# Muskingum Routing Method The Muskingum routing method models the storage volume in a channel length as a combination of wedge and prism storages (Figure 7:1-3). ![Figure 7:1-3: Prism and wedge storages in a reach segment (After Chow et al., 1988)](https://1348478613-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MTFpIHUB1K63DLhLJaf%2Fuploads%2FCxKYn37n0I22QXpJMhoe%2Fa1.jpg?alt=media\&token=789358d9-5ab2-466c-8076-062925d92a0c) 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, $$K*q\_{out}$$, where $$K$$ 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 $$K*X\*(q\_{in}-q\_{out})$$, where $$X$$ 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 $$V\_{stored}=K*q\_{out}+K*X\*(q\_{in}-q\_{out})$$ 7:1.4.1 where $$V\_{stored}$$ is the storage volume (m$$^3$$ H$$*2$$O), $$q*{in}$$ is the inflow rate (m$$^3$$/s), $$q\_{out}$$ is the discharge rate (m$$^3$$/s), $$K$$ is the storage time constant for the reach (s), and $$X$$ is the weighting factor. This equation can be rearranged to the form $$V\_{stored}=K\*(X\*q\_{in}+(1-X)\*q\_{out})$$ 7:1.4.2 This format is similar to equation 7:1.3.7. The weighting factor, $$X$$, 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 $$X=0.0$$. For a full-wedge, $$X=0.5$$. For rivers, $$X$$ 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 $$q\_{out,2}=C\_1*q\_{in,2}+C\_2*q\_{in,1}+C\_3\*q\_{out,1}$$ 7:1.4.3 where $$q\_{in,1}$$ is the inflow rate at the beginning of the time step (m$$^3$$/s), $$q\_{in,2}$$ is the inflow rate at the end of the time step (m$$^3$$/s), $$q\_{out,1}$$ is the outflow rate at the beginning of the time step (m$$^3$$/s), $$q\_{out,2}$$ is the outflow rate at the end of the time step (m$$^3$$/s), and $$C\_1=\frac{\Delta t-2*K*X}{2*K*(1-X)+\Delta t}$$ 7:1.4.4 $$C\_2=\frac{\Delta t+2*K*X}{2*K*(1-X)+ \Delta t}$$ 7:1.4.5 $$C\_3=\frac{2*K*(1-X)- \Delta t}{2*K*(1-X)+\Delta t}$$ 7:1.4.6 where $$C\_1+C\_2+C\_3=1$$. To express all values in units of volume, both sides of equation 7:1.4.3 are multiplied by the time step $$V\_{out,2}=C\_1*V\_{in,2}+C\_2*V\_{in,1}+C\_3\*V\_{out,1}$$ 7:1.4.7 To maintain numerical stability and avoid the computation of negative outflows, the following condition must be met: $$2*K*X<\Delta t<2*K*(1-X)$$ 7:1.4.8 The value for the weighting factor, $$X$$, is input by the user. The value for the storage time constant is estimated as: $$K=coef\_1*K\_{bnkfull}+coef\_2*K\_{0.1bnkfull}$$ 7:1.4.9 where $$K$$ is the storage time constant for the reach segment (s), $$coef\_1$$ and $$coef\_2$$ are weighting coefficients input by the user, $$K\_{bnkfull}$$ is the storage time constant calculated for the reach segment with bankfull flows (s), and $$K\_{0.1bnkfull}$$ is the storage time constant calculated for the reach segment with one-tenth of the bankfull flows (s). To calculate $$K\_{bnkfull}$$ and $$K\_{0.1bnkfull}$$, an equation developed by Cunge (1969) is used: $$K=\frac{1000\*L\_{ch}}{c\_k}$$ 7:1.4.10 where $$K$$ is the storage time constant (s), $$L\_{ch}$$ is the channel length (km), and $$c\_k$$ 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 $$c\_k=\frac{d}{dA\_{ch}}(q\_{ch})$$ 7:1.4.11 where the flow rate, $$q\_{ch}$$, is defined by Manning’s equation. Differentiating equation 7:1.2.1 with respect to the cross-sectional area gives $$c\_k=\frac{5}{3}\*(\frac{R\_{ch}^{2/3}\*slp\_{ch}^{1/2}}{n})=\frac{5}{3}\*v\_c$$ 7:1.4.12 where $$c\_k$$ is the celerity (m/s), $$R\_{ch}$$ is the hydraulic radius for a given depth of flow (m), $$slp\_{ch}$$ is the slope along the channel length (m/m), n is Manning’s “$$n$$” coefficient for the channel, and $$v\_c$$ 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 | $$X$$: weighting factor | .bsn | | MSK\_CO1 | $$coef\_1$$: weighting factor for influence of normal flow on storage time constant value | .bsn | | MSK\_CO2 | $$coef\_2$$: weighting factor for influence of low flow on storage time constant | .bsn |