JOURNAL OF GEOPHYSICALRESEARCH
VOLUME 84, No. 8
AuGusT, 1959
A Note on the Muskingum Flood-RoutingMethod
J. E. NAs•
Hydraulics Research Station
Department o• Scientific and Industrial Research
Walling]ord, Berks., England
Abstract--An exact method of solution of the flood-routing equation, when the storage is a
linear function of weighted inflow and outflow, is developed. This operation is shown to be
equivalent to routing a multiple of the inflow through reservoir storage and subtracting the
excess inflow. Modified coefiqcientsfor the Muskingum equation are developed which do not
depend on the routing interval being small relative to K.
fled equations for the Muskingum coefficients
Introduction--The Muskingum method, which
is a finite difference method of solution of the
are derived.Theseequations
are true evenwhen
flood routing equation, under the assumption T is not small relative to K.
that storageis a linear function of a weighted
Notation-mean of inflow and outflow, S - K[xI nu (1 -x)Q], is widely used, both in its original form
I(t) - inflow, fta/sec
[McCarthy, 1938] and as the basis of a number
Q(t) = outflow,fta/sec
of graphicalor semi-graphicalmethods.In the
x
= a numerical parameter
use of these methods, however, it is sometimes
S(t) - storage,ft ahrs/sec
overlooked that an essential requirement, to
K ----a time parameter, hours
insure accuracyin such finite differencecalcuk
=
x)
lations,is that the finite interval T must be small
D = the differential operator d/dr
relative to the other time elements involved.
Ct C2 Cathe Muskingum coefficients
This fact was emphasizedby Clark [1945] in
c
- exp-- T/K(1-- x)
discussing
the Muskingumflood-routingmethod.
q
= the outflow from routing I through
Nevertheless,it still happensthat values of T
S = K(1- x)q
sensiblyequal to K are recommendedfor use
T = the routing interval, hours
in actual calculation.
m -- slopeof inflow curve
Whereas it is possiblethat in practice the
The exactsolution--The fundamentalequations
inaccuraciesso introduced are generally not
are
significant relative to the inaccuraciesintroducedby the basicstorageassumption,and the
usual inaccuraciesof the data, it may happen,
I -- Q-I-dt
(1)
particularly in theoretical work, that a high
relative accuracyis required.The failure of the
S = K(xI -]- (1 -- x)Q)
(2)
Muskingum method when T/K is not small is
from which
demonstratedby the widely acceptedbelief that
routing through linear storage with x = 0.5
dI
operates as a pure delay. This conclusionis
dt
I --xK•-= Q-]-(1- x)K
d_Q-
based on the fact that the substitution of T = K
and x = 0.5 in the Muskingum equation yields
Q• = Io, the other coefficientsbeing zero. That
1 -
xKD
Q(t)= 1+ (1- x)KDI(t)
this conclusion is erroneous is demonstrated in
this note. An exact method of solution under
When x =
the storageassumptionis developed,and modi-
reservoir case
1053
(3)
zero we have the corresponding
J. E. NASI-I
1054
X--O
to obtain Q (Fig. 1). Clearly this involves a
further shift of the center of area (1 -- x)K so
5
that the total shift is K. However I and Q are
/
not otherwise identical
\
/
\
even when x =
0.5 as
shownin Figure 1. It shouldbe noted that the
negative initial values of I' result in negative
initial values of Q.
If we divide out the operatorin (3) we obtain
xx
Q=E--1--
(7)
Fro. 1--Routing through storage with x _-- 0.5
q-(1-- x){1q-(1-- x)KD}I
1
Q(0= I q-KDI(t)
I
I
xI
(4) Q= 1+ (1- x)KD(1-- x)-- 1-- x (8)
which has the solution
We see, therefore, that the outflow consistsof
the sum of two parts, the first of which we shall
Q= • e
e at
(5) call q, being the result of routing I/(1 -- x)
through S - K(1 -- x)q, and the second
Now (3) may be looked upon as the result of
part being simply the inflow multiplied by
operating on I(t) successivelywith i -- xKD
x).
and 1/[1 q- (1 -- x)KD]. The operationI -- xKD
There are various ways of routing through
merely involves differentiationof the inflow and
reservoirstorage.Equation (5) may always be
1/[1 q- (1 -- x)KD] represents
reservoirrouting
integrated graphically, or mathematically if I
with S = (1 -- x)KQ. Therefore(3) is equivalent
is in a suitable form. A simple graphical solution
to subtracting xK times the first derivative of
not involvingintegrationhas beendemonstrated
I from I and routing the remainder through
by Nash and Farrell [1955]. It frequently hapreservoirstoragewith S = (1 -- x)KQ. These
pens, however, that a coefficient solution is
operationsgenerally can be carried out graphi- desired. Formulas for the coefficients are calcucally, and mathematically as well, when I is a lated in the next section.
simple function of time. We can learn more
from (3). Let us define
Modificationo! the Muskingum coefftcients--
1 -t/a:
f t/•I
I'(t) = (1 -- xKO)I(t)
(6)
This means that I' is the result of routing I
backwards (that is, calculation of inflow from
outflow) through linear reservoirstorageS =
--xKI. The effectof the negativexK is achieved
by taking the routing procedurefrom right to
left; that is, in the negative direction of time
(rig. 1).
When we cometo t• at which I' becomeszero,
I would fall off logarithmicallyand never actually reach zero unlessI' took negative values.
This means that when I starts from zero and
rises at a finite rate, I' must always take
negative values initially.
It is clear, too, that the interval between the
centersof area of I'(t) and I(t) is xK. We must
now route I' forwardsthroughS -- (1 -- x)KQ
Q, = Clio + C2II + CaQo
(9)
We shall use (8) to obtain the expressionfor
the C's. In expressingQ as a functionof Io I• and
Q0 only, we must neglect secondand higher
derivatives of I; that is, we must assumeI to
consistof straight-line segments.If the second
or higher derivativesare required, we must use
three or more values of I in (9). However, by
choosingtime intervals which are sufficiently
short, the calculationusing only I0, I•, and Q0
can be made as preciseas is desired.The only
differencebetween the present calculation and
the usual developmentof the Muskingum coefficientequationis that we are not limited to
values of the
time
interval
which
are small
comparedwith K.
The solutionof (8) whenI is a seriesof straight
MUSKINGUM
FLOOD-ROUTING
segments
is obtainedas follows.Let m -- (I• -Io)/T be the slopeof a segment.
METHOD
1055
Q, - Io[K/T(1 -- c) -- c]
-•- I,[--K/T(1
--c) -•- 1]-•-Qoc
(13)
Let
This is the modified form of the Muskingum
equation when T is not small relative to K. If
T/K is taken very small the coefficientsin (13)
and in the Muskingum equation converge.
1
q(t)-- I -•-(1-- x)KDI(t)
then
Conclusiom--We
Q = q/(1 - x) - xI/(1 - x)
(10)
Let k - (1 -- x)K and c -- exp[-- T/K(1 -- x)]
to simplifythe notation. From (5)
q--like
-•/•f (Io-•-mt)e
•/•dt
q = 1/ke-•/•[kloe
•/•
q-mkø'e•/•(t/k- 1)q- A]
q-
Io-]- mk(t/k-
1)-]- A/ke-•/•
(11)
We solvefor the arbitrary constantA by letting
q - qoat t - 0 and obtain qo- Io -- mk -•- A/K.
Substitutingin (11) we obtain
q -- Io + mk(t/k -- 1)
Io)/T
Io q- mk)e
-•/•
for m and letting
q• -- Io -•- k/T(T//k -- 1)(I, -- Io)
ß
+ [qo-
+
o)]C
q, = Io[k/T(1 -- c) -- c]
-•- I,[--k/T(1
seen that
the
Musk-
it would seem more reasonable to abandon
q-(qoSubstituting (I•t = T, we obtain
have
ingum method is equivalent to either routing
the inflow backwards (that is, calculating I
from Q) through storageS - --xKI and subsequently forwards through S ---- (1 -- x)KQ,
or routing a multiple of the inflow I/(1 -- x)
through S -- K(1 -- x)q and subtracting
xI/(1 -- x). We have also developedthe values
of the coefficientsto be used when T/K is not
small. The negative outflow, which may occur
when the inflow risessteeply, has been explained
as being essentiallyassociatedwith the storage
assumption,and not with any particular method
of solution. This rather unrealisticconsequence
of the storage assumptionsuggeststhat some
modification is desirable.As it is necessaryin
practice to determine x and K by experiment,
-- c) -•- 1] + qoc
(12)
the
storage assumption and consider the linear
operation to consistof two parts, a pure delay
and a single reservoir routing [Hopkins, 1956],
the two parametersto be determined by experiment. The pure delay plus the storagefactor K
would be equal to the lag between the centers
of area of inflow and outflow, and the ratio of
the storagefactor to the lag would form a dimensionlessparameter which might be constant as
a first approximation or reflect some characteristics, at present unknown, of the channel. It
might be possibleto determine this relation by
.
means of a statistical correlation.
,.
whenceby (10)
Qi=Io
Ikl--c c
+I11kl--c] 1 •x 1
Tl--x
1--x
T1
--x
1--x
Acknowledgment--This note is published by
permission of the Director, Hydraulics Research
Station, Department of Scientific and Industrial
Research, Itowbery Park, Wallingford, Berkshire,
England.
1--x
REFERENCES
c
-•-qo1_ x
But q0/(1- x) -- Qo- xlo/(1- x) which when
substituted in (12), bearing in mind that k =
K(1x), gives
C•,Aa•r, C. O., Storage and the unit hydrograph,
Trans. Am. $oc. Civil Engrs., 110, 1419-1446,
1945.
McC.aaTnY, G. T., The unit hydrograph and flood
routing, unpublished manuscript presented at a
conference North Atlantic Div., Corps of Engi-
1056
J.E.
neers, War Dept., June 24, 1938 (see also Engineering construction--flood control, The Engineer School, Fort Belvoir, Va., pp. 147-156,
1940).
NAs•, J. E. A•D J.P. F•RRSLL,A graphical solu-
NASH
tion of the flood routing equation for linear
storage-dischargerelation, Trans. Am. Geophys.
Union, 36, 319-320, 1955.
I-IorI•Ns, C. D., Discussionon previous refereno•,
Trans. Am. Geophys.Union, 37, 500-501,1956.