Appendix B - Link Storage Routing

Storage Routing links model the storage and movement of water through a length of a river (drainage link) using a hydrologic routing method. They can represent the travel time of water through a reach, the attenuation of flow rates due to channel shape, and roughness and reach processes.

Storage Routing links offer a choice of several different hydrologic routing methods:  

• 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.

Storage Routing Theory

This section covers aspects of the theory relevant to routing

Background

Link routing in MUSICX is carried out using the Muskingum method, which is a lumped kinematic approach to flow routing (Brutsaert, 2005: p.224). It is based on mass conservation and the assumption of a monotonically increasing storage (stage) and discharge relationship in a link. The latter is a simplification of the full momentum equation for river reaches and assumes that diffusion and dynamic effects are negligible. In this approach, the reach is considered to be compressed to a single, representative point or, in the case where the reach has been subdivided, a number of representative points.

The law of conservation of mass then lets us say that for a reach:

Equation 1

where:

S is the reach storage in (m³)

t is time (s)

Put another way: the volume of water within a reach changes at a rate equal to the difference between the inflow and outflow rates to and from the reach.

For flow routing MUSICX uses an implicit Euler scheme which can be expressed as (Clark and Kavetski, 2010):

Equation 2

Which is to say the flux over a time period n to n+1 is a function of the storage at the end of the time-step (Sⁿ⁺¹). Other hydrological models use an implicit Heun scheme (also known as the Crank-Nicholson method):

Equation 3

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):

Equation 4

where:

S is the reach storage (m³)

I is the inflow (m³/s)

O is the outflow (m³/s)

K is a constant (s)

x is a weighting factor denoting the importance of inflow relative to outflow

To simplify the following description an index flow (q) can be defined as:

Equation 5

From which the Muskingum storage function (Equation 4) can be rewritten as:

Equation 6

MUSICX offers a form of variable parameter Muskingum routing in which the value of x is a constant but the value of K is allowed to vary with flow (Koussis, 1978). From the Kleitz-Seddon rule (Brutsaert, 2005: p.190) the speed that a small monoclinal wave will move through a reach is:

Equation 7

where:

c_w is the wave speed (m/s)

Q is the flow rate (m³/s)

A_c is the channel's cross-sectional area (m²)

Equation 7 can be rearranged to give a relationship for the time this wave would take to travel the length of a reach by noting that the volume of water in the reach is equal to the reach length times the cross-sectional area:

Equation 8

where:

T_w is the time the wave takes to pass through (s)

From Equations 6 and 8, it can be seen for a fixed K that T_w=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:

Equation 9

To convert this equation into a storage function it is necessary to integrate it:

Equation 10

Where S(x) is the link storage at flow rate x.

Relationship in Equation 9 in MUSICX is defined using a functional relationship (a power function). The storage function has the following form:

Equation 11

where:

a is an arbitrary constant

b is an arbitrary exponent

Integrating Equation 11 gives a storage function (Equation 10) of:

Equation 12

Defining:

Equation 13

and

Equation 14

allows Equation 12 to be rewritten in the more familiar form:

Equation 15

where:

k is a storage-delay constant

As S has units of m³ and q has units of m³/s, k has the units m^3(1-m)s^m.

In terms of k and m, Equation 11 is:

Equation 16

By choosing the appropriate functional form and appropriate values of m and x the following routing methods can be replicated:

• Linear Muskingum routing (power function with m=1)
• Non-linear Muskingum routing (power function with m≠1).

Stability Criteria

The routing formulation implemented in MUSICX uses an implicit Euler scheme with the assumptions that:

• the inflow and outflow rates are constant over the time-step; and
• the storage function relates the storage at the end of the time-step to the average flow rates during the time-step.

If x=0 then for this formulation the routing is unconditionally stable. If any other value of x is used then for hydrological stability there are two criteria that must be met, and these are:

Equation 17

Equation 18

where:

is q

Rearranging Equation 18, the second criterion is that:

Where K() is the slope of the storage curve at .

These criteria are illustrated in the Figure below, and they apply to each reach division. It should be noted that these constraints are much more liberal than the equivalent for the implicit Heun scheme.

The criterion expressed in Equation 18 is enforced as follows:

• For the power function with m=1, the function K() is in fact a fixed value of K and the original version of the Muskingum method is being used. A check is made to enforce the requirement that K ≤ dt/x; or
• For the power function with m≠1 (and x>0), the value of K() can range between zero and infinity, presenting a potential stability problem. The flow/storage relationship must be modified to avoid this problem and the point at which this should be done is at the largest allowed slope dt/x. This gives:

Equation 19

The corresponding storage value is S(_limit).

If m < 1, a linear section is inserted into the start of the flow/storage curve starting from the point (0,0). For this case, the required equation of the linear section is:

Equation 20

where:

K_max is the routing constant required for the linearised section, and:

Equation 21

However, if the linear section is inserted from the point (0,0) to the point (_limit, S(_limit), then from Equation 15 and Equation 19, the resultant slope is:

Equation 22

This means the slope of the linear segment from the point (0,0) to the point (_limit, S(_limit)), K>K_max, which is too steep, and further adjustment is required. The approach adopted is to offset the power curve so that it meets a linear segment starting from the point (0,0) with a slope of K_max. The offset required is then:

Equation 23

Therefore:

Equation 24

The storage function then becomes:

Equation 25		when
		when

As the routing characteristics of the storage methods are a function of the slope of the storage curve, shifting the curve like this will not affect results.

If m>1, a linear section replaces the section of the flow/storage curve after the point (_limit,S( _limit)) which continues the slope (K_max) 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 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:

• Linear Muskingum routing (power function with 0 ≤ x < 1 and m=1)
• Non-linear reservoir routing (power function with x=0 and m>0)
• Non-linear Muskingum routing (power function with 0 < x ≤ 1 and m>0)

Values of m such that m>1 are possible for natural channels (see for example, Bates and Pilgrim, 1982; however, such values would typically only occur for overbank flows). Other studies (eg. Wong and Laurenson, 1983, 1984) show that values of m such that m<1 also occur for natural channels, and highlight that the validity of assuming constant values of m and k apply over the full flow range is fairly tenuous for natural channels. The assumption is reasonable for regular geometric shapes though (eg. trapezoidal and triangular cross sections), as found by Mein et al (1974).

From the above list it may be seen that the valid range of values of x is 0 ≤ x ≤ 1. More information on the valid range of values of K is given in the section on "Stability Criteria".

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 K_max = 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.

Brutsaert, W. (2005) Hydrology - an introduction. Cambridge University Press, Cambridge.

Clark, M.P. and Kavetski, D. (2010) Ancient numerical daemons of conceptual hydrological modeling: 1. Fidelity and efficiency of time stepping schemes, Water Resources Research, 46, W10510, doi:10.1029/2009WR008896.

Koussis, A.D. (1978) Theoretical estimation of flood routing parameters. J. Hydraul. Div. Am. Soc. Civ. Eng., 104(HY1): 109-115.

Koussis, A.D. (2009) Assessment and review of the hydraulics of storage flood routing 70 years after the presentation of the Muskingum method. Hydrological Sciences Journal, 54(1): 43-61. February.

Laurenson, E.M. (1959) Storage analysis and flood routing in long rivers. Journal of Geophysical Research, 64(12): 2423-2431, doi:10.1029/JZ064i012p02423.

Linsley, R.K., Kohler, M.A. and Paulhus, J.L.H. (1949) Applied Hydrology. McGraw Hill, New York.

McCarthy, G.T. (1938) The unit hydrograph and flood routing. Manuscript presented at a conference of the North Atlantic Division, US Army Corps of Engineers, 24 June 1938 (unpublished)

Mein, R.G., Laurenson, E.M. and McMahon, T.A. (1974) Simple non-linear model for flood estimation. J. Hydraul. Div. Am. Soc. Civ. Eng., 100: 1507-1518.

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.

Ladson, A.R. (2008) Hydrology: an Australian introduction. Oxford University Press, South Melbourne, Vic., Australia. 304 p. ISBN: 0195553586.

Linsley, R.K., Kohler, M.A. and Paulhus, J.L.H. (1982) Hydrology for Engineers. 3rd Ed., McGraw Hill, Auckland.