River / Channel Routing Worksheet
Muskingum Hydrologic Routing
Peak Inflow
0.0 m³/s
Hr 0.0
Peak Outflow
0.0 m³/s
Hr 0.0
Attenuation
0.0%
Peak Reduction
Translation Lag
0.0 hr
Time Difference
Routed Hydrograph (Inflow vs Outflow)
Reach Storage vs Outflow (Wedge Effect)
Routing Time-Series Results
| Time (hr) | Inflow I (m³/s) | Outflow O (m³/s) | Storage S (m³) |
|---|
Hydrologic Basis: Muskingum Method
1. The Storage Concept (Prism and Wedge)
In natural channels, storage is not just a function of outflow (like in a level-pool reservoir). It depends on both inflow and outflow. During the rising limb of a flood, a "wedge" of water forms as inflow exceeds outflow. The Muskingum storage equation accounts for this:
$$S = K [X I + (1 - X) O]$$
Where $K$ is the travel time through the reach, and $X$ is a weighting factor expressing the relative importance of inflow vs outflow ($0 \le X \le 0.5$).
2. The Routing Equation
Combining the storage equation with the continuity equation $\left(\frac{I_1 + I_2}{2} - \frac{O_1 + O_2}{2} = \frac{S_2 - S_1}{\Delta t}\right)$ yields the finite-difference routing equation:
$$O_2 = C_0 I_2 + C_1 I_1 + C_2 O_1$$
Where the routing coefficients are:
- $C_0 = \frac{\Delta t - 2KX}{2K(1-X) + \Delta t}$
- $C_1 = \frac{\Delta t + 2KX}{2K(1-X) + \Delta t}$
- $C_2 = \frac{2K(1-X) - \Delta t}{2K(1-X) + \Delta t}$
3. Numerical Stability Criteria
To prevent the routing coefficients from becoming negative and causing the calculated hydrograph to oscillate wildly (or dip below zero), the time step $\Delta t$ must fall within a specific mathematical window:
$$2KX \le \Delta t \le 2K(1-X)$$
If $\Delta t$ is too small, $C_0$ becomes negative. If $\Delta t$ is too large, $C_2$ becomes negative.
4. Reference Parameter Guidelines
Weighting Factor (X) Guidelines
| Channel Type / Condition | Typical X Value |
|---|---|
| Reservoir / Level Pool | 0.00 |
| Natural River w/ Wide Floodplains | 0.10 - 0.20 |
| Standard Natural Stream | 0.20 - 0.30 |
| Man-made / Concrete Channel | 0.40 - 0.50 |
| Note: X = 0.5 implies pure translation (no peak flow attenuation). Average natural river default is typically 0.20. | |
Estimating Travel Time (K)
Formula: $K \approx \frac{L}{V_w}$ (Reach Length ÷ Wave Velocity)
Flood wave velocity ($V_w$) travels faster than average water velocity ($V$) due to kinematic wave effects.
| Channel Geometry | Wave Vel. ($V_w$) |
|---|---|
| Wide Rectangular | 1.67 × V |
| Wide Parabolic | 1.44 × V |
| Triangular / V-Ditch | 1.33 × V |
| Rule of Thumb: $K$ should roughly equal the time difference between the peak of the inflow hydrograph and the peak of the outflow hydrograph. | |
References & Sources
- Chow, V. T., Maidment, D. R., & Mays, L. W. (1988). Applied Hydrology. McGraw-Hill. (Definitive derivation of the Muskingum method and stability criteria). [Search on Google Scholar]
- US Army Corps of Engineers (USACE). (1994). EM 1110-2-1417: Flood-Runoff Analysis. (Guidelines for estimating K and X in natural river reaches). [View USACE Manual]
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