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- linear Muskingum routing
- non-linear Muskingum routing (using a power function)
Scientific provenance
The storage (hydrologic) routing and lagged flow techniques are all well established; an excellent review of storage routing techniques is provided by Koussis (2009) and discussion on this is provided by Perumal (2010). The Muskingum technique dates back to a 1938 study on the Muskingum River in the US (McCarthy, 1938); it has been used in literally thousands of practical applications and discussed in thousands of journal papers since. Non-linear routing also has a long history (eg. Linsley et al, 1949; Laurenson, 1959; Mein et al, 1974).
Dependencies
Each routing link requires a node at its upstream end to define inflows, and a node at the downstream end to process outflows as appropriate.
Assumptions and Constraints
The Storage Routing link can be partitioned into divisions of equal length. Other assumptions and constraints are listed in the Table below. Further details on assumptions and constraints are given in the Theory section below.
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No | Assumption/Constraint |
|---|---|
1 | The routing methods are based on lumped conceptual storages (ie. they are not spatially distributed). This is appropriate in river reaches where the friction slope is approximately equal to the bed slope. |
2 | Hence, it is assumed the cross-sectional geometry for a routing link can be represented via a single cross-section with a single representative bed/friction slope between the upper and lower ends of the link. Results from the routing methods may be very sensitive to the assumptions made about this representative cross-section. |
3 | The velocity is uniform, the water surface is horizontal across any section perpendicular to the longitudinal flow axis, and flows are gradually varying. |
4 | The relationship between storage and discharge is a monotonically increasing function so that the storage theory is applicable. Also, the same relationship is assumed to apply to both the rising and falling limbs of hydrographs. |
Storage Routing Theory
This section covers aspects of the theory relevant to routing
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In Equation 3, the flux over a time period n to n+1 is an average of the functions of storage at the start of the period and the end. Clark and Kavetski (2010) describe the implicit Euler scheme as being ‘ubiquitous in "industry-standard" engineering software’. The truncation error for the implicit Heun scheme is more sensitive to the length of the time-step than the Euler scheme. The implicit Euler scheme produces hydrologically sensible results for a wider range of parameter values than the implicit Heun scheme.
Form of the Routing Equation
To make use of the continuity equation (Equation 1) it is necessary to relate the storage in the reach to the flow entering and leaving that reach. MUSICX uses the Muskingum storage function (Koussis, 2009):
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From Equations 6 and 8, it can be seen for a fixed K that Tw=K. That is, the time a wave takes to pass through a reach is K. This means that the K in the linear Muskingum routing scheme has two interpretations: as a constant in the storage relationship or as the wave travel time.
Extending the method for variable K
For the convenience of the model users (it being easier to think in terms of the wave travel times) MUSICX defines a variable K which uses the wave travel time interpretation. Combined with the assumption that the flow rate in the reach is represented by the index flow q:
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- Linear Muskingum routing (power function with m=1)
- Non-linear Muskingum routing (power function with m≠1).
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The routing formulation implemented in MUSICX uses an implicit Euler scheme with the assumptions that:
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If m>1, a linear section replaces the section of the flow/storage curve after the point (
limit,S( limit)) which continues the slope (Kmax) at this point.
Reach subdivision
As noted in the Assumptions and Constraints section, above, MUSICX can divide the storage representing the flow routing in a routing link into a number of equal divisions. This means that the flow in a routing link passes through a cascade of storages. The user inputs a value of the storage-delay constant, k, which applies to each division on the link for use in the K() relationship (Equation 16) for each division.
Input data
The information that users may provide is summarised in the Table below. Note that there is also a global requirement to specify the model time-step, dt. In addition, of course, data on the inflow to the routing link is required for each model time-step.
Parameters |
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Length of routing reach |
Initialisation Type (governs whether an initial value of storage or an initial value of flow is entered) |
Initial value of storage |
Initial value of flow (note this is mandatory for Lagged Flow) |
Number of routing divisions in the link |
Representative flow rate (to calculate link delivery time) |
x , the weighting factor in routing |
The storage delay constant, k, if m≠1 or the Muskingum K, if m=1, in the routing equation |
m, the exponent in the routing equation |
Flow versus travel time relationship |
Governing data for fluxes (reach processes) - details below |
Data needs for lateral fluxes (reach processes)
For the net evaporation flux, FluxNE, the additional data requirements are:
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For the general purpose (time series) flux, FluxTS, a time series of losses and/or gains is required for each Owner in the system being modelled. Data required for Owners is discussed in the chapter on Ownership.
Parameters or settings
Information on the meaning and function of each parameter, whether it is a "physical" parameter or otherwise, and its units can be found in the Theory section above. Where applicable, information on default values and the range of valid values can also be found in the Theory section.
Sensitivity of parameters
Clearly, which parameters are the most sensitive depends on the routing option being used. For Storage Routing, the parameters that are relevant from the point of view of sensitivity depend on whether the stream is flowing or it has ceased to flow and, if it is flowing, whether the power function or piecewise routing is being used.
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As for the case where the stream has ceased to flow, when the stream is flowing results may also be sensitive to values of the parameters defining the fluxes being modelled. The degree of sensitivity will depend on the magnitudes of these fluxes relative to the magnitudes of the flows and reach storage. A circumstance where the results could be expected to be most sensitive to these fluxes would be near cease to flow.
Valid ranges of values of parameters
Reiterating the section on "Form of the Routing Equation", by choosing the appropriate functional form and appropriate values of m and x the following hydrologic routing methods can be replicated:
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Information on evaluating k (or travel time or K), x and m is given in a number of sources such as Pilgrim (1987). It is worth noting that when discussing storage-discharge relationships, Pilgrim (1987: eg equation 7.4) refers to the term being a "representative discharge for a reach" and this is the same as the index flow used here. Pilgrim (1987: Figure 7.21) also recasts the wave speed-discharge relationship developed by Wong and Laurenson (1983, 1984) into a relationship between time of travel of flood peak and discharge, which is more relevant to the way routing in MUSICX would be typically applied. However, note that the discussion of aspects such as stability criteria by Pilgrim (1987) applies to the implicit Heun scheme and not the implicit Euler scheme used in MUSICX.
Output data
Outputs include time series of link outflows and modelled fluxes.
References
Bates, B.C. and Pilgrim, D.H. (1982) Investigation of storage-discharge relations for river reaches and runoff routing models. Proc. Hydrology and Water Resources Symposium. Melbourne, 11-13 May: 120-126. Institution of Engineers, Australia.
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Wong, T.H.F. and Laurenson, E.M. (1984) A model of flood wave speed-discharge characteristics of rivers. Water Resources Research, 20(12): 1883-1890, doi:10.1029/WR020i012p01883.
Bibliography
Gill, M.A. (1978) Flood routing by the Muskingum method. Journal of Hydrology, 36: 353-363.
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