Flood Routing

Flood Routing

Techniques to predict the flood hydrograph at a downstream section, including Reservoir and Channel Routing methods like Muskingum.

What is Flood Routing?

The technique of determining the flood hydrograph at a downstream section of a river or reservoir by utilizing the data of flood flow at one or more upstream sections.

It is used to predict two main properties of a flood wave:

Magnitude (Attenuation)

The reduction of the peak flow downstream.

Timing (Lag)

The delay in time of the peak flow.

Lumped vs. Distributed Routing

Hydrologic Routing (Lumped): Uses the continuity equation and storage-discharge relationships (e.g., Muskingum, Modified Puls). It treats the reach as a black box. Hydraulic Routing (Distributed): Solves the continuity and momentum equations (St. Venant Equations) simultaneously. It describes flow at every point along the channel.

Hydraulic Routing (St. Venant Equations)

While hydrologic routing methods (Muskingum, Level Pool) are lumped models based only on continuity, Hydraulic Routing is a distributed method. It computes flow as a function of both space (

x

) and time (

t

) by simultaneously solving the Saint-Venant Equations of 1D unsteady open channel flow.

Saint-Venant Equations

A pair of partial differential equations:

Continuity Equation: Conservation of Mass ( $\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = q$ ).
Momentum Equation: Conservation of Momentum, balancing forces of gravity, friction, pressure gradient, and convective/local acceleration.

Simplified Hydraulic Models

Depending on the channel slope, some terms in the Momentum equation can be neglected: Kinematic Wave assumes friction and gravity forces balance exactly (ignoring pressure and acceleration). Diffusion Wave includes pressure gradients, allowing the wave to attenuate (flatten) over time, unlike the purely translating kinematic wave.

Reservoir Routing (Level Pool Routing)

Used for routing floods through reservoirs or lakes where the water surface is horizontal. The inflow hydrograph (

I

) is modified by the reservoir's storage (

S

) to produce an outflow hydrograph (

O

Continuity Equation:

Continuity Equation

I - O = \frac{dS}{dt}

Discretized for a time interval

\Delta t

Discretized Continuity Equation

\frac{I_1 + I_2}{2} \cdot \Delta t - \frac{O_1 + O_2}{2} \cdot \Delta t = S_2 - S_1

This is often solved using the Storage-Indication Method or Modified Puls Method by constructing a curve of

(2S/\Delta t + O)

O

Muskingum Method (Channel Routing)

Used for routing floods in river channels. Unlike reservoirs, storage in a channel is a function of both inflow (

I

) and outflow (

O

) due to the wedge storage effect.

Muskingum Storage Equation

S = K [xI + (1-x)O]

Variables

$K$ : Storage time constant (has units of time). It approximates the travel time of the flood wave through the reach.
$x$ : Weighting factor (dimensionless, 0 to 0.5). Represents the relative importance of inflow vs. outflow on storage.
$x=0$ : Reservoir-type storage (Linear Reservoir).
$x=0.5$ : Pure translation (wedge storage equals prism storage). Typical natural streams have $x \approx 0.2$ .

Muskingum-Cunge Method

The original Muskingum method requires historical flood data to calibrate the parameters

K

and

x

. The Muskingum-Cunge method overcomes this limitation by linking the routing parameters directly to the physical hydraulic characteristics of the channel (geometry, slope, roughness).

Physical Basis of Muskingum-Cunge

By matching the Muskingum finite difference scheme with a simplified form of the Saint-Venant momentum equation (diffusion wave), Cunge demonstrated that

K

and

x

can be calculated directly.

$K$ is related to the travel time through the reach length $\Delta x$ at celerity $c$ ( $K = \Delta x / c$ ).
$x$ is derived from the channel width ( $B$ ), slope ( $S_0$ ), and discharge ( $Q$ ), acting as a diffusion term. This makes Muskingum-Cunge a pseudo-hydraulic method, highly powerful for ungauged rivers.

Note

For numerical stability in the Muskingum method, the routing interval

\Delta t

should be chosen such that

2Kx \le \Delta t \le K

Routing Equation

The outflow at the next time step (

O_2

) is calculated from known values (

I_1, I_2, O_1

Muskingum Routing Equation

O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1

Where the coefficients are derived as:

Routing Coefficients

C_0 = \frac{-Kx + 0.5\Delta t}{K(1-x) + 0.5\Delta t} \\ C_1 = \frac{Kx + 0.5\Delta t}{K(1-x) + 0.5\Delta t} \\ C_2 = \frac{K(1-x) - 0.5\Delta t}{K(1-x) + 0.5\Delta t}

Note

Check: The sum of coefficients must always equal 1:

C_0 + C_1 + C_2 = 1

Muskingum Channel Routing Simulation

Adjust the Storage Time Constant (K) and Weighting Factor (x) to see how the flood wave is attenuated and delayed.

K (Storage Constant): 12 hours

x (Weighting Factor): 0.2

Δt (Time Step): 6 hours

C0: 0.0476

C1: 0.4286

C2: 0.5238

ΣC: 1

Loading chart...

The two primary types of flood routing are Reservoir Routing (lumped modeling, assumes horizontal water surface) and Channel Routing (distributed modeling, accounts for wedge storage as water slopes downstream).

Reservoir vs. Channel Routing

Reservoir Routing uses the Storage Equation (

\Delta S = I - Q

) assuming a level pool (Level Pool Routing). Storage is a function of outflow alone (

S = f(Q)

). Channel Routing accounts for both prism storage and wedge storage, meaning storage is a function of both inflow and outflow (

S = f(I, Q)

Hydraulic Routing and the St. Venant Equations

While hydrologic routing uses the continuity equation, hydraulic routing incorporates both continuity and momentum.

The St. Venant Equations

Hydraulic routing (e.g., dynamic wave routing) is based on the 1D shallow water equations, known as the St. Venant equations. These consist of a Continuity Equation (conservation of mass) and a Momentum Equation (conservation of momentum). They account for backwater effects, tidal influence, and rapid flow changes that hydrologic methods like Muskingum cannot handle.

Kinematic Wave Routing

A simplified form of the St. Venant equations that assumes the friction slope equals the bed slope, neglecting pressure and inertial terms. It is widely used in overland flow and simple stream routing where backwater effects are negligible.

Key Takeaways

Flood Routing tracks how a flood wave changes magnitude and speed as it travels downstream.
Attenuation is the flattening of the hydrograph peak, reducing flood severity.
Translation (Lag) is the delay in the arrival time of the peak flow.
Hydrologic routing uses simplified continuity, whereas Hydraulic routing solves the 1D Saint-Venant equations (continuity and momentum) for highly accurate distributed flow modeling.
Reservoir Routing assumes a horizontal water surface where storage is a function solely of outflow ( $S = f(O)$ ).
It relies on the fundamental continuity equation: $\text{Inflow} - \text{Outflow} = \text{Change in Storage}$ .
The Modified Puls Method is the standard numerical technique for solving level pool routing.
Flood Routing predicts the changes in a flood wave as it moves downstream.
Reservoir Routing assumes a level pool where storage depends only on outflow. Channel Routing must account for wedge storage, depending on both inflow and outflow.
Attenuation is the reduction of the peak discharge. Lag is the delay in time of peak.
Reservoir Routing assumes a level water surface (storage depends only on outflow).
Muskingum Method models storage as a function of both inflow and outflow, accounting for wedge storage.
The sum of Muskingum coefficients ( $C_0, C_1, C_2$ ) is always unity.
The parameter $K$ represents travel time, and $x$ weights the inflow vs outflow effects.

PreviousHydrographs - Examples & Applications

Quiz Me

NextFlood Routing - Examples & Applications