Channel Routing Hydrograph - Muskingum-Cunge Method

The Muskingum-Cunge routing method is based on the combination of the conservation of mass and the diffusion representation of the conservation of momentum. It is sometimes referred to as a variable coefficient method because the routing parameters are recalculated every time step based on channel properties and the flow depth. It represents attenuation of flood waves and can be used in reaches with a small slope.

Although popular and easy to use, the Muskingum model includes parameters that are not physically based and thus are difficult to estimate. Further, the model is based upon assumptions that often are violated in natural channels. An extension, the Muskingum-Cunge model, overcomes these limitations.

The continuity equation is (with lateral inflow qL, included):

    ∂A∂t+∂Q∂x=qL

and the diffusion form of the momentum equatiand the diffusion form of the momentum equation:
    Sf=So−∂y∂x

Combining these and using a linear approximation yields the convective diffusion equation (Miller and Cunge, 1975):

    ∂Q∂t+c∂Q∂x=μ∂2Q∂x2+cqL

where c = wave celerity (speed); and μ = hydraulic diffusivity. The wave celerity and the hydraulic diffusivity are expressed as follows:

    c=dQdA

and

    μ=Q2BSo

where B = top width of the water surface. A finite difference approximation of the partial derivatives, combined with Equation 80, yields:

Ot=C1It−1+C2It+C3Ot−1+C4(qLΔx)

The coefficients are:

    C1C2C3C4=ΔtK+2XΔtK+2(1−X)=ΔtK−2XΔtK+2(1−X)=2(1−X)−ΔtKΔtK+2(1−X)=2(ΔtK)ΔtK+2(1−X)

The parameters K and X are (Cunge, 1969; Ponce, 1978):

    K=ΔxcX=12(1−QBSocΔx)

But c, Q, and B change over time, so the coefficients C₁, C₂, C₃, and C₄ must also change. The program recomputes them at each time and distance step, Δt and Δx, using the algorithm proposed by Ponce (1986).
Again, the choice of these time and distance steps is critical. The steps are selected to ensure accuracy and stability. The Δt is selected as the minimum of the following: user time step from the control specifications; the travel time through the reach; or 1/20^th the time to rise of the peak inflow with the steepest rising limb, rounded to the nearest multiple or divisor of the user time step. Once Δt is chosen, Δx is computed as:

    Δx=cΔt

The value is constrained so that:

    Δx<12(cΔt+QoBSoc)

Here Qo = reference flow, computed from the inflow hydrograph as:

    Qo=QB+12(Qpcak−QB)

where QB = baseflow; and Qpeak = inflow peak.

Please refer to the HEC-HMS for the detail of the Muskingum-Cunge Model

where QB = baseflow; and Qpeak = inflowSelect Muskingum-Cunge Model from Watershed > Hydrograph Routing > Channel Routing menu. Specify the inflow hydrograph and enter the channel parameters and routing interval. The routing interval is as same as the time increment of the inflow hydrograph.  Once the parameters are entered, click on the Calculate button to compute the coefficients. OK button accepts the values and proceeds to the previous dialog.

After entering all information, click Routed Hydrograph button to generate the routing hydrograph. A dialog is opened with the tabular and graphic hydrograph data, from there you can draw the hydrograph on screen and save it to a file.

Channel Routing Hydrograph - Muskingum-Cunge Model

Channel Routing Hydrograph - Routed Hydrograph Dialog