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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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Number of routing divisions in the link |
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 |
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.
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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It is also worth noting that as the flow rate approaches zero, travel time is going to get longer (potentially approach the infinite). However, when the power function is used the maximum travel time (encapsulated in K) allowable in the routing method is governed by the stability criteria, where Kmax = dt/x for 0 < x ≤ 1, as discussed in the section on "Stability Criteria" (also recalling particularly that when m=1, travel time (K) is constant over the full flow range).
Output data
Outputs include time series of link outflows.
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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Perumal, M. (2010) Discussion of "Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method". Hydrological Sciences Journal, 55(8): 1427-1430, doi:10.1080/02626667.2010.491260.
Bibliography
Gill, M.A. (1978) Flood routing by the Muskingum method. Journal of Hydrology, 36: 353-363.
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