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Hydrologic River Flood Routing. Basic Equation. Muskingum Method. For a non-uniform flow , S t  f(O t ) Because I t =f(y 1 ) and O t = f(y 2 ) . Therefore, S t =f(I t ,O t ) . Prism storage = f(O t ); Wedge storage = f(I t , O t ).

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Hydrologic River Flood Routing

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Hydrologic river flood routing l.jpg

Hydrologic River Flood Routing

Basic Equation

Routing


Muskingum method l.jpg

Muskingum Method

  • For a non-uniform flow, Stf(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

Routing


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Continuity Equation in Difference Form

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

Routing


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Derivation of Muskingum Routing Equation

  • 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.

Routing


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Muskingum Routing Procedure

  • 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.

Routing


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Muskingum Routing - Example

Routing


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Estimating Muskingum Parameters (1)

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.

Routing


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Example (Graphical Method)

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Estimating Muskingum Parameters (2)

  • 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 = AIt+BOt

    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)

Routing


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Example (Gill’s Method)

Routing


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Estimating Muskingum Parameters (3)

  • 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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Example stephensen method l.jpg

Example (Stephensen Method)

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Disadvantages of Hydrologic Routing

(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.

Routing


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Nonlinear Muskingum Model

  • 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.

Routing


Nonlinear s t vs xi t 1 x o t relation l.jpg

Nonlinear St vs XIt + (1-X)Ot Relation

Routing


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Muskingum Crest Segment Routing

  • 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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Distributed Flow Routing

  • 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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Governing eqs for flow routing saint venant equations l.jpg

Governing Eqs. For Flow Routing(Saint-Venant Equations)

Routing


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Steady vs Unsteady / Uniform vs Non-uniform

Routing


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