Geologic stability of Mystic Lake dam, Gallatin County, Montana, and computer simulation of
potential flood hazards from the failure of the dam
by Graham Stephen Hayes
A thesis submitted in partial fuIIfiIlment of the requirements for the degree of MASTER OF SCIENCE
i n Earth Science
Montana State University
© Copyright by Graham Stephen Hayes (1981)
Abstract:
The failure of Mystic Lake dam poses a major threat to the residents of the Bozeman Creek drainage.
Outdated engineering practices used in the construction of the dam coupled with an unstable geologic
setting create a potentially hazardous situation. The east abutment of the oam is founded in the toe of a
Quaternary landslide. Water seeps through the landslide debris and ponds in a depression at the foot of
the dam. Unvegetated slumo scarps in the landslide directly below the dam site are attributed to the
increased pore pressure from the seepage water. The potential for liquefaction in the event of an
earthquake is extrerne Iy high.
A mathematical model is programmed in FORTRAN IV to simulate the failure of the dam and the
movement of the floodwave. The hypothetical failure is induced by overtopping the dam with a
rain-storm discharge greater than the spillway capacity (780 cfs). The breach is assumed to erode as an
exponential function of time, producing an estimated peak discharge of 83,500 cfs in approximately 7.5
minutes.
A hydraulic routing method utilizing the complete equations of unsteady flow is solved numerically by
a four-point implicit finite difference method.
Changes in the flow regime of Bozeman Creek make the computation of the initial water surface
profile and the establishment of intermediate boundary conditions impossible. Until sufficient gaging
data are available# trie routing portion of the model is not applicable to Bozeman Creek and the extent
of flooding from the failure of Mystic Lake dam cannot be simulated.
An estimate for the extent of flooding from the failure of the dam is approximated by plotting the
percentage attenuation of the breach hydrograph against the depth of flow.
A 60 percent attenuation yields a depth' of flow of 10.5 feet, at the canyon mouth # 4.5 feet deeper than
the 5 CO year flood. STATEMENT
In
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presenting
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PERMI SSI ON
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inspection..
for
Dat e.
OF
thesis
for
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It
is
thesis
written
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understood
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shall
GEOLOGI C S T A B I L I T Y OF MYSTI C LAKE DAM, GALLA TI N COUNTY
MONTANA, AND COMPUTER S I MU L A T I O N CF POT ENT I AL
FLOOD HAZARDS FROM THE FAI L URE OF THE DAM
by
GRAHAM STEPHEN HAYES
A t h e s i s su bm itte d in p a r t i a l
fuI I f i Ilment
of the r e q u i r e m e n t s f o r t he d eg r ee
of
MASTER
OF
SCI ENCE
i n
Earth
Science
Approved?
Chairman,Graduate
Committee
He a.d, i / Ma%p r & epai rf c^ment
Graduate
Dean
MONT ANA STATE U NI V E RS I T Y
Bozeman, Montana
'
Apri I ,
1981
A/3'72
c T ' - 5'
i i
VI TA
G r a h a m S t e p h e n H a y e s was b o r n i n C h a t h a m , O n t a r i o , C a n ­
ada on D e c e m b e r 2 8 , 1 9 5 5 .
H i s p a r e n t s a r e W i l f r e d H. H a y e s
a nd K a t h e r i n e V . H a y e s .
He a r a d u a t e d f r o m C y c r e s s L a k e
S e n i o r High S c h o o l , F o r t M y e r s , F l o r i d a , i n June 1973.
He
r e c e i v e d a B a c h e l o r of S c i e n c e d e g r e e i n g e o l o g y f rom
Wh e a t o n C o l l e g e , W h e a t o n , I l l i n o i s , i n J u n e 1 9 7 7 .
After
w o r k i n g o n e s u mme r f o r t h e G e o l o g i c a l S u r v e y o f C a n a d a , He
m a r r i e d K a r e n S. M a r k e l l o on J u n e 1 7 , 1 0 7 8 .
He e n t e r e d t h e
g r a d u a t e s c h o o l at M o n t a n a S t a t e U n i v e r s i t y t h e f o l l o w i n g
S e p t e m b e r , a nd h e l d b o t h r e s e a r c h and t e a c h i n g a s s i t a n t s h i p s
w h i l e w o r k i n g t o w a r d a M a s t e r of S c i e n c e Degr ee i n E a r t h
S c i ences.
ACKNOWLEDGMENT
The
Custer
ing
author
(committee
committee),
tinual
Raleigh,
ings.
De l
for
My
Georgia
U.S.
Zi emba
of
discussion
of
unsteady
and
Dr.
Donald
L.
Smith
study
A m e i n,
and
U n i v.
indebted,
allowing
Bergan t i n e ,
stream
Glen
to
Mr .
No r m
the: U. S.
for
Tom G r a d y
head-scratching
the
and
anc
of
to
the
the
dam
Conservation
data;
to
advice
Profes­
concerning
the. c o u n t l e s s
concerning
Dr.
program­
Buhl#
Soil
their
for
l i s t ­
Hinrichs
access
cross-section
Wyatt
con­
Carolina
Faulkner
with
vehicle
of
Ncrth
program
Mark
help
to
providing
Cf
and
to
(read­
the'investigation.
Martin
invaluable
for
and
Stephan
the
hours
equations
flow.
Finally,
the
is. also
and
Dr.
Professor
thesis.
problems,”
of
constant
to
the
and
to
correspondence
their
Taylor
thanks
throughout
Michael
providing
his
the
help
helpful
Ralph
for
Robert
drafting
Dr.
Service,
t o Mr .
Service,
a nd
for
writer
Forest
.
site;
and
(reading’ committee),
aspect
the
to
his
extend
in itia tin g
appreciation
Dyreson
ming
for
thanks
to
chairman),
encouragement
Special
sor
wishes
a
note
s uppo r t ,
computer,
and
of
thanks
to
pa t.i en c e w i t h
for
typing
the
my
my
wife
long
final
Karen,
and
copy
for
her.
I at e h o u r s
of
the
on.
thesis.
TABLE
OF
CONTENTS
Pa g e
i i
V I T A ............................ ..... .......................................................................
.............................................
i i i
.........................................................................................
vi
ACKNOWLEDGMENT.................................. ...
LIST
OF
FI GURES
ABSTRACT.
.
.
.
.' .
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.
.
I N T R OD U C T I ON .
.
. .
Location
.
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...
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I
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.
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.
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.
.
I
of
t h e Dam .
.
PROBLEMS
Geology
ASSOCI ATED
■Engineering
Geologic
...
WI TH
and
.
.
.
Construction
.
.
.
.
.
.
. .
Hazards
.
.
.
Dam
.
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%
.
.8
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I 2
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15
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■
21
?5
•
25
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. .
2o
.
27
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.
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3(J
.. .
.
.
. .
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.
. .
39
. .
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.
• .
39
.
40
Routing
.
.
.
Introduction.
.
.
.
........................................
. . . .
a routing
. .
.
11
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
hydrograph
solution
.
.
.
Breach
Numerical
.
.
.
development .
of
.
.
. .
Breach
Selection
.
.
. .
..................................................
.
.
........................................................ .....
.
.
Downstream
.
.
Introduction
Failure.
.
DAM.
Flood
DESCRI PTI ON
.
THE
Stability
Downst ream
I
.
History
Site
.
.
.
. .
.
.
the Study
.
.
.
of
.
vii
.
.......................
.
.
.
.
«
.
.
Purpose
Dam
MODEL
.
.
.
. .
method
. .
.
.
.
.
43
Pa q e
Dat a
COMPUTER
0 0 0
.
...
.
. •
•«
.
.
.
.
.
MODEL.
.
.
.
.
0 0 0 .
.
0
. .
•
•
.
Z>PP-END I C E S .
CITED.
.
0
.
.
.
.
.
•
.'.
. . .
.
.
-
•. . . . .
I.
Program
Fl ow
Chart.
I I „
Program
Listing .
Exampl e
Dat a
File
. . . .. .
.
.
.
.
. .
.
.
,
•
» -
•
53
6?
«.
66
^
71
AND CONCLUSI ONS
REFERENCES
III.
.
R E PR ES E'NT A T I ON OF THE
RESULTS ■.
SUMMARY
requirements
•' •
•
•
73
82
•
« •
•
• *
« «
.
.
.
« «
- 83
•. . .
•
« « •
84
»
=
1 16
LIST
OF
FI GURES'
Fi g u r e
Pa g e
1m I. o c a t i on Map
2.
Engineering
3.
Geologic
4«
5.
*
Map
Scarp
6 . Breach
of
the
Map
.
Development
Effective
Area
8.
Depth-Volume
9.
Rating
Curves
10.
Breach
Outflow
11.
Discrete
12.
Matrix
13.
Channel
14.
Depth
Area
15.
Matrix
to
of
«
of
*
*
.
©« ■ © .
for
for
and
Diversion
© .
Depth
around
17.
Cross-Section
of
18©'
Depth
19.
Cross-sectional
.
ah
Channel
Discharge
© © .
22
.
.29
©
.
. .
.
Ti me
.
.
w
Outlet
33.
© © ©
34
Gates©.
.
.
. . .
.
Rand
S .
©
. .
A,
..
©
.
© . .
Re a c h
. ■.
© .
.
Length
A tte n u a t io n Curve.
Leverich
>
©
© .
Coefficients.
Island.
.
© .©
. ©. . . . . .
.
9
. . .
.
© .
P r o f i l e ' at
.©■
. © .
„
Area
..
1
.
- Matrices
.
a
4
© © ©
M y s t i c Lake
Configuration.
^
© © •
« © «©
Flow t h r o u g h
T i m e - D i s t a n c e Grid©
solve
Region
* « •■©
Spi I I wa y
.
© •• © ©
D a m . . © . . . . .
Hydrography
Curve
*
t h r o u g h Time.
Breach
Curve
« •
the
*
Configuration
-
*
M y s t i c Lake
«
Ma p .
7.
16. . F l o w
« *
Drawings
S e i Sm i c Ri . sk
Sl ump
@ *
36
38
-
46
©
51
.
.
'
«
54
54
. . .
56
. ' ..•©/ « .'
.57
Ax
.
.
.
60
©
.
.
.
69.
Road
;..
.
.'
, 70' .
vi i
ABSTRACT
T h e f a i l u r e o f M y s t i c L a k e dam p o s e s a m a j o r t h r e a t t o
t h e r e s i d e n t s o f t h e B o z e ma n C r e e k d r a i n a g e .
Outdated
e n g i n e e r i n g p r a c t i c e s u s e d i n t h e c o n s t r u c t i o n o f t h e dam
c o u p l e d w i t h an u n s t a b l e g e o l o g i c s e t t i n g c r e a t e a p o t e n - .
tia lly
hazardous s i t u a t i o n .
Th e e a s t a b u t m e n t c f t h e dam i s
founded in the to e of a Q u a t e r n a r y l a n d s l i d e .
W a t e r s eeps ,
t h r o u g h t h e l a n d s l i d e d e b r i s and p o n d s i n a d e p r e s s i o n a t .
t he. f o o t o f t h e . d a m .
U n v e g e t a t e d s I u mo s c a r p s i n t h e l a n d ­
s l i d e d i r e c t l y b e l o w t h e dam s i t e a r e a t t r i b u t e d t o t h e
i n c r e a s e d por e p r e s s u r e from the seepage w a t e r .
Th e p o t e n ­
t i a l f o r l i q u e f a c t i o n i n t h e e v e n t o f an e a r t h q u a k e i s
e x t r erne I y h i g h .
A m a t h e m a t i c a l m o d e l i s p r o g r a m m e d i n FORTRAN I V t o
s i m u l a t e t h e f a i l u r e o f t h e dam a n d t h e m o v e m e n t o f t h e
floodwave.
The h y p o t h e t i c a l f a i l u r e i s i n d u c e d b y o v e r t o p ­
p i n g t h e dam w i t h a r a i n - s t o r m d i s c h a r g e g r e a t e r t h a n t h e
s p i l l w a y c a p a c i t y (730 c f s ) .
Th e b r e a c h i s a s s u m e d t o e r o d e
as an e x p o n e n t i a l f u n c t i o n o f t i m e # p r o d u c i n g an e s t i m a t e d
p e a k d i s c h a r g e o f 8 3# 5 00 c f s i n a p p r o x i m a t e l y 7 . 5 m i n u t e s .
A h y d r a u l i c r o u t i n g method u t i l i z i n g t h e c o m p l e t e e q u a t i o n s
o f u n s t e a d y f l o w i s s o l v e d n u m e r i c a l l y by a f o u r - p o i n t
i m p l i c i t f i n i t e d i f f e r e n c e method.
C h a n g e s i n t h e f l o w r e g i m e o f B o z e ma n C r e e k make t h e
c o m p u t a t i o n o f t h e i n i t i a l w a t e r s u r f a c e p r o f i l e and t h e
e s ta b lis h m e n t of i n t e r m e d i a t e boundary c o n d i t i o n s impos­
sible.
U n t i l s u f f i c i e n t ga g in g data are a v a i l a b l e # toe
r o u t i n g p o r t i o n o f t h e m o d e l i s n o t a p p l i c a b l e t o B o z e ma n
C r e e k and t h e e x t e n t of f l o o d i n g f r o m t h e f a i l u r e o f M y s t i c
L a k e dam c a n n o t b e s i m u l a t e d .
An e s t i m a t e f o r t h e e x t e n t o f f l o o d i n g f r o m t h e f a i l u r e
o f t h e dam i s a p p r o x i m a t e d b y p l o t t i n g t h e p e r c e n t a g e a t t e n ­
u a t i o n of the breach hydr og ra ph a g a in s t the depth of f l o w .
A 6 0 p e r c e n t a t t e n u a t i o n y i e l d s a d e p t h ' o f f l o w , o f 1 0 . 5 f eet ,
a t t h e c a n y o n m o u t h # 4 . 5 f e e t d e e p e r t h a n t he 5 CO y e a r
flood.
I NTRODUCTI ON
Location
Bozeman
flanks
of
oer
4.9
tic
Lake
man
in
Ra n g e
ship
Creek
the
G allatin
square
I).
and
South,
(Fig.
I).
canyon
mouth
and
floodplain
steady
(less
have
downtown
the
According
Engineers
is
western
East
than
Fr om
of
on
Lake,
the
I
up-'
Mys­
of
Boze­
South,
30,
Town­
B o z e ma n
Creek
north west-trending
stream,
River,
between
occupies
which
residential
Th e
into
section
the
wide),
drain
Township
forested,
of
the^northern
southeast
Mystic
reach
mile
Ma n y
(Fig.
half
G allatin
one
25,
on
Montana.
basin
12.miles
section
lower
established
B o z e ma n
Purpose.of
of
The
miles
southwestern
a narrow,
urbanization.
opments
of
7 East.
the
square
drainage
lake
the
through
canyon
the
half
Range
flows. 7 .m iles
of
in
52.4
Range,
Th e
eastern
6 East,
3
miles
(Fig.
the
drains
a nd
floodplain
is
the
a narrow
undergoing
commercial
including
devel-.
part
of
1).
Study
to
guidelines
(.1 9 7 5 ) ,
a high
downstream
Foster
( C H 2M H i l l ,
Mystic
hazard
1980,
established
Lake
dam
potential.
p.
iv)
by
the
U . S.
is
classified
I n.
an
Corps
as
executive
.
having
s u mma r y
stated:
B a s e d on v i s u a l r e c o n n a i s s a n c e a n d e n g i n e e r ­
i n g j u d g e m e n t , t h e dam i s . l o c a t e d s u c h t h a t i t s
f a i l u r e c o u l d c a u s e e x t e n s i v e p r o p e r t y d a ma g e a n d
p o s s ib le loss of l i f e .
2
111*os'
noTse
4 5 *4 3 '
I
S
O
I .8 O
Contour Intervel IOOO Feet
HOTS3'
Fiqure I.
L o c a t i o n map.
Note the
b e t w e e n M y s t i c L a k e , B o z e ma n C r e e k
spatial
and t h e
relationship
t own o f Bozeman.
3
In
his
recommendations
Creek
Reservoir
Co mp a n y
stressed
the
need
geologic
stab ility
the
extent
of
ure
of
dam.
the
The
the
available
of
cally
the
of
the
Bo z e ma n
this
dam
potential
of
and
section
Art
dam was
to
city
of
B o z e ma n
Th e
lake
in
height
ches
in
of
the
associated
is
Lake
along
to
dam
establish
the
compiling
a geologic
Creek
and
f a il­
firs t,
by
the
to
all
inves­
demonstrate
B o z e ma n
the
to
with
by
he
dam's
twofold:
dam
second,
B o z e ma n
Engineers,
routing
conducting
of
of
the
the
ex­
numeri­
movement
of
Dam
Engineer,
Lake
Th e
and
failure
property
waters.
hydraulic
data
site,
the
Corps
evaluation
Mystic
Service
pany.
U „ S„
of
Forest
the
a
Bozeman,
investigation
flooding
following
Mystic
the
of
downstream.
the
City
for
stab ility
f Ibodwave
The
and
of
simulating
History
and
city
further
engineering
tigation,
tent
for
the
downstream " f l o o d i n g
purpose
evaluate
to
is
original
from
the
diameter),
summar i zed
Va n ' t
through
a
I
to
the
store
e a rth -fill
the
dam
from
in
1903
special
and
use
irrigation
structure
(one
crest
a report
by
(1 9 80 ) .
Bozeman. Creek
o u t l e t "pipes
to
Hu l
constructed
and
used
wa s
on
permit
and
the
U.S.
granted
Reservoir
Com­
municipal
measured
16, a n d
(Fig.
1904
43
other
2 - A , B )«
feet
12. i n ­
Both
---- 20 —
crest elev. 6401.9
I
___________________
"
a
rX
a
#
#
— .......... * ;—
12 a 16 in. conduits^
...........................
—
:
sssWiasonry
cere wall
A. Cross Section
spillway
bedrock ...
crest elev. 6401.9
original crest
■
pipes
B. Profile - Longitudinal
F i g u r e 2.
E n g i n e e r i n g d r a w i n g s o f M y s t i c L a k e d a m.
C r o s s - s e c t i o n A i s drawn
t h r o u g h t h e dam a l o n g t h e o u t l e t p i p e s .
Note the e x t e n t of t h e c o r e w a l l i n
t he l o n g i t u d i n a l p r o f i l e B.
Wh e r e t h e c o r e w a l l i s a b s e n t / s e e o a a e w a t e r p o n d s
i n a d e p r e s s i o n a t t h e t o e o f t h e d a m/ p l a n v i e w C.
Mystic Lake
C. Plan View
Figure
2.
(continued)
6
upstream
the
and
upstream
had
face
A masonry
thick
at
long,
extended
line
the
d o w n s t r e a m, f a c e s
of
2 feet
riprapped.
base,
the
from
d a m.
below
There
2 feet
The
the
have
the
west
top
been
in-1 903.
ted
in
the
near
spillway
leaving
c i a Is
only
a .9
to
raise
the
s Di U w a y
cover
slab.
feet
long
Fl ume
dam
at
and
the
crest
feet.
1190
at
the
this
dam
hand
operated
the
new
.1967,
at
dismantled
a
two
slide
In
floor.
3
the
fa ll
of
the
since
construc­
original
dam,
v r cmpt-.eti
feet
to
its
height
with
At
foot
crest,
'1964,
was
installed.
the
made
s o i l Iway,
.1.8.5
long
time,
dam
1520
old
Three
in
the
s ame
outlet
developed
of
height
and
the
by
of f i -
a Parshall
the
present
4 40
were
the
concrete
gate
repairs
dam
the
concrete
The
the
B),
runoff.event
a new
crack
2-A,
high
2-C).
to
center
approximately
(Fig.
to
.
feet
a
installed
(Fig.
1 40
the
replace
and
longitudinal
spillway
on
spillway
the
to
anew
wa s
feet
the
dam
4 feet
s p i l l w a y ' was
1932,
by
elevation,
crest. '
wa s
In
1959,
pipes
a nd
was
modifications
freeboard
wide
raised
acre-feet
ture
in
feet
top,
the
abutment
aam c r e s t
outlet
was
At
ed
20
of
1 with
w all,
wall
core
A concrete
In
core
the
2-C).
foot
the
2 to
through
crest
west
(Fig.
at
of
abutment
several
installation
timber
of
original
its
1919
thick
slopes
of
the
48
impound­
acre-feet
gate
struc­
control
with
years
later,
lower
end
injecting
of
a
a
7
mixture
series
of
of
j " "I n
in
the
loud
masonry
holes
July*
drilled
1977*
upstream
roar
observed
from
end
of
the
sound
the
from
hole
the
and
pipes
within
the
dam
,
. .... .
B o z e ma n
clusion
that
the
embankment
dam
spec t i o n s
inch
r ev ea l e d
vertical
wall*
and
cast
locally, o.ut-of-round
removal
of. m a t e r i a l
from
into
the
outlet
pipe
as
pairs
could
the
12
at
conducted
in
the
level
the
1977).
by
of
of
inch
conduits
had
order
until
(NortherhTesting
had
face
of
long
bends
prevent
plpgs
ends
mor e
of
con^and
pipe.
pipe
and
outlet
the
wall
outlet
piping#,
and.downstream
Testing
the
to
core
to
expanding
of
led
the
upstream
measure
water
. D rill
Northern
inch, d i a m e t e r
dam b y
from
decreased*
flow
monitoring
12
In
$
we r e
downstream
1977-78#
the
by
.
(Williams#
the
iron
algae
.
settlement
a temporary
b e . made
'
mechanical
ups t ream
300.feet
lessened.and
sections.
sinkhole, development#
serted
reservoir
sheared
that
both
and
television
displacement
that
water
a nd
had
sinkhole
M u d dy
tests
differential
a
accompanied
the
a
floor,
.2-0*
to
o fficia ls
into
(Fig.
dye
city
bentonite
entering
200
diminished
(1977)*
and
spillway
observed
dam.
a pool
sand*
the
dam
As
fluorescein
by
the
spillway.
sinkhole
Laboratories
wa s
w it h in , the
.
through
of
from
washed
through
water
face
issuing
the
cement*
the
a
In3
core-
a nd
further
additional
wer e
the
12
complete
La bo r a t p r i e s *
in­
inch
re-
I 9 7,7 ) .
,
8
It
was
a nd
recommended
grouted
upstream
ries,
face
1977;
have
not
that
of
the
pressive
forces
Mystic
compressive
of
these
forces
1950).
the
be
be
To
reshaped,
applied
(Northern
to
Testing
this
date,
history
of
50
and
m illion
the
1970).
Further
north
formed
Lake
over
150
of
this
study
(1964a;
ago,
of
the
ago
lined
the
Laborato­
final
repairs
the
of
culmination
1 9 r, 5 ;
Aram,
Range
the
volcanic
breccias
(Fig.
wer e
ring
map
5).
the
was
of
the
rock
evolved
1979;
of
flows
produced
(tree
faults
a period
Gallatin
rocks
displacing
Nor mal
andesite
the
c o m­
northwest-trending
developed,
dam a t
region,
deformed
the
a*ter
the
geologic
1964b)
At
(McMannis,
a natural
A Generalized
ago.
vertically.
erosion
years
a series
extensive
end
orogeny
faults
years
involving
and
years
diminished
covered
east
into
thrust
development,
Landslide
Laramide
area
phase,
horizontally
Roberts
wall
geologic
m illion
units
poses
curtain
1977).
the
Lake
50-60
stream
core
pipes
undertaken.
summarize
folds#
outlet
Ge o l o g y
f To
the
both
a grout
Williams,
been
Dam S i t e
in
and
that
Hughes,
erosion
and
as
and
breccias
(Chadwick,
present
topography.
flowed
southern
end
from
the
of
Mystic
count).
produced
for
A
the
ma p p e d
pur­
The
rock
units
by
l u mp e d
into
undifferentiated
9
?!*:
+
+
+
Tv
+
LI
Landslide Deposits
MPa
Alluvium
—
m m
Volcanic Breccia
m
Amsden Formation
Mississippian
Undifferentiated
Devonian-Ordovician
Undifferentiated
Cretaceous
Undifferentiated
Cambrian
Undifferentiated
Jurassic
Undifferentiated
Precambrian
Undifferentiated
' V 1X-V
Contact
Fault
X
Anticline
X
"
Syncllne
Dashed w here
Approxim ate
Quadrant Quartzite
D o tte d where
C overed
F i g u r e 3.
G e n e r a l i z e d g e o l o g i c map o f t h e M y s t i c L a k e
region.
Th e dam i s f o u n d e d i n t h e u p p e r d o l o m i t e o f t h e
A ms d e n F o r m a t i o n , t h e Q u a d r a n t Q u a r t z i t e a n d a Q u a t e r n a r y
landslide deposit.
( A f t e r R o b e r t s , 1964a/ * 1 9 6 4 b ) .
10
units
the
with
age
distinction
foundation.of
Quartzite
a nd
the
status
assigned
ma p p e d
separately
slide
(1964b)
the
landslide
by
as
it
Ams d e n
deposit,
Roberts.
check
of
incorrectly
the. C r e t a c e o u s
ternary
Lake.
landslide
. Outcrops
rison,
basal
The
is
of
contact
the
also
at
ma p p e d
first
the
the
units
which
Formation,
retain
volcanic
source
the
the
the
for
act
as
Quadrant
original
breccia
rock
in
of
stream
of
the
w a s ' a I so
the
and
slide,
Th e
the
land­
Jurassic
where
of
the
rich
Morrison
the
shore
of
of
Mystic
Mor­
lacustrine
from
extended
Paleozoic
Qua­
upper
conclusion.
downstream
was
Roberts
part
shale
this
contact
confluence
as
gastropod
support
that
eastern
carbonaceous
the
error.
showed
Formations
along
Kootenai
toe
site
portions
Kootenai
deposit
of
dam
ma p p e d
conglomerate
limestones
the
d a m,
Three
debris.
A field
a nd
the
only*
Th e
the
dam
was
to
the
south
rocks
are
ex­
posed.
The
dam
anticline
that
portion
of
cent
the
of
white
ish
to
site
the
white,
located
plunges
Vtn.sden
west
huff
is
to
the
with
medium-grained,
is
the
northeast
southeast
Formation,
abutment,
dolomite
on
which
(Fig.
of
3).
The
underlies
a resistant,
interbedded
q u a r t zose
fimb
and
an
13-15
upper
per­
fine-grained,
stringers
of
calcareous
yellow­
sandstones.
tween
the
According
Ams d e n
boundary
is
drawn
sandstone.
Quadrant
at
the
sandstone
of
The
the
site
dam
grain
angular
Both
the
units
it
the
is
is
founded.in
within
to
breccia.
Th e
a distinctive
rust
s ilt
the
size
breccia
becomes
W and
resistant
toe
of
deposit
coloration
to
the
the
foundation.
NE.
forming
cement.
The
from
landslide
ouartz
remainder
deposit.
boulder
from
texture
The
unit,
buff,
lancslide
derived
a blocky
The
dcminantely
grained,
range
be­
50'
c lif f
the
particles
has
contact
dip
of
me d i u m
calcareous
the
the
gradational.
25 percent
a
by
is
N 10'
a friable,
together
blocks
up
is
(1955),
section
strike
held
sizes
Quadrant
ma k e s
Quadrant
dam
McManni s
the
where
Quartzite
Elsewhere
but
and
to
and
size
the
ande­
imparts
deposit.
PROBLEMS
■
With
Mystic
76
the
Lake
years.
■
-
dam
has
pear
significant;
ing
and
w ill
■
of
the
relatively
are
'
.
.
outlet
well
for
pipe.
the
last
many
problems
associated
these
or o h l e m s
do
presented
warrant
as
careful
discussed
in. t h r e e
geologic
stability
a
not
ap­
composite
of
consideration. .
sec t i o n s T
and
engineer­
downstream
hazards.
and
Building
considered
to
1959;
Sowers,
years
into
principal
as
the
Williams
earth
be
a
dam
safe
1962).
means
as
a core
( 1959,
reducing
wall
p.
to
the
providing
pressures
with
a core
engineering
literature
of
.
In. f a c t ,
.O riginally,
well
However,
Construction
an
the
wall.
stand.
be
TH E DAM
'
.failure
wh e n
they
WI TH
'
singly
however,
construction,
Engineering
dam
there
viewed
interrelated.problems
problems
the
functioned
dam..'When
The
'
.
of
Nevertheless,
the
core
.
exception
with
flood
ASSOCI ATED
one
even
core
the
2
exerted
4) . a I so
to
find
wall
3
bn
the
of
Icok
back
of
thought
water
dam
thick
Lonoer
(Williams,.
twenty
a masonry
to
be
through
structural
feet
reported
no
mention
was
flow
to
is
practice
has
additional
even
wall
can
the
the
strength.
nqt
( W i i I i a ms ,
with­
1959)=
that:..
I n . the. c a s e o f M y s t i c L a k e , t h e c o r e w a l l i s
k n o w n t o h a v e c r a c k e d b e f o r e c o n s t r u c t i o n was c o m­
p l e t e d . .• T h e e x t e n t o f c r a c k i n g i s n o t . k n o w n .
.
h o w e v e r , t h e c o r e w a l l m u s t be d i s c o u n t e d , as f a r
13
as
impart inq
The
"additional
the
engineers
any
of
a much
Mystic
at
safety
requirements
does
(Fig.
not
2-B ) .
abutment,
dam
at
Wh e r e
accumulates
dam
(Fig..
tories,
ponding.in
varies
(W illiams,
CH2M H i l l ,
such
a way
foundation
and
ure
of
( CH2M
Part
qf
dam
of
the
"zonation"
A properly
with
the
the
least
more
I 97.7;
Wi I.s o n
and
seepage
tests
conducted
Testing
a no
dam
through
downstream
the
Labora­
water
pressure
lead
the
and
from
that
p.o.r e - w a t e r
core
east
level
feared
material
may
also
dam
should
at
the
material
Bureau
Marsal ,
be
materials
material
by
is
the
the
mo v e s
moder n
the
of
near
reservoir
the
by
to
the
.
in
fa il­
1980),
earth
Barron,
emb an k me n t
height
Northern
problem
U.S.
or
immediately
raise
prompted
Furthermore,
Wat er
the
wall
allowed
exists.
with
permeable
( W i l l i a m s , 1959;
than
absent
embankment
permeable
the
is
embankment
seepage
core
build
length
198,0)'. • I t
H ill,
the
1959).
1959;
will,
constructed
progressively
face
of
to
wall
problem
the
the
core
by
(2:1)
total
a depression
2- C )
1977;
angle
the
the
which
in
dam
(W illiams,
extend
rate
Lake
steeper
a seepage
a
strength.
strength", provided
slopes
wall
structural
of
to
the
(W illiams,
be
built
core
built
up
and
out
to
Reclamation,
1979).
Northern
due
D rill
Testing
lack
1 95 9 ) .
in
zones
Iqyers
of.
t h e . dam
1974;
cores
and
Laboratories
I 4
revealed
within
"permeable
the
Testing
embankment
failure
hole
development
ment
of
the
breakage
d a m.
of
the
failure
Williams,
of
If
the
storm
in
events
the
U.S.
Wilson
Lake
and
dam,
is
of
Marsal,
largest
flood
that
most
tions' possible
Reclamation,
the
4.9
inches
in
and
72
in
square
in
6 hours,
hours;
that
the
(Northern
settle­
may
lead
the
assumed
of
a nd
to
■
,1 9 7 7 ;
a
a nd
be
and
1980).
in
that
on
A PMF
1964;
is
from
U.S.
the
a combin­
condi­
Bur eau
precipitation
estimated
hours,
the
rain
to
1 977;
Mystic
me t e o r o l o g i c
12
snowmelt
failures
routed, a probable
The
was
dam
study
expected
( Ch o w,
basin
of
high
Harbauqh,
reservoir.
region
inches
unusually
cause
Fread
hydrologic
contribution
sink­
subsequent
a nd
by
common
estimated
drainage
15.7
resultinq
Laboratories,
conducting
CH2M H i l l ,
I t . wa s
dam
continues,
Testing
reasonably
a given
the
continue
I 953;
through
mile
Lake
differential
spillway
In
severe
and
the
w ill
most
(1 9 8 0 )
can
1977;
to
dam o r
1,979).
( PMF )
channels"
I '980).
second
CH2 M H i l l
the
pipes
(Northern
(Middle-brooks,
flood
of
pipes
the
the
erosion
Mystic
the, s e t t l e m e n t
dam
ma x i mu m
ation
outlet
CH2M H i l l ,
Overtopping
of
attributed
outlet
197 7 ;
internal
1977).
of.the
is
or
materials
Laboratories,
Th e
the
lenses
and
fell
runoff
to
be
18.1
on
of
in
11.8
inches
snowpack
was' e q u a l
to
'
the
r a t e ’o f
,ted
vo I u me
cfs.
780
23
percent
15
in filtra tio n .
of
5x600
Because
of
.
c f Sx
the
the
of
The
acre-feet
spillway
PMF
be
had
PMF
and
a peak
the
dam h a s
on
dam w o u l d
the
resulting
the
an
discharge
overtopped
entered
had
estirna—
of
a ma x i mu m
after
36x800
capacity
approximately
reservoir
(CH2M
Hillx
,1980).
In
r e cbecking
tained
in
the
C H2M H i l l ' s
U.S.
Bureau
( 1 9 7 7 ) > t h e PMF
Da ms "
inches
of
rain
ac r e - f e e t .
i n . 72
Even
change, in
storm
this
even
be
without
istic
storm
with
Geo I o g i c
S t a b i I i ty
' Three
V
dam a r e
a nd
ma s s
the
factors
seismic
be
by
a . much
of
town
that
greater
than
a
sig­
that
of
the
smaller.and
14. 5/ . .
1x820
represent
the
Small
only
Recognized
d a m; ..and
discharge
controlling
hazardsx
movement
United
difference
devastate
the
produce
figures
must
con­
a
B o z e ma n
dam ■c o u l d
mo r e
the
real­
spillway
cfs).
. , . .
Though
of
a peak
(780
it
would
breached
capacity
new
data
"Design.of
should
a volume
t h e , PMFx.
failure
and
area
these
magnitude
the.
overtopped
hours;
agairst
Recla m a t i d n x
this
though
nificant
of
in
of
figures
only
States
.
the
condition
been
'
. of
stability
,
the
of
the
.
f oundat ion x
site.
small, p e r c e n t a g e
have
geologic
.
n e a r . t h e .dam
a
the
caused
of
t he.
dam
failures
by. e a r t h q u a k e s ; :
the'
in
;
.
16
pr obl em- must
be
tive
area.
Th e
four
levels
of
States
based
seismic
would
risk
zones
bIe
in
past
shocks
■ ■'
0.2g
(CH2M
sliding
of
water
pressure
'
cracks induced
the
of
dam. d u e
landslides
major -earthquakes
the
country
( A l g e r m i s sen
and
Per­
Lake
dam
is
d a ma g e
lifetim e#
1959.
and
has
in
largest
bedrock
1980).
for
was
d a ma g e
the'shock
The
a zone
being
acceleration
estimated
from.an
to
to
entering
the
Mystic
greater
earthquake
depends
hazards
faulting
due
tectonic
seiches,
at
be
to
an
to
of
produced
reservoir.
an
and
ground
(Sherard
epicene
earth
in
' - ■
:
d a m.
pore-
through
freeboard
:
on
I
overtopping
motion
a nd
on
foundation#
increase
Movement #
by
the
'
the
(liq u e fa c tio n )# piping fa ilu re s
- -' - . -. .
b y g r o u n d m o t i o n # a- d e c r e a s e i n
or
Hebgen
-
principal
dam e m b a n k m e n t
several
the
and. t h e , p r o x i m i t y of
■ .
-
include:
su'scepti--.
experienced
-
H ill#
site.
located
the
Th e
event#
dam b y . s e t t l e m e n t
the
no.
of
activity
the
.Although
rest
size
seismic
map#
4).
United-
same
the
dam
t he
the
the
potential
the
in
delineated
in
The
to
on
this
of
d a ma q e
activity.(Fig.
'
than
Survey
ac­
U . S.. t h a n
of
during
seismi c a l l y
the .western
■
dam#
a
in
its
Lake
from
shown
earthquake
in
Geodetic
earthquake
seismic
in
often
the
La k e .',ea r t h qua ke
ter
and
Coast
Mystic
t o -maj or
both
U« S .
mo r e
1976).
major
especially
frequency -is
occur
kins,
addressed#
potential
on
'
or
others#
|
I 0 6 3;
'
I
Major Damage
Figure
damage
4.
S e i s m i c r i s k map o f t h e U n i t e d S t a t e s s h o w i n g
from e a r t h q u a k e a c t i v i t y .
( A f t e r A l g e r m i s s e n and
the p o t e n t i a l f o r
Perkins, 1976).
,Housnerz
1977).
The
fact
that
seismic
events
ability
to
tive
the
It
X,
been
(7.3
on
being
Two
of
the
location
with
in
a nd
of
50 y e a r s
section
hazards.
by
material
embankment
the
i h
a re
regarding
the
n ex 't.
could
the
Th e
unknown
a 10
(A I q e r m i s s e n
as
are
earthquake.
Mercalli
next
a
intensity
TOO y e a r s ,
percent
and
its
c um u I a - .
experience
of
ano
chance
Perkins,
is
or
a. f l u i d
problem,
ground
previously.in
a-
of
1976;
worth
repeating
Seepage
water
ponding
a. p o t e n t i a l l y
pqre-water
subjected
greatly
foundation
(Seed,
material
1973;
induced
by
Wilson
in
at
pressure
the
the
the
hazardous
to-severe
reduces
differential
motion
the. e n g i n e e r i n g
are
creates
liquefaction
other
area
0 . 4 g . has
mentioned
earthquake,high
The
of
Modified
scale),
previous
earthquake.
timing
a ma x i mu m
During
as
major
this
2-C)
behave
complacency
earthquakes
dam - ( F i g .
the
to
survived
1 98 0 ). .
seismic
sulting
lead
that
Richter
construction
saturated
dan.has
previous
problems
an
Lake
another,
acceleration
exceeded
CU? M H i l l ,
not
estimated
event
horizontal
a nd
from
magnitude,
seismic
should
withstand
effects
has
Mystic
toe
of
develops
strength
causes
and
Mar sa I > 19 7 9 ) .
earthquakes.
is
when
The.re­
and
settlement,
the
situation.
shocks..
shear
context
it
of
to
intensified
■Sherard
19
(1973,
p.
343)
reported:
F o r a dam t h a t i s u n d e r g o i n g r e l a t i v e l y l a r g e b u t
s lo w d i f f e r e n t i a l s e t t l e m e n t , even mo d er a te e a r t h ­
q u a k e s h a k i n g c o u l d be s u f f i c i e n t t o c a u s e a b r u p t
opening of c ra c k s ,
A com bination of moderate d i f ­
f e r e n t i a l s e t t l e m e n t and a m o d e r a t e e a r t h q u a k e
c o u l d r e s u l t ih la r g e c r a c k s .
This
may
implies
not
only
amplify
the
The
a dam
ern
Standard
ples
a nd
of
which
lies
the
that
ma k e s
deeol.y
the
Northern
up
upper
15
remainder
Testing
in
into
and
pump
of
The
the
in
of
the
t wo
Laboratories
earthquake.
stability
performed
holes
by
conducted,
unit
of
the
Quartzite,
abutment,
p.
was
2)
s am­
cored,
These
Ams d en
abutment,
four
d am .
wer e
performed.
west
(1977,
the
material
North­
and
tests,
were
of
foundation
cf
Quadrant
west
an
dam
may
foundation
were
of'the
the
Lake
but
of
d r ill
foundation
tests
Mystic
the
program
1977,
dolomite
percent
weathered.
event
stability
resistance
embankment
the
the
a d rillin g
bored
and
in
of
a ctivity,
consideration
Laboratories
were
settlement
earthquake
geologic
penetration
the
by
hazards
During
permeability
reveal
and
caused
important
1 975 ) .
holes
differential
potential
is
Testing
probe
be
most
site
(AS CE,
that
is
tests
Formation,
fractured
which
under­
reported
t o. b e :
p a r t i a l l y t o c o m p l e t e l y decomposed t o a medi um. g r a i n e d q ua r t z o s e s a n d a t t h e c o n t a c t , b e c o m i n g
s o m e w h a t mor e m a s s i v e w i t h d e p t h .
Th e d e c o m p o s e d
a n d f r a c t u r e d a r e a s a r e v e r y p e r m e a b l e as
by
20
e v i d e n c e d b y pump
tio n bedrock, ■
The
remaining
the
entire
Seepage
east
near
of
the
in
solid,
slide
dam s i t e
50
to
60
the
east
permeability.
is
of
Th e
dam’ s
fractured,
in
dam's
in
the
the
ASCE
foundation
the
and
p.
and
debris.
permeability
is
materials
(1975,
founda­
landslide
foundation
weathered,
to
the
a tte s ts , to
bedrock;
According
the
located
abutment
d e b ris ...
all
conducted
percent
abutment
impermeable
are
tests
net.
anchored
present
exhibit
at
.
the
high
88):
l e a k a g e i n t h e f o u n d a t i o n and e mb a n k me n t a r e t he
m o s t f r e q u e n t ■c a u s e o f f a i l u r e s a nd a c c i d e n t s ,
e v en t o moder n dams.
Since
slide,
it
over
is
half
the
imperative
dam
to
recent.mass
of
the
dam a p p e a r s
to
in
the
deposit
straight
are
However,
from
t h e . dam a n d
dam,
are
movement.
slump
caused
features
is
not
numerous
in
are
surface
oresent
full
in
re la ti vely
the
the
and
toe
body
blocks
visible
creep
on
the
in
a small
equilibrium .
dam
of
the
site
for
evidence
east
abutment
the
Trees
signs
of
landslide
recent,
downstream
of
scale
'the
ma s s
along
as. " p i s t o l - h u t t e d "
trees
scale,
leaning
recent
east
small
growing
uphill
slide
The
land­
landslide
trees
well
no
the
the
of
at
a recent
stable .
show
of
with,
as
in
landslide
indications
Rotated
scarps
by
in
The
be
founded
examine
of
movement.
movement.
is
deposit.
All
indicate, that
sl ump
scarps
of
the
below
these
slide,
the
da nr
21
range
ure
in
5.
height,
Tree
from
2 to
coring
to
to
the
scarp
several
of
the
vegetated
roadway
and
I f t h i s
is
lack
vegetation
of
gests
fact
true/
that
that
below
pressure
the
mass,
latest
directly
below
stability
a nd
Dow n s t r e a m
. ' Wi t h
the
Flood
the
the
on
year
given
a
the
read
dam
.5)
the
in
a nd
m?y
of
the
dam.
turn
east
of
Mystic
side
flood,
(U.S.
situated
increased
have
triggered
these
letter
to
developments
Creek,
Lake
town
(one
Soil
the
The
scarps
as
to
the
d a m.
the
that
also
dwellings
are
Corps
of
for
damage
and
2 busines- "
the
of . o c c u r i n g
Service,
on
increasing.
a f f e c t e d . - by
p e r c e n t , chance
Conservation
increasing
potential
dam i s
residential
of
sug­
that
serious.questions
housing
50
The
scarps.
are
implies
presence
of
construction.
vegetated
the
with
trend
( F i g . 5)
the
trees
"
B o z e ma n
are
year)
In
of
there
frequency
raises
of
number
of
the
8
fig ­
' However,
younger.than
(Fig.
The
the
in
Hazards
failure
,Presently,
ses
safety
are
to
slide; material
mo v e me n t . .
dam
of
the
related
than
the
depicted
age
follow
A and
pond
are
inconclus ive.
B parallel
in
the
floodplain
from
younger
water
pore-water
be
scarps
A and
seepage
was
scarp's
on
are
to
and
the
scarps
appear
scarps
the
mo v e me n t
these
they
feet
determine
respect
would
25
100Vn
19R0).,
E n g i n e e r s , included
in.
*
\
spillway
F i g u r e 5.
L o c a t i o n map o f s l u m p s c a r p s i n t h e t o e o f t h e
l a n d s l i d e b e l o w t h e dam .
S c a r p s A and B a r e u n v e q e t a t e d #
a l l o t h e r s are v e g e t a t e d .
Hachures denote downthrown
blocks.
Sl ump s c a r p s ar e a t t r i b u t e d t o i n c r e a s e d p o r e p r e s s u r e from the seepage wa ter pond.
2?
appendix.
4 of
the
Engineer,
Art
would
cause
not
CH2M H i l l
Va n 1 t
Hul
report ' M 980)/
proposed
extensive
that
property
Rozeman- C i t y
the
d a ma g e
failure
of
the
or
of
Iite
loss
dam
because:
t h e a r e a i m m e d i a t e l y d o w n s t r e a m o f t h e m o u t h - of
t h e - c a n y o n o p e n s up o n t o f l a t a l l u v i a l f a n s t h a t
w o u l d s p r e a d any w a t e r s o v e r a f a i r l y
wi de area..
Several
fans,
t em
aI
canyons
but
has
Ro z e ma n
but
of
the
floodplain
allow
it
to
- ma n
the
by
water
canyon,
are
the
through
the
cause
constrictions
at
flow
wer e
would
at
to
he
the
the
into
sys­
aI luvir
Fr om
water
that
flow
capacity
not
b e e n ' an
the
a narrow
rather
convey
constructed
not
made
stream
such.-
of
alluvial
than
has. p r o p o s e d .
are
( U. S .
Th e
have
as
have
incised
culverts
4 do
flood
Soil
is
the
and
flow
d a m ag e
(U.S.
Creek
front
t h e m. -
may
V a n 11 H u l
only
restrictions
flood
what
bridges
estimates
culverts
Any
major
9
frequency
These
debris.
or
of
unrecognizable
as
17,
event,
1.972).
into
bridges
roads
Range
one
restrict
out
Of.the
s ame
not
B o z e ma n
would
17
where
100-year
is
itse lf
spread
Creek.
Gallatin
presently
that
There
of
is
the
Creek
entrenched
fan,
mout h
along
Soil
at
and
across
Ho z e
bridges
that
by
or
the
ice
during
overtopped
Conservation
blocked
Conservation
flow
capacity
4 are
assuming
areas
the
or
Service,
flow
other
culverts
ups tre am- from
Service,
could
these
1972).
Several
major
east-west
structed
above
culverts
and
handle
area
the
elevation
bridges
large
flood
upstream
the
until
the
number
the
flow
tia lly
lack
of
of
low
of
of
situation
ity
of
Mystic
Lake
property
damage
failure.
The
ing
a model
ure
of
the
to
dam
and
of
.dam a n d
loss
remainder
predict
of
of
the
the
life
this
the
and
designed
up,
flooding
south
car,yen
culverts
would
to
the
of
mout h#
and
conveying
create
points
severe
in
of
to
a poten­
is
the
instabil­
potential
the. e v e n t
paper
extent
net
the
dam f a i l e d .
evidence
to
back
at
Cr e e k ,
the
are
con­
overtopped.
bridges
if
been
Since
developments
fan
in.Bozeman
body
is
have
floodplain.
will
housing
alluvial
town
crossings
roadway
the
the
water
capacity
A significant
s o u th , of
of
these
the
an
water
hazardous
at
events,
A combination
town,
roads
of
devoted
flooding
tit
the
to
from
dam s
describ­
the
fa il­
MODEL
DE S CRI P T I ON
I n t r o d u c t i on
The
of
the
implementation
dam
urements
of
the
or
be
system
cost
desired
of
insures
data
accuracy
model
the
best
Models
symbolic,
the
ically
and
real
the
real
the
idealized
can
1976).
and
and
system
for
never
into
mus t
set
of
data
of
basic
mo d e l
look-alike
is
model
with
the
scale).
real
by
a "mathematical
situation
real
system"
that
shares
the
An
analog
mo d e l
is
a hybrid
ic
model.
The
real
system
circuits
(M iller
arranged
in
and
some
real
between
of
and
the
the
best
requirements,
the
real
groups :
system.
physical
a representation
that
A symbolic
shares
mo d e l
structural
< us u
approxi­
of
an
properties
. 1975 ,
a physical
dynam­
system
description
Wool h i se r ,
between
is
the
natural
in
can
a model,
reached
combination
a reduction
system
reason
be
meas
possible)
implementation,
three
characteristics
failure
system
Th e
A. p h y s i c a l
a
not
simulate
approximation
broken
by
that
fully
model
realistic
of
'tronic
real
The
the
reouires
system.
model.
possible
be
f loodwave
A balance
the
a
the
simulate
are
collection,
of
to
measurements
completely,
analog.
sim ilar
with
mat es
can
of
(when
how d e t a i l e d ,
available
ally
of
(Zeigler,
a model
mov e me nt
estimates
measured
no. m a t t e r
of
the
behavior
never
the
a nd
of
p.
and
i67).
a symbol
depicted
by
a network
of
elec
accordance
to
ecuat ions
from
.26
electronic
ic
model
theory,
all.threp
drawbacks,
what
heavily
of
time
money.
and
for
the
an
reproducing
The
ent
equations,from
Th e
study
analog
the
a nd
lack
the
a symbol­
the
downstream
model"
previous
the
the
experience
and
broken
dam.
with
equipment
into
of
a
two
in
compon­
The
floodwave
second
through
■ y'
Dam F a i l u r e
This
portion
of
sections.
The
firs t
breach
the
second
through
and
the
breach.
t h.e m o d e l
deaf s w ith
is
also
t he
approximates
subdivided
d e v e I o p me n t
the
to
time-depend­
hydrograph.
the
in
model.
development
of
ex­
best
reasons:
previous
is
de­
implementation
following
outflow
the
and
f o r see n d i f f i c u l t i e s
model
mo v ement
from
of
considered
physical
the
merits
equipment,
experience
the
scale
resulting
the
the
of
and
simulates
simulates
channel
of
is
both
av ai l a b l e
cost
computer;
mo d e l , "
accurate
of
the
pn
have
best
mode I
based
a digital
first
breach
model s
and
A symbolic
of
an
of
availability
description
component
the
the
programming;
generate
ents.
types
modeller,
this
availability
FORTRAN
the
c o ’h s t i t.u' t es. " t h e
on
perience
choice
to
(Chow, ' 1 9 6 4 ) . .
Since
pends
analogous
of
into
t wb.
, t he
o u t f I oiw - hy d r o g r a,ph
Breach
the
development.
f a i l u r e , of
There
are
t wo
(Yevjevichz
in t o . t h e
take
( Chow,
crest
an
Surface
t h e . dam b y
material
failure
storm
is
spillway
capacity
•to
accurately
to
take
on
Even
erosion
is
at
are
of
simulated
by
( 780
the
most
(Yevjevich,
with
breaches
of
works
of
pressure
The
of
fa il­
of
the
Since
side
1978),
The
greater
reservoir
embank­
breaches,
is
only
hypotheti­
the.dam,by
lake
the
the
dr -
(Yevjevich,
deep
of
to
interface)
overtopping
piping
develop
1975 1 Fr e a d ,
flow
mode
this- s t u d y .
Th e
eroded
reservoir.
the
.
deep
induced, s e tt le m e n t
by
of. f a i l u r e .
form
by
with
overtopping
into
and
under
the
by
escapes.
air-water
of
nature
t h e . shape, o f
a triangular
water
opening
flow
outlet
in
c.f s ) .
time
predict
though
the
created
free-surface
determined
caused
an
the
surface
complex
a discharge
capacity
at
the
induced
p roducing
the
failure
breach
being
seismically
breaches
Because .of
allows
are.associated
storms,
or
is
breach, involves
generally
opening
impounded
breach
dam t h a t
breaches
rain
the
b r e a c h e s 1— s u r f a c e
free-surface
is
Deep
a surface
of
op. eni n.g b e l o w
ure.
1975)..
the
A deep
of.breach
seiches.
types
is
which
A surface
of
(the
type
cal
major
1964).
through
me n t
a dam- t h r o u g h
1975).
place,
A breach
a
rain
than
a s s u me d
the
to
be
it
is
impossible,
breach
it
is
slope
by.
of
assumed
I. to. I.
progressive
the
majority
of
28
modellers
complete
have
assumed
the
breach
to
(encompassing
the
entire
dam),
Dressier,
1954;
U. S .
19 6 1 ;
Vasi I i ev ,
19 7 3 ;
Yevjevich,
Thomas,
1 977;
1979).
The
breach,
while
is
not
This
19 7 0 ;
Glass
1978, *
assumption
being
of
a nd
U . S.
a
the
true
of
(Stoker,
Sakkas
others,
Soil
1957;
and
complete
case
I960;.
Gundlach
a nd
Service,
instantaneous
for. m a t h e m a t i c a l
the
1 95 7;
Strelkoff,
1 976, '
a nd
and
19 4 F ;
Conservation
simulation
in
instantaneous
Engineers,
1972;
attractive
for
especially
Corps
Thomas,
1975;
Ra j a r ,
realistic
is
A r my
be
sim plicity,
of
most
dam
failures.
of
e a rth -fill
dams
"
which
require
attain
its
a
final
•(■B a l l o f f e t
a nd
Har b a u g h ,
1 973. ) ,
the
development
finite
amount
size
CF r e a d ,
others,
while
1 97 4 ;
of
time
for
1978).
Fre a d ,
recognizing
the
Other
I 978;
breach
to
investigators
I 980;, Fread
and
the
importance
of
a constant
rate
of
the
breach,
assumed
mooel
the
rate
of
breach
as
function
time
in
of
erosion.
In
this
increase
exponentially
illustrates
surface
may
be
the
breach.
a
development
The
area
of
of
a
the
erosion
of
time.
is
Figure
time-dependent,
breach
at
any
a s s u me d
to
6
triangular
time
step,
expressed.as:
Area
"
•
'= B r e a c h
Z
Depth'
(1)
no
<sD
T = Time
BDn= Breach Depth
BAn= Area
F i g u r e 6.
E r o s i o n a l dev elopment of a t r i a n g u l a r s u r f a c e breach thr oug h t i m e .
The b r e a c n d e p t h ( B D ) i n c r e a s e s l i n e a r l y w i t h t i m e w h i l e t h e b r e a c h a r e a ( BA)
i n c r e a s e s e x p o n e n t i a l l y as a f u n c t i o n o f d e p t h , ( BA = BC2 ) .
30
The
breach
function
area,
of
increases
therefore,
depth.
linearly
Breach
where
rate
the
of
ratio
change
dimensions
of
the
depth
the
breach
eral
to
The
breach
reached
time
quickly
on
structure
s e e ms
t wo
wh e n
lines
reasonable.
that
of
hand,
range
failed
in
of
was
Foster
of
the
of
the
factor
used
is
in
Cr eek
min­
dam,
overtopped
5 min­
( CH2M H i l l ,
materials
by
or
Lat­
in
5 t o. 10
Buffalo
to
ma x i mu m
base
approximately
overtopped
evidence
d a m,
size,
the.
cease.
this
in
evidence.
the
velocity
Lake
(2)
dam h e i g h t
will
flow
the
T i me
Th e
once
Mystic
constructed
fa il
the
sim plicity,
is
and
by
the
according
dam
feet/sec.
comparable
and
other
establishes
dbwncutting
Also
"the
these
of
flood
1978).
erode
be
following
in
as s umed
for
to
Ti me
governed
for
the
v ),
is
the
a
time:
depth
However,
on
p.
on
as
Dam H e i g h t / F a i l u r e
until
based
( Fr e a dy
of
continue
failure
induced
x
bedrock
estimated
a rain
are
It
is
by
time
breach
2,
e a rth -fill
Based
the
would
an
1980,
of
depth,
function
= Ti me
exponentially
Dam H e i g h t / F a i l u r e
decreased.
equation
ute's
of
breach
a
bedrock.
has
ignored.
utes,
as
Depth
the
erosion
breach
Th e
increases
that
woulo
flood-waters."
estimated
failure
31
Breach
plotted
as
hydrograph.
a function
hydrograph.
the
In
failure
of
order
Mystic
. The
discharge
of
time
is
to
show
the
Lake
dam,
from
known
as
breach
dam
a breach
effects
a
a breached
of
flooding
hydrograph
from
mus t
be
generated.
Previous
breached
ing
investigators
dam
could
broad-crested
I 960,'
1 96 1 , '
weir
creases
well;
flow
in
depth
to
breach
time,
attaining
a
(1957,
p.
the
the
Corps
through
Harbaugh,
involves
the
not
1 9 5 7 ,'
1 97 3 ) . .
length
only
of
in­
increases
in
length
greater
than
?00
length
a
describ
Engineers,
channel
but
flow
equations
of
and
assumption
The
through
Tracy
(U.S.
by
1 980, ' . F r e a d
this
channel.
eventually
According
flow
1978;
with
as sumed
approximated
weir
Fre a d,
A major.problem
the
be
have
as
feet.
2):
f o r very small h e a d - t o - l e n g t h r a t i o s , , the weir
c r e s t becomes a r e a c h o f o pen c h a n n e l
in which
frictional
r e s i s t a n c e p r e d o m i n a t e s , and f o r which
t h e d i s c h a r g e i s mor e p r o p e r l y e v a l u a t e d by one o f
t h e open c h a n n e l f l o w f o r m u l a s t h a n by a w e i r
formula.
Th e
ually
to
flow
through
rapidly-varied,
varies
with, time
steady
flow
the
flow
a breached
can
and
be
conditions
unsteady
space.
treated
over
as
the
dam. i s
flow
classified
because
However,
for
steady
flow
time
step
is
the
as
velocity
sim plicity,
if
the
grad­
un­
change
negligible.
in
(Chow,
32
1959).
model
Manning's
to
equation
approximate
the
for
steady
outflow
flow,
from
the
used
in
breach#
this
is
written
as:
1.49
.. n
Q
where
is
Q is
the
the
the
discharge#
hydraulic
Manning’ s
Using
3 S1Z2
radius#
roughness
Manning's
as
a function
of
As
the
area
ing
breach
through
level
the
the
. i n. t h e
level
tween
the
(AL)
Th e
tion
of
ume
the
of
(Fig.
the
increases#
breach
of
the
as
a
and
lake
pressing
area
to
breach
the
8)
the
in
any
the
discharge.
level
lake
Th e
of
P
and. n i s -
discharge
of
water
well#
as
level
a
2
( BD )
area
in
of
in
in
of
the
be­
the
the
depth
( F D2 ) .
7)
is
a .func­
step.
i?y e x ­
reservoir
vol­
time
the
turn#
lake
drop
defines
be d e t e r m i n e d
discharge#
the
base
(Fig.
previous
escap­
difference
flow
time
the
the
7)#
of
function
can
and
and
breach..
7 illustrates
Th e
(Fig.
the
causing
Figure
level
point
calculated
the
time.
time
is
in
as
of
lake
discharge, during
lake
area#
flow
area
effective
at
slope#
discharge
decrease.
function
the
level
the
increases
point
(FD)
flow
the. channel
effective
a given
of
S is
cross-sectional
the
of
at
the
coefficients.
elevation
breach#
A is
equation#
reservoir
development
lake
the
(3)
is
as
a
a. f u n c t i o n
function
of
Il
Ti,
CO
CO
T =Time
BDn = Breach Depth
FDn = Flow Depth
Ln= Lake Level
AL = Change in Lake
Level
F i g u r e 7.
As t h e b r e a c h a r e a i n c r e a s e s w i t h t i m e , t h e d i s c h a r g e t h r o u g h t h e
Th e d e p t h o f f l o w ( F D )
b r e a c h i n c r e a s e s c a u s i n g a d r o p i n t he Lake L ev el ( A L ) .
s
t
e
p
i
s
d e f i n e d by t h e l a k e
and t h e e f f e c t i v e a r e a o f f l o w ( FD > a t a n y t i m e
level
(L) and t h e b r e a c h d e p t h ( B D ) .
34
16-00
24 0 0
32-00
40-00
LAKE DEPTH
RESERVOIR
(FT)
DEPTH-VOLUME
F i g u r e 8.
Curve d e f i n i n g
f u n c t i o n of the r e s e r v o i r
the volume of
depth.
(From
CURVE
M y s t i c L a k e as a
C H 2M H i l l , 1 9 8 0 )
35
the
effective
breach
lake
area
area.
level
order
to
Th e
is
the
the
example
in
by
subtracting
Lj
and
is
then
Lg ,
flows
the
Th e
gates
over
the
lake
the
level
in
take
the
step
to
l ake,
~ Lake
lake
7,
the
the
and
the
can
be
approximated
level
(AL),
the
Leve I
in
(Fig,
lake
-
AL
7)
level
at.
level
Lg
can
The.
discharge
the
are
in
illustrated
calculated
at
the
p r e v i o u s . time
be
for
computed.
the
sum o f
the
and
the
spillway
By
time
the
level
breach.
gates
9.
The
of
the
the
present
step,
take, l e v e l s
f l o w . and. t h e
from
figure
the
projected
area
outlet
curves
be
lake
dam i s
spillway,
( A)
"known"
t he. e f f e c t i v e
from
the
Lg .
i=$,3...n
<5)
level
lake
can
^
Between
discharge
step
discharge
k n o wn
difference
calculate
the
the
be
calculate
figure
by
ano
‘
Leve I ^ g
the
level
must
level
d' ep t h - d i sc ha rg' e
from
time
the
instantaneous
discharge
discharge
lake
Level
to
total
between
L e v e l . = Lake
contributed
breach.
outlet
time
(A U ,
corresponding
the
step:
from
used
The
for
= Lake
Lake
by
discharge.
change
time
AL
defined
the
the
value
previous
the.present
For
that
calculate
projecting
from
flow
relationship
such
A "known"
by
of
and
the
a v e r a g i n g ; t he'
step
volume
resulting
with
of
the
outflow
decrease
■
8
36
DISCHARGE
SPILLWAY
(CFS)
RATING
CURVE
8
DISCHARGE
12
&
16"
OUTLET
f i g u r e 9.
Rating curves of
f o r t h e s p i l l w a y and o u t l e t
(CFS)
PIPES
RATING
CURVE
d i s c h a r g e as a f u n c t i o n of d e p t h
gates.
( A f t e r CH2M H i l l , 1 9 8 0 ) .
37
.in
the
Lake
level
(Fig.
8).
level
comouted
the
An
fa ll
the
from
tested
lake
a
the
for
Mystic
Lake
conditions:
step
5 seconds,
imated
and
t o - be
the
.047,
inflow
discharge,
from
an
ting
the
arrived
the
sc heme
is
used
from
and
into
83,500
eouation
calculated
lim it
difference
in itia l
of
from
dam
of
0.01
using
the
failure
time
R
the
lake
in itia l
the
lake
by
ma x i mu m
discharge
through
straight
from
line
actual
through
levels
foot,
lake
8)
I f . ' t he
against
difference
does
not
AL
is- ....
a new
(Fig.
7.5
to
breach
at
with
the
(1977)
and
a time
est­
dam c r e s t ,
Th e
peak
obtained,
for
dam^
dam h e i g h t
Th e
genera­
channel
results
breached
plotting
is
assumptions
1 0 0 0 ' . c-fs.
Kirkpatrick
a
10)
minutes,
level
dam f a i l u r e s .
the
5.
the
closely
developed
discharge
of
for
equal
agrees
equation.by
the
(Fig.
following
equation
at. t h i s
test
curve
i s negligible.
a Manning's
cfs,.
to
lake
outflow - hydrograph
a
depth-volume
depth-volume- curve
tolerance
until
read
level
A hypothetical
ted
the
projected
within
be
iterative
projected
between
can
compu­
He
versus
equation.for
peak
the
points:
1.85
Qmax
yields
an
= 65
estimated
x Dam H e i g h t
peak
(6)
discharge
of
83,800
'
Lake
dam,
with
a
crest
height
of
48
feet.
cfs
for
Mystic
DISCHARGE IN CFS
38
35.00
TIME IN MINUTES
F i g u r e 10.
Breach o u t f l o w h y drog ra p h from the h y p o t h e t i c a l
o v e r t o p p i n g of M y s t i c L a k e dam.
The p e a k d i s c h a r g e f r o m t h e
f a i l u r e , 83,500 c f s , i s reached in 7.5 m inutes.
39
Downstream
Routing
.
Introduction.
face
the
breach
Having
a n d ' an
remainder
of
outflow
the
downstream.
35)
routing
is
a time-dependent
hydrograph
model
f loddwave
flood
developed
for
description
According,
defined
to
Mystic
involves
Chow
is
enumeration
beyond
reason,
in
the
directed
Yevjevich,
routing
(1964,
sec.
Thomas,
1964;
of
all
flood
purpose
of
w ill
discussed
an
be
in-depth
this
and
routing
study.
For
and
25,
p.
1950)
Comer,
t he
C h o w,
I 965 )
a
.
methods
this
compared
evaluate on,
19 3 4 J G i l c r e s t ,
Brakensiek
downstream
divided
choosing.a
description
routing
For
the
and
.
only
reader
1959,
is
1964,
M iller
and
1975.
The
is
methods
terms.
to:
and
d a m,
as:
comparison
scope
routing
general
Cunge,
and
sur­
Lake
t h e p r o c e d u r e w h e r e b y t h e t i m e a nd m a g n i t u d e o f
f l o o d wa v e a t a p o i n t o n a s t r e a m i s d e t e r m i n e d
f r o m t he k n o w n o r a s s u m e d d a t a a t o n e . o r m o r e .
p oints upstream.
The
' .
into
two
flood
of
routing
portions.
routing
the
enuations.
section
First;
method
numerical
of
are
the
the
model
criteria
discussed,
techniques
description
used,
to
for
followed
solve
by
the
a
40
Selection
fall
into
The
t wo
a
routing
basic
hydraulic
unsteady
of
flow
methods
are
derived
by
arise
from
the
mo me n t u m
applied
to
open
The
and
as
such/
continuity
expressed
Q is
is
the
or
outflow
de
laws
for
+ M
the
discharge/
per
is
t
un i t
is
the
is
incompressible/
the
in
(Amei n
and
of
Matter
laws
of
of
These
mass, a n d
is
a form
of
conservation.
of
ma s s ' i s
the
distance
mu s t
change
system
area
in
to
9 A / 9 1 can
Fang/
the
1970).
is
with
the
area
is
is
equation
expressed
a
change
( C h ow /
as
7 then
a
the
the
at
?erp/
as
and
dis­
cross-sec­
rate
in
and
x
inflow
po s i t i v e
that
time
area/
lateral
90/9 x /
net
unchanged
Equation
the
( i n.f l o w
rate
re-write
be
q
7 states
a
change
remains
convenient
at
cross-sectional
and
Fouation
water
plus
the
length
flow
change
equations
(1871).
conservation
t i me. /
stream
negative).
with
A is
of
I t i s
the
..
(7)
3t
area
of
hydrologic.
complete
flow.
to
met hods
o
- q
9x
tional
ma s s
the
and
conservation
the
varies
charge
of
channel
charge
Since
on
Sairit-.Venant
conforms
equation
distance/
outflow
based
routing
as:
A
where
Flood
c a t e g o r i e s -- h y d r a u l ic
equations
energy/
method.
the
the
9A/3t.
distotal
1959).
7 so
that
function
becomes:
the
of
depth
41
where
y
is
the
depth
defined
as
the
d erivative . dA/dy.
Not
only
the
mo me n t u m
the
change
effects
must
as
in
of
the
g
frictio n
bottom
is
and
slope,
Manning's
the
of
top
the
width
system
of
be
the
system
acceleration,
d ue
and
stream,
conserved
mo me n t u m e q u a t i o n
of
the
accounts
to
is
the
but
for
the
expressed
a s :.
+
ax
the
Th e
energy
30 + 3Q2M
where
B is
ma s s
well.
the
at
a nd
f
gravitational
and
is
(9)
+ s o + sf
the
constant,
friction
Sq
is
slope
the
channel
evaluated
using
equation:
n 2 Q2
2 . 2 A2
The
derivation
of
references;
de
1959;
and
Amei n
While
equations
mo r e
ity
real
system.
Fang,
1970;
hydraulic
unsteady
sim plified
equation
equations
Saint-Venant,
the
of
( 10)
RV T
is
7-9
18 7 1 ;
found.in
the
the
G itcrest,
Liggett,
methods
flow,
is
are
1950;
Ch o w, .
the
complete
1975.
based
on
hydrologic
methods
approach
to
t h e . p r o b l e m . ■' O n l y
utilized
to
approximate
The
effects
of
following,
the
acceleration
the
present
continu­
behavior
and
of
frictional
the
a
4?
resistance
ignored
(M iller
Th e
governed
the
following
the
ease
of
equationz
the
effects
it
with
(M iller
and
these
u tilizin g
the
hydraulic
methods
increased
system
tions.
are
remaining
flow
yield
but
at
computer
non-linear
The
x and
Cungez
t z the
terms
are
arriving
at
a
to
met hod
uses
only
in
has
described
1975).
The
a mor e
is
of
of
continu­
where
resistance
are
unsteady
appreciable
flow
accel­
any
hydrologic
only
way
to
with
8-9
are
constants
or
The
y
of
computations
constitute
differential
contained
variables
flow.
method
representation
rigorous
partial
success­
a hydraulic
unsteady
Equations
variables
the
simula­
by
accurate
cost
the
and
situations
rapidly-varied
f loodwaves
be
only
r e a l ' s y 3-
solutionz
perform
frictional
equations
either
of
in
and
time.
accuracy
the
required
the
the
be
of
conditions
dependent
are
should
behavior
hyperbolic
independent
methods
the
Th e
cannot
complete
systemz
of
19 8 0 ) .
and
routing
applicable
dam-break
effects
a nd
time
acceleration
associated
real
be
equation
considerations:
hydrologic
can
( Freadz
the
t wo
computations
of
model
the
of
the
mo me n t u m
1975).
reproducing
negligible
fully
the
in
computer
ity
met hod
Cungez
by
Because
eration
in
between
amount
tion.
a nd
for
choice
equations
t e mz t h e
the
accounted
in
a nd
the
Qz
functions
a
equa­
equations
the
of
xz t z y
43
and/or
Q„
lytical
This
system
methods;
u tiliza tion
Fre ad,
of
the
of
conations
only
numerical
,
cannot
alternative
for
be
solved
solution
t e c h n i q u e s . ( Ame i n
a nd
solution.
partial
differential,
ics
fin ite
difference
methods
of
co mp I e x i t y
of
Because
finite
the
difference
Further'
to
two
are
equations,*
methods
information
applications
only
are
methods,
and
but
There
merical
is
ana­
the
Fang,
flood
the
method
(Liggett
I 9 7 C;
analyzed
numerical
routing
number.of
to
of
may.be
c h a r a c t e r i st-.
and
and
in
196?;
Brakensiek., J 966;
Liggett
Wo o l h i s e r ,
1. 967;
Stre lko ff,
A b b o t t , .1975;
others,
Of
Liggett
and
1,9 7 0 ;
Cunge,
1975).
method,
only
compared.
■
a nd
the
Richtmyer,
and
Cupae,
methods
found
nu­
hyperbolic
characteristic
be
concerning
large
anpli cable
the
w ill
a
references:
most
by
I 980).
Numerical
1970;
,:
their
following
Amei n ,
1967;
Vasi I i e v ,
1 97 5 ;
Vasi t i e v
and
1976.
the
available
commonl y
used
explicit
and
im plicit
method
in
im plicit
was
(Abbott
Liggett
Wo o l h i s e r ,
C h a u d h r y . and
difference
numerical
flood
methods.
Th e
based
vestigators
. and
fin ite
and
on
the
!ones cu,
1967;
Contractor,
routing
the
are
t wo.
the
decision
to
e mp l o y ,
conclusions
of
previous
1 967. '
Amei n
1973;
schemes,
and
Price,
Ameir,
Fang,
1974. ;
19.6.7;
1969;
Amei n
an
■
in­
1963;
19 7 0 ;
and
Ch. u,
44
1 9 7 5/
Greco
Hi I I e r
and
19 7 8 ) .
handle
to
mo r e
are
197 5 /
time
in
into
a
methodz
their
steps
Stability
here
solution
small
like
errors
"blows-up".
Th e
stablez
allows
governed
only
convergence..
implies
tends
t h. at .
as
towards
Cungez
19 7 5 ) .
roundoff
Of
error
the
Ax
an
and
increases
various
a nd
by
Amei n
Greco
considered
the
and
the
as
Ax
and
are
required
to
the
A
the
methodz
concept
zerbz
At
a
method
Fang
s c hemes
developed
time
(1970)z
explicit
until
time
im plicit
( 1975) .
method
and
for
and
s ma 11e r z t h e
for
1981).
us e z t h e
(1961)
and. C o n t r a c t o r
Fread . (1978;
this
the
step
Liggett
Preissmann
Chaudhry
step
computation
available
by
ex­
convergence
1974a;
b e c o me
the
being, u n c o n d i­
of
the
restric­
introduction
large
of
. .
for
exponentially
' ( . Fr eadz
and
not
Sz
can
im plicit
( D. z D y re. son z o r a l . c o m m o n . z
Pahattoni
best
The
selection
approach
im plicit
and
refers
The
solution
Howeverz
four-point
(I 9 7 3 ) z
for
At
exact
weighted
used
a nd
im plicit
tionally,
by
flexible
those
grow
o t h e r Sz
very
s c h e me z ' l i k e
to
a nd
method
procedures.
stable
Ponce
1 975/ '
im plicit
stable
time
Qun q e z
the
unconditionally
conditionally
computation
usez are
and
1980/
steps..
the
causes
1978/
findingsz
to
and
Liqqett
distance
small
schemes.
errors
to
1975;
Freadz
d iffic u lt
also
very
p licit
■u s e d
Cungez
unequal
methods
of
Panattoniz
According
although
ted
and
study
198. 0)
because
is
. 45
of
its
and
a bility
bed
to
four-point
equations
8-9
by
corresponding
sions.
im plicit
replacing
set
of
The.solution
over
a discrete
Ti me
lines
spacing
variable
cross-sections
The
subscript
i =1
is
the
boundary.
with
algebraic
the
need
Th e
distance
the
the
boundary
The
points
where
the
lines
or
in
figure
depth
designated
to
be
location;
at
the
not
the
the
the
along
constant.
Location
downstream
intersections
points
and
11.
with
parallel
need
i =N• i s
nodes
are
sought
cross-sections
also
represent
solution s for
and
lines,
spatial
and
are
distance-axis,
constant,
which
a
a lg e b ra ic .expres­
I o c a t i on' s o f
Ax,
with
expressions
the
be
upstream
time-distance
to
solves
derivatives
grid, i l l u s t r a t e d
not
identifies
grid
partial
of
spacing
i
numerically
difference
superscript, j .
stream
the
drawn - p a r a l l e l
which
the
sc heme
time-distance
are
At,
the
space
with
fin ite
time-ax i s> represent
the
streams
slope.
Th e
by
model
in
time
discharge
of
and
are
sought.
Derivatives
are
represented
by
■K r e p r e s e n t s
any
' imated
a
, nodes
using
i
and'i+1
in
the
the
weighted
following
variable.
forward
along
f c u r - p o i n t . . i mp l i c i t , s c h e m e
The
algebraic
time
difference
the
distance
expressions
derivatives
quotient
axis:
are
centered
where
approxbetween
46
t
«----- Ax----- »
i»j+i
At
M
i+i»j
M
Distance
F i g u r e 11.
D i s c r e t e t i m e - d i s t a n c e a r i d f o r t h e s o l u t i o n of
the f o u r - p o i n t
i m p l i c i t method.
T i m e l i n e s a r e d e s i qn a t e d
by i a n d d i s t a n c e l i n e s b y i .
The s o l u t i o n o f t h e c o n t i n u ­
i t y a n d mo me n t u m e q u a t i o n s i s s o u g h t a t p o i n t ^ .
47
+ K
w
SK
91
The.
2At
spatial
derivatives
wa r d , d i f f e r e n c e
cent
time
factors
line
influence
When
yields
of
IiO
results
box
equally
s c h e me
weights
the
examined
t hat
approaches
in sta b ility
between
by. t h e
the
accuracy
1.0.
a
t wo
for­
adja­
weighting
( F r e a d , .1 9 74 a ;
reported
by
by
of
197 8 ) .
and
by
a nd
the
Amei n
on
0 on
as.0
0 .6 ;for
and
adjacent
j +1
or
0
and
both
the
time
backward
A weighting
others
pff
t wo
on
Ch a u d h r y . a n d
1978),
Chaudhry
the
im plicit
values
drops
A value
fully
factor
and- P o n c e
on
values
proposed
was
( 1 97 4 a.)
a
( Fread,
computations
Fread
the
a nd
weighting
accuracy
by
(I 2)
values,
only
the
concluded
a nd
the
of
(1 9 73) ,
position
is. c o n t r o l l e d
the.solution
the
which
of
0 is
s c h e me
The. . i n f l u e n c e
the
its
approximated
+ ( I -0 )
influence
lines.
(1970)
j +1,
also
AX
the
difference
a nd
but
0
time
0.5
j
are
1-0:
BK
BX
of
quotient,
lines,
0 and
weights
M I)
-
factor
Fang
time
lines.
accuracy
of,
Contractor
( 1 9 7 8 ) . - They
departs
from
minimizes
the
reduces
the
Contractor,
0.5
loss
pseudq
(1973).,
48
Variables
same
location
other
in
than
time
When
8 and
9
the
are
and
space
e q u a tio n s ' 11. through
finite
- Qit l '
by
and
the
1.3/
( 1- 0 )
;
at
derivatives
by:
are
N n
equations
d i f f e r e nc e o p e r a t o r s f r o m
four-point
im plicit
obtained:
-
oil
[ B f 1 ♦ B f 1I
'+ « 0
AX
:.
in
i'
-
-
4
."2
Iil
• . J
( 14)
q .= 0
2At
- Q'i+1
oin -
'(QVA)^} - (Q2ZA)J+1;
+ 0
2&t
(Q2/ A ) J +1 - (Q2ZA)J'
( 1- 0 )
Ax' '
v
AX.
.;
■
r
j
.1
•
Sr
rj + i
:j + b
yW - - h ,
2
rH
1- 0 )
I
V
^
I
.. (
i+1
[A f1 ♦ A fh
+ 9 ' : .0 ■
+
>T
Ajj ,+ A'
the
(13)
variables
following
2
J +I
evaluated
Ki + K i n '
fin ite
y f1
Bj + M n
the
other
the
+ (1-0)
AX
are
.
difference, equations
P in
as
+ ( 1.-0)
derivatives
replaced
. -
derivatives
S ’1 * ■£}
2
'
l \
+ (1-0)
Ax
;
• 49
d
where
all
those
with
variables
the
from
cit
solves
method
1980).
the
previous
are
time
j
values
equal
are
blocks.
direct
tested
t wo
because
However,
N-I
the
interior
the
four-point
the
next
the
by
(Amei n
from
known
and
15
cannot
only
neighboring
each
f o r ; the
solution
there
ap. pl y i no
of
grid
2N .
l i n e , by
'
node
unknowns
j .
a't'j+1
continuity
each
be
solved
the
in
N nodes
N-I
four
share
or
cross-
grid
equations
t.-x
a
and
blocks
.unknowns
'
and
of
equations
block,
or
These, " i m ­
M in
are
im pli­
conditions
grid
the re -are
point
M in
t wo
ex­
1969;
each
at
the
14
Hy
2N - 2 e q u a t i o n s '
values
point
p o i n t s - M..
at
the
at
distance-axis,
to
for
15,
are
while
are
time
Fang,
initial,
a nd
there
values
and
variables
evaluating
any. t w o
simultaneously
on
values
u n k n o w ns ’ i n . c o lit mo n . : S i n c e
■sections' alo ng
with
14
Equations
manner
unknowns.
tria l
to
step,
all
known
known,
unknown
sc heme
that
are
where
unknowns
are
computations,
moment um’ e q u a t i o n s ,
grid
the
j
unknown.
time
iterative
line
are
system
Assuming
established
plied''
for
(I S).
= 0
superscripts
j +1
a previous
a Newton-Raphson
along
the
explicit
trapolated
Fread,
with
superscripts
Unlike
J
Sf i+.5
So + 8
blocks
14
a total
result.
ari d
15
of
Two
additional
tionships
the
equations
at
system
flow
the
upstream
o f . ZN
ous.
boundary
es I ,
breach
the
upstream
boundary
as
a
If
c. urve
the
function
fu lfills
stream
superflu­
required
flow
is
time
at
the
subcri t i ­
satisfies
a h e cua. t i o n
the
complete
t r a v e l , up­
becomes
of
while
the
cannot
c o n d i t ion s are
condition,
rating
When
boundary
( T r e a d , . I 9, 80) „
discharge
pounds,r i e s
disturbances
downs t r eam
ZN s i m u l t a n e o u s
representing
19 6 9 ;
1- 970;
(Fig.
12)
from
i n
the
point
S' c a n
the
Fang,
taught
be
them
in
generated,.
derivatives
of
t he equati on.
solution
M. i n
obtained
Gaussian
1969).
of
each
the
of
by
equations
a matrix
F r e a d , ' 1 9. 78/
are
partial
unknown
as
the
flow
rela­
for.' , t h e
downstream
bound­
condition.
The
at
equations.
boundary
depth-discharge
of
simultaneous
two
upstream
ary
downstream
therefore
Instead
the' d e p t h - d i s c h a r g e
and
is- s u p e r c r i t i c a l ,
s t r e a m «'
the
describing
the
19 8 0 ) .
readily
Three
solved
( Amei n
and
m a t r i c e s ' A,
a coefficient
matrix
each
equation
respect
Matrix
R is
continuity
grid
or
blocks.
of
for
vector
of
S-
composed
to
each,
residuals
a nd ' mo men t u nr . e qua. t i o ns
The
matrix
solution
the
vector
manipulations
ma t r i . x ' i n v e r s i o n
e l i m i n a t i o n , i's
a lgebra
a
with
by
Fang,
R and
A. i s
elimination
i n- b e g i n n i n g
be
configuration
a variety
Gaussian
can
firs t
t he. s o l u t i o n o f
(Arne i n
such
and
method
simultaneous
Dj .
Hi
Hj
'
al2
v
a 22
*23
a 32
a 31
to ­
ll:
r
'
i
si
*24
r2
a 33
a 34
r3
a 43
a 44
a 45
a 46
a 53
a 54
*5 5
*5 6
'
S2
53
.
r4
R =
T5
.
S4
S =
55
U l
H -*
“
a 2 N - 2 , 2 N - 3 a 2 N - 2 , 2 N - 2 a 2 N - 2 , 2 N - l a 2 N - 2 ,2 N
r 2N-2
S2N-2
■ 3 2N - I , 2N - 3 a 2 N - l , 2 N - 2 a 2 N - l , 2 N - l a 2 N - l , 2 N
r 2N-1
S2 N - 1
r 2N
S2N ■
a2 N ,2 N -l
a 2N ,2N
F i g u r e 12.
M a t r i c e s A , R and S f o r t h e s o l u t i o n of s i m u l t a n e o u s e q u a t i o n s .
The
e l e m e n t s o f m a t r i x A a r e t h e p a r t i a l d e r i v a t i v e s o f t h e c o n t i n u i t y a n d mo me n t u m
e q u a t i o n s w i t h r e s p e c t t o t h e u n k n o w n s Q and y o n t h e t i m e l i n e j + 1 .
Matrix
R i s a v e c t o r o f r e s i d u a l s f r o m t h e s o l u t i o n o f t h e c o n t i n u i t y and mo me n t u m
equations at p oint M (F ig. 11).
Th e s o l u t i o n v e c t o r S c o n t a i n s t h e r e s i d u a l s
from the t r i a l
v a l u e s o f Q a n d y on t h e t i m e l i n e j + 1.
52
equations
(although
Unknowns
are
reduced
to
k n o wn s
others^
c ,
'3
sections
so
in
values
Sp .' o
.5
.
and
one/
within
by
back
The
on
trices
A/
S. a r e
and
on/
are
depth
three
the
the
name).
system
is
remaining
un­
( J a me s
a nd
The
lim it
If
of
any
of
0.05
of
values
S a nd
to
the
then
On c e
line
been
t h e y , now
a nd
the
the
time
the
time
step
step/
breach
procedure
is
a new
the
value
for
at
is
s4 '
values
depth/
fa ll
or
by
f he
At.
The
the
tested
and
50.0
the
until
j 4-I
each
is
the
time
values
change
calculated
boundary.
values/
value.
new m a -
'!known"
discharge
tria l
'
subtracted
With
upstream
*
for
outside
generate
the
,
*
are
repeated
b e c o me
the
establishing
to
from
cross
residuals
u n k n o w n s ' on
incremented
hydroqraph
of
used
procedure
"zero-out".
solved/,
S
for
. added
at
c o r r e s p o n d i n g . unknown
residuals
have
foot
is
its
are
these
the. values
residual
sign)
residuals
discharge,
range
residuals
discharges.
respectively,
corresponding
each,
its
R- a n d
vector
so
the
corrected
solution
depths
and
range/
(depending
whole
the
substitution
unknown
discharge.
specified
from
omit
until
unknown.
the
a tolerance
for
These
one
of
two
on/, are
the
same c r o s s - s e c t i o n s .
cfs
and
calculated
values
tria l
a nd
mathematically
equation
then
m ercifully
1977).
The
the
teachers
eliminated
one
are
mo s t
The
generating
in
53
matrices*
"zeroing-out”
time
i.s
step
through
the
Data
unsteady,
stream*
known
, The
the
map
data
with
Fr e a d ,
both
is
prder
the
solve
mov e me nt
their
to
to
of
the
routeo
or*
wh e n
foot
contour
In
survey
this
scale
were
data
13)
to
wave
stream
define
the
geometry
are
The
prohibative*
profiles
and'
from
from
U.S.
canyon
the
at
(Fig.
14)
and
is
drawn
acquired
Soil
using,
the
taken
mout h
Conservation
c o m m u n . #■ I 981 ) .
"in fle c­
geometry.
plotted
represented
from
were
canyon
p oints, of
in
perpendic­
1976;
established
is
are
stream
others*
were
cross-section
the
a nd
personal
changes
be
scale
Bergantine*
any
must
1:?4*0G0
( R.
record
down­
a
wer e,
.in. t h e
those
of
p.rcfiles
(Glass
of
channel.
profiles
interval
obtained
at. each
(f(y,))#
the
the
study*
ma p *
equations
along
Cross-sections
cross-sections
area
costs,
the
inter-relationships
flow.
Service
order
been
of
B o z e ma n
depth
has
flood
direction
through
of
the
cross-sectional
1:24*000
The
incrementing
of
a 20
(Fig.
and
points
data.sources.
tion"
and
series
19 7 8 ) .
from" a
The
at
step
a
In
simulate
variables
first
ular
survey
and
assumed
with
residuals
channel.
flow
channel
to
until
requirements.
all
or
repeated
the
as
by
a function
a
third
polynomial:
I
54
F i g u r e 13.
Channel c o n f i g u r a t i o n showing c r o s s s e c t i o n l o c a t i o n s at p o i n t s of " i n f l e c t i o n " .
DEPTH
F i g u r e 1 4.
C r o s s - s e c t i o n a l a r e a f ( y ) as a f u n c t i o n
f o r any c r o s s - s e c t i o n A - A 1z B - B 1 o r C- C' ( F i g . 1 3 ) .
of
depth
55
:
Area
where
y
=
i ' s t he
depth
The
coefficients
the
form
oration
the
of
= a + by
f (y)
. and
are
15).
coefficients
16,
b/
? .
+d y .
c
d are
by
values
b,
and
c
and
in
d,
the
(16)
coefficients.
solving
s i m u l t a n e o u SIy
The
a,
2
calculated
equation
(Fig.
a,
+ cy
4 equations,
in
a matrix
solution
obtained
by
in
config-
vector
S,
Gaussian
are
e lim i­
nation.
Wh e r e
the
stream
-A,
B) ,
the
total
and
02,
and
the
Bal tze' r. and
to
the
treated
single
have
by
as
total
the
of
broken
Al
allows
the
curve.
On c e
the
reaches
island.
Th e
A2
(Chow,
whole
bifurcated
length
Ax
length
which
16
Ql
1959:
being
equal
channel
to-be
and. r e p r e s e n t e d
stream
(Fig,
components
t he . c r o s s - s e c t i o n
total
with
an
into
a nd
1975).
.cross-section
determined,
a series
around
A' i n t o '
Cunge,
parts
a single
Q is
area
1968;
depth-area
been
divides
discharge
Lai ,
sum o f
flow
by
a
positions
is
need
represented
not
be
constant.
The
failure
rapidly
(Fig.
convergence
(Fread,
overcome
10)
even
19 7 8 ).
by
rising
the
can
when
hydrograph
cause
an
.. H o w e v e r ;
proper
associated
probl ems;
im plicit
these
of
in stab ility
technique
computational
selection
of
the
with
time
is
the
dam’ s
and.nan-
employed.
problems
and
c a n . be
distance
r
' ^
a
^ y 1)
.
yI
3
Y
=. '
y2
.
b
(y2) 2
'
y3
y4
( y 3) 2
S =
A =
f f y y .
Iil4) 3
C
d
F i g u r e 15,
M a trix c o n f i g u r a t i o n fo r the s o l u t i o n of the
death-area c o e f f ic ie n ts .
Th e e l e m e n t s o f m a t r i c e s Y a n d A
a r e t he v a I u e s o f d e p t h , a n d a r e a f r o m t h e d e p t h - a r e a c u r v e
in f i g u r e 14.
The v a l u e s i n t h e s o l u t i o n v e c t o r S a r e t h e.
c o e f f i c i e n t s f o r a t h i r d , o r d e r p o l y n o m i a l d e f i n i n g the
d e p t h - a r e a c u r v e i n f i g u r e 14.
V -,
;
..i
,sland
cross - sec tio n s
B. Profile
F i g u r e 16.
Map a n d p r o f i l e
discharge Q is d iv id e d in to
v i e w o f f l o w d i v i s i o n a r o u n d an i s l a n d .
QI a n d Q2 , a n d t h e t o t a l a r e a A i n t o Al
Thet
an c A 2
58
steps.
too
If
the
large
depths
time
to
the
step
and/or
time
fails
to
con v e r g e .
Because
the
increase
time
"blow-Op".
these
made
the
By
increasing
wa v e
step
problems
too
negative
of
distance
by
front
the
inserting
disappear.
large,
the
step
How-'
procedure
the
hydrograph,
flatten
the
out
time
and
downstream
for
= Ti me
of
the
as
it
very
moves
distance
direction..
selection
Peak
is
Arrival,
of
/
peaked,
downstream.
steps
are
Fread
the
it
allowed
( I 974 a ) ' p r op­
time
step:
20
(17)
At
increases
because
the
arrival
the.peak
increases
at
locations
further
and
further
step,
feet
Th e
Th e
to
the
in itia l
unsteady
increase
increase
near
cross-section
the
is
discharge,
vicinity
the
distance
advances,
downstream.
1200
'to
the
the
floodwave
of
tance
and
a guideline
the
the
or
'
case,
in
At
As
in
dam-break
dampen
being
posed
the
step
small,
change.in
cross-sections,
the
to
too
decreasing
if
This
is
the
develop
ever,
to
to
computation
additional
tends
step
relative
tend
causing
time
the
time
step
gradually
from
130
downstream
conditions
must
flow
in
be
known
equations.
allows
feet
at
the
dis­
the
dam t o
boundary.
of
depth
at
time
and
t =0
discharge
in
Th e ' d i s c h a r g e
order
at
to
each
at; e a c h
solve
cross-
59
section
i
where
the
or
is
calculated
Qi
= Qi-1
is
the
spillway
outflow
nificant,
to
be
discharge
and
by
outlet
and
dry
before
dam
is
the
areas
The
erative
CC h o w ,
energy
only
the
along
depth
1959;
a
at
of
the
at
1 96 4,-
the
end
is
Th e
over land
flow
the
of
due
w ill
the
reach
of
sections
may
Ray,
be
a nd
2
w ill
be
Z1 + V^/2g = Z2 + V^/2g + hf
are
are
(Fig.
con­
large
If
saturated.and
(Cunge,
Figure
Equating
17),
the
the
ra in fall,
by
step-backwater
Ax .
sig-
as s umed
calculated
19 6 6 ) .
written
be
1975).
a heavy
is
inflow
is
areas
be n e g l i g i b l e
length
I
it
from
standard
and
would
from
inflow
lateral
(Cunge,
cross-section
as
failure
inundation
inundated
channel
(18)
lateral
data,
to
arrives
the
the
i+1.
discharge
Bailey
system
qj
a lack
wave
stream
known
channel
the
wh e n
the
each
and
losses
flood
by
dam b e f o r e
and
of
in filtra tio n
s c h e me
strates
heads
because
overtopped
d ue. t o
i
or
In filtra tio n
important
the
gates,
tributaries
sidered
losses
at
sections
however
zero.
i=2,3,4...n
+ q 1-1
between
contributed
as:
the
1 9 75 ) .
an
i t ­
method
17.i l l u ­
the
total
total
a s:
( 19)
60
' / / / /
Datum
F i g u r e 17.
L o n g i t u d i n a l c r o s s - s e c t i o n of channel l e n g th
to i l l u s t r a t e the te rm s i n the s t a n d a r d s t e p -h a c k water
method.
( F r o m C h o w, 1 9 5 9 ) .
61
where
Vj
and
constant,
Vg
are
h^ is.the
Zj
and
Zg
are
and
Zg
A known
.? a r e
corrected
b e c o me s
the
section.
ried
the
is
stream;
is
if
surface
elevations:
(22)
the
at
channel
section
equation
the
and
by
bottom
I
and
19,
residual
in
the
which
the
supercritical
is
the
of
is
computations
for
the
are
carried
section
depth
at
computations
in
at
tested
The
solved
flow
they, ar e
depth
deoth
disappears.
regime
subcritical
elevations.
a tria l
The
a new d e p t h
direction
governed
flow
water
+ =2
are
known
The
as:
<21)
into
until
expressed
+ Z1
depth
entered
gravitational
(20)
the
= ^
Zj
loss
the
AX
Z1 = Y1
where
v e l o c i t i e s ■/ g i s
frictio n
hf = Sf
a nd
the
and
y g then
the
are
next
car-
channel.
carried
downstream
.
If
up­
(Chow,
1959).
The
backwater
calculation
calculation,
coefficient
estimate
19 5 9 ;
of
the
1 964 ) .
'n.
the
requires
Manning's
resistance
friction
n is
slope'
S^,
an e s t i m a t e
an. e m p i r i c a l
to
flow
in
a given
Furthermore,
Chow
( 1959,
p.
of
used
the
in
roughness
coefficient
channel
I Cl)
the
to
(Chow,
states:
62
• the g re a te st d i f f i c u l t y
l i e s , in the d e t e r m i n a t i o n
of the roughness c o e f f i c i e n t
f o r t h e r e i s no
exact , method of s e l e c t i n g t he n v a l u e .
At t h e
p r e s e n t s t a g e o f k n o w l e d g e , t o s e l e c t a v a l u e of
n „ . . i s r e a l l y a m a t t e r o f i n t a n g i b l e s . . . »'
f o r b e g i n n e r s , i t c a n be n o mo r e t h a n a g u e s s . „
Reasonable
tables
from
field
contained
Barnes
slope
that
reached
and
export
1968),
bed
of
the
slope
ting.the
reach
S^
R.,
M is
the
extremely
the
stream
slope
than
a stream
term
(D.
R.,
and
slope
friction
frictio n
state,
can
and
rrust
d ue
to
loss
i.e .,
energy
be
are
More.energy
be
a m ild,
accounted
Reichmuth,
oral
the
to
steepness
dissipated
for
in
c o mmu n . ■,
import
equal
1980).
bed
to
By
the
set­
slope.
3 for
upstream
each
reaches
irregularity
in
a stream
(waterfalls
uniform
of
(Morisawa,
be
and
the
a. s t r e a m
rates
the
the
comparisons
In
to
from
y. a. l ues, f o r
equation
for
"drop-structure's",
with
low
balanced
equal
using
is
photo
commun.,
n' v a l u e s
the
(.04 5 -.0 8 5 ),
h^ .
assumed
oral
slope
The
and
unreasonably
Reichmuth,
high
bed.
loss
the
1964)
back-calculated
a steep
energy
M a n n i n g ' s w- ,
(1959;
steady
stream.
are
in
and
a
of
yielded
friction
in itia l
of
Chow
material
CD.
Manning's
in
(1967),
friction
has
estimates
.
slope.
Th e
t he .Manni ng
1980).
and
of.
with
pools),.,
increased
friction
COMPUTER REPRESENTATI ON OF
The
mathematical
described
model
are
program
listed
in
ability
of
code
model/
the
the
forms
a
module/
as
variable
Fang/
the
a
FORTRAN
In
order
was
3
of
MAI NLI NE/
read
the
BREACH
is
the
in/
while
NOAH
modified
of
or
the
read­
the
subprograms:
these
subprograms
the
"housekeeping"
failure
the
per­
conditions
are
subprograms
are
stored/
and
definitions.
from
insure
modules
and
routes
I V computer
structure
concerning- input
and
to
in itia l
generated
lists
previously
a n d . NOA H
t i me-dependent
hyd r o q r a p h ,
( NOAH
in
into
are
data
the
Ea c h
In
provided
simulates
stream.
NOAH.
MODEL
from
retain
divided
to
output
II.
to
task.
calls
in f o r m a ti o n , is
breach
is
variables
regulated/
ule
and
and
specific
all
well
Appendix
BREACH
established/
as
represented
program
MAINLINE/
equations
THE
of
and
output
formats
The
BREACH m o d ­
the
dam ari d
f Icodwave
a program
user
by
the
down­
Amei n
and
1968).
In
these
order
to
modules
subroutines
AREA -
BACKWATER
of
require=
to
solve
function
area
as
-
execute
a
the
following
subroutine
function
and
specific
r e p e t a t i ve
subroutine
depth,
their
to
of
to
or
assignments/
s u b r o u t i n e s : a nd
specialized
calculate
the
each
function
calculations.
cross-sectional
depth.
determine
discharge
at
each
the
of
in itia l
conditions
cross-section.
DRCDY
-
function
rating
subroutine
curve
at
the
to
compute
downstream
the
derivative
boundary
with
depth.
DRDY -
subroutine
hydraulic,
LAKE
radius
function
top
-
width
the
to
with
with
respect
to
as
a
-
subroutine
OUTLET
-
function -subroutine
RADI US
-
function
radius
RATI NG
-
-
S P I L L WA Y . . -
function
function
the
derivative
the
derivative
ol.the
the
of
the
elevation
lake
volume.
coefficient
matrix.
calculate
as
of
depth.
water, s u r f a c e
the
to
subroutine
boundary
to
matrix
the
depth.
gates
solve
Gaussian
the
through
each
to
the
a function
discharge
of
lake
level.
to
determine
the
hydraulic
to
calculate
the
discharge
as
a
breach.
subroutine
solve
TOPW-
the
downstream
modified
to
subroutine
function
the
Si MEQ
of
take
to.generate
outlet
to
function
MATRI X-
the
the
to
calculate
reservoir
through
take
respect
subroutine
subroutine
of
respect
.
function
DTWDY
of. t h e
the
function
coefficient
elimination.method
in
place
subroutine
spillway
subroutine
cross-section
as
as
a
to
the
cf
and
to
function
calculate
(Gauss-Jordan)
storage
the
of
of
a
to
space.
discharge-
the
the
derivative
depth .
matrix.using
conserve
compute
at
lake
top
the
level.
width
at .
cross-
sectional
VOLUME
-
area
function
reservoir
WP -
function
the
The
of
depth.
function
calls
is
to
as
of
the
compute
a
reservoir
the
function
through
the
illustrated
wetted
of
a
perimeter
of
depth.
program
in
of.the
and
flow
the
sequence
c h a r t . in
App-
I .
includes
width,
water
these
stream.
collected
user)
a
channel
Information
of
•
volume
progression
subroutine
tion
respect to depth.
'
s u b r o u t i n e to dete rm ine the
subroutine
stream
logical
pendix
as
with
the
at
interest
to
maxi mum, v a l u e s
surface
ma x i mu m
In
of
elevations,
values
at
each
be
of
time
various
cross-sections
record
(to
be
the
simula­
discharge,
cross-section
a. c o n t i n u o u s
hydrographs..
from
depth,
and. t he
addition,
for. plotting
gained
of
of
top
occurrence
along
the
discharge
selected
by
is
the
r
RESULTS
The
program
defining
an
a uniform
bed
Creek
fully.
The
flow
The
and
the
which
nel
regime
extreme
bed
area
flow
Creek.
the
from
alters
of
the
is
the
water
of
flow
to
create
to
producing
the
the
"control
regime
from
in
sections"
the
subcritical
an
in
cross-sedions
a nd
S im ilarly,
or
and
success­
changes
constriction,
gentle,
data
defining
run
A constriction
constriction.
to
data, f i l e
failed
d ue
with
cross-sections
channel
Creek
regime.
steep
the
of
B o z e ma n
upstream
at
bed- s l o p e ,
B o z e ma n
flow
uniform
program
this
of
successfully
wh e n
believes
the. f l o w
the
the
variablity
slope
impedes
channel,
in
run
with
However,
input,
author
a backwater
the
was
and
stream
slope.
regulate
critica l
tested
imaginary
B o z e ma n
the
wa s
chan­
flow
super­
a change
expansion, in
super
to
and
in
the
subcritical
( C h o w , 1959).
These
lishing
the
flow
surface
lation.
in itia l
separate
the
"control
the
in itia l
regime
profile
depths
"control
constant
may
be
be
chance
known
profile
sections".
of
over
computed
w i t h: a
must
backwater
present
conditions
is
However,
sections"
at
a
flow.
the
problem
In
total
in
estab­
a stream
length,
where
the
by
a.single
backwater
in
the
regime,
each
computed
A network
flow
"control
for
of
each
gaging
calcu­
the
section"
reach
water
and
between
stations
along
a
67
B o z e ma n
Creek
s t r i c t ions
the
flow
stream
in
may
not
the
stream
under
under
hiqh
.for
the
establishment
"control
sections"
and
as
such
the
matrix
Without
reouire
rating
( M.
Because
of
is
model
the
magnitude,
ure
of
the
at
the
percentage
depth, of
tion
Fo r
canyon
as
total
a
of
function
( R. ,
discharge
conveyance
the
total
can
and
be
of
calculated
r egi me- i s
of
these
conditions
1981).
this
routing
Creek.
task
is
portion - .
Therefore,
from
th e .fa il­
.
of
the
by
plotting
hydrograph
depth
the
U.S.
slope
as:
Soil
common.,
S of
of
the
against
K at- L e v e r i c h
-personal
a channel
data,
the
estimated
by
realistic
flow
common.,
flooding
breach
depth
Bergan t i n e ,
Q,
be
the
incorporated- into
B o z e ma n
conveyance
of
be
problems
in
boundary
approximation
can
gaging
allow
Eac h
gaging
simulated.
mout h
The
to
to
on
event.
changes
the
extent
firs t
attenuation
Service
any
rough
flow.
tabulated
these
be
not
Cons-
control
Therefore,
personal
applicable
dam c a n n o t
a
curves
from
timing.and
However,
flow
not
exert
conditions.
Amein,
of
only
intermediate
curves
impossible.
the
as
rating
solution
the
boundary
the
completely.
may
discharge
involving
act
may
conditions
a high
of
problem
conditions.
flow
Another, problem
the
channel
water
normal
extrapolations
solve
Road
the
is
Conserva­
1981).
0.01,
t he .
For
6.8
(23)
K = Q / Y F
The. d e p t h
corresponding, to
SCS
and
table
figure
18.
plotted
Figure
sideration:
the
the
against
18
worst
conveyance
the
presents
percentage
all
possible--0%
p o s s i b l e — 100% a t t e n u a t i o n . . B a s e d
the
Teton
and
attenuation
mo u t h
file
is
at
flooding
is
over
Leverich
for
in
the
all
Creek
dam
possible
(Fread,
Ro a d
cases
figures
magnitude
(Fig.
on
of
18
of
the
19
illustrates
as
cases
a 60
dam
a
The
for
best
from
percent
the
canyon
pro­
the
extent
500
year
flood.
in
con­
and. t h e
and
reference
the.estimated
the
attenuation
A cross-sectional
attenuation.
and
from
t he. h y d r o g r a p h s
failures,
1980).
19)
read
attenuation,
t h e . 7. m i l e s , b e t w e e n
reasonable
denoted
indicate
Buffalo
is
point
of
flood,
to
69
20
O 40
first
approximation
60
80
5 0 0 year
f lo o d .
DEPTH
F i g u r e 18.
D e p t h o f f l o w a t t h e c a n y o n m o u t h ( L e v e r i c h Rd . )
as a f u n c t i o n o f t h e p e r c e n t a g e a t t e n u a t i o n o f t h e b r e a c h
o u tflo w hydrograph (F ig. 10).
A 60 p e r c e n t a t t e n u a t i o n ,
b a s e d on p r e v i o u s dam f a i l u r e s , p r o d u c e s a d e p t h o f f l o w
o f 1 0 . 5 f e e t ; 4 . 5 f e e t d e e p e r t h a n t h e 500 y e a r f l o o d .
®
O
O
C
Vert. Exag. IO O X
JC
JC
10
5 0 0 yr flood
9 8 .5 %
S- 5 ■
1000
2000
D istance
3000
4000
in fe e t
F i g u r e 19.
C r o s s - s e c t i o n a l p r o f i l e a t L e v e r i ch R d . t o c e m o n s t r a t e
t h e e x t e n t o f f l o o d i n g f o r 3 c a s e s o f a t t e n u a t i o n : 0% ( w o r s t
p o s s i b l e c a s e ) , 60% ( f i r s t a p p r o x i m a t i o n ) , a n d 9 8 . 5 % ( 5 0 0 y e a r
flood).
( P r o f i l e d a t a f r o m R . , B e r g a n t i n e , p e r s o n a l r o mmu n . ,
1981).
SUMMARY
1)
.
practices
Mystic
that
are
requirements,
masonry
core
total
length
water
seeps
toe
of
wall
of
flood
The
cent
landslide
mite
and
ponding
from
zone.
Sl ump
below
sure
to
of
toe
are
outlet
the
in
pond.
pioes
in
a
the
to
a nd
at
handle
the
a
in
the
in
Geologic
toe
of
a
weathered
from
the
the
setting.
re­
dolo­
seepage
risk
of
high
seismic
risk'
landslide
deposit
direct­
the
dam
has
absent,
unstable
already
The
which
to
dam e s c a l a t e s
an
is
the
7 R.O c f s .
abutment
attributed
settlement
inadequate
The
through
depression
Pore-pressure
in
present
seepage
the
of
safety
steep."
extend
a
engineering
moder n
wall
in
founded
western
using
overly
core
than
is
sandstone..
the
is
to
not
ponds
located
dam
earthquake
dam
the
is
the
a nd
greater
the
scarps
differential
failure
dam
are
does
Wh e r e
spillway
.the
at
an
the
from
the
of
friable,
d ama g e
ly
d a m.
dam
and
Co mp a r e d
and
cracked
The
half
constructed
is
The
.
eastern
water
outdated.
a discharge
2 )
was
slopes
through
with
dam
embankment
the
the.dam.
Lake
now
the
AND CONCLUSI ONS'
in
near
increased
is
the
oore-pres-
a ls o .undergoing
past
failure
lead
of
the
to
the
dam' d u e
piping.
3 )
produce
.
a
peak
If
overtopped,
discharge
of
the
83,500
dam w o u l d
cfs,
fai I
creating
rapidly
a
severe
and. ■
72
potential
for
4)
break
„
and
met hod
.
difference
be
the
because
time
Changes
conditions
of
Without
stream
B o z e ma n
Creek
7).
in
flow
of
the
routing
method.is
solution
a d a m-
frictional
unsteady
of
unconditionally
and
life ,
with
and
by a h y d r a u l i c
im plicit
steps#
of
associated
equations'
is
loss
acceleration
numerical
it
attenuation
mout h#
t he.
and
and
the
A firs t
by
flow,
the
the
best
fin ite
unsteady
stable#
distance
4.5
feet
regime
in
in
not
steps
data#
the
extent
of
plotting
of
a depth
deeper
the
the
of
re­
need,
model
is
flooding
of
the
depth
flow
the
of
Creek
the
boundary
breach
than
B o z e ma n
establishing
intermediate
approximation
attenuation
yields
flow
problems
gaging
achieved
percentage
yon
complete
for
flow
modelled
four-point
insurmountable
be
be
the
small
possible
constant.
6 ).
can
only
s c heme
to
appreciable
The
equations
stricted
sent
can
damage, a n d
rapidly-varied
has
u tilizin g
5)
not
The
f loodwave
effects
flow
property
not
in itia l
conditions.
applicable
cannot
extent
flow
be
of
10.5
500
feet
year
to
modelled.
flooding
against
hydrograph.
of
pre­
the
A 60%
at
flood.
the
can­
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She r a r d , J . L . , ' W o o d w a r d , R . J . , G i z i e n s k i , S „ F; , and
C l e v e n g e r , W. A . , 1 9 6 3 , E a r t h a n d e a r t h - r o c k d a m s ) e n g ­
i n e e r i n g p r o b l e m s o f d e s i g n and c o n s t r u c t i o n : New Y o r k ,
W i l e y , 725 p .
S o w e r s , G.
York,
F ., 1962, Earth
Asia P u b l i s h i n g
Stoker, J .
Comm,
J . , . 1 9 4 8 , Th e f o r m a t i o n o f
on P u r e a n d A p p l i e d M a t h . ,
---------- 1 9 5 7 ,
Water
waves:
New
a n d r o c k f i l l dam e n g i n e e r i n g :
H o u s e , 283 p .
York,
breakers
v . 1, p.
a nd b o r e s :
1-87.
Inter-science,
507
p.
S t r e l k o f f , T ., I 970, Numerical s o l u t i o n . o f
Saint-Venant
e q u a t i o n s : J o y r . : H y d r a u l i c D i v . , Am. S e c . C i v . E n q . ,
9 6 , no, . HYl , p . 22 3 - 2 5 2 .
Sn ,
New
v.
S . T . , and B a r n e s , A . H . , I 97 0 , G e o m e t r i c a n d f r i c t i o n a l
. . e f f e c t s on . s u d d e n r e l e a s e s : J o u r . H y d r a u l i c D i v . , Am.Soc. C i v . E n g . , v . 96, no. H Y l I , p . 2 I 85-2 200.
T h o m a s , H. A . , 1 9 3 4 , The h y d r a u l i c s o f f l o o d m o v e m e n t s i n
r i v e r s : P i t t s b u r g , P e n n . , C a r n e g i e LrVs t . o f T e c h . ,
70 p.
T r a c y , H. J . , 1 9 5 7 ,
crested weirs:
Discharge c h a r a c t e r i s t i c s of
U . S . G e o l . Survey Ci r e . 397,
broad15 p .
30
U.S.
Ar my
d. am:
U.S.
A r my C o r p s o f E n g i n e e r s # I 9 6 0 / F l o o d s
s u d d e n l y b r e a c h e d dams — c o n d i t i o n s o f
t a n c e # h y d r a u l i c model i n v e s t i g a t i o n :
R e p o r t I / 1 76 p .
r e s u l t i n g from
mi ni mum r e s i s Mi s c . P a p e r 2 - 3 7 4
U.S.
A r my C o r p s o f E n g i n e e r s # 1 9 6 1 # F l o o d s
s u d d e n l y b r e a c h e d d a m s - - c o n d i t i o n s of
t a n c e # h y d r a u l i c model i n v e s t i g a t i o n :
R e p o r t 2# 121. p .
r e s u l t i n g from
ma x i mu m r e s i s ­
Mi s c . P a p e r 2 - 3 7 4
'U.S.
C o r p s o f E n g i n e e r s # 1 9 5 7 / F l o w t h r o u g h a. b r e a c h e d
U . S . ' A r my C o r p s o f E n g . / M i l . H y d r o . B u l l . / v . 9 .
A r my C o r p s o f E n g i n e e r s # 1 9 7 5 # N a t i o n a l p r o g r a m o f i n ­
s p e c t i o n o f d a m s : W a s h i n g t o n 0. C. # D e p t , o f t h e A r my #
v. I-V.
U . S . B u r e a u o f R e c l a m a t i o n # 1974# D e s i g n
■ e d . ) : U.S. Dept, o f I n t e r i o r # Water
■816 D „
U.S.
Department of A g r i c u l t u r e # Soi l C o n s e r v a t i o n S er vic e #
197 2 # E a s t G a l l a t i n R i v e r and u p p e r t r i b u t a r i e s f l o o d
h a z a r d a n a l y s e s # G a l l a t i n C o u n t y # M o n t a n a : B o z e ma n #
M o n t a n a # U . S . D e p t , o f A g r i c u l t u r e # S C S # 112 p .
U.S.
D e pa rt me nt of A g r i c u l t u r e #
1 979# S i m p l i f i e d , d a m - b r e a k
R e l e a s e No . 6 6 # 36 p .
U.S.
,
o f s m a l l darns ( 2 n d
R e s . T e c h . . Pub. #
Soil Conservation Service#
routing procedure: Technical
Department of A g r i c u l t u r e # S oi l C o n s e r v a t i o n S er v ic e #
' 1 9 8 0 # P r e l i m i n a r y i n v e s t i g a t i o n r e p o r t , B o z e ma n C r e e k
w a t e r s h e d # G a l l a t i n C o u n t y # M o n t a n a : B o z e ma n # Mo n t a n a . #
U . S . D e p t , o f A g r i c u l t u r e # S C S# 37 p.
/
. . -Van' t
'
'
H u l # A . # 1 9 7 9 # S u mma r y o f d e s i g n and c o n s t r u c t i o n h i s ­
t o r y o f M y s t i c L a k e dam# i n C H2 M. H i l l # ' 1 9 8 0 # A p p e n d i x I
P h a s e I i n s p e c t i o n r e p o r t # n a t i o n a l dam s a f e t y p r o g r a m .
M y s t i c L a k e dam# B o z e m a n # M o n t a n a # MT- 3 8 0 : u n p u b l i s h e d
r e p o r t # p. 2 0 - 2 6 .
V a s i I i e v , 0 . F. , 1 9 7 0 # N u m e r i c a l , s o l u t i o n o n t h e n o n - l i n e a r
p r o b l e m s o f u n s t e a d y f l o w i n o p e n c h a n n e l s : 2nd I n t e r .
C o n f . on N u m e r i c a l M e t h o d s i n F l u i d D y n a m i c s , B e r k e l e y # '
C a l i f . # 1 9 7 0 # P r o c - . # p . 41 0 - 4 22 .
81
V a s i t i e v / . 0. F . / V o y e v o d i n , A. F . , a n d A t a v i n , A. A . , 1 9 7 6 ,
N u m e r i c a l methods f o r c a l c u l a t i o n of u n s t e a d y f l o w in
s y s t e m s o f o p e n c h a n n e l s a n d c a n a l s : S y m p o s i u m on u n ­
s t e a d y f l o w , N e w c a s t l e - u p o n - T y n e , E n g l a n d , 1 9 7 6 , Pr oc- .
p . E 2 - 1 5 - E2 - 2 2 .
W h i t m a n , 6 . B . , 1 9 5 5 , The e f f e c t s
t h e d am- b r e a k p r o b l e m : Royal
399-407.
of h y d r a u l i c
Soc. London,
resistance
v . 227, p.
in
Wi I I i am s , T. T . , 1 9 5 9 , R e p o r t on M y s t i c L a k e f o r t h e B o z e ma n
C r e e k R e s e r v o i r A s s o c i a t i o n ' : u n p u b l i s h e d r e p o r t , 13 p .
----- - - 1 9 7 7 , S i n k h o l e i n s p e c t i o n r e p o r t . f o r t h e Bo z e ma n C r e e k
R e s e r v o i r A s s o c i a t i o n , i n CH 2 M- H i I I , 1 9 8 0 , A p p e n o i x 3 ,
P h a s e I i n s p e c t i o n r e p o r t , n a t i o n a l dam s a f e t y - p r o g r a m .
M y s t i c L a k e d a m , B o z e m a n , M o n t a n a , MT- 8 8 0 : u n p u b l i s h e d
r e p o r t , p. 4 5- 50 .
W i l s o n , S . D . , a n d Ma r s a l , R. . J . , 19 7 9 , C u r r e n t t r e n d s i n
• d e s i g n a n d c o n s t r u c t i o n o f e m b a n k m e n t dams : New Y o r k ,
Am. So. c. C i v . E n g . , 125 p.
Y e v j e v i c h , ,V. M. , 1 9 6 4 , B i b l i o g r a p h y a n d d i s c u s s i o n ' - o f f l o o d
r o u t i n g m e t h o d ' s a n d u n s t e a d y f l o w i n c h a n n e l s : L1. S .
G e o l . S u r v e y W a t e r S u p p l y P a p e r 1 6 9 0 , 235 p.
---------- 1 9 7 5 , S u d d e n w a t e r r e l e a s e , i n Mahmoo d , K . , and
Y e v j e v i c h , V. M . , e d s . , U n s t e a d y f l o w i n o p e n c h a n n e l s :
F o r t C o l l i n s , C o l o . , W a t e r Res. P u b . , p. 5 87- 66 5 .
7e i g I e r , B . P . , 1 9 7 6 ,
New Y o r k , W i l e y ,
Theory
4 35 p .
of
modelling
and
simulation:
APPENDI CES.
APPENDI X I
• PROGRAM FLOW CHART
/ READ ■I N
INITIALIZE'
DATA/
V ARI A B L ES
AND CONSTANTS'
GO SUB
PROJECT
BACKWATER
" KNOWN" VALUES
COMPUTE NUM LOOPS .IN BREACH
NUM = ( T / 2 Q ) / TSTEP
BREACH
T =T
+ DT
— :— (
ASSI GN
NEW " KNOWN"
COMPUTE
/
output
T <
no a h )
VALUES
MAX VALUES
HYDROGRAP h s /
TF I NAjZ>—N O - / O U TPU T MAX VALUES^
APPENDIX. I I
■ PROGRAM L I S TI NG
*******> ****************************************************
C
C .
M A I N L I N E PROGRAM
C'
.
■■
,
'
C HOUSEKEEPI NG- MODULE TO I N I T I A L I Z E CONDI T I ONS AND REGULATE
C CALLS TO THE BREACH AND NOAH SUBPROGRAMS. .
C
COMMON/ARRAY/A(200,2Q0),N
COMMON/ BREACHS/ M U M , T S T E P , L O , L I , L2 , DE P T H , BD, HDAM, '
* A R E A S , F L A G , I N LAKE ,.T Ql , V O L , MNB ^T " FAI L .
C OMMON/ PAR AME TERS / Y. ( 1 0 0 , 2 ) ,Q (1 0 0 , 2 ) , Z- ( I CO ) , MN ( I 0 0 ) ,
* I NF L O (1 OO).,. D I ST ( 1 0 0 ) - , QMAX < 100 ) , YMAX M O O ) , T MA X ( I O O ) ,
* ' XSEC (5 ) , DT , T
- CQMMON/ COE F S / A C T ( i - 00) , AC2( I 00) < AC3 ( I O O , RC1 , R C 2 , RC 3COM MON/ CO N ST A. NTS/ G, T F I N A L , TH FT A , M N , Q T OL , YTOL .
REAL L A K E , L O , L I , L 2 , I N F L 0 , M N 8 , I N L A K E , MN
I NTEGER F L A G, XSEC ' .
C
.
/
•
*■ A
* *
A
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
. *
*
C
-■
.
C USER I N FORMAT I ON :
c•
- '
.
*
*
*
*
. . .
*
*
*
*
*
*
*
*
*
*
f t *
*
*
*
*
* - *
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
:
A*********************. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
SUBROUTI NES REQUI RED
-' M A I N L I N E
,
-
.
.
.
•
.
BREACH
LAKE -
BACKWATER
C
..
-
-
.
NOAH
.MATR I X
' • S I ME Q.
-
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * **
c
.
.
■
. :
...
:
C FUNCTI ON SUBROUTI NES REQUI RED
> '■
BREACH
. MAI NL I NE
C
C
OUTLET!)
OUT L E T X )
C
RADIUS!) ■
SPI L L WAY ( )
c.
S
PI L L WAY! )
C • ■ . . VOL UME! ) .
V OL UME ! ) .
C
• ••
:
C
C
.C
C
C
■■; .
.
NOAH ■
AREA!)'
DRCDY( )
DRDY O
. DTWDY.! )
. RATI NG! )
.RCY ( )
' TOPW( )
• WPO
■
85
************************************************************
C
"
1
C I NPUT FORMAT:
C
'
.
C
THE I NPUT WI L L BE DI SCUSSED I N TWO S E C T I O N S , ONE FOR
C EACH SUBPROGRAM - BREACH- AND NOAH.
ALL I NPUT WI L L BE REAC
C. I N I N AN ' F ' FORMAT UNLESS S P E C I F I E D OTHERWI SE.
THE 6 F ' .
C F ORMAT -CONTROL . REQUI RES ONLY THAT THE DATA BE SEP EPATED BY
C COMMAS O R . S P A CE S . '
E I T H E R . RE AL OR I NTEGER VALUES CAN BE ■
C READ I N WI TH NO OTHER FORMATTI NG S P E C I F I C A T I O N S .
C
EG. 3 . 5 0 6 , I 5 , 0 . I 7 6 3 E - 2
5
C
.
•
************************************************************
c
'
C BREACH V ARI A B L ES
C
C V ARI A B L E
C
C
C
C
C
C
C
C
C
C
'
;
- ■
.
•
’
. ■
.
:
DEFI NI TI ON
= BREACH. AREA
= E F F ECT I VE - AREA OF BREACH FLOW
-- = ■BREACH DEPTH AT EACH TI ME STEP
DI SCHARGE THRU THE BREACH
= I N I T I A L LAKE DEPTH
= CHANGE I N LAKE LEVEL FROM THE LAST TI ME STEP,
USED AS A CORRECTOR TO PROJECT THE CHANGE I N
'
LAKE LEVEL FOR THE NEXT TI ME ,STEP
DT' = CHANGE IN- TI ME PER 'NlJM ' LOOPS THRU BREACH
XFD
-'=
DEPTH OF FLOW I N THE BREACH CHANNEL
C
HDAM
= DAM ---HEI GHT ( 4 8 F EE T )
C
I NL AKE
= I NFLOW I NTO THE LAKE ( C F S )
C
- LO
.= LAKE LEVEL AT THE TI ME STEP
T-2
C
Li
''
, = LAKE LEVEL AT THE TI ME STEP
T-I
C
|_2
'
■'= LAKE LEVEL CALCULATED AT
PRESENT TI ME S T E p T
C
LDUM
= PROJECTED LAKE LEVEL FOR
PRESENT TI ME STEP T
C
MNB
= MA NNI NG' S -N FOR THE BREACH CHANNEL
C
NUM '
= NUMBER OF LOOPS THRU B REACH
FOR EVERY CALL TO
C
NOAH
C
SLOPE
. = F R I C T I O N SLOPE OF THE BREACH CHANNEL
C
T
= TI ME STEP COUNTER
C
TFAIL
. = ESTI MATED. TI ME OF FAI LURE
C ■ TQ I
= SUM DI SCHARGE FROM THE S P I L L W A Y , O U T L E T S , .
C
. AND BREACH ( CF S ) - START OF EACH CALL TC THE
C
BREACH
SUBROUTI NE C
TQ 2
'= SUM DI SCHARGE FROM THE S P I L L W A Y , OUTL ETS,
C;
AND. BREACH ( CF S ) - , AFTER ' N. U M' LOOPS I N THE
C
BREACH
SUBROUTI NE
AREAB
AREAF
BD . BQ
DEPTH
DL
86
C
C
C
C
TSTEP
TZERO
' VOL
= TI ME STEP I NCREMENT ( s e c o n d s )
= STARTI NG TI ME FOR EACH CALL TO
= VOLUME OF THE LAKE
PREACH
************************************************************
C
c . BREACH
DATA
REQUI REMENTS:
<
.
C
C
C
..
CARD OR RECORD I : COMMENT CARD. ( SEE EXAMPLE
CAhD OR RECORD 2 : I N I T I A L CONDI TI ONS OF THE
READ ( 1 0 5 , 1 > D E P T H , I N L A K E , TST Ep ,M N B , T F A I L
I FORMAT( / , 5 ( F ) )
C
:
DATA)
BREACH
.
************************************************************
CC I N I T I A L I Z E BREACH CONSTANTS AND VARI A B L ES
T = FLAG = O. O
. H D A M= 4 8 . O . ■
LO=LI =DEPTH
VOL = VOL UME( D E P T H )
B D= OeO
T Q I = S P I L L W A Y ( D E P T H ) + OUTLET ( DEPTH)
C
-
************************************************************
C
C NOAH
•
V ARI A B L E S
cC
C
C.
C
C
C
C
:
VARI ABLE'
AA( I , J )
Al
A2
A3
A4
ACI,2 ,3
c.
C
C
C
C
C
C
C
C
-
c.
C ■
C ■'
•
AVA
AVR .
AVQ ■
DIST(I)
DQ
DQDT
DQDX
DT
DT D X .
DXDT
DEFI NI TI ON.
S
C O E F F I C I E N T MATRI X
= AREA AT. NODE
I, J
= AREA A T . NODE. ' I , J+1
S
AREA AT NODE
1+1,J
S
AREA A T . NODE- ' 1 + 1 , J + 1
S
C O E F F I C I E N T S OF THE . DEPTH- A RE A ' CURVE AT EACH
X - S E C T I ONS
AVERAGE AREA BETWEEN A2 AND A4
= AVERAGE HYDRAULI C RADI US BETWEEN R 2 AND R 4
AVERAGE. DI SCHARGE BETWEEN. Q 2 A N D • Q4
'= DI STANCE BETWEEN; NODE-S I AND 1+1
r
CHANGE I N DI SCHARGE
= CHANGE I N DI SCHARGE WI TH CHANGE IN. TI ME
= CHANGE I N DI SCHARGE WI TH CHANGE ' I N DI STANCE
r
CHANGE I N TI ME ( TI ME- STEP)
S
CHANGE. I N TI ME WI TH. CHANGE I N DI STANCE
Z
CHANGE I N DI STANCE- WI TH CHANGE I N T I MF
87
DY
DYDT ■
DYDX
C
■G
C . INFL 0.( I )
MN(I).
C
PEAKT
C
PC EQ2
C
CPCEQ4 .
c.
C
C . PC EY2
C
PC EYA
C
C
PMEQ2
C
C
PMEQ4
C
C
C . PMEY2'
C
PMEY 4
C
C
Ql
C
C ■ Q2
Q3
C
Q4
C
QMAX.( I ) •
C
C ' QTOL
C
C
C
.
C
C
C
C
C
C
C'
C
C
C ..
C
C
C
C
C
C
C
R2
PA­
RCI / 2 / 3
RCE
RESY
RESQ
RME
SF
SLOPE.
T
TF I NAL
THETA
TMAX(I)
TW2 '
■ TW'4
XSEC(L)
= CHAN,GE IN DEPTH .
CHANGE IN DEPTH WITH CHANGE IN TIME
= CHANGE IN DEPTH WITH CHANGE IN DISTANCE
= GRAVITATIONAL -CONSTANT
S LATERAL INFLOW ALONG CHANNEL LENGTH DX(I)
MANNING'S N FOR EACH REACH '
= TIME OF ARRIVAL OF THE PEAK
-= PARTIAL DERIVATIVE OF THE CONTINUITY EQUATION
WITH RESPECT TO Q2
S PARTIAL DER IVATIVF OF THE CONTINUITY EQUATION
WITH RESPECT TO Q4
= .■PARTIAL DERIVATIVE OF THE CONTINUITY EQUATION
WITH RESPECT TO Y2
= PARTIAL DERIVATIVE OF THE CONTINUITY EQUATION
WITH RESPECT TO Y4
S ' PARTIAL DERIVATIVE OF THE MOMENTUM EQUATION
WITH RESPECT TO Q2
=■ PARTIAL DERIVATIVE OF THE MOMENTUM EQUATION
WITH RESPECT 10. Q4
S PARTIAL DERIVATIVE OF THE MOMENTUM EQUATION
WITH. RESPECT TO Y2
= . PARTIAL DERIVATIVE OF THE MOMENTUM EQUATION
WITH RESPEC T TO Y4
= DISCHARGE AT NODE I / J
r DISCHARGE. AT NODE
I/J+1
DISCHARGE AT NODE 1+1/J
= DISCHARGE AT NODE.' I+1/ J+1
•r MAXIMUM DISCHARGE AT EACH X-SECTION
= TOLERANCE LIMIT FOR ERROR IN DISCHARGE ( 5 0 . 0
. CFS )
s • HYDRAULIC RADIUS AT NODE
I/ J+1
S HYDRAULIC RADIUS AT NODE
I +I/ J+1
= COEFFICIENTS OF THE DOWNSTREAM RATING CURVE
= RESIDUAL OF THE CONTINUITY EQUATION
S DEPTH RES IDUAL
= DISCHARGE RESIDUAL ■
S RESIDUAL OF THE MOMENTUM EQUATION
S FRICTION SLOPE TERM .
S • WATER ,.SURFACE SLOPE.
■= TIME STEP COUNTER
S TIME FOR TERMINATION ,
' S WEIGHTING COEFFICIENT ;
' S TIME OF PEAK EVENT. AT EACH X-SECTION/
= TOP WIDTH AT NODE I/ J+1 •
"
S TOP WIDTH AT NODE I+1/ J+1
S X-SECTIONS AT WHICH HYDROGR APHS ARE GENERATED
S
S
S
88
C
C
C
C
C
C
C
C
Yl
■ Y2
Y3
Y4
YMAX ( I )
YTOL
Z(I)
DEPTH AT NODE
IzJ
DEPTH AT NODE
IzJtI
r- DEPTH A F ' NO D E
I +IzJ
=
DEPTH AT NODE
1 + 1 z J +1
.
=■ MAXI MUM DEPTH AT EACH X- SEC T ION
S
TOLERANCE L I M I T FOR ERROR I N DEPTH ( . 0 5 FEET)
= CHANNEL BOTTOM ELEVATI ON Al EACH X - S E C T I ON
=
5
2
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *,* * * * * * * * * *
C
C NOAH
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DATA
REQUI REMENTS:
CARD OF RECORD 3: COMMENT CARD (SEE EXAMPLE DATA)
CARD OR RECORD 4; ROUTING CONSTANTS FOR NOAH
READ(I 05, 1 ) NzTF INAL,THETAzQTOL,YTOL
CARD OR RECORD 5: X-SECT IONS FOR HY DRO GRAPHS (LIMIT = A)
XSEC(I)=DAMSITE
READ( I O5 z ? ) ( XS EC( I ) z I = 2 z 5 )
2 FORMAT( 4 ( F))
CARD OR RECORD 6: COMMENT CARD (SEE EXAMPLE DATA)
CARDS OR RECORDS 7 - N + 7 : VARIABLES AT EACH NODE
READ( I O5 z 3 ) ( Z ( I ) z DI S T ( I ) , I N F LO( I ) , I = I , N)
3 FORMAT( / , 3 ( F) )
CARD.OR RECORD N+8: COEFFICIENTS OF THE DOWNSTREAM
RAT ING CURVE
READ ( 1 0 5 , 4 ) RCI z RC2 z RC.3 '
4 FORMAT(3(F))
CARDS OR RECORDS N+9 - 2N+9; DEPTH-AREA COEFFICIENTS
READ(I 0 5 , 4 ) ( AC1( I ) , AC2 ( I ) z,AC3(I ) , I = I , N)
C
************************************************************
C
C INITIALIZE NOAH CONSTANTS AND VARIABLES
G= 32. 2
NN= 2. * N
XSEC(I)=I
C
C COMPUTE THE INITIAL VALUES OF DEPTH AND DISCHARGE AT EACH
C CROSS SECTION
CALL BACKWATER
c
•
C SET THE MAX VALUES TO ZERO
89
DO 5 I = 1 , N
Q M A X ( I ) =YMAX ( . D = T M A X ( I ) = O , O
5 CONTI NUE
C
•
******.****** W. *********************^******* * * * * * * * * * * * * * * * * * *
c
'
C START THE S I MUL AT I ON
C
C TEST RUN TI ME
6 'I F (T . GE . T F I NAL ) GO TO 1 V
C
C PROJECT THE VALUES AT T HI S TI ME
DO 7 I = I z N
Q (I,2)= Q (I,I)
Y ( I , 2 ) = Y ( I , I)
7 CONTI NUE
c •
C
C
C
C
C
-
STEP
TO T HE NEXT
STEP
.
COMPUTE "NUM" LOOPS I N BREACH AS A FUNCTI ON OF THE ARRI VAL
TI ME OF THE WAVE.
WHEN T > T F A I L , THE T I ME . OF ARRI VAL O F
THE PEAK AT ANY CROSS- SECT I ON I S EQUAL TO THE PRESENT
TI ME;
T I S TREATED AS THE PEAK ARRI VAL T I M E .
.
.
PEAKT=T
I F ( P E A K T . L T . T F A I ' L ) PEAKT = T F AI L
NUM = ( P E A KT / Z O. O ) / TST-EP + . 5 .
C
C CALCULATE THE NEW OUTFLOW FROM T HE. BREACH
CALL BREACH
C
'
C ROUTE THE FLOO DWAVE DOWNSTREAM
CALL NOAH
C
.
.
C RE - A S S I GN "KNOWN" VALUES FOR THE NEXT TI ME
DO 8 I = I , N
Q(IzI)=O(IzZ)
Y(IzI)=Y(IzZ)
8 CONTI NUE
C
C TEST
STEP
FOR MAX DEPTH AND DI SCHARGE AND SET TI MES OF PEAK.
DO 9 I = I z N
I F ( Y ( I , Z ) . GT. YMAX ( I ) ) ' Y M A X ( I ) = Y ( I z Z ) ) T M A X ( I ) = T - D T
TF (Q ( I z-Z ) . GT . QMAX ( I ) ) QMAX ( I )=Q ( I z Z) ; TMA X ( I ) = D D T
9 CONT I NUE
C ' .
,
C OUTPUT DATA FOR HYDROGRAPHS AT SELECTED X- SEC TI ONS
WRI TE( 1 0 1 , 1 0 ) T , ( 0 ( XSEC( L ) z Z ) z L = 1 , 5 )
90
10 F O R M A T ( 6 F 9 . 1 )
C
C LOOP T HRU . A GA I N I F
TF I NAL
HAS
NOT BEEN REACHED
»
GO TO 6
C
************************************************************
C
C OUTPUT FI NAL RESULTS
I I W R I T E ( 1 0 8 , I 2)
12 F O R M A T ! / / ' STATI ON
MAX- DEPTH
S URF - E L E V
TOP WI DTH
',
. * ' MAX- Q
PEAK-TIME' / )
WRITE(108,13)(I,YMAX(I),Z(I)+YMAX(I),T0PW(YMAX(I),I)
* , QM A X ( I ) , T M A X ( I ) , I = 1 , N )
13 FORMAT( 3 X , I 3 , 5 ( F 1 0 . 1 ) )
END
91
***<r********A***********-* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
SUBROUTI NE BACKWATER
C
C SUBROUTI NE TO CALCULATE THE I N I T I A L CONDI T I ONS OF FLOW
C ( DEPTH AND DI SCHARGE) AT EACH CROSS S E C T I O N .
A STANDARD
C STEP- BACKWATER METHOD I S U T I L I Z E D .
C
COMMON/ ARRAY / A ( 2 0 0 * 2 0 0 ) *N
' C 0 M M 0 N / B R E A C H S / N U M , T S T E P , L 0 , L 1 , L 2 , DEP T H , BDf HDAM,
* A R E A B , F L A G , I NL A K E , T Q 1 , V O L , MN B , T F A I L
C O MM ON /P ARA ME TER S/ Y( 1 00 ,2 ) ,Q (1 0n , 2) ,Z (1 00 ) ,MN(IOO),
* I N F L O C 1 0 0 ) , D I S T ( I O O ) , Q M A X ( I 0 0 ) , YMAX ( 1 0 0 ) , T M A X ( I O O ) ,
* XSEC ( 5 ) , D T , T
C 0 M M O N / C O N S T A N T S / G , T F I N A L , T H E T A , nJ N , Q T O L , Y T O L
REAL I NF L O, MN
C
************************************************************
C
C I N I T I A L I Z E THE UPSTREAM CONDI TI ONS BETWEEN THE DAM AND XC SECTI ON NUMBER ONE
C
C DI SCHARGE AND DEPTH
QDAM = S P I L L WAY ( DE P T H) + OUTLET( DEPTH )
Q ( I , I ) = QDAM
Y DAM = DEPTH - 4 2 . 1
Y ( I , I ) - Y DAM .
C '
C E LEVA TI ONAL HEAD
HU = 6 3 9 6 . 0 + Y D A M
. I HD = Z ( I ) + Y ( I , 1 )
C
C V E L OCI T Y HEAD
VHU = ( Q D A M / ( Y D A M * 2 0 . ) ) / ( 2 . 0 * G )
VHD = ( Q (1 , 1 ) / AREA(Y ( I , 1 ) , 1 ) ) / ( 2 . 0 * G )
C
C F R I C T I O N SLOPE
SF = 6 3 9 6 . 0 - Z ( I )
CC CALCULATE RESI DUAL FROM USI NG Y ( I , T )
RESI D = ( HU+ VHU) - ( HD+ VHD+ SF)
C
C TEST THE RANGE QF THE RESI DUAL
I
F ( RESI D . G T . 0 . 0 1 ) Y ( I , I ) =Y ( I , I ) + RE S ID ,* GO TO I
I F ( RES I D . L T . - O . 01 ) Y ( I , I ) =Y ( 1 , 1 ) +R ESI D ,'GO TO I
C
1
92
C COMPUTE THE I N I T I A L CONDI TI ONS AT EACH CROSS SECTI ON
DO 3 I = I , N - I
I TER=I
C
C DI SCHARGE
QU = 0 ( 1 , 1 )
QD = Q ( 1 + 1 , 1 ) = Q U + I N F L O ( I )
C
C DEPTH ( YD I S A T RI AL VALUE THAT IS CHANGED EACH I T E R A T I O N )
YU = Y ( I , I )
YD = YU
C
C RESTART NEXT I T E RAT I ON
2 CONTI NUE
I F ( I T E R . G T . 2 0 ) OUTPUT "TAI LS' - TO CONVERGE A T ' , 1+1 I STOP
C
C STAGE
HU = Z ( I ) + YU
HD = Z ( 1 + 1 ) + YD
C
C AREAS
AU=AREA (YU , I )
A D = A RE A ( Y D , 1 + 1 )
C
C HYDRAUL I C RAD I I
RU=AUZWP( Y U , I )
RD= ADZW P ( Y D , 1 + 1 )
C
C COMPUTE MA NNI NG' S N FOR EACH REACH ASSUMI NG THE F RI CT I ON
C SLOPE I S EQUAL TO THE RED SL OPE.
SF=( Z ( I ) - Z ( 1 + 1 ) ) ZDIST(I)
MN Cl ) = ( I . 4 9 * ( (AU + A D ) Z 2 . 0 ) * (.(RU + RD) Z2 . 0 ) * * . 6 7 * ( S F
* * * . 5 ) ) Z ( ( QU+QD) Z2. 0 )
C
C CALCULATE THE VEL OCI TY HEAD TERM
VHU = ( G U Z A U ) * * 2 Z ( 2 . 0 * G )
VHD = ( Q D Z A D ) * * 2 / ( 2 . 0 * G)
C
C CALCULATE THE RESI DUAL FROM USI NG YD
RE S I D = ( HU + VHU) - (H.D+VHD + SF)
C
C TEST THE RANGE OF THE RESI DUAL
i f ( r e s i d . g t . O o Q 1 ) y d = y d + r e s i d ; i t e r = i t e r + i ; go t o 2
L F ( R E S I D „ L T <, - 0 . 0 1 ) Y D = Y D + R E S I D ; I TE R=I T ER + I ; G0 TC 2
C
93
C ASSI GN FI NAL. DEPTH VALUE
Y ( 1 + 1 , 1 ) = YD
C
3 CONTI NUE
C
C OUTPUT THE I N I T I A L CONDI TI ONS FOR EACH CROSS- SECTI ON
WRITEd 08,4 )
4 F O R M A T ( 5 X ' S E C - # ' 3 X ' D E P T H ' 4 X ' D I S C ' S / ' DX ' 9 X ' M N ' X
DO 6 1 = 1 , N
W R I T E ( I 0 8 , 5 ) I , Y ( I , I ) , Q ( I , 1 ) , D I S T ( I ) , MNCI )
5 FORMAT ( 6 X , I 2 , 4 ( F 9 . 2 ) )
6 CONTI NUE
C
RETURN
END
94
************************************************************
C
SUBROUTI NE
BREACH
C
C. SUBPROGRAM TO CALCULATE THE UPSTREAM CONDI T I ONS OF D I S C CHARGE AT THE DAM S I T E AS A FUNCTI ON OF THE RATE OF F A I L C URE AND. THE LAKE L E V E L .
C
C O MMON / B R E A C H S / N U M/ T S T E P , L O , L I , L 2 / D E P T H , 8 D , H D A M ,
* A R E A S , F L A G , I NL AKE, TQ I , V O L , M N B , T F A I L
COMMON/ PAR AMETERS/ Y ( 1 0 0 , 2 ) , Q d 0 . 0 , 2 ) , Z( 1 0 0 ) , MN.( 1 0 0 ) ,
* I N F L O ( I O D ) , D 1 S T ( 1 0 0 ) , QMAX( I 0 0 ) , Y MAX( 1 0 0 ) , T MAX( I 0 0 ) ,
* XSEC( 5 ) , DT, T
REAL L O , L I , L 2 , M N B , I N L A K E , LDUM
I NTEGER FLAG
C
************************************************************
C
'
■
C SET TI ME FACTOR
TZERO=T
C
C LOOP THRU BREACH 1 NUM1 TI MES
DO 3 I = 1 , N U M
C
C I NCREMENT TI ME COUNTER
T = T + - T STEP
C
C PROJECT THE CHANGE I N LAKE LEVEL
I F d . E Q . I . ANDe L O . ' EQ. DEPTH) DL = 0 . 0 0 1 I GO TO I
D L = ( L O - L l )
I LDUM = L I - DL
C
.
C DETERMI NE THE DI MENSI ONS OF THE BREACH
RD = T / ( T F A I L / H D A M )
I F ( B D . G E . HDAM) BD = HDAM
AREAS = B D * * 2
C
C COMPUTE THE EFFECTI VE. AREA OF FLOW THRU T HE BREACH
FD = BD - ( DEPTH - L DUM)
AREAF = F D * * 2
C
C DETERMI NE T H E . F R I C T I O N SLOPE FOR THE BREACH CHANNEL
SL OP E = ( ( 6 4 0 1 . 9 - B D + F D ) - (.Z ( I ) + Y ( I , 1.) ) ) / 5 O O.
C
C COMPUTE THE BREACH DI SCHARGE.
BQ = 1 . 4 9 / MNB * SQRT( S L OP E )
* AREAF
*
(RADIUS(FD,
95
*
A R E A F ) ) * * . 67
C TOTAL DI SCHARGE FROM THE D AM - S P I L L WA Y , OUT L ET , S BREACH
T Q2 = S P I L L WA Y ( L D U M) + OUTL ET ( L DUM) + B Q
C
C COMPUTE THE NEW LAKE VOLUME AND LEVEL AFTER THE TI ME STEP
VOL = VOL + T S T E P * ( I NLAKE —, ( ( T Q 2 + TQ1 ) / 2 . C) )
CALL LAKE ( V O L , L 2 ) .
C
C TEST I F DL WAS AN APPROPRI ATE PREDI CTOR OF LAKE CHANGE
C I F NOT, PREDI CT NEW DL AND LOOP THRU AGAI N
C OV E R- E S T I MA T E
I F ( L 2 - L D U M „ G T . . 0 „ 0 5 ) D L = L 1 - L 2 ; OUTPUT
* ' DL TOO L ARGE'
GO TO I
C UNDER- E ST I MA T E
I F ( L D U M - L 2 . G T . 0 . 0 5 ) DL = L . 1 - L 2 ,* OUTPUT
* ' DL TOO SMALL" ,* GO TO I
C
C R E - A S S I G N VALUES FOR NEXT TI ME STEP
TQI = T Q2
LO = L i
L I = L2
DT = T - T Z E RO
C
C OUTPUT THE BREACH HYDROGRAPH
WRI T E ( I 0 1 , 2 ) T , TQ 2
2 FORMAT( 2 ( F 1 2 . 1 ) )
3 CONTI NUE
C
C ASSI GN NEW DI SCHARGE VALUE FOR NOAH
Q ( 1 , 2 ) = TQ2
OUTPUT DT , T Q2
RETURN
END
96
********* AA* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
SUBROUTI NE
C
C
C
C
C
C
NOAH
SUBPROGRAM TO ROUTE THE FLOODWAVE DOWNSTREAM FROM THE DAM
U S I N G A FOUR P O I N T I M P L I C I T F I N I T E D I F F E R E N C E METHOD FOR
THE N U M E R I C A L S O L U T I O N OF THE COMPLETE E Q U A T I O N S OF U N STEADY FLOW.
*
*
COMMON/PARAMETERS/Y(100z2)zQ-(100#2)#Z(100)zMN(1UO),z
INFLO(130)zDIST(100)zQMAX<100)zYMAX(100)zTMAX(TOO)z
XSEC(5)zDT,T
COMMON/APRAY/A<200z200)zN
COMMON / C ON S T A NT S / Gz T F I NA L z T HE T A z NNz OT OL z Y T OL
C
************************************************************
C
O U T P U T ’ I NTO NOAH1
I T E R = I/* FLAG = O
C SET ARRAY TO ZERO
1 DO 2 I' = I , N N
DO 2 J = I z NN+1
A(IzJ)=O.O
2 CON T I N U E
C
C GENERATE THE C O E F F I C I E N T M A T R I X
OUTPUT ’ I N T O M A T R I X . . . ’
CAL L M A T R I X
C
C SOLVE E Q U A T I O N S
C AL L SI MEQ
3
SI MUL TANEOUSL Y
WRI T E ( 1 0 8 z 3)
F O R M A T ( X ’ X- SEC
* 1
Ql
DO 5 I = I z N
J =( I * 2 ) - 1
CORRECT THE I M P L I E D DEPTHS
RESY=A( J zNN+1)
RE S Q= A ( J + 1 z N N + 1 )
Y ( I z2)=Y <I z2)-RESY
Q(I,2)=Q(Iz2)-RESQ
C
C SET
FLAG
I F
FOR
ADDITIONAL
Y2
RESQ')
Yl
02
AND
DI SCHARGES
WI T H
RES Y '
RESI DUALS
ITERATION
( ABS(RESY)-YTOL. GT.O.OR.ABS(RESQ)-QTOL.GT.O)
F L A G= I
C
C OUTPUT THE RES ! D U A L S
WRI T E ( I 0 8 , 4 )
I,Y<I,1),Y(I,2),RESY,Q(I,1),Q(I/2),RESQ
4 F O R M A T ( 2 X , 1 3 , 6< F l 2 . 3 ) )
5 CON T I N U E
C
C TEST FOR A D D I T I O N A L I T E R A T I O N
I F ( F L A G . EQ. 0 ) GO TO 6
C
'
C TEST FOR CONVERGENCE
I F ( I T E R . G E . 2 5 ) OUTPUT 6 F A I L S TO C O N V E R G E . . . I T E R = 2 5 ' ;
* STOP
I TER=ITER + ! ;
FLAG = O/' GO TO I
C
6 OUTPUT I T E R , T , DT
W R I T E ( I 0 8 , 7) ■
7 F O R M A T ( X • X- SEC
Yl
Y2
011
*
8
9
'
02'
)
DO 9 1 = 1 , N
WRI T E ( 1 0 8 , 8 )
I , Y ( I , I ) , Y ( I , 2 ) , 3 ( I , I ) , Q ( I , 2)
F O R M A T ( 2 X , 1 3 , 4 ( F I 2 . 2) )
CONT I NUE
C
RETURN
END
98
************************************************************
C
SUBROUTI NE MATRI X
C
C SUBROUTI NE TO. GENERATE THE MATRI X OF C O E F F I C I E N T S FROM
C THE EQUATI ONS AT EACH X SECTI ON AND THE UP AND DOWNSTREAM
C . BOUNDARI ES.
•
C
‘
COMMON/ CONST ANT S/ G, T F I N A L , T H E T A , N N , QT OL f YT OL
COMMON/ PARAME. TERS/ Y ( I 0 0 , 2 ) , Q ( I 0 0 , 2 ) , Z ( I C O ) , MN( I OO) , •
* I NF L O ( I 0 0 ) , D I S T d P O ) , QMAX ( I 0 0 ) , YM AX ( 1 0 0 ) , TMAX ( I 0 0 ) ,
*
XSEC( 5 ) , P T , T
COMMON/ ARRA Y / A( 2 0 0 , 2 0 0 ) , N
REAL I NF L Of MN
************************************************************
Q
C ELEMENTS OF THE
C UPSTREAM
A(IfI)=O.O
MATRI X
FROM THE
BOUNDARY
A d , 2) = 1 . 0
C DOWNSTREAM
A(NN,NN~1)=-DRCDY(Y(N,2)>
A(NN,NN)=1.O
C RESI DUALS
A d ,N N-M ) = 0 . O
A (NN , NN +1 ) = Q ( N , 2 ) - R A T I N G ( Y ( N , 2 ) )
C
DO I
C
C SI MPLIFY
I=1,N-1
NOTATI ON
C
M= 2 * I
L=M-I
C
D X D T = D I S T ( I ) / DT
DTDX=DTZDI ST( I )
C
C DEPTHS
Y1=Y(I,1)
Y2 = Y ( 1 , 2 )
Y3=Y(1+1,1)
Y 4 =Y (1 + 1 , 2 )
C
C TOP WI DTHS
TW2 = T 0 P W ( Y 2 , I )
CONDI TI ONS
99
TWA = TOPW( Y 4 » I > 1 )
C
C AREAS
A l =AREA(Y1,I)
AZ=AREACY 2 , I )
A3 = A R E A ( Y 3 , I + 1 ).
AA=AREA( Y 4 , 1+1)
C
C HYDRAULI C R A D I I
. R Z = A Z Z WP C Y 2 , I )
RA= A Z Z WP ( Y A , 1 + 1 )
C
C DI SCHARGES
Q I=Q ( I ,1 )
QZ=QC1 , 2 )
Q3 = Q C 1+ 1 , I )
QA = Q ( I + 1 , 2 )
C
C COMPUTE THE AVERAGE VAL UES
BETWEEN
TH E NODES
C
C AREAS
AVA= CA 2 + A A ) Z 2 . 0
C
C HYDRAULI C R A D I I
AVR= CR2 + RA ) Z 2 . 0
C
C DI SCHARGES
AVQ= CQ2 + G 4 ) Z 2 , 0
C
C C A L C U L A T E THE F R I C T I O N
.
SLOPE
.
.
TERM
SF=CMNCI)*AVQZAVA)**2ZCZ.Z*AVR**1.33)
C
C COMPUTE
C RESPECT
THE D E R I V A T I V E S OF THE
TO DEPTH AND DI SCHARGE
FRICTION
SLOPE
WI T H
DSFDY2=-2„0*SF*DRDY(Y?,I)ZC3„0*AVR)-SF*TW2ZAVA
D S F D Y 4 = - 2 o O * S F * D R D Y CY A , I + 1 ) Z C3, 0 * A/VR ) - S F * TW A Z AVA
DSFDQ2=MNCI.)**2*AVQ/C2„2*AVA*AVA*AVR**1.33)
DSF DQA= DSF DQZ
C
C CONTINUITY
E QU A T I ON
C
C CHANGE I N Q
DQ= CQA- Q 2 ) * T H E T A + CQ3- Q1 ) * Cl . O - T H E T A)
C
C CHANGE I N Y
DY=0.5*CCY2-Y1)+CTW4ZTW2)*CY4-Y3))
I OO
C
C RCE
I S THE R E S I D U A L FROM THE C O N T I N U I T Y EQUAT I ON
. RCE = DY + DQ / ( TW. 2, * DXDT) - I NFLO < I ) * DT / TW2
C
C P A R T I A L D E R I V A T I V E S OF THE C O N T I N U I T Y EQi I A T I O N ( C F ) WI
C RESPECT TO DEPTH AND D I S C H A R GE AT THE T I M E L I N E J +1
C
PC E Y^ = O . 5 - ( DT WDY( Y 2 , I ) / ( T W 2 * T W 2 ) ) * ( Q „ 5 * T W 4 * ( Y 4 - Y 3 )
* D Q * D T D X - I N F L O ( I ) * DT)
C
PCEYA=O.5 * T W 4 / T W 2 + 0 . 5 * D T W D Y ( Y 4 , 1 + 1 ) * ( Y 4 - Y 3 ) / T W 2
C
.
PCEQ2=- DTDX*THETA/ TW2
C
.
PCEQA=DTDX* THFTA/ TW2
C
C MOMENTUM EQUAT I ON
C
C CHANGE I N Q WI T H CHANGE I N T I ME
D QD T = - 0 . 5 * D X DT * ( Q 2 7 A 2 - Q I / A I + G A / A 4 - Q 3 / A 3 ) / G
C
C CHANGE I N Q / A WI T H CHANGE I N DI S T A NCE ( D X I S FACTORED
DQADX=0.5*(((Q4/A4)*+2-(Q2/A2)**2)/G
* T HET A +
* ( (Q3/A 3 ) * * 2 - < Q 1 / A I ) * * 2 ) / G * ( 1 . 0 - T H E T A ) )
C
C CHANGE I N E L E V A T I O N AL HEAD
HEAD=THETA*(Y2+Z(I ) - Y 4 - Z ( 1 + 1 ) ) + ( 1 . 0 - T H E T A ) * ( Y 1 + Z ( I
* Y 3 —Z ( 1 + 1 ) )
C
C RME I S THE R E S I D U A L FROM T HE MOMENTUM E QU A T I ON
RME = DQDT + D QADX+ H E A D - ( D I S T ( I ) * S F )
C
C P A R T I A L . D E R I V A T I V E S OF THE MOMENTUM E Q U A T I O N ( ME) WI TH
C RESPECT TO DEPTH AND DI S CHA RGE AT THE T I M E L I N E J+1
TH
+
OUT)
)-
C
*
PMEY2=0.5*DXDT*(Q2/A2)*(TW2/A2)/G-((Q2/A2)**2*TW2*
THETA)/(A2*G)+THETA-DSFDY2*D IST(I)
*
PMEY4=0.5*DXDT*(G4/A4)*(TW4/A4)/G +((G 4/A 4)**2*TW 4*
THETA)/(A4*G)-THETA-DSFDY4*DIST(I)
C
C
PMEQ2 = - 0 . 5 * D X D T / ( A 2 * G ) + . ( Q 2 / A 2 ) * T H E T 4 / ( A 2 * G ) - D S F D Q 2 *
*
DIST(I)
C
*
P M EQ4=-0.5 *DX DT/(A 4*G )-(Q 4/A 4)*T HETA / (A4*G)-DSFDQ4*
DIST(I)
C
C GENERATE THE C O E F F I C I E N T
C
C C O N T I N U I T Y E QU A T I ON
A ( M» L> =PCE Y2
A ( M , L + T) =PCEQ2
A ( M/. L + 2 ) = P C E Y 4
A(M,L+3)=PCEQ4
C MOMENTUM EQUAT I ON
A (M + 1 / L ) = P M E Y 2
A(M+1,L+T)=PMEQ2
A ( M + 1 / L + 2 ) =P' MEY4
A ( M + 1 / 1 + 3 ) - PMEQ4
C RESI DUALS
A CM, N N+1 ) =RCE
A( M + 1 , N N + I ) = R ME
C
I ■CONT I NUE
C
RETURN
END
MATRI X
I 02
*** *************************************** ******************
cSUBR OUT I NE
SI MEQ
C
C SUBROUTI NE TO SOLVE THE C O E F F I C I E N T MATRI X USI NG THE GAUSS
C - JORDAN E L I M I N A T I O N METHOD FOR SI MULTANEOUS EQUATI ONS
CCOM. MO N / ARRAY / A( 2 0 0 / 2 0 0 ) *N
C
************************************************************
C
C I N I T I A L I Z E PRGM
N N= 2 * N
C
C SEARCH FOR LARGEST
DO 9 K = 1 , N N
I
I
-
K
PI VOT
ELEMENT
-
I F ( K . E Q . N N ) GO TO 5
BIG=ABS(A(K,K))
DO 3 I = K + 1 f N N
BBIG=ABS(AdzK))
I F ( B I G . L t e BBI G)
3 CONTI NUE
BI G=BBI G;
II=I
C
C TEST
•
FOR
REPL ACEMENT
I F ( I I e EQe K)
GO TO 5
C
C SWI TCHEROO
DO 4 J = I f N-N + I
SUB=A(IIfJ)
A(IIfJ)=A(KfJ)
A ( K z J)=SUB
4 CONTI NUE
C
C CALCULATE NEW MATRI X WI T H I N SAME SPACE
C ( D I V I D E EQUATI ON BY THE LEADI NG C O E F F I C I E N T S )
5 DO 6 J=K + 1 fNN+1
A(KfJ)=A(K,J)/A(KfK)
6 CONTI NUE
C
C BACK
S UB S T I T U T I ON
DO 8 I = I f N N
I F ( I eEQeK) GO TO 8
DO 7 J = K + I f NN + 1
A(IfJ)=A(IfJ)-(A (IfK )*A (K fJ))
7 CONTI NUE
8 C ON T I N U E
9 CONT I NUE
RETURN
END-
***************************************************** *******
C
REAL
F U N C T I ON
AREA( Y , I )
C
C F U N C T I O N S U B R O U T I N E TO C A L C U L A T E THE AREA OF FLOW AT EACH
C CROSS S E C T I O N FROM THE DEPTH / AREA CURVES .
C
COMMON/COEFS/AC1(100),AC2(100),AC3(100),RC1,RC2,RC3
REAL Y
I NT E GE R I
C
************************************************************
C
*
AREA = ( A C I ( I )
(Y**3))
C
RETURN
END
*
Y)
+
( AC 2 ( 1 )
*
( Y**2> )
+
( AC 3 ( 1 )
*
105
******************************** ****************************
C
REAL F U N C T I ON D R C D Y ( Y )
C
C F U N C T I O N S U B R OU T I N E TO C A L C U L A T E THE D E R I V A T I V E OF THE
C D E P T H - D I S C H A R G E R A T I N G CURVE ( DOWNS TR E AM BOUNDARY)
C
C0MM0N/C0EFS/AC1(100),AC2(100),AC3(10U),RC1,RC2,RC3
REAL Y
C
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ft^ * * * * * * * * * * * * * * * *
C
DRCDY
C
RETURN
END
=
RC I
+
2 * RC.2 * Y + 3 * R C 3 * Y * * 2
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C
REAL
F UNCT I ON
C
DR D Y ( Y, I )
-
C F U N C T I O N S UBROUT I NE TO COMPUTE THE D E R I V A T I V E
C H Y D R A U L I C R A D I U S WI TH RESPECT TO DEPTH
C
OF
THE
■ COMMON/ COE F S / ACI ( I 0 0 ) „ AC2< I 0 0 ) , AC 3 ( I 0 0 ) , R C I , RC2/ . RC 3
REAL Y
I NTEGER I
C
* * A AAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAA
C■
DRDY = T 0 P W < Y , I > / W P ( Y , I > - A R E A ( Y , i ) / ( W P ( Y , I ) * * 2 ) * ( - 0 „ 5 )
*
*
* ( ( TOPWC Y , I ) + 2 . 0 * Y ) * * 2 ) * * ( - 1 . 5 ) * 2 . 0 * ( T O P W ( Y , I ) + 2 „ 0 * Y )
* (DTWDY(YfI)+2.0)
C
RETURN
END
I 07
************************************************************
C
REAL
FUNCTI ON
DTWDY ( Y , I )
C
C FUNCTI ON SUBROUTI NE TO COMPUTE THE D E R I V A T I V E OF THE TOP
C WI DTH WI TH RESPECT TO DEPTH
C
COMMON/ COE FS / A C l ( 1 0 0 ) # A C 2 ( 1 0 0 ) y A C 3 ( T O O ) v R C 1 * R C 2 / R C 3
REAL Y
I NTEGER
I
C
************************************************************
C
DTWDY = 2 . 0 * A C 2 ( I )
C
RETURN
END
+ 6 . 0 * A C 3 ( I ) *Y
108
************************************************************
C
SUBROUTI NE
C
L AKE ( V 0 L / - L 2 )
-
C F U N C T I O N S U B R OU T I N E TO CAL C U L A T E THE LAKE DEPTH FROM T HE
C D E P T H - V O L U ME ' CURVE E Q U A T I O N S .
THE CURVE I S TREATED AS
C TWO CURVES FOR GREATER A C C U R A C Y .
C
REAL
VOL,L2
C
************************************************************
C
I F ( VOL . L E . 2 3 4 3 5 2 8 0 )
GO TO I
C
L 2 =' ( . 1 6 8 7 8 I E 01 + ( . I 1 5 4 8 0 E - 0 5 * V O L ) + ( - . 9 6 2 1 86
* E - 1 4 * ( V O L A A g ) ) + ( . 4 I 45 51 E - 2 2 * (V 0 L * * 3 ) ) )
RETURN
C
I
a
L2
+
= ( ( . 1 4 8 0 2 5 E - 0 5 * V OL ) + ( - . 2 6 7 I 7 4 E - I 3
( .3 0948 3E-21 a ( V 0 L * * 3 ) ))
C
RETURN
END
*
( VOL * * 2 ) ) '
109
* * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * ** * ********************
C
REAL FUNCTI ON OU T L E T ( Y )
C
C FUNCTI ON SUBROUTI NE TO CALCULATE THE DI SCHARGE
C OUTLET GATES AS A FUNCTI ON OF LAKE LEVEL
C
REAL Y
C
THRU THE
************************************************************
C
*
OUTLET = ( ( . 3 5 0 0 0 0 E + 01 * Y ) +
2 ) ) + ( . 7 3 2 4 2 2 E - 0 3 * ( Y * * 3 ) ))
C
RETURN
END
8203 13E-01
*
( Y* *
110
****************************** ******************************
C
REAL F U N C T I ON R A D I U S ( Y , A )
C
C F U N C T I O N SUB ROUT I NE TO C A L CUL AT E
C ( R = A R E A Z WE T T E D P E R I M E T E R )
C
REAL Y » A
C
THE
HYDRAULI C
RADI US
************************************************************
C
P = S QR T ( Y * * 2 * 2 . 0 )
RADI US = A / P
C
•
RETURN
END
*
2.0
■Ill
************************************************************
C
REAL
F UNCT I ON
RATING(Y)
C
C F U N C T I O N S U B R OU T I N E
C DOWNSTREAM BOUNDARY
TO C A L C U L A T E THE D I S C H A R GE AT T HE
FROM THE D E P T H - D I S C H A R G E RAT I NG CURVE
C
COMMON/ COE F S / A C 1 ( I O O ) / A C Z ( I O O ) , AC 3 ( I 0 0 ) , RCl , R C 2 , R C 3
■ PEAL Y
C
************************************************************
C
RATI NG
C
RETURN
END
=
RCI * Y + R C 2 * ( Y * * 2 )
+ R C3 * (Y * * 3 )
1 1
2
************************************************************
C
REAL
FUNCTI ON
S P I L L WA Y ( Y )
C
C FUNCTI ON SUBROUTI NE TO CALCULATE
C A FUNCTI ON OF LAKE DEPTH
C
REAL Y, SD
C
THE
SPI LLWAY
DI SCHARGE
AS
************************************************************
C
IF(Y.LE.42.1)
S P I L L W A Y = O . O;
SD = Y - 4 2 . 1
S P I L L W A Y = ( ( . 1 3 4 3 1 0 E 02 *
* ( S D * * 2 ) ) + ( - . 1 0 7 5 1 6 E 01 *
RETURN
SD) * ( . 2 6 4 7 4 4 E
(SD**3)>) ■
RETURN
C
‘
END
■
02
*
113
* * * A * * * * * * * * f t * * * A * i r * A * * * * * * * A * A * * * * * A * * * * * * * * * * * * * * * * * * * * A * A
c
-
REAL F UNCT I ON T O P W C Y , ! ) .
C
C F U N C T I O N S U B R OU T I N E TO C A L C U L A T E THE TOP WI DTH OF FLOW AT
C EACH C R O S S - S E C T I O N AS THE D E R I V A T I V E OF THE A PEA WI TH
C RESPECT TO D E P T H }
C
COMMON/ COE F S / A C 1 ( 1 0 0 ) , A C Z C i n O ) , A C 3 ( 1 0 0 ) , R C 1 , R C 2 , RC3
REAL Y
I NTEGER
I
C
AAAAAAAAAAAAAAAAAAAAAAAAAA- AAAAf r AAAAAAAAAAAAAAAAAAAAAAAAAAAAA
c
TOPW
= ACI(I)
C
RETURN
END
+ 2 * A C 2 ( I > *Y
+ 3 * AC 3 ( 1 ) *Y * * 2
I I4
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * £ * * * * * * * * * * * * * * - * * « * * * * * *
C
REAL
C
C
C
C
C
C
F U N C T I ON
V O L U ME ( Y )
F U N C T I O N S U B R OU T I N E TO C A L C U L A T E THE I N I T I A L VOLUME CF T HE
L AKE FROM THE DE P T H- V OL UME CURVE.
THE CURVE I S BROKEN
I N T O TWO S E C T I O N S AND T REAT ED WI TH TWO E Q U A T I O N S FOR I NCREASED A C C U R A C Y . '
REAL
Y
C
■
****************************************** *** ***************
C
I F (DEPTH.L E . 24)
GO TO
I
C
*
VOLUME = - . 2 3 2 3 2 0 E OA + ( „ 7 9 1340E 06 *
( . 3 8 8 2 0 8 E 04 * ( Y W ) , ) + ( . 1 76 4 5 SE 0 3
Y) +
* (Y**3))
RETURN
C
I
*
VOLUME = ( . 6 7 4 2 7 3 E 0 6 *
( . 1 8 4 3 3 6 E 03 * ( Y * * 3 ) )
C
RETURN
END
Y)
+ ( . 8T6750E
04
*
( Y* * ?) )
+
************************************************************
C
REAL
F U N C T I ON
WP(YzI)
C
C F U N C T I O N S U B R OU T I N E TO C A L C U L A T E THE WETTED
C EACH X - S EC T I O N AS A F U N C T I O N OF TH E DEPTH
P E R I ME T E R
AT
C
COMMON/ COE F S / A C 1 ( I 0 0 ) / A C 2 ( T 0 0 ) z A C 3 ( I 0 0 ) / R C l z RCZ z RC3
REAL Y
I NTEGER
I
C
************************************************************
C
WP =
SQRT ( TO- PWt Yz I ) * * 2
C
RETURN
END
+
Y**2)
+
Y
A P P E N D I X I 11
EXAMPLE DATA F I L E
DEPTH I N L A K E T ST EP
MNB
0.047
48
1000
5 .0
N T F I N A L THETA QTOL YTOL
97 3 6 0 0 . 0
0.6
50.0
. 0 5.
X- SEC T I ONS
20,50,70 ,97
DIST(I)
Z(I)
INFLO(I)
100.0
.0
6330.0
.0
6310.0
100.0
.0
125.0
6 2 90. 0
6265.0
125.0
.0
.0
6244.2
150.0
.0
6226.6
I 50.0
I 75.0
6210.3
.0
.0
6198.0
175.0
6185.8
20 0 . 0
•
.0
200.0
.0
6175.6
.0
2 25.0
6166.0
225.0
• .0
6155. I
.0
250.0
6144.3
.0
250.0
6132.2
275.0
6120.2
.0
6109. 7
275.0
.0
6102.3
300.0
.0
.0
6093.6.
300.0
6084.9
.0
325.0
.0
325.0
6075.6
350.0
.0
6066.7
.0
60 5 7 . 3
350.0
.0
6049. 6
375.0
.0 .
375.0
6042.3
6035. 0
400.0
■
»0
.0
6027.2
400.0
6019.4
.0
425.0
425.0
.0
6011.2
6002.9
450.0
.0
.0
5994. I
450.0
4 75.0
.0
5985.4
.0
475.0
5976.I
5966.9
500.0
.0
5953.9
500.0
.0
.0
525.0
5938.9
5924.4
550.0
.
.0
.0
5909.2
550.0
,0
5894.I
575.0
TFAIL
4 50.
FOR HYDROGRAPHS ( I = D A M S I T E )
X-SEC=I
( NOT READ I N )
I
• 2
3
4
'5
6
7
8
9
10
I I
I 2
I 3
I 4
I 5
16 .
17
I 8
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
I I7
5 75.0
600.0
600.0
5829.2. 6 2 5 .0
5811.9 . 625.0
5797. 3
6 5 0. 0 ■
5784.8
650.0
5772.4
675.0
5759.5
675.0
700.0
700.0
725.0
5746.6
57 3 3 . 2
5718.8
5703.2
5687.5
5671.0
5654. I
5636. 7
5619.3
56 01.3
5581.6
5558.6
55 3 5 . 5
5511.8
5488. I
5463.6
54 3 9 . 2
5414.0
725.0
825.0
5346.3
5324.9
5305. 7
5286.I
975.0
5267.I
I 000.0
I 000.0.
1025.0
1025.0
1050.0
1050.0
1075.0
1075.0
1100.0
I 100.0
1125.0
1125.0
1150.0
5391.1
5368.7
5247.6
5230.2
5217.1
■5204.0
5190.6
5177.2
5163.4
5145.9
5127.3
5110.1
5093.8
5075.1
,0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
. .0
.0
.
.0
.0
.6
.0
750.0
750.0
775.0
775.0
800.0
800.0
825.0
850.0
850.0
875.0
875.0
90 0 . 0
900.0
■ 925.0
9 25.0
■9 5 0 . 0
950.0
975.0
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
.0
.0
.0
..0
.0
5878. I
58 62.3
5845.7
.
-.0
.0
56
57
58
59
60
61
62
63
.
64
65
66
. 67
68
69
70
71
. 72
73
74
75
76
77
pO
'
■
'
.
■
.0
,0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
.0
■ .0
.0
.0
78
.
79
80
81
82
83
84
1I 8
5049.5 1150.0
.0
.0
5030. 7 1 1 75.0.
5013.4 1175.0
.0
.
0
4 9 9 7 . 5 1 20 0 . 0
4982. 1 I 200.0
.-O
,0
4965.9 1225.0
«0
.
4 9 4 8 . 4 'I 2 2 5 . 6 '
.0
4931.1 1250.0
.0
4913.8 I 2 50.0
.0
4897.0 1275.0
;.o
4881.3 1275.0
4865.6
875.0
. -o
.0
4855.1
, 0.0
I 3 16 02E + 03
. 242019E+03
. 5 9 8 7 5 0 E+01
- . 246667E+01
. 3 74 2 75 E+01
. I 18736E+02 •
. I 49800E+01
. 2621 4 0 6 * 0 2 '
— . 1 3 0 7 9 4 E + 01
. 441394E+02
- . 981 6 8 5 E +00
.495098E+02
' . 504776E+01
. 333553E+02
.983465E+01
. I 75051 E+ 02
. 8 1 7 1 1 3 E+01
. 7.8 78 4 9 E + 0 0
. 6 5 0 7 6 2 E + 01
- . I 59294E+02
.613183E+01
- . I 27807E+02
.
.
5 9 7 3 9 5 E + 01
-.645286E+01
. 579634E+01
. 665988E+00
. 5 6 1 873E+01
. 778483E+01
. 5 4 2 I 3 8 F +0 I
• . I 5 6 9 4 7 E +0 2
. 5 2 2 4 0 4 E + 01
. 2 3 6 0 4 5 . E + 02
.201692E+01
. 3 3 1 I 0 1 E + 02
- . 4 1 4576.E + 01
. 4 2 8 9 1 0E+02
- . 359250E+01
. 428749E+02
- . I I 5 7 1 3E+01
. 270954E+02
. 1 481 2 0 E + 01
. I 0 0 0 0 9 E+ 0 2
. 3 5 1 339E+01
. 279490E+01
. 5 2 9 3 9 4 E +0 I
. I 6 9 0 3 5 E.+0 I
. 6 1 5 1 5 0E+ 0 1
- . I 0 8 0 2 8 E + OO
.6520926+01
- . 2 4 4 7 8 . 6 E + 01
.689034E+01
- . 4 7 8 7 6 9 E + 01
.7284396+01
- . 728351E+01
.7678446+01
-.977933E+01
.8097126+01
- . I 2431 I E+0 2
.8515796+01
I 5 08 2 9 E + 02
. 8 9 5 9 1 OE+01
- . I 78907E+02
.
. 9 4 0 2 4 0 E+01
- . 206985E+02
.9870346+01
2 3 6 6 2 3 E + 02
85
86 .
87
88
89 .
90
91 .
92
93
94
95
96
97
.2174466+02
- . 490 83 3 E - 0 1
.1123856-01
. 71 5 6 0 3 6 - 01
. 1469636+00
. T3 9 5 6 5 6 + 0 0 .
-.1835236-01
-.1442546+00
- . 104385 6+00
- . 64 5 1 6 4 6 - 0 1
- . 5565686-0 I
- . 5204086-01
- . 4 7 9 7 2 9 6 - 01
- . 4-3 9 0 4 9 6 - 0 1
- . 3938496-0 I
-.3486496-01
.5898276-01
. 2569486+00
.3879776+00
. 232 51 4 6 + 0 0
.6409536-01
. 5939776-01
.1645356+00.
. 18931 2 6 + 0 0
.1680256+00. 14 6 73 9 6 + 0 0
.1240336+00
' . 101 32 7 E+ OC
.7720216-01
. 5307726-01
. 275 3 3 2 6 - 0 I
.1989216-02
- i 24 9 73 9. 6 - O I
RATI NG- CURVE
1
2
3
4
5
6
.
7
8
9
.10
■
11
12 ■
13
14
15
16
17
18
19
20
21 •
22
23
24
25
26
27
28
29
30
31
32
Q
119
2 662 6 1 E+ 02
1 6 79 6 0 E +0 2
I 0 8 1 6 7 E + 01
9 378 0 0 E+00
7 8 7 0 8 1 E+00
6 36362E+00
4 78792E+00
321222E+00
I 5 6 8 0 2 E +0 0
. 7 6 1 9 I 6E - 0 2
. 1 7 8 8 9 1 E+00
. I 1 6 3 1 2E + 01
. 310264E+01
. 5 0 4 2 I 6E + 0 I
. 7 0 5 6 2 8 E + 01
. 9 0 70 4 0 E + 0 I .
. I I 1 5 9 1 E + 02
- . 5 79 79 8 E +0 T
-.361247E+02
- . 6 6 4 5 1 3E+02
-.709898E+02
- . 5 20 48 7E +0 2
-.324763E+02
- . I 2 90 3 9 E + 0 2
.729996E+01
. I 90207E+02
.165269E+02
. 1 4 0 3 3 1 E+02
. I 14638E+02
. 8 8 94 4 6 E +0 I
. 6 2 4 9 5 6 E + 01
. 3 6 0 4 6 6 E + 01
. 8841 9 3 E + 0 0
. I 198 9 6 E +02
. 3 1 4 3 5 1 E+02
. 508805E+02
. 557764E+02
.399437E+02
.297665E+02
. 4 3 1 1 90E +02
. 5 6 8 1 39E+02
- . I 0 8 8 2 4 E +03
- . 707507E+03
- . I 3 0 6 1 9E + 04
- . 1 91947E+04
- . 2 5 3 2 7 6 E +04
.
. 1 0 3 3 8 3 E+02
. 7 2 1 348E+01
. 267432E+01
. 3 8 4 1 51E+01
. 506428E+01
. 6 2 8 7 0 5 E+ 0 1
. 7 5 6 5 4 1 E+01
. 884376E+01
. I 0.1 7 7 7 E+ 0 2
. I I 5 I I 6 E +0 2
. I 2 9 0 1 1 E+ 0 2
. I 28073E+02
. I I 0 3 8 5 E+02
. 9 2 6 9 73 E +01
. . 7 4 3 2 9 0 E + 01
■ 5 5. 9608E + 01
. 3691Z2E+01
. 6 9 6 7 1 8E+01
. 1 38322E+02
. 2 0 6 9 7 1 E+02
.219970E+02
. 1 82201E+02
.143173E+02.
.104145E+02
. 6 3 8 5 8 6 E +01
.376349E+01
. 3 4 7 6 0 8E+01
. 3 1 8 8 6 7E+01
. 289256E+01
. 2 5 9 6 4 4 E +0 I
■ . 2 2 9 1 6 1 E+01
. 1 98679E+01
. 1 67325E+01
. 3 I 4 1 4 9E +0 I
. 568554E+01
.822960E+01
.877772E+01
. 6 4 8 6 8 6 E +01
. 518734E+01
. 7 9 6 2 84 E + 0 1
. I 0 8 0 9 5 E+ 0 2
.551594E+02
. I 99880E+03
. 344600E+03
.492850E+03
. 6 4 1 1 0 0 E +03
51 9 3 7 0 E- 0 1
33
- . 384399E-0I
34
35
5 2 8 0 4 4 E- 0 2
195932E-01
36
37
345876E-01
—. 4 9 5 8 1 9 E - 01
38
—. 65 2 5 7 9 E - 01
39
-» 809338E-01
40
41
- . 9 7 2 9 1 4E-01
42
11 3 6 4 9 E + 0 0
- . 130688E+00
43
44
1 2 9 6 4 9E+ OC
- . 108200E+00
45
- . 8 6 7 51 2 E - 0 I
46
47
64 4 77 5 E- 0 I
48
- . 4 2 2 0 3 7 E - 01
49
- . 191 0 5 0 E - 0 1
- . 497944E-0 I
50
- . 1 1 7628E + 00
51
- . 1 8 5 4 6 1 E+OG'
52
- . 1 9 7 4 4 5 E+ 00
53
- . 1 5 8 5 1 4 E+00
54
- . 11 8 2 8 5 E+OG
55
- . 78 0 55 9E - 0 1
56
57
- . 365293E-01
- . 100859E-01
58
- . 8 7 4 0 9 1 E - 02
59
- . 739592E-02
60
- . 6 0 1 0 1 6 E- 0 2
61
—. 46 2 4 4 1 E ^ O 2
62
- . 31 9 79 0 E- 0 2
63
64
- . 177138E-02
- . 304 1 I 0 E - 0 3
65
- . 1 53 54 I E - O I
66
67
- . 4 0 4 I 7 4 E-Ol
-.654807E-01
68
69
- . 71 2 89 9 E - 0 1
—. 4 9 6 9 3 3 E - 0 1
70
- 71
- . 3 7 5 9 9 2 E - 01
-.645299E-01 .
72
- . 9 2 1 513E-01
73
74
- . 180467E + 0 1
- . 7 5 8 9 7 2 E + 0.1
75
76
- . 13 3 74 SE + 0 2
77
193010E+02
- . 25 2 2 7 2 E+ 0 2
78
I 20
. 3 1 6 0 6 5 E +04
.2051452+04
.5708312+03
. I 6 7 6 0 9 2 + 03
.2902912+03
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79
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M ONTANA STA TE
U N IV E R S IT Y - B O Z E M A N
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