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

Question 5: Channel or river routing and 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.

(i) What channel routing does, and the equations behind it (7 marks)

Channel routing takes a hydrograph known at an upstream section and predicts the hydrograph at a section some distance downstream. Two things happen to a flood wave as it travels, and routing quantifies both. Translation moves the whole hydrograph later in time by the travel time of the wave, which is what the engineer needs in order to say when the crest will arrive at a town, a bridge or a confluence. Attenuation reduces the peak and spreads the wave, because a reach stores water on its rising limb and gives it back on the falling limb; a wide floodplain stores a great deal and flattens the crest strongly, a confined rock channel stores little and passes the crest almost intact. Routing is therefore the tool that answers the two design questions how high will the crest be here and how long have we got.

The practical uses follow directly. In flood forecasting the routed crest and its arrival time set the warning lead time. In design, routing a flood from a proposed dam or from a tributary confluence establishes whether two flood waves will superimpose — a hydrograph that is delayed by twelve hours may miss the tributary peak entirely, or may coincide with it and produce a crest neither branch could produce alone. In floodplain mapping the routed peak at each section is the input to the water-surface profile computation that draws the flood line.

The fundamental equations. All routing rests on the one-dimensional unsteady flow equations of Barré de Saint-Venant, which are continuity and momentum for a channel:

$$\frac{\partial A}{\partial t}+\frac{\partial Q}{\partial x}=q_L \qquad\text{(continuity)}$$

$$\frac{1}{A}\frac{\partial Q}{\partial t} +\frac{1}{A}\frac{\partial}{\partial x}\!\left(\frac{Q^{2}}{A}\right) +g\frac{\partial y}{\partial x}-g\bigl(S_0-S_f\bigr)=0 \qquad\text{(momentum)}$$

in which $A$ is flow area, $Q$ discharge, $q_L$ lateral inflow per unit length, $y$ depth, $S_0$ bed slope and $S_f$ friction slope. Solving both is dynamic routing, and it is what HEC-RAS unsteady and MIKE 11 do. Dropping the two inertia terms leaves the diffusion wave; dropping the pressure term as well leaves the kinematic wave, for which $S_f=S_0$ and the discharge is a single-valued function of depth. Hydrologic routing methods such as Muskingum abandon the momentum equation altogether and replace it with an empirical storage relation, retaining only the lumped continuity statement

$$I-O=\frac{\mathrm{d}S}{\mathrm{d}t}$$

which is why they are cheap, why they need calibration data, and why they cannot represent backwater.

Travel time comes from the celerity of the flood wave, which is not the water velocity but $c=\mathrm{d}Q/\mathrm{d}A$. For a wide channel under Manning friction this evaluates to $c=\tfrac{5}{3}V$, so a reach in which the mean velocity at the flood stage is 1.2 m/s propagates its crest at

$$c=\tfrac{5}{3}(1.2)=2.0\ \text{m/s} \quad\Longrightarrow\quad t_{\text{travel}}=\frac{18\,000\ \text{m}}{2.0\ \text{m/s}}=9000\ \text{s}=\boxed{2.5\ \text{h}}$$

for an 18 km reach. The crest therefore arrives two and a half hours after it passes the upstream gauge — noticeably sooner than the water itself, which is the practical reason a flood warning cannot be based on tracking a parcel of water downstream.

(ii) Two assumptions in a typical routing method, and their design consequences (7 marks)

Take the Muskingum method as the typical case, since it is the one most often used by hand and the one embedded in most hydrologic packages.

Assumption 1 — reach storage is a linear, single-valued function of weighted discharge, with $K$ and $X$ constant over the whole range of flows. The method writes $S=K[XI+(1-X)O]$ and then uses one pair of $K$ and $X$ for the entire event. In reality the wave celerity increases with discharge, so $K$ shortens as the flood rises; and once the flow leaves the main channel and spreads across the floodplain the storage-discharge relation changes character completely, because a small rise in stage now recruits an enormous area of shallow storage. Design consequence. Parameters calibrated on in-bank events under-predict attenuation and mis-time the crest for the overbank design flood — typically the routed peak is too high and too early, which is conservative for structure sizing but dangerously optimistic for warning lead time. The engineer's response is to calibrate on the largest observed events available, to state the range of discharge over which the parameters are valid, and to move to a diffusion-wave or full dynamic model when the design event is well outside that range. It also means routing parameters cannot simply be transferred to a reach after channelisation, dyking or floodplain development, all of which change the storage relation.

Assumption 2 — the flow is one-dimensional with no backwater, so stage and discharge are uniquely related at each section and the wave travels downstream only. Hydrologic routing has no momentum equation, so it cannot represent a downstream control propagating an effect upstream: a tidal boundary, a reservoir drawing its pool up into the reach, a bridge or a culvert choking the section, ice jamming, or a tributary in flood holding back the main stem. It also cannot represent the looped rating discussed in Problem 3(iii), since it assumes discharge depends on stage alone. Design consequence. Wherever a downstream control exists, the routed answer is simply wrong, and the engineer must recognise the situation and switch to a dynamic model — this is the usual reason a floodplain study on a low-gradient river is done in HEC-RAS unsteady rather than by Muskingum. It also drives the layout decision: if a proposed embankment or bridge opening creates a backwater effect, its consequences cannot be assessed with the tool that produced the original design flows, and the study must be redone. A further practical assumption worth naming is that lateral inflow within the reach is negligible; where it is not, the reach must be subdivided at the tributaries and the local hydrographs added, or the routed crest will be too small.

Both assumptions push the engineer in the same direction: state the range of validity, calibrate inside it, and treat any application outside it as requiring a different method rather than a different parameter.

(iii) The Muskingum-Cunge method (6 marks)

The classical Muskingum method has a serious practical weakness: its two parameters have no physical meaning and must be found by trial from observed inflow and outflow hydrographs, so it cannot be used on an ungauged reach or on a channel that does not yet exist. Cunge's contribution was to show that the Muskingum routing equation is a finite-difference approximation to the diffusion-wave form of the Saint-Venant equations, and that matching the numerical diffusion of the scheme to the physical diffusion of the flood wave fixes $K$ and $X$ from channel geometry and hydraulics alone. The method is therefore physically based but computationally as cheap as Muskingum, and it is the routing engine inside HEC-HMS and many design packages.

The computational equation is the ordinary Muskingum form,

$$O_2=C_0I_2+C_1I_1+C_2O_1$$

with the same three coefficients, but the parameters are now evaluated as

$$K=\frac{\Delta x}{c},\qquad X=\frac{1}{2}\left(1-\frac{Q}{B\,S_0\,c\,\Delta x}\right)$$

where $\Delta x$ is the reach sub-length, $c=\mathrm{d}Q/\mathrm{d}A$ the kinematic wave celerity, $Q/B$ the discharge per unit width, $B$ the top width and $S_0$ the bed slope. Because $c$, $Q$ and $B$ all vary with stage, the parameters are recomputed at every time step and every sub-reach in the variable-parameter form, which is what lets the method reproduce the steepening of the rising limb that fixed-parameter Muskingum cannot.

For a representative reach with $c=1.5$ m/s, $\Delta x=5000$ m, unit-width discharge $Q/B=2.0$ m²/s and $S_0=0.001$,

$$K=\frac{5000}{1.5}=3333\ \text{s}=0.926\ \text{h},\qquad X=\frac{1}{2}\left(1-\frac{2.0}{0.001\times 1.5\times 5000}\right)=\boxed{0.367}$$

values that would otherwise have required a gauged event to obtain.

Two key assumptions. First, the flood wave is a diffusion wave: the local and convective acceleration terms of the momentum equation are negligible, so the friction slope is balanced by the bed slope and the water-surface slope alone. This holds on steep and moderate channels with gradually rising hydrographs and fails on very flat rivers with rapidly varying flow, and — as with all hydrologic routing — it excludes backwater and any downstream control. Second, the channel is prismatic within each sub-reach and the flow is one-dimensional with a single-valued rating, so that a representative $c$, $B$ and $S_0$ can be assigned to the sub-reach; lateral inflow is neglected unless added explicitly. A practical corollary is the resolution constraint: $\Delta x$ and $\Delta t$ must be chosen together, usually so that the Courant number $c\,\Delta t/\Delta x$ is near unity and $X$ stays non-negative, otherwise the scheme's numerical diffusion no longer matches the physical diffusion and the routed hydrograph either over-smooths or oscillates.