361

APPENDIX B

DEVELOPMENT OF THE MUSKINGUMROUTING EQUATIONS

To develop a storage relationship for a channel, it is assumed that channelinflow, I, channel outflow, O, and channel storage, S, varies with flow depth,y, according to similar power functions:

cI � O � ay (B.1)

dS � S � by (B.2)I O

If the hydraulic properties of the channel are constant then it is reasonableto assume that the coefficients a, b, c, and d remain constant along the channelreach as well. The flow Equation B.1 can be solved for y and substituted intothe storage Equation B.2 to yield respective equations for inflow (upstream)and outflow (downstream) sections. For the inflow section

1 / c d d / c d / cI I 1 d / cS � b � b � b I (B.3)�� � � � � � �I a a a

and thus

d / cS � KI (B.4)I

where

Stormwater Management for Land Development: Methods and Calculations for Quantity Control.Thomas A. Seybert © 2006 John Wiley & Sons, Inc. ISBN: 978-0-471-72177-2

362 APPENDIX B

d / c1K � b (B.5)� �a

Similarly for the outflow section

d / cS � KO (B.6)O

The Muskingum method assumes that the average storage in the channelreach is a weighted function of the storage at the upstream and downstreamcross sections. Using a simple weighting parameter, X, the average storagerelationship can be expressed as

S � XS � (1 � X)S (B.7)I O

If Equations B.5 and B.6 are substituted into B.7, the average storagerelationship takes the form of

d / c d / cS � XKI � (1 � X)KO (B.8)

or

d / c d / cS � K[XI � (1 � X)O ] (B.9)

For simplicity, the ratio of d /c is assumed to be unity, which reduces therelationship for K in Equation B.8 to b /a. With these assumptions, the storagerelationship of B.9 takes on the familiar form of the Muskingum relationship

S � K[XI � (1 � X)O] (B.10)

The conservation of mass relationship, also known as the continuity equa-tion, is combined with Equation B.10 to solve for the routed hydrograph.Conservation of mass is expressed as

�SI � O � (B.11)

�t

In differenced form, with n representing the beginning of a time step andn�1 representing the end of a time step, Equation B.11 becomes

I � I O � O S � Sn n�1 n n�1 n�1 n� � (B.12)� � � � � �2 2 �t

The differenced form of Equation B.10 is also necessary and is

APPENDIX B 363

S � S � K[X(I � I ) � (1 � X)(O � O )] (B.13)n�1 n n�1 n n�1 n

Equations B.12 and B.13 can be combined to eliminate the storage terms,Sn and Sn�1.

I � I O � O K[X(I � I ) � (1 � X)(O � O )]n n�1 n n�1 n�1 n n�1 n� �� � � � � �2 2 �t

(B.14)

Equation B.14 can be algebraically manipulated to create a working formof the equation. By combining like subscripts and solving for outflow at theend of the �t time step, the equation takes the form of

�KX � 0.5�t KX � 0.5�tO � I � I� � � �n�1 n�1 nK � KX � 0.5�t K � KX � 0.5�t

K � KX � 0.5�t� O (B.15)� � nK � KX � 0.5�t

The leading coefficient of every term on the right side of Equation B.15is expressed in terms of the routing parameters K, X, and �t. These parametersare considered known values since they can be estimated or calculated priorto the routing process.

Equation B.15 can be written in a simplified form as

O � C I � C I � C O (B.16)n�1 0 n�1 1 n 2 n

where

�KX � 0.5�tC � (B.17)� �0 K � KX � 0.5�t

KX � 0.5�tC � (B.18)� �1 K � KX � 0.5�t

K � KX � 0.5�tC � (B.19)� �2 K � KX � 0.5�t

These three routing coefficients, C0, C1, and C2, are dimensionless. There-fore, the time units of K and �t must be the same. Also, mathematical integrityrequires the sum of the three coefficients to be equal to unity.