

Problem 5: Channel and river routing, flood-wave behaviour (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.

Part (i) — The convolution method and its assumptions (6 marks)

Convolution routes a hydrograph through a reach by treating the reach as a linear system characterised by a single impulse-response function. If a unit volume of water were released instantaneously at the upstream end, the discharge it would produce with time at the downstream end is the reach’s response function $u(t)$, sometimes called the channel unit hydrograph or the instantaneous unit hydrograph of the reach. Any arbitrary inflow hydrograph is then decomposed into a sequence of such impulses, each impulse produces a scaled and lagged copy of the response, and the outflow is their sum. In continuous form,

and in the discrete form used for computation, with a time step $\Delta t$,

identical in structure to the rainfall-runoff convolution of Problem 2(ii), with inflow replacing effective rainfall. The response function is obtained either from a pair of observed inflow and outflow hydrographs by deconvolution (solving the linear system for $u$, usually by least squares because direct back-substitution amplifies noise), or analytically from a conceptual model — a single linear reservoir gives $u(t) = (1/K)e^{-t/K}$, and the Nash cascade of $n$ identical linear reservoirs gives the gamma-shaped $u(t) = \frac{1}{K\Gamma(n)}(t/K)^{n-1}e^{-t/K}$.

Two underlying assumptions. First, linearity: the response of the reach is proportional to the input, so doubling the inflow doubles every outflow ordinate, and the responses to separate inputs add without interaction. This is what licenses both the scaling and the summation in the convolution sum, and it is the assumption that fails hardest in practice, because a river that spills onto its floodplain routes a large flood quite differently from a small one — the celerity drops and the attenuation rises. Second, time invariance: the response function does not change with time, so the same $u(t)$ applies whether the impulse arrives at the beginning or the end of the event and whether the event occurs in spring or autumn. This fails where the channel scours or aggrades during the flood, where ice affects conveyance, where vegetation growth changes roughness seasonally, or where a downstream control such as a tidal boundary or a reservoir varies during the event.

Two further assumptions are implicit and worth naming: that there is no significant lateral inflow along the reach (or that it is handled as a separate input), and that backwater effects are absent, so that the outflow depends only on what has entered upstream and not on the downstream water level.

Part (ii) — River reaches and rating curves in channel routing (7 marks)

The reach as the unit of computation. A river is not routed as a single object. It is divided into reaches: lengths of channel over which the geometry, slope, roughness and conveyance may reasonably be taken as uniform, bounded at each end by a node. The nodes are placed where something changes — at a tributary confluence, at a major change of slope or cross-section, at a bridge, weir or dam, at a gauging station, and at the boundaries of a flood plain that begins to convey flow. Each reach is then routed independently by a storage method such as Muskingum, or by a hydraulic method solving the Saint-Venant equations, and the outflow computed at the downstream node of one reach becomes the inflow to the next, with any tributary hydrograph added at the confluence node. This cascade structure is what allows a long river to be modelled with a manageable number of parameters, and it is the architecture of HEC-HMS, HEC-RAS and every operational river-forecast model.

Reach length is not arbitrary. It must be short enough that the flood wave takes at least one routing interval to traverse it — the condition $\Delta t \le K = L/c$, with $c$ the wave celerity — and short enough that lumping the storage over the reach is defensible. A convenient rule for Muskingum–Cunge is to subdivide until each sub-reach satisfies both the Courant condition and the requirement that $X$ remain non-negative.

The role of the rating curve. The rating curve is what converts between the discharge that routing computes and the stage that engineering decisions require, and it enters channel routing in four distinct ways.

1. Supplying the storage-discharge relation. Hydrologic routing needs $S$ as a function of $O$. For a reach, the stage-discharge rating at the downstream node, combined with a stage-volume relation obtained from surveyed cross-sections, gives exactly that: for each stage, the rating gives the discharge and the cross-section geometry gives the storage volume in the reach, and eliminating stage between them yields $S = f(O)$. Without a rating there is no storage-outflow curve and no hydrologic routing.
2. Providing the boundary condition. A hydraulic routing model solving the Saint-Venant equations requires a downstream boundary condition, and the normal choice is a rating curve at the outlet section — either a measured rating, or a normal-depth rating computed from Manning’s equation, $Q = \frac{1}{n}A R^{2/3}S_0^{1/2}$. The looped rating discussed below is the physical reason this choice matters.
3. Calibrating the routing parameters. Concurrent inflow and outflow hydrographs, both obtained by applying ratings to recorded stages at the two ends of the reach, are what allow $K$ and $X$ to be determined by the storage-loop method described in Problem 3(iii). The quality of the routing parameters is therefore inherited directly from the quality of the two ratings.
4. Converting the routed result into stage and inundation. The output of routing is a discharge hydrograph, but flood damage depends on water level. The rating at each node converts the routed peak discharge into a peak stage, and the peak stages along the river define the flood profile that is intersected with the terrain model to map the inundated area. This is the step that produces the deliverable a client actually uses.

The looped rating and its consequence. During the passage of a flood wave the water-surface slope is not the bed slope: on the rising limb the surface is steeper than the bed, so for a given stage the discharge is greater than the steady-flow rating predicts, and on the falling limb it is less. Plotting stage against discharge through an event therefore traces a counter-clockwise loop rather than a single curve, and the peak discharge occurs before the peak stage. The magnitude of the effect is given by the Jones formula,

where $Q_n$ is the steady (normal) discharge from the single-valued rating, $S_0$ the bed slope and $c$ the wave celerity. The loop is negligible on steep rivers, where $S_0$ is large, and pronounced on flat ones. Its practical consequence is that hydrologic routing methods, which assume a single-valued storage-discharge relation, are unsuitable for very flat rivers and for reaches under backwater influence, and a full dynamic model is required instead.

Part (iii) — Predicting the downstream effect of a flood wave (7 marks)

The question asks for the solution in principle for peak flow rate, maximum wave height and time to crest at points downstream. The following is the sequence a practising hydrotechnical engineer would set out.

1. Generate the inflow hydrograph at the upstream boundary. The sudden large rainfall event is converted to a runoff hydrograph at the head of the study reach by the rainfall-runoff procedure of Problem 2: design or observed hyetograph, loss model to obtain effective rainfall, unit-hydrograph convolution or kinematic-wave overland routing, plus baseflow. If a gauge exists at the upstream boundary, its rated record is used directly in preference to any model. Tributary hydrographs are prepared in the same way for insertion at each confluence.
2. Assemble the physical description of the river. Survey or extract from LiDAR a series of cross-sections at every significant change of geometry, typically every few hundred metres, extending laterally across the full flood plain; assign Manning roughness to the channel and to each overbank zone from field inspection and calibration; measure the bed slope; and locate every hydraulic structure — bridges, culverts, weirs, dams — with its own head-discharge relation. Divide the river into reaches at the nodes described in part (ii).
3. Choose the routing method appropriate to the river and the question. The choice is governed by which terms of the momentum equation matter. $$\underbrace{\frac{1}{g}\frac{\partial V}{\partial t}}_{\text{local}} + \underbrace{\frac{V}{g}\frac{\partial V}{\partial x}}_{\text{convective}} + \underbrace{\frac{\partial h}{\partial x}}_{\text{pressure}} + \underbrace{S_f - S_0}_{\text{friction, gravity}} \;=\; 0$$ Retaining only the last group gives the kinematic wave, adequate for steep rivers with no backwater; adding the pressure term gives the diffusion wave, which reproduces attenuation and is the basis of Muskingum–Cunge; retaining everything gives the full dynamic wave, which is required here because the question asks for wave height and for the effect of a sudden event, and because backwater at confluences and structures is likely. In practice this means an unsteady one-dimensional solver such as HEC-RAS or MIKE 11 applied to the Saint-Venant system, with the continuity equation $\partial A/\partial t + \partial Q/\partial x = q_L$ solved simultaneously with the momentum equation above.
4. Set boundary and initial conditions. Upstream, the inflow hydrograph from step 1; downstream, a rating curve, a normal-depth assumption, or a known water level if the river discharges to a lake, reservoir or the sea; laterally, the tributary and local inflows. The initial condition is a steady-flow water-surface profile computed for the base flow prevailing before the event.
5. Calibrate and verify against a past flood. Roughness coefficients are adjusted until the model reproduces an observed event — the recorded downstream hydrograph, the observed peak stages, and surveyed high-water marks — and the calibrated model is then verified on a second, independent event, exactly as in Problem 3(ii). No prediction should be presented without this step.
6. Run the design event and extract the three required quantities. At each cross-section of interest the model produces a full time series of discharge and stage, from which:
     • the peak flow rate is the maximum of the routed discharge hydrograph, which decreases downstream as storage in the channel and flood plain attenuates the wave;
     • the maximum wave height is the maximum stage, converted to elevation and to depth above the pre-event water surface;
     • the time to crest is the clock time of that maximum, and the difference between crest times at successive sections divided by the distance between them gives the observed wave celerity.
7. Cross-check the model with an independent hand calculation. This is what distinguishes an engineered answer from a software output. For a wide channel with Manning friction, the kinematic celerity of the flood wave is $c = \frac{5}{3}V$, so with a mean flood velocity of $V = 1.8$ m/s the wave travels at $c = 3.0$ m/s and covers a 24 km reach in $24\,000/3.0 = 8000$ s = 2.2 h. If the model reports a crest lag of two to three hours over that reach, it is behaving correctly; if it reports twenty minutes or twenty hours, something is wrong with the geometry or the roughness. Attenuation can be checked against the reach storage: the volume held between the inflow and outflow hydrographs must equal the change in reach storage computed from the cross-sections.
8. Present the result with its uncertainty, and translate it into action. Report the peak discharge, peak stage and crest time at each location with a stated confidence band reflecting the uncertainty in the design rainfall, the loss parameters and the roughness; map the inundation extent by intersecting the peak profile with the terrain model; and convert the crest times into available warning lead time for the flood alert system of Problem 4(ii).