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).

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.