Encyclopedia of Environmental Science and Engineering, Volume I and II

474 HYDROLOGY

From a logical point of view, it is probably easier to

develop the routing relationship by considering storage, or

volume changes. In a fi xed time interval ∆ T, the reservoir

infl ow volume is VI ( J ), where J indicates the current time

interval. The corresponding outfl ow volume is VO ( J ) and the

reservoir storage volume is S ( J ). If the current infl ow volume

VI ( J ) were to equal the previous outfl ow value VO ( J −1), then

the reservoir would be in a steady state and no change in res-

ervoir storage would occur. Using the hypothetical steady

state as a datum for the current time interval, we can defi ne

changes in the various fl ow and storage volumes, where ∆

indicates an increment,

∆ VI ( J ) 
 VI ( J ) − VI ( J − 1) (15)

∆ VO ( J ) 
 VO ( J ) − VO ( J − 1) (16)

∆ S ( J ) 
 S ( J ) − S ( J − 1). (17)

To maintain a mass balance for the current time interval,

∆ VI ( J ) 
 ∆ S ( J ) + ∆ VO ( J ). (18)

Using the relationship for a linear reservoir,

S ( J ) 
 K * QO ( J ) (19)

where QO ( J ) is the outfl ow which is equal to VO ( J )/ ∆ T. The

corresponding equation for the previous time interval is

S ( J − 1) 
 K * QO ( J − 1). (20)

Subtracting this equation from Eq. (5) we obtain

∆ S ( J ) 
 K * ∆ QO ( J ). (21)

Substituting that ∆ QO ( J ) 
 ∆ VO ( J )/ ∆ t,

∆ S ( J ) 
 ( K / ∆ t )* ∆ VO ( J ). (22)

Substituting in equation (4) for ∆ S ( J )

∆ VI ( J ) 
 ( K / ∆ t + 1)* ∆ VO ( J ) (23)






VO 

Kt

VI J

1

1 

∗ (). (24)

This equation can be rewritten for fl ows by substituting

∆ VO ( J ) equals ∆ QO ( J )* ∆ T and ∆ VI ( J ) equals ∆ QI ( J )* ∆ T,

i.e.,






QO J 

Kt

( ) ∗ QI J( )

1

1 ^

(25)

where ∆ QI ( J ) 
 QI ( J ) − QO ( J − 1). (26)

Then QO ( J ) 
 QO ( J − 1) + ∆ QO ( J ). (27)

Equation (25) to (27) represent an extremely simple reser-

voir or lake routing procedure. To achieve this simplicity,

the change in infl ow, ∆ QI ( J ), and the change in outfl ow,

∆ QO ( J ), must each be changes from the outfl ow, QO ( J − 1),

in the previous time interval, as defi ned in a similar manner

to Eqs. (15) and (16). The value of K is determined from

the storage-discharge relationship, where K is the gradient,

d S /d Q. This storage factor, K, which has dimensions of time,

can be considered constant for a range of outfl ows.

When the storage–discharge relationship is non-linear,

which is usual, it is necessary to sub-divide into linear seg-

ments. The pivotal values of storage, S ( P,N ), and discharge,

QO ( P,N ), where N refers to the N th pivot point, are tabu-

lated. Calculations proceed as described until a pivotal value

is approached, or is slightly passed. The next value of K is

calculated, not from the two new pivotal values, but from the

latest outfl ows QO ( J ) and from the corresponding storage

S ( J ). The current value of storage is calculated from,

SJ SPN S

SPN QOJ

QO P N K N N

() ( , )

(, ) ( ( )

(, ))* ( , ).






 

1

11

11

Σ

(28a)

Then the next K ( N,N + 1) value is calculated,

KNN

SPN SJ

QO P N QO J

,

,

,


.





1

1

1

( )

( ) ( )

( ) ( )

(28b)

This procedure maintains continuity for storage and discharge,

and is easy to program because no iterations are required.

It will be noted that the routing procedure can be carried

out without calculating the latest storage value: only infl ows

and outfl ows need be considered. The storage value at any

time can be calculated from Eq. (19), which states that there

is always a direct and unique relationship between storage,

S ( J ), and outfl ow, QO ( J ).

In summary, the factor 1/(1 + K / ∆ t ) in Eq. (25), repre-

sents the proportion of the infl ow change, ∆ QI ( J ) which

becomes outfl ow. The remaining infl ow change becomes

storage. The process is identical for increasing or decreas-

ing fl ows: when fl ows decrease, the changes in outfl ow and

storage are both negative. Eqs. (25) and (27), the heart of the

matter, are repeated for emphasis,






QO J 

Kt

( ) ∗ QI J( )

1

1 

(25)

QO ( J ) 
 QO ( J − 1) + ∆ QO ( J ). (27)

CHANNEL ROUTING

The assumptions of reservoir routing no longer hold for chan-

nel calculations. The channel system takes time to respond

to an input. Also, storage is a function of conditions at each

end of the length of channel being considered, rather than

just the conditions at the outfl ow end.

The simplest channel routing procedure is the so-called

Muskingum method developed on the Muskingum River

(G.T. McCarthy,^24 Linsley^2 ).

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