9.2 Flood routing methods

Flood Routing Methods

Flood routing predicts how a flood wave changes shape as it moves through a river reach or reservoir. The wave doesn't just translate downstream unchanged; it gets attenuated (the peak drops) and delayed (the peak arrives later) because of storage effects in the channel, floodplains, or reservoir pool. Understanding this behavior is essential for flood forecasting, issuing evacuation warnings, and operating dams during flood events.

Two broad categories of routing methods exist: hydrologic routing (simpler, based on the continuity equation alone) and hydraulic routing (more complex, solving the full or simplified Saint-Venant equations). This section covers the most common method in each category, plus reservoir routing.

Principles of flood routing

Flood routing answers a practical question: given a known inflow hydrograph at an upstream point, what will the outflow hydrograph look like at a downstream point?

As a flood wave travels downstream, two things happen:

• Attenuation: The peak discharge decreases because water spreads into channel storage, floodplains, and wetlands.
• Translation (delay): The peak arrives later at the downstream point because the wave takes time to travel through the reach.

The degree of attenuation and delay depends on channel geometry, slope, roughness, and floodplain characteristics.

Two main approaches:

• Hydrologic routing treats flow as a function of time only (at discrete points along the reach). It uses the continuity equation plus an empirical storage-discharge relationship.
  • Examples: Muskingum method, Muskingum-Cunge method
• Hydraulic routing solves the Saint-Venant equations, which describe unsteady open-channel flow in both time and space.
  • Examples: Dynamic wave model (full Saint-Venant), Diffusion wave model (simplified)

Muskingum method for rivers

The Muskingum method is the most widely used hydrologic routing technique for river reaches. It works by relating the storage in a reach to the inflow and outflow using two parameters.

Storage concept: Total storage in a reach is split into two components:

• Prism storage: The volume of water that would exist if inflow equaled outflow (uniform flow depth along the reach).
• Wedge storage: The additional (or deficit) volume caused by the difference between inflow and outflow. During a rising flood, inflow exceeds outflow, creating a positive wedge; during recession, the wedge is negative.

Storage equation:

$S = K[XI + (1-X)Q]$

where:

• $S$ = storage in the reach
• $I$ = inflow to the reach
• $Q$ = outflow from the reach
• $K$ = storage constant, approximately equal to the flood wave travel time through the reach (units of time)
• $X$ = weighting factor that controls how much inflow vs. outflow influences storage ($0 \leq X \leq 0.5$)

When $X = 0$, storage depends only on outflow (reservoir-like behavior with maximum attenuation). When $X = 0.5$, the flood wave translates downstream with no attenuation. Most natural rivers have $X$ values between 0.1 and 0.3.

Routing procedure (step-by-step):

1. Determine parameters $K$ and $X$ from observed inflow-outflow data (or calibration).
2. Choose a time step $\Delta t$. A good rule: $\Delta t$ should satisfy $2KX \leq \Delta t \leq 2K(1-X)$ to keep coefficients positive and the solution stable.
3. Calculate the routing coefficients:

$C_0 = \frac{\Delta t - 2KX}{2K(1-X) + \Delta t}$

$C_1 = \frac{\Delta t + 2KX}{2K(1-X) + \Delta t}$

$C_2 = \frac{2K(1-X) - \Delta t}{2K(1-X) + \Delta t}$

Check that $C_0 + C_1 + C_2 = 1$. This serves as a quick verification.

1. For each time step $j$, compute the outflow at the next time step:

$Q_{j+1} = C_0 I_{j+1} + C_1 I_j + C_2 Q_j$

1. Repeat step 4 through the entire inflow hydrograph to generate the routed outflow hydrograph.

Reservoir routing techniques

Reservoir routing determines how an inflow hydrograph is transformed into an outflow hydrograph as it passes through a reservoir. Because reservoirs have large storage volumes relative to their throughflow, they typically produce significant attenuation of the flood peak.

Continuity equation (discrete form):

$\frac{dS}{dt} = I - Q$

where $S$ is reservoir storage, $I$ is inflow, and $Q$ is outflow.

The storage-outflow relationship $S = f(Q)$ is unique to each reservoir and depends on its physical characteristics (area-elevation curve) and outlet structures (spillways, gates, orifices). This relationship is typically developed from the reservoir's stage-storage and stage-discharge curves.

Modified Puls method (level pool routing):

This is the standard approach for reservoirs where the water surface can be assumed approximately horizontal (the "level pool" assumption). It works well for most reservoirs where the pool length is short relative to the flood wave length.

The routing equation rearranges the continuity equation into:

$\frac{2S_{j+1}}{\Delta t} + Q_{j+1} = (I_j + I_{j+1}) + \left(\frac{2S_j}{\Delta t} - Q_j\right)$

Step-by-step procedure:

1. Develop the storage-outflow relationship $S = f(Q)$ from the reservoir's stage-storage and stage-discharge data.

2. Construct an auxiliary curve (or table) of $\frac{2S}{\Delta t} + Q$ versus $Q$. This is the key working relationship.

3. At each time step, compute the right-hand side of the routing equation. The known quantities are $I_j$, $I_{j+1}$, and the value of $\frac{2S_j}{\Delta t} - Q_j$ from the previous step.

4. Use the auxiliary curve to look up $Q_{j+1}$ corresponding to the computed value of $\frac{2S_{j+1}}{\Delta t} + Q_{j+1}$.

5. Update $\frac{2S_{j+1}}{\Delta t} - Q_{j+1}$ for the next time step by subtracting $2Q_{j+1}$ from the value found in step 4.

6. Repeat through the entire inflow hydrograph.

Comparison of routing methods

Choosing a method: Start with the simplest approach that fits your situation. If you have a reasonably uniform river reach and limited data, the Muskingum method is often sufficient. If backwater effects, tributaries, or complex geometry matter, hydraulic routing is necessary despite the added effort. For reservoir operations, the Modified Puls method is the standard unless the reservoir is so large that the level pool assumption breaks down.