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)
# $ %&
' # $#
'% ##*
* ' #&$
& **' '
+ &*+ '*
+ *'
% +% &$
$ +$# &%
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# +& +'
## +* +*'
#' &# +&
# * +
#* $ &*+
#& '$ *$
#+ '* *#
# #%* *#
#% #* '*
#$ #% '#&
' $ #
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.
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(. )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.#&
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Inflow1 I
(m!s)
"utflow1"
(m!s)
Ave.
Inflow
(m!s)
Ave.
"utflow
(m!s)
torage
(m)
9eighted Average :lu7
7I ; (#/7)"
(m!s)
$ %&
#& % %.% %&.% %+.'#'*
# $# ##& %% %#'' #'.& #.# $&.+ $.$+$''
'% ##* #'.& #'.& #+*' #.& #*+.$ #'.* #'%.'*#*
' #&$ '+* #+.& #&'' #$$.'& '#&.& #&.# #%.$'
**' ' %# #$+ '#%$' '%&.'& +.#& '&.$ '+*.++*&
&*+ '* *$* '%.& '$+& $.& *#. *+.' &.+*
+ *' &%% ' *'+ *'.& *$.& **# *&#.%#+
+% &$ +&* *+*.& **'*% &&#.'& &+%.#& &'&.$ &*.+*
+$# &% +%*.& &*.& *$'* ++.'& +#.&& &%$. &$&.#'#
+& +' +% +.& &''$* ++ +*#.' +'%.' +.%%'
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+* +*' +&*.& +'.& &%+ +* +$.' +*#.' +*.%%
&# +& +'.& +%.& &#$ +#$ +#'.+ +'%.+ +'&.
* + &'* +#$ *% &#.& &&%.$ &$.* &%.$#*
$ &*+ *.& &*.& *'$* & *$#.* &.* &''.+&
'$ *$ &$.& &#'.& %+ **#.& *'+.& *+* *&+.'*
'* *# '%% **+ '$% #.& &*.$ $+.* %.%&'
#%* *# '#&.& '+'%* #.& '%+.& '&. #.'#%
#* '* #&$ .& '$% '$ ''& '+ '&'.%$*
#% '#& #'# '**.& #+*$' #%%.'& #.&& '*. #$%.%$
$ # $$ #$'.& ##'+ #& #*' #+' #&.%$+
<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.
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<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)
$ %& #& ''& %+*$ $& +& ++*$&
# $# %#'' %'%# #%+$ #'*+ $#' ###'#*
'% ##* #+*' #'$$+ *'+* '#' #'##%% ''#&+
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' #&$ #&'' '&'%# #'* &%% '*'#%%% *%*'*
**' ' '#%$' &*'%$ #$&+* #'$%+ &#%+ $++'+*
&*+ '* '$+& #*$+ '$%##+ #+$* $+++ #+#%%$
+ *' *'+ #+* $+$ '+*+ #&#%$' '&%%
+% &$ **'*% '&$%# *&$+%* *&#' ''&'''' #**
+$# &% *$'* *%* **%# $$$% '%&$'' *%'%%*
+& +' &''$* %%#'$ *&&+'& *'&'& '&$#+' &'$%*&
+* +*' &%+ *#'#+* *#$&+ *'% *%#'#' +&+&'*
&# +& &#$ *''& '+*# +'&%& '%%++& '$&'$
* + *% ++$ ''&'$ '%+# '$#+## ''*$
$ &*+ *'$* '$%##+ #&'# '#'$* '+%&'* #+%%*++
'$ *$ %+ ''$**# #%'*# #&&$# #%$$*$* #'*#&$*
'* *# '$% #&+$ +#$ #'## #'&+** $'%'+
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#%* *# '+'%* ##+'%# %&+ +'** %$+'%** *%+'&+
#* '* '$% &+ #$&+ +#+ &#' '%&+$'
#% '#& #+*$' *+''& ##++* ''' &*&% #%##+
$ # ##'+ '%$ %# #& ''#*' ##%#*
Σ"' 8
&#**%
ΣI' 8
%**
ΣΙ" 8
*''*&
Σ" 8
'$+'&*
ΣI 8
'$#%**&
3sing the above e2uations yields6
A 8 #'&&.+'+#+* s
> 8 '$.#&# s
k 8 A;> 8 %'%*.'++ s 8 '. h
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x 8 A!(A ; >) 8 .#&#&&$#&*
"bserve that these results for k and x are the same as those of the graphical
procedure. :or comparison purposes1 the observed outflow hydrograph and that predicted using the estimated values of k and x are graphed below.
%. "us#ingum Routing
3se the ,us-ingum routing procedure to route the hydrograph tabulated belowthrough the same river reach of ?art A
elect a ∆t 8 # h1 as suggested by the inflow data. @owever1 chec- that with the
selected ∆t1 parameter values meet restrictions6
x 4 .& ∆t/k 4 # / x
:or this case6 .#&#& 4 (.&) (+)!%'%*. 4 # / .#&#& Thus1 ". ?roceed withrouting1 by obtaining <o1 <#1 and <'.
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This yields6 <o 8 .+#%B <# 8 .*+*B and <' 8 .&$'#$. 3sing these values
in the ,us-ingum routing e2uation6
obtain the outflow hydrograph as tabulated below. The resulting hydrographs are
also graphed below..
Time (h) Inflow
(m!s)
<o 7 Ii;#
(m!s)
<# 7 Ii
(m!s)
<' 7 "i
(m!s)
"utflow
(m!s)
# &
&
' # +.#% #. '$.++$& &.%$&
' #'.&* *.+* #.*+' %.*#
* '& '.%% +$.'#*% *+.*'* #&.#$$
& *& '.%*#& ##'.**# %.+&& ''.+*
+ + .'' #&&. #.+&#& '.*&
*.'&$ '.+*** #$#.&#& **'.*'+%
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% % *%.#$%+ '*'.'&#% '+#.$%' &&'.*'%
$ $ *%.%## '+$.$ '.### +*&.%+##
# & *.%%*$ '.$%& %'.*$+ .'
## & *+.*'& '+%.'* *#+.#% #.'*$*
#' +% *'.#&#+ '&$.&&&& *.# *.&'
# &$ +.*&* '&. **.$+% +.&*
#* & .%$& '*.#% *#%.*$+' +&.&*
#& *' '&.$&&* #. %.+ &%&.$$%
#+ & '#.+'&*& #*&.&## *+.$%$% &#.$++
# #%.&+# #'#.#'&$ *.$& ***.#&
#% '& #&.**+& #.%''' '+'.$#+ %'.#$+
#$ ''& #.$'% %+.&#%& ''+.* '+.'*
' ' #'.&* .%+++& #$.*++ '%.+$#
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