muskingum routing - example

7/25/2019 Muskingum Routing - Example

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http://www.engr.colostate.edu/~ramirez/ce_old/classes/cive322-

Ramirez/CE322_We/E!ample_"us#ingumRouting.htm

CE322 %asic &'drolog' Jorge A. Ramirez

"us#ingum Routing - E!ample

The inflow and outflow hydrographs of a river reach are tabulated below.

Time (h) Inflow

(m!s)

"utflow

(m!s)

# $ %&

' # $#

'% ##*

* ' #&$

& **' '

+ &*+ '*

+ *'

% +% &$

$ +$# &%

http://www.engr.colostate.edu/~ramirez/ce_old/classes/cive322-Ramirez/CE322_Web/Example_MuskingumRouting.htm






# +& +'

## +* +*'

#' &# +&

# * +

#* $ &*+

#& '$ *$

#+ '* *#

# #%* *#

#% #* '*

#$ #% '#&

' $ #

The ,us-ingum routing procedure is used for systems that have torage /

0ischarge relationships that are hysteretic. That is1 for systems for which theoutflow is not a uni2ue function of storage. The S vs. Orelationship for the river

reach under consideration is graphed below.



(. )arameter Estimation

3se these observations to obtain the ,us-ingum routing parameters k and x forthis river reach. The initial storage in the system is #&1 m.

*raphical )rocedure:

The graphical procedure consists in generating graphs of [xI + (1-x)O] vs. S for

different values of x1 arbitrarily selected such that 4 x 4 .&. The optimal valueof x is selected as that which produces the narrowest and straightest loop graph

of [xI + (1-x)O] vs. S . The slope of the least s2uares linear fit to the resulting points is the estimate of k .

a) 5enerate accumulated storage in the system. 3se continuity e2uation asfollows6

# ' * & + % $

78.'& 78.& 78.# 78.#&



Inflow1 I

(m!s)

"utflow1"

(m!s)

Ave.

Inflow

(m!s)

Ave.

"utflow

(m!s)

torage

(m)

9eighted Average :lu7

7I ; (#/7)"

(m!s)

$ %&

#& % %.% %&.% %+.'#'*

# $# ##& %% %#'' #'.& #.# $&.+ $.$+$''

'% ##* #'.& #'.& #+*' #.& #*+.$ #'.* #'%.'*#*

' #&$ '+* #+.& #&'' #$$.'& '#&.& #&.# #%.$'

**' ' %# #$+ '#%$' '%&.'& +.#& '&.$ '+*.++*&

&*+ '* *$* '%.& '$+& $.& *#. *+.' &.+*

+ *' &%% ' *'+ *'.& *$.& **# *&#.%#+

+% &$ +&* *+*.& **'*% &&#.'& &+%.#& &'&.$ &*.+*

+$# &% +%*.& &*.& *$'* ++.'& +#.&& &%$. &$&.#'#

+& +' +% +.& &''$* ++ +*#.' +'%.' +.%%'



+* +*' +&*.& +'.& &%+ +* +$.' +*#.' +*.%%

&# +& +'.& +%.& &#$ +#$ +#'.+ +'%.+ +'&.

* + &'* +#$ *% &#.& &&%.$ &$.* &%.$#*

$ &*+ *.& &*.& *'$* & *$#.* &.* &''.+&

'$ *$ &$.& &#'.& %+ **#.& *'+.& *+* *&+.'*

'* *# '%% **+ '$% #.& &*.$ $+.* %.%&'

#%* *# '#&.& '+'%* #.& '%+.& '&. #.'#%

#* '* #&$ .& '$% '$ ''& '+ '&'.%$*

#% '#& #'# '**.& #+*$' #%%.'& #.&& '*. #$%.%$

$ # $$ #$'.& ##'+ #& #*' #+' #&.%$+

<olumns # = ' are given.

<olumns = * are the average inflow flu7 (I i;# ; Ii)!' and outflow flu7 ("i;# ;

"i)!'1 respectively.

<olumn & is the cumulative storage in the system obtained using the continuity

e2uation below.



<olumns + / $ are the values of the weighted average flu7 [xI + (1-x)O] fordifferent values of x. The graph of <olumns + / $ vs. <olumn & is shown below.

>ased on these results1 a value of x 8 .#& is selected. The best least s2uares fit to

the corresponding points yields a value of k 8 '. h.

+east ,uares )rocedure

Inflow

(m!s)

"utflow

(m!s)

torage "'

(m!s)'

I'

(m!s)'

"I

(m!s)'

"

(m+!s)

I

(m+!s)

$ %& #& ''& %+*$ $& +& ++*$&

# $# %#'' %'%# #%+$ #'*+ $#' ###'#*

'% ##* #+*' #'$$+ *'+* '#' #'##%% ''#&+



' #&$ #&'' '&'%# #'* &%% '*'#%%% *%*'*

**' ' '#%$' &*'%$ #$&+* #'$%+ &#%+ $++'+*

&*+ '* '$+& #*$+ '$%##+ #+$* $+++ #+#%%$

+ *' *'+ #+* $+$ '+*+ #&#%$' '&%%

+% &$ **'*% '&$%# *&$+%* *&#' ''&'''' #**

+$# &% *$'* *%* **%# $$$% '%&$'' *%'%%*

+& +' &''$* %%#'$ *&&+'& *'&'& '&$#+' &'$%*&

+* +*' &%+ *#'#+* *#$&+ *'% *%#'#' +&+&'*

&# +& &#$ *''& '+*# +'&%& '%%++& '$&'$

* + *% ++$ ''&'$ '%+# '$#+## ''*$

$ &*+ *'$* '$%##+ #&'# '#'$* '+%&'* #+%%*++

'$ *$ %+ ''$**# #%'*# #&&$# #%$$*$* #'*#&$*

'* *# '$% #&+$ +#$ #'## #'&+** $'%'+



#%* *# '+'%* ##+'%# %&+ +'** %$+'%** *%+'&+

#* '* '$% &+ #$&+ +#+ &#' '%&+$'

#% '#& #+*$' *+''& ##++* ''' &*&% #%##+

$ # ##'+ '%$ %# #& ''#*' ##%#*

Σ"' 8

&#**%

ΣI' 8

%**

ΣΙ" 8

*''*&

Σ" 8

'$+'&*

ΣI 8

'$#%**&

3sing the above e2uations yields6

A 8 #'&&.+'+#+* s

> 8 '$.#&# s

k 8 A;> 8 %'%*.'++ s 8 '. h



x 8 A!(A ; >) 8 .#&#&&$#&*

"bserve that these results for k and x are the same as those of the graphical

procedure. :or comparison purposes1 the observed outflow hydrograph and that predicted using the estimated values of k and x are graphed below.

%. "us#ingum Routing

3se the ,us-ingum routing procedure to route the hydrograph tabulated belowthrough the same river reach of ?art A

elect a ∆t 8 # h1 as suggested by the inflow data. @owever1 chec- that with the

selected ∆t1 parameter values meet restrictions6

x 4 .& ∆t/k 4 # / x

:or this case6 .#&#& 4 (.&) (+)!%'%*. 4 # / .#&#& Thus1 ". ?roceed withrouting1 by obtaining <o1 <#1 and <'.



This yields6 <o 8 .+#%B <# 8 .*+*B and <' 8 .&$'#$. 3sing these values

in the ,us-ingum routing e2uation6

obtain the outflow hydrograph as tabulated below. The resulting hydrographs are

also graphed below..

Time (h) Inflow

(m!s)

<o 7 Ii;#

(m!s)

<# 7 Ii

(m!s)

<' 7 "i

(m!s)

"utflow

(m!s)

# &

&

' # +.#% #. '$.++$& &.%$&

' #'.&* *.+* #.*+' %.*#

* '& '.%% +$.'#*% *+.*'* #&.#$$

& *& '.%*#& ##'.**# %.+&& ''.+*

+ + .'' #&&. #.+&#& '.*&

*.'&$ '.+*** #$#.&#& **'.*'+%



% % *%.#$%+ '*'.'&#% '+#.$%' &&'.*'%

$ $ *%.%## '+$.$ '.### +*&.%+##

# & *.%%*$ '.$%& %'.*$+ .'

## & *+.*'& '+%.'* *#+.#% #.'*$*

#' +% *'.#&#+ '&$.&&&& *.# *.&'

# &$ +.*&* '&. **.$+% +.&*

#* & .%$& '*.#% *#%.*$+' +&.&*

#& *' '&.$&&* #. %.+ &%&.$$%

#+ & '#.+'&*& #*&.&## *+.$%$% &#.$++

# #%.&+# #'#.#'&$ *.$& ***.#&

#% '& #&.**+& #.%''' '+'.$#+ %'.#$+

#$ ''& #.$'% %+.&#%& ''+.* '+.'*

' ' #'.&* .%+++& #$.*++ '%.+$#

muskingum routing - example

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