Muskingum Routing Method
Last updated
Last updated
The Muskingum routing method models the storage volume in a channel length as a combination of wedge and prism storages (Figure 7:1-3).
When a flood wave advances into a reach segment, inflow exceeds outflow and a wedge of storage is produced. As the flood wave recedes, outflow exceeds inflow in the reach segment and a negative wedge is produced. In addition to the wedge storage, the reach segment contains a prism of storage formed by a volume of constant cross-section along the reach length.
As defined by Manning’s equation (equation 7:1.2.1), the cross-sectional area of flow is assumed to be directly proportional to the discharge for a given reach segment. Using this assumption, the volume of prism storage can be expressed as a function of the discharge, , where is the ratio of storage to discharge and has the dimension of time. In a similar manner, the volume of wedge storage can be expressed as , where is a weighting factor that controls the relative importance of inflow and outflow in determining the storage in a reach. Summing these terms gives a value for total storage
7:1.4.1
where is the storage volume (m HO), is the inflow rate (m/s), is the discharge rate (m/s), is the storage time constant for the reach (s), and is the weighting factor. This equation can be rearranged to the form
7:1.4.2
This format is similar to equation 7:1.3.7.
The weighting factor, , has a lower limit of 0.0 and an upper limit of 0.5. This factor is a function of the wedge storage. For reservoir-type storage, there is no wedge and . For a full-wedge, . For rivers, will fall between 0.0 and 0.3 with a mean value near 0.2.
The definition for storage volume in equation 7:1.4.2 can be incorporated into the continuity equation (equation 7:1.3.2) and simplified to
7:1.4.3
where is the inflow rate at the beginning of the time step (m/s), is the inflow rate at the end of the time step (m/s), is the outflow rate at the beginning of the time step (m/s), is the outflow rate at the end of the time step (m/s), and
7:1.4.4
7:1.4.5
7:1.4.6
where . To express all values in units of volume, both sides of equation 7:1.4.3 are multiplied by the time step
7:1.4.7
To maintain numerical stability and avoid the computation of negative outflows, the following condition must be met:
7:1.4.8
The value for the weighting factor, , is input by the user. The value for the storage time constant is estimated as:
7:1.4.9
where is the storage time constant for the reach segment (s), and are weighting coefficients input by the user, is the storage time constant calculated for the reach segment with bankfull flows (s), and is the storage time constant calculated for the reach segment with one-tenth of the bankfull flows (s). To calculate and , an equation developed by Cunge (1969) is used:
7:1.4.10
where is the storage time constant (s), is the channel length (km), and is the celerity corresponding to the flow for a specified depth (m/s). Celerity is the velocity with which a variation in flow rate travels along the channel. It is defined as
7:1.4.11
where the flow rate, , is defined by Manning’s equation. Differentiating equation 7:1.2.1 with respect to the cross-sectional area gives
7:1.4.12
where is the celerity (m/s), is the hydraulic radius for a given depth of flow (m), is the slope along the channel length (m/m), n is Manning’s “” coefficient for the channel, and is the flow velocity (m/s).
Table 7:1-3: SWAT+ input variables that pertain to Muskingum routing.
Variable Name | Definition | File Name |
---|---|---|
MSK_X | : weighting factor | .bsn |
MSK_CO1 | : weighting factor for influence of normal flow on storage time constant value | .bsn |
MSK_CO2 | : weighting factor for influence of low flow on storage time constant | .bsn |