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Hydrologic River Flood Routing

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Basic Equation

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- For a non-uniform flow, Stf(Ot)
- Because It=f(y1)and Ot= f(y2). Therefore, St=f(It,Ot).
- Prism storage = f(Ot); Wedge storage = f(It, Ot).
- Assume: Prism storage = KOt; Wedge storage = XK (It – Ot)
- Total storage volume at any time instant, t, is
St = KOt + XK (It - Ot) = K [X It + (1 - X)Ot]

where K = travel time of flood wave in a reach, [hrs];

X = weighting factor, 0 ~ 0.5

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- Referring to figure, the continuity equation in difference form can be expressed as

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- By Muskingum Model,
at t = t2, S2 = K [X I2 + (1 - X)O2]

at t = t1, S1 = K [X I1 + (1 - X)O1]

- Substituting S1, S2 into thecontinuity equation and after some algebraic manipulations, one has
O2 = Co I2 + C1 I1 + C2 O1

- Replacing subscript 2 by t +1 and 1 by t, the Muskingum routing equation is
Ot+1 = Co It+1 + C1 It + C2 Ot, for t = 1, 2, …

where ; ; C2 = 1 – Co – C1

Note: K and t must have the same unit.

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- Given (knowns): O1; I1, I2, …; t; K; X
- Find (unknowns): O2, O3, O4, …
- Procedure:
(a) Calculate Co, C1, and C2

(b) Apply Ot+1 = Co It+1 + C1 It + C2 Otstarting from t=1, 2, … recursively.

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Graphical Method:

- Referring to the Muskingum Model, find X such that the plot of XIt+ (1-X)Ot vs St behaves almost nearly as a single value curve. The corresponding slope is K.

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- Gill’s Least Square Method (Gill, 1978, “Flood routing by Muskingum method”, J. of Hydrology, V.36)
St = K[XIt+(1-X)Ot] = KX It+K(1-X)Ot = AIt+BOt

where A = KX, B = K(1-X)

- Model: St’ = So + A It + B Ot, t = 1, 2, …, n
where St’ = cumulative relative storage at time t;

So = initial storage of channel before flood

- Use least squares method to solve for A and B and So. Then, K and X can be obtained as
K = A + B, X = A / (A + B)

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- Stephensen’s Least Squares Method (Stephensen, 1979, “Direct optimization of Muskingum routing coefficient, “ J. of Hydrology, V.41)
- Ot+1 = Co It+1 + C1 It + C2 Ot, t = 1, 2, …, n-1
- Use least squares method to estimate the values of Co, C1 and C2 directly. (Does not guarantee satisfying Co + C1 + C2 = 1)
- Alternatively, Co + C1 + C2 =1, C2 = 1 – (Co + C1)
(Ot+1 - Ot) = Co (It+1 - Ot) + C1 (It - Ot), t = 1, 2, …, n

;

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(1) Ignore the dynamic effect of flow;

(2) Assume stage and storage is a single–valued function of

discharge - implying flow is changing slowly with time.

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- In some stream reaches, St and XIt + (1-X)Ot reveals pronounced nonlinear relation.
- To accommodate the non-linearity, the Muskingum model can be modified as
St = K[XIt + (1-X)Ot]m

or

St = K[XItp + (1-X)Otq]

- Using the nonlinear relationship, the routing becomes more difficult. A procedure using state-variable technique was developed by Tung (1985) to perform channel routing using nonlinear Muskingum model.

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- Purposes:
- solve for single outflow rate or
- route only a segment of inflow hydrograph

- Basis:for t =1, 2, …
Since

By substituting Ot into the routing equation for Ot+1 and, after some algebraic manipulations, one has

From recursive substitution of outflow at one time in terms of that at the previous time, one could have

where K1=C0, K2 = C0 C2 + C1; Ki = Ki-1 C2, i≥3

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- Estimate flow rate or water surface elevation at different locations and time simultaneously, rather than separately at different locations, so that the unsteady, non-uniform nature of actual flow phenomena are more closely modeled.
- Modeling of flow movement can be in 1-D, 2-D, or 3-D in space and in time, depending on the dominant flow velocity field to be modeled.

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