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Muskingum-Cunge Method
Explained


Victor M. Ponce
and
Janaina Da Silva


XX January 2020


Abstract. The


1.  INTRODUCTION

The


2.  RATIONALE

The Muskingum Method

The Muskingum method is a hydrologic routing method used to model the storage volume of a flood in a river channel. As a flood wave advance through the river, the volume entering the system (inflow) is greater than the volume leaving it (outflow), producing a positive wedge storage and a negative prism storage (Fig. 1a). After the flood wave have passed, recession is installed. During this period, the outflow is greater than the inflow, producing a negative wedge storage and a positive prism storage (Fig 1b).

forces acting in a control volume

Fig. 1  Definition sketch.

The cross-sectional area of the flood flow is assumed to be directly proportional to the discharge at the section. Therefore, the volume of the wedge storage is K X (I -O ), and the volume of the prism storage is K O (Fig. 2).

forces acting in a control volume

Fig. 1  Definition sketch.

The total storage is defined as the sum of these two components, resulting in:

V = K X (I -O ) + K O (1)

Which can be rearraged as:

V = K [X I + (1 - X ) O ] (2)

In which I is the inflow volume, O is the outflow volume, K is a proportionality coefficient which express the time of travel of the flood wave through the channel reach, and X is a weighting factor which depends on the shape of the wedge storage.

The routing equation for the Muskingum equation is then defined as:

                                                        
Q n + 1     =   C0 In + 1  +  C1 In   +   C2On

(3)

in which n is a temporal index, and:

              Δt - 2KX
C0 = __________________
           2K (1 - X ) + Δt
(4)

              Δt + 2KX
C1 = _____________________
           2K (1 - X ) + Δt
(5)

           2K (1 - X ) - Δt
C2 = _____________________
           2K (1 - X ) + Δt
(6)

in which Δt is the routing period (time interval).

The Muskingum flood routing is performed by solving Eq. 3, with the coefficients C0, C1 and C2 calculated by Eqs. 4, 5 and 6, respectively. The parameters K and X are assumed to be constant throughout the range of the flow and are determined by calibration using measured inflow and outflow hydrographs.

The Muskingum-Cunge Method

The modified version of the Muskingum method, created in 1969 by Cunge, is a viable alternative for cases where hydrologic data are not available, but where hydraulic data can be determined. The Muskingum-Cunge method is also an alternative to the more complex dynamic wave models, which lack robustness and have significant data requirements (Ponce and Chaganti, 1994).

Cunge (1969) rewrites Eq. 2 for the discharge at x = (j + 1) Δx and t = (n + 1) Δt as:

   n + 1                 n + 1            n             n           
Q j + 1     =   C0Qj       +  C1Qj   +   C2Qj + 1

(7)

in which j is a spatial index, n is a temporal index and C0, C1 and C2 are calculated by Eqs. 4, 5 and 6.

In the Muskingum-Cunge version, the parameters K and X are calculated by:

          Δx
K  =  _____
           c
(8)

          1                   q   
X  =  ___ ( 1  -  __________ )
          2              So c Δx
(9)

in which Δx = reach length (space interval); c = flood wave celerity; q = unit width discharge; and So = channel bed slope.

The Muskingum-Cunge flood routing is performed by solving Eq. 7, with the coefficients C0, C1 and C2 calculated by Eqs. 4, 5 and 6, respectively. The parameters K and X are calculated using the formulas derived by Cunge (Eqs. 8 and 9) and may vary in time and space.

The method has linear and nonlinear modes. In the linear mode, average flow values are used to calculate the routing parameters at the start of the computation, and these are kept constant throughout the computation in time. In the nonlinear mode, the routing parameters are recalculated for every computational cell as a function of local flow values (Ponce and Chaganti, 1994).

Ponce and Yevjevich, 1978 have shown that the way of calculating the parameters has a definite bearing on the overall accuracy of the Muskingum-Cunge method. They assessed the mass conservation properties of the method with variable parameters when compared with the results obtained by Thomas, 1934 in his application of the method with constant parameters. The parameters were calculated directly by using a two, three and four-point average of the values. The three and four-point showed better results, which fall within the range encompassed by the constant parameter calculations.

Later, in 1994, Ponce and Chaganti demonstrated that better results can be attained by

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3.  DESCRIPTION

the

forces acting in a control volume

Fig. 1  Definition sketch.


4.  EXAMPLE

The


5.  DISCUSSION

The


6.  SUMMARY

The


REFERENCES

Ponce, V. M., and V. Yevjevich. 1978. Muskingum-Cunge method with variable parameters. Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, December, 1663-1667.


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