Transcript ESTABLISHING STAGE-DISCHARGE RELATION

ESTABLISHING STAGE-DISCHARGE
RELATION (1)
• WHY A STAGE-DISCHARGE RELATION?
– FLOW IS THE VARIABLE OFTEN REQUIRED FOR
HYDROLOGICAL ANALYSIS
– CONTINUOUS MEASUREMENT OF FLOW
USUALLY IMPRACTICAL OR PROHIBITIVELY
EXPENSIVE
– STAGE OBSERVATIONS CONTINUOUSLY OR AT
REGULAR SHORT TIME INTERVALS
– STAGE OBSERVATION COMPARATIVELY EASY
AND ECONOMICAL
– RELATION BETWEEN STAGE AND DISCHARGE
CAN BE ESTABLISHED:
* THE DISCHARGE RATING CURVE
OHS - 1
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ESTABLISHING STAGE-DISCHARGE
RELATION (2)
GENERAL:
– RATING CURVE ESTABLISHED BY
CONCURRENT MEASUREMENTS OF STAGE h
AND DISCHARGE Q COVERING EXPECTED
RANGE OF RIVER STAGES AT SECTION OVER A
PERIOD OF TIME
– IF Q-h RATING CURVE NOT UNIQUE, THEN
ADDITIONAL INFORMATION REQUIRED ON:
* SLOPE OF WATER LEVEL (BACKWATER)
* HYDROGRAPH h(t) (UNSTEADY FLOW)
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– Q-h EXTRAPOLATION MAY BE REQUIRED TO
COVER FULL RANGE OF STAGES
– RATING EQUATION IS USED TO TRANSFORM
h(t) INTO Q(t)
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Stage-Discharge Relationship
599
598
597
596
595
594
200
OHS - 3
400
600
800
1,000
1,200
Discharge (cumecs)
1,400
1,600
1,800
2,000
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Analysis of stage-discharge data
Station name : CHASKMAN
Data from 1997 1 1 to 1997 12 30
Single channel
Gauge Zero on 1997 7 29 =
.000 m
Number of data = 91
Power type of equation q=c*(h+a)**b is used
Boundaries / coefficients
lower bound upper bound
594.00
595.19
595.95
595.19
595.95
600.00
a
b
-592.170
-593.866
-594.025
9.709
2.770
2.531
c
.7147E-03
.1507E+02
.2263E+02
Number
W level
M
Q meas
M3/S
Q comp
M3/S
DIFf
M3/S
Rel.dIFf
0/0
Semr
0/0
1
2
3
4
5
6
594.800
595.370
596.060
596.510
598.080
597.700
9.530
36.480
127.820
231.400
738.850
583.340
8.541
46.661
136.679
226.659
783.019
610.359
.989
-10.181
-8.859
4.741
-44.169
-27.019
11.58
-21.82
-6.48
2.09
-5.64
-4.43
3.75
2.12
2.90
2.06
3.63
3.03
Overall standard error =
5.904
Statistics per interval
Interval Lower bound Upper bound Nr.of data Standard error
1
594.000
595.192
38
7.20
2
595.192
595.950
27
5.24
3
595.950
600.000
26
4.84
OHS - 4
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THE STATION CONTROL
GENERAL:
– THE SHAPE, RELIABILITY AND STABILITY OF
THE Q-h RELATION ARE CONTROLLED BY A
SECTION OR REACH OF CHANNEL AT AND/OR
D/S OF GAUGING STATION = STATION
CONTROL
– ESTABLISHMENT OF Q-h RELATION REQUIRES
UNDERSTANDING OF NATURE AND TYPE OF
CONTROL AT A PARTICULAR STATION
– ESTABLISHING A Q-h RELATION IS NOT SIMPLY
CURVE FITTING
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TYPES OF STATION CONTROLS
• CHARACTER OF RATING CURVE DEPENDS
ON TYPE OF CONTROL, GOVERNED BY:
– GEOMETRY OF THE CROSS-SECTION
– PHYSICAL FEATURES OF THE RIVER D/S
• STATION CONTROLS CLASSIFIED IN MANY
WAYS:
– SECTION and CHANNEL CONTROLS
– NATURAL and ARTIFICIAL CONTROLS
– COMPLETE, COMPOUND and PARTIAL
CONTROLS
– PERMANENT and SHIFTING CONTROLS
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CONTROL CONFIGURATION IN
NATURAL CHANNEL
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SECTION CONTROL
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CHANNEL CONTROL (1)
OHS - 9
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FITTING RATING CURVES (3)
MAIN CASES:
– SIMPLE RATING CURVE
* SINGLE CHANNEL
* COMPOUND CHANNEL
– RATING CURVE WITH BACKWATER
CORRECTION:
* NORMAL FALL
* CONSTANT FALL
– RATING CURVE WITH UNSTEADY FLOW
CORRECTION
– RATING CURVE WITH SHIFT ADJUSTMENT
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FITTING SINGLE CHANNEL SIMPLE
RATING CURVE (1)
TO BE CONSIDERED:
–
–
–
–
–
EQUATIONS USED
PHYSICAL BASIS EQUATION PARAMETERS
DETERMINATION OF DATUM CORRECTION
NUMBER AND RANGE OF RATING SEGMENTS
DETERMINATION OF RATING CURVE
COEFFICIENTS
– ESTIMATION OF UNCERTAINTY IN RATING
CURVE
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FITTING SINGLE CHANNEL SIMPLE
RATING CURVE (2)
• EQUATIONS:
– PARABOLIC TYPE:
Q = c2(h + a)2 + c1(h + a) +c0
– POWER TYPE:
Q = c(h + a)b
log Q = log c + b log(h + a),
Y
= A + BX
OHS - 24
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FITTING OF SINGLE CHANNEL SIMPLE
RATING CURVE (3)
RELATION BETWEEN POWER TYPE RATING
CURVE AND MANNING EQUATION
MANNING:
Q = KmAR2/3S1/2
FOR RECTANGULAR X-SECTION:
A = B.H
RH
MANNING: Q  KmBS1/2.H5/3
POWER:
Q = c(h + a)b
OHS - 25
SO: c = KmBS 1/2
h+a=H
and
b = 5/3
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FITTING OF SINGLE CHANNEL SIMPLE
RATING CURVE (4)
• POWER b IN POWER TYPE RATING CURVE
VARIES WITH SHAPE OF CROSS-SECTION:
–
–
–
–
–
OHS - 26
RECTANGULAR:
TRIANGULAR:
PARABOLIC:
IRREGULAR:
COMPOUND:
b = 1.7
b = 2.5
b = 2.0
1.2 <b<3 (TYPICALLY)
b>5
(
,,
)
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FITTING OF SINGLE CHANNEL SIMPLE
RATING CURVE (5)
DATUM CORRECTION a:
Q = c(h + a)b
so: Q = 0 for a = - h
METHODS TO DETERMINE a:
– TRIAL AND ERROR
– ARITHMETIC PROCEDURE
– COMPUTER-BASED OPTIMISATION
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FITTING OF RATING CURVES IN
HYMOS
FOLLOWING STEPS ARE REQUIRED:
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* SELECT THE REQUIRED PERIOD AND STATION
* CHECK THE MAXIMUM RANGE OF WATER LEVELS IN
THE TIME PERIOD
* INSPECT THE AVAILABLE STAGE DISCHARGE DATA
TOGETHER WITH A REPRESENTATIVE CROSS-SECTION
OF THE CONTROL
* IDENTIFY THE BREAKS IN THE SCATTER PLOT
* ELIMINATE OUTLIERS IF UNRELIABLE (MIND OTHER
REASONS FOR SCATTER!!!)
* SELECT EQUATION TYPE AND ‘a’ FORCED OR FREE
* SELECT THE INTERVALS WITH OVERLAPS TO FORCE
INTERSECTIONS
* INSPECT THE PLOT AND THE TABULAR OUTPUT
* REPEAT IF RESULT IS UNSATISFACTORY
* SAVE THE CURVE PARAMETERS IF ACCEPTABLE
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COMPOUND CHANNEL RATING CURVE
(1)
hf
hr
Br
B
Qriver = (hrBr)(Kmrh2/3S1/2
and
Qfp = hf(B-Br)(Kmf hf 2/3S1/2
Qtotal = Qriver + Qfp
OHS - 45
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COMPOUND CHANNEL RATING CURVE
(2)
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COMPOUND CHANNEL RATING CURVE
(3)
• COMPUTATIONAL PROCEDURE (1):
– FIRST THE RATING CURVE IS FITTED FOR THE
MAIN CHANNEL UP TO BANKFULL LEVEL
– THIS CURVE IS EXTENDED TO RIVER STAGES
ABOVE BANKFULL LEVEL = Qr
– ABOVE BANKFULL LEVEL:
OBSERVED FLOWS Qobs ARE CORRECTED FOR
MAINCHANNEL FLOW Qr TO OBTAIN FLOOD
PLAIN FLOW ONLY = Qf:
Qf = Qobs - Qr
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COMPOUND CHANNEL RATING CURVE
(4)
COMPUTATIONAL PROCEDURE (2):
– LAST WATER LEVEL RANGE IS USED TO FIT
THE CURVE FOR THE FLOOD PLAIN FLOW Qf
ALONE
HENCE:
– h < BANKFULL:
Q = c1(h + a1)b1
– h  BANKFULL
Q = c1(h + a1)b1 + c2(h + a2)b2
OHS - 48
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Rating curve (compound channel)
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45
0
50
100
150
200
250
300
350
400
450
500
550
600
Discharge (m3/s)
Rating Curve
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Measurements
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RATING CURVE WITH BACKWATER
CORRECTION
NO UNIQUE STAGE-DISCHARGE CURVE
WHEN
STATION CONTROL IS AFFECTED BY OTHER
CONTROLS DOWNSTREAM
CAUSES:
* FLOW REGULATION D/S
* LEVEL IN MAIN RIVER OR TRIBUTARY AT
CONFLUENCE
* WATER LEVEL IN RESERVOIR D/S
* VARIABLE TIDAL EFFECT
* D/S CONSTRICTION WITH VARIABLE CAPACITY DUE
TO WEED GROWTH
* RIVERS WITH RETURN OF OVERBANK FLOW
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BACKWATER EFFECT
hx
h0
Lx
S
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CHANNEL CONTROL
EXTENT OF CHANNEL CONTROL:
– FIRST ORDER APPROXIMATION OF
BACKWATER EFFECT (rectangular channel):
at x = 0:
h0 = he + h0
at x = Lx:
hx = he +hx
hx = h0.exp[(-3.S.Lx)/(he(1-Fr2)]
Fr2 = u2/(gh) often << 1
Q = KmBhe5/3S1/2
he = {q/(KmS1/2)}3/5
ln(hx/h0) = -3.S.Lx/he
at: hx/h0 = 0.05:
Lx = he/S
Backwater:
Froude:
Manning:
So with q = Q/B:
OHS - 52
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BACKWATER
VARIABLE BACKWATER:
CAUSES VARIABLE ENERGY SLOPE FOR THE
SAME STAGE
HENCE:
DISCHARGE IS A FUNCTION OF BOTH STAGE
AND OF SLOPE:
SLOPE-STAGE-DISCHARGE RELATION
GENERALLY:
OHS - 53
ENERGY SLOPE APPROXIMATED BY WATER
LEVEL SLOPE
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BACKWATER CORRECTION (1)
Q  Km R
Qm
Qr
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 Sm 
  
 Sr 
p
2/ 3
S
1/ 2
• FALL BETWEEN MAIN
AND AUXILIARY
STATION TAKEN AS
MEASURE FOR
SURFACE SLOPE
A
 Fm 
  
 Fr 
p
•
•
•
•
•
m = MEASURED
r = REFERENCE
S = SLOPE
F = FALL
VALUE OF POWER P
THEORETICALLY 0.5
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BACKWATER CORRECTION (2)
• TWO PROCEDURES FOR BACKWATER
CORRECTION:
– CONSTANT FALL METHOD
* STAGE-DISCHARGE RELATION IS AFFECTED BY
BACKWATER AT ALL TIMES
– NORMAL (OR LIMITING) FALL METHOD
* STAGE-DISCHARGE AFFECTED ONLY WHEN THE
FALL REDUCES BELOW A GIVEN VALUE
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CONSTANT FALL METHOD
• MANUAL PROCEDURE
– SELECT AN AVERAGE FALL, CALLED THE
REFERENCE FALL Fr
– CREATE A RATING CURVE h-Qr WHERE:
Qr = Q/(Fm/Fr)
– CREATE A SECOND RELATION FOR
Qm/Qr = f(Fm/Fr)
– USE SECOND RELATION TO UPDATE Qr AND
THE STAGE-DISCHARGE RELATION h-Qr, etc.
• USE:
Q = Qr(Fm/Fr)p
OHS - 56
with Fm from observations
Fr from procedure
Qr from rating curve
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CONSTANT FALL METHOD
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CONSTANT FALL RATING
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CONSTANT FALL COMPUTATIONAL
PROCEDURE
FITTING:
Qr
Q
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 Fr 
 Qm  
 Fm 
p
 Fm 
 Qr  
 Fr 
p
– FIRST A REFERENCE
FALL IS SELECTED
– A RATING CURVE IS
FITTED BETWEEN h
AND Qr
– VALUE OF p IS
OPTIMISED
USE:
– FOR GIVEN h AND FALL
Fm, Qr AND Fr FROM
THE STORED
INFORMATION
– DISCHARGE FROM
SECOND RELATION
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CONSTANT FALL METHOD WITH
HYMOS (1)
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CONSTANT FALL METHOD WITH
HYMOS
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION (1)
MANUAL PROCEDURE:
– PLOT STAGE AGAINST DISCHARGE AND MARK
THE BACKWATER FREE MEASUREMENTS
– FIT A RATING CURVE FOR THE BACKWATER
FREE MEASUREMENTS: Qr-h RELATION
– PLOT FALL VERSUS STAGE AND DRAW A LINE
FOR THE NORMAL OR LIMITING FALL Fr
– COMPUTE Qm/Qr AND Fm/Fr FOR EACH
OBSERVATION AND DRAW AVERAGE CURVE
– ADJUST THE CURVES BY HOLDING TWO
CONSTANT AND PLOTTING THE THIRD, ETC.
OHS - 62
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION (2)
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION (3)
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION (4)
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION (5)
USE OF THE PROCEDURE WITH h AND Fm
GIVEN:
–
–
–
–
–
OHS - 66
READ Fr FROM Fr - h CURVE
CALCULATE Fm/Fr
READ Q/Qr FROM Qm/Qr - Fm/Fr RELATION
READ Qr FROM Qr - h RELATIONSHIP
MULTIPLY Q/Qr WITH Qr TO COMPUTE Q
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NORMAL FALL METHOD FOR
BACKWATER CORRECTION
• COMPUTATIONAL
PROCEDURE:
1/ p
Fr
 Qr 

 Fm 
 Qm 
Fr
 a  b h  c h2
– COMPUTE BACKWATER
FREE RATING CURVE
– DERIVE Fr FROM Fm,
Qm AND Qr
– FIT PARABOLA TO Fr - h
DATA
– OPTIMISE PAR. p
• USE:
Q
OHS - 67
 Fm 
 Qr  
 Fr 
p
– WITH ABOVE
REATIONS FOR Qr-h
AND Fr-h APPLY LAST
EQUATION
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RATING CURVE WITH UNSTEADY
FLOW CORRECTION (1)
• NOTE:
– WATER SURFACE SLOPE ON FRONT SIDE OF
FLOOD WAVE STEEPER THAN ON BACK SIDE
– DISCHARGE PROPORTIONAL WITH ROOT OF
SLOPE
• HENCE:
– FOR THE SAME STAGE, THE DISCHARGE IS
LARGER FOR RISING STAGES THAN FOR
FALLING STAGES
– RATING CURVE HAS TO BE ADJUSTED TO
ACCOMMODATE FOR THESE EFFECTS
OHS - 68
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RATING CURVE WITH UNSTEADY
FLOW CORRECTION (2)
Qm
1
c S0
1
c S0
OHS - 69
 Qr

1 

1 dh 

c S 0 dt 
(Qm / Qr ) 2  1

dh / dt
 a  b h  ch2
Qm = measured discharge
Qr = steady state discharge
c = flood wave celerity
S0 = bed slope (energy slope
for steady flow)
dh/dt = change of h per unit of
time
Procedure:
– trial Qr - h relation is
established from
measurements where
dh/dt = 0
– compute 1/cS0 and fit a
relation for 1/cS0 = f(h)
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RATING CURVE WITH UNSTEADY
FLOW CORRECTION (3)
• CORRECTION REQUIRED IF FACTOR
(1+1/cS0.h/t)1/2 < 0.95 OR >1.05
• CORRECTION FACTOR HIGH WHEN:
– BED SLOPE IS SMALL
– CELERITY IS SMALL
– h/t IS LARGE
• USE:
–
–
–
–
OHS - 70
OBTAIN Qr VIA Qr-h FROM OBSERVED h
OBTAIN 1/cS0 VIA 1/cS0-h FROM OBSERVED h
OBTAIN h/t FROM HYDROGRPAH
APPLY JONES FORMULA TO COMPUTE ACTUAL
(UNSTEADY) FLOW
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EXAMPLE UNSTEADY FLOW
CORRECTION(1)
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EXAMPLE UNSTEADY FLOW
CORRECTION(2)
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EXAMPLE UNSTEADY FLOW
CORRECTION(3)
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EXAMPLE UNSTEADY FLOW
CORRECTION(4)
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EXAMPLE UNSTEADY FLOW
CORRECTION(5)
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UNSTEADY FLOW WITH HYMOS
(BEFORE CORRECTION)
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UNSTEADY FLOW WITH HYMOS
(WITH CORRECTION)
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SHIFTING CONTROL (1)
• CONSIDERATION:
– A STABLE CONTROL IS A DESIRABLE
PROPERTY OF A GAUGING STATION
– ALLUVIAL STREAM-BEDS ARE NOT STABLE
DUE TO SILTATION AND SCOUR (MOVING
DUNES AND BARS)
– AS A CONSEQUENCE THE STAGE-DISCHARGE
RELATION WILL VARY
– EXTENT AND FREQUENCY OF VARIATION
DEPENDS ON:
* TYPICAL BED MATERIAL SIZE
* FLOW VELOCITIES
OHS - 78
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OHS - 79
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SHIFTING CONTROL (3)
INDETERMINATE Q-h
OHS - 80
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SHIFTING CONTROL (4)
ALTERNATIVE: u-R PLOT
OHS - 81
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SHIFTING CONTROL (5)
APPROACHES
• FOUR POSSIBLE APPROACHES:
– FITTING A SIMPLE RATING CURVE BETWEEN
SCOUR EVENTS
– VARYING THE SHIFT PARAMETER
– APPLICATION OF STOUT’S SHIFT METHOD
– FLOW DETERMINED FROM DAILY GAUGING
OHS - 82
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SHIFTING CONTROL (6)
SIMPLE RATING BETWEEN EVENTS
• USE:
– WHERE RATING SHOWS LONG PERIOD OF
STABILITY
– WHERE SUFFICIENT GAUGINGS PER PERIOD
ARE AVAILABLE
– WHERE SHIFTS IN RATING ARE EASILY
IDENTIFIABLE:
* PLOT DATA WITH DATE
* FLOOD EVENTS CAUSE CHANGE
* NOTES IN THE FIELD RECORD BOOK ON REASONS
FOR SHIFT
OHS - 83
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SHIFTING CONTROL(7)
VARYING SHIFT PARAMETER
Q=c1(h+a1
)b1
n
a (hr hm )/n
i 1
Q = c1(h+a1+a)b1
OHS - 84
• USE:
– WHERE RATING SHOWS
PERIODS OF STABILITY
BUT INSUFFICIENT
DATA ARE AVAILABLE
FOR NEW RATING
– THEN PARAMETER “a”
IS ADJUSTED AS
SHOWN LEFT:
hr = rated h for Qm
hm = observed stage for
Qm
CHECK APPLICABILITY
OF a FOR FULL OR
PARTIAL RANGE OF h
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SHIFTING CONTROL (8)
STOUT’s METHOD (1)
PROCEDURE:
hr =(Qm/c)1/b - a
h = hr - hm
ht = f(hi, hj)
Qt = c1(ht+ht+a1)b1
OHS - 85
– FIT A MEAN RELATION
FOR ALL POINTS IN
PERIOD
– DETERMINE hr FROM Qm
– DETERMINE h FOR
INDIVIDUAL MEAS.
– DETERMINE ht BY
LINEAR INTERPOLATION
BETWEEN h’s
– ht ARE USED TO
CORRECT RATING
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SHIFTING CONTROL(9)
STOUT’s METHOD (2)
OHS - 86
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SHIFTING CONTROL (10)
STOUT’s METHOD (3)
• WHEN:
– GAUGING IS FREQUENT
– MEAN RATING IS REVISED PERIODICALLY
– IF PREVIOUS METHODS DO NOT APPLY
• ASSUMPTION:
– SHIFTS GRADUAL CHANGES IN RATING
• DRAWBACK:
– ERRORS IN MEASUREMENT ARE MIXED
DEVIATIONS DUE TO SHIFTS IN CONTROL
– INDIVIDUAL MEASUREMENT ERRORS HAVE
SEVERE CONSEQUENCES DIFFERENT FROM
ORDINARY RATING CURVE
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OHS - 88
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OHS - 89
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SHIFTING CONTROL (11)
DAILY GAUGING
• WHEN:
– IF BROAD SCATTER IS AVAILABLE NEITHER
FROM BACKWATER NOR FROM SCOUR
– CALCULATED SHIFT IS ERRATIC
– HENCE WHEN NON OF OTHER PROCEDURES
APPLY
• NOTE:
– IMPORTANT PARTS OF THE HYDROGRAPH MAY
BE MISSED
– BETTER TO RELOCATE THE STATION UNLESS
URGENT NEED
OHS - 91
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