The Stage-Discharge Rating
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Transcript of The Stage-Discharge Rating
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The Stage-Discharge RatingThe Stage-Discharge Rating
D. Phil Turnipseed, P.E.D. Phil Turnipseed, P.E.HydrologistHydrologist
USGS-FERC Streamgaging SeminarUSGS-FERC Streamgaging SeminarWashington, D.C.Washington, D.C.
June 6-7, 2006June 6-7, 2006
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Ratings developed by making discharge Ratings developed by making discharge measurementsmeasurements
1
10
100
1 10 100 1000 10000 100000Discharge (cfs)
Sta
ge
(ft)
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A straight line on rectilinear paper is of the form:A straight line on rectilinear paper is of the form:
y = mx + by = mx + bwhere: m = slope of line where: m = slope of line and: b is the y interceptand: b is the y intercept
y = mx + b, or y = 0.5x +5
0
5
10
15
20
25
30
-20 -10 0 10 20 30 40 50
x
y
y
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Logarithmic Coordinate SystemLogarithmic Coordinate System
• Many hydraulic relations are Many hydraulic relations are linearlinear in in log formlog form
• Examples include:Examples include:– Discharge equations for weirsDischarge equations for weirs– Open-channel flow equations, with Open-channel flow equations, with
simplifying assumptionssimplifying assumptions
• This means This means SEGMENTSSEGMENTS of ratings may of ratings may be be linearlinear in log space in log space
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Stage-Discharge Relations for Stage-Discharge Relations for Artificial ControlsArtificial Controls
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Equations commonly used to relate water discharge to Equations commonly used to relate water discharge to hydraulic head (h)hydraulic head (h)
RECTANGULAR WEIRRECTANGULAR WEIR
Q = C B hQ = C B h 1.5
where:where:
C = a discharge coefficientC = a discharge coefficient
B = top width of weir or lengthB = top width of weir or length of weir crest normal to flow of weir crest normal to flow
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Equations commonly used to relate water discharge to Equations commonly used to relate water discharge to hydraulic head (h)hydraulic head (h)
V-NOTCH WEIR (90 degrees)V-NOTCH WEIR (90 degrees)
Q = 2.5 hQ = 2.5 h 2.52.5
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Relation between water discharge (Q) and Head (h) for Relation between water discharge (Q) and Head (h) for a v-notch weir a v-notch weir
(pzf at gage height = 0.0)(pzf at gage height = 0.0)
0
4
1
2
3
Water SurfaceWater Surface
h
Q = 2.5 hQ = 2.5 h 2.52.5
Gag
e H
eigh
tG
age
Hei
ght 2.502.50
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RATING CURVE FOR A V-NOTCH WEIRRATING CURVE FOR A V-NOTCH WEIR(PZF at GH = 0.0, therefore h = ght)(PZF at GH = 0.0, therefore h = ght)
0.1
10
0.01 0.1 1 10 100 1000
Q = 2.5h 2.5
Gag
e H
eigh
t (e
=0)
Discharge
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Relation between water discharge (Q) and head (h) for a v-notch Relation between water discharge (Q) and head (h) for a v-notch
weir (pzf at gage height = 1.0)weir (pzf at gage height = 1.0)
44
33
22
11
55
00
Q = 2.5 hQ = 2.5 h 2.52.5
3.503.50
h = GH - eh = GH - e
hh
eeWill be scaleWill be scaleoffsetoffset
Water SurfaceWater Surface
Gag
e H
eigh
tG
age
Hei
ght
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0.1
10
0.1 1 10 100 1000
Rating if offset used(Head plotted againstdischarge)
Rating for a V-notch weir when PZF = 1.0 ft.Rating for a V-notch weir when PZF = 1.0 ft.G
age
heig
ht
Discharge
Rating if no offset used (Gage heightplotted against discharge)
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Example of relation between PZFExample of relation between PZFand gage heightand gage height
Gage Height
PZF
Head = GH - PZFor about 0.37 (2.55 - 2.18)
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Measuring Point of Zero FlowMeasuring Point of Zero Flow
Gage PoolGage Pool
Include Velocity HeadInclude Velocity Head
Deepest Point on Control
Control Section Perpendicular to FlowControl Section Perpendicular to Flow
Control SectionControl Section
Gage PoolGage Pool
FlowFlow
FlowFlow
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Stage-Discharge Relations for Stage-Discharge Relations for Natural ControlsNatural Controls
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Section ControlsSection Controls
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Section ControlsSection Controls
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Common equations used to relate water Common equations used to relate water discharge to channel conditionsdischarge to channel conditions
Section ControlSection Control
Q = a(GH-e)Q = a(GH-e)bb
where:where:a = coefficienta = coefficientb = slope of the relationsb = slope of the relations(b is almost always greater than 2)(b is almost always greater than 2)
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Rating curve shapesRating curve shapes
<2
1>2
1
>21
SectionControl
ChannelControl
Overbank
Gh
- e
Section ControlSection ControlQ = a(GH-e)Q = a(GH-e)bb
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Channel ControlsChannel Controls
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Common equations used to relate water Common equations used to relate water discharge to channel conditionsdischarge to channel conditions
Channel ControlChannel Control
Q = Q = 1.49 1.49 A R A R 2/32/3 S S 1/21/2 n n
Where:Where:A = cross section areaA = cross section areaR = hydraulic radius (area/wetted perimeter)R = hydraulic radius (area/wetted perimeter)S = energy slopeS = energy slopen = Manning’s “n” (roughness coefficient)n = Manning’s “n” (roughness coefficient)
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Rating curve shapesRating curve shapes
<2
1>2
1
>21
SectionControl
ChannelControl
Overbank
Gh
- e
Gh
- e
Channel Control Q = CD 1.67
(Manning’s Eq.)
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Different Controls, Same SiteDifferent Controls, Same Site
Channel control orpartial channel control
Section control
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Rating curve shapesRating curve shapes
<2
1>2
1
>21
SectionControl
ChannelControl
Overbank
Gh
- e
Gh
- e
Overbank ControlOverbank Control Q = CD Q = CD (>2)(>2)
(Manning’s Eq.)(Manning’s Eq.)
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Open-Channel Flow: Open-Channel Flow:
• Types of FlowTypes of Flow
• States of FlowStates of Flow
• Regimes of FlowRegimes of Flow
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Open-Channel Flow: Open-Channel Flow:
• Types of FlowTypes of Flow• States of FlowStates of Flow• Regimes of FlowRegimes of Flow• Basic equationsBasic equations
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Temporal flow classifications Temporal flow classifications
Depth and velocity areDepth and velocity are constantconstant with timewith time
SteadySteady UnsteadyUnsteady
Depth and velocity changechange with time
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Spatial flow classifications Spatial flow classifications
ConstantConstant depth and velocitydepth and velocity along the channel lengthalong the channel length
UniformUniform VariedVaried
ChangingChanging depth and velocity along the channel length
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Spatial flow classifications Spatial flow classifications
water-surface slope = channel water-surface slope = channel
SlopeSlope SSww = S = Soo
UniformUniform Gradually VariedGradually Varied
water-surface slope = friction water-surface slope = friction
SlopeSlope SSww = S = Sf f
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Gradually Varied FlowGradually Varied Flow
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Flow-Classification Flow-Classification Summary: Summary:
A.A. Steady flowSteady flow1.1. Uniform flowUniform flow2.2. Varied flowVaried flow
a)a) Gradually varied flowGradually varied flowb)b) Rapidly varied flowRapidly varied flow
B.B. Unsteady flowUnsteady flow1.1. Unsteady uniform flow (rare)Unsteady uniform flow (rare)2.2. Unsteady flow (i.e., unsteady varied flow)Unsteady flow (i.e., unsteady varied flow)
a)a) Gradually varied unsteady flowGradually varied unsteady flowb)b) Rapidly varied unsteady flowRapidly varied unsteady flow
From Chow, 1959
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Open-Channel Flow: Open-Channel Flow:
• Types of FlowTypes of Flow
• States of FlowStates of Flow• Regimes of FlowRegimes of Flow
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State of Flow: State of Flow:
• State of flow governed by effects of viscosity State of flow governed by effects of viscosity and gravity relative to the inertial forces of the and gravity relative to the inertial forces of the flowflow
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States of Flow: States of Flow:
• Viscosity vs. inertia: Reynold’s NumberViscosity vs. inertia: Reynold’s Number
RR = VL/ = VL/עע
where V = velocity of flowwhere V = velocity of flow L = hydraulic radiusL = hydraulic radiuskinematic viscosity of water= kinematic viscosity of water = עע
• Laminar flow: Laminar flow: RR << 500 500• Turbulent flow: Turbulent flow: RR >> 2000 2000
• Laminar flow rare in open channelsLaminar flow rare in open channels
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States of Flow: States of Flow:
• Gravity vs. inertia: Gravity vs. inertia: Froude NumberFroude Number
F F = V/(gL)= V/(gL)1/21/2
where V = velocity of flowwhere V = velocity of flow L = hydraulic radius (depth)L = hydraulic radius (depth) gg = acceleration of gravity = acceleration of gravity
• FF = 1: V = = 1: V = (gD)(gD)1/21/2 Critical flow Critical flow Equilibrium Equilibrium
• FF < 1: V < < 1: V < (gD)(gD)1/21/2 Sub-critical flow Sub-critical flow Gravity dominates Gravity dominates
• FF > 1: V > > 1: V > (gD)(gD)1/21/2 Super-critical flow Inertia dominates Super-critical flow Inertia dominates
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States of Flow: States of Flow:
• Critical velocity (gD)Critical velocity (gD)1/21/2 known as the “ known as the “wave celeritywave celerity”” – – velocity of a gravity wave generated by a local disturbance invelocity of a gravity wave generated by a local disturbance in
shallow watershallow water
• Ability of a gravity wave to propagate upstream is a criterion for Ability of a gravity wave to propagate upstream is a criterion for identifying sub-critical or super-critical flowidentifying sub-critical or super-critical flow
• Flow in most channels is controlled by gravityFlow in most channels is controlled by gravity Sub-criticalSub-critical
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States of FlowStates of Flow
F < 1.0 F >1.0Sub-critical (tranquil) flowSub-critical (tranquil) flow Supercritical (rapid) flowSupercritical (rapid) flow
critical flowcritical flow
F = 1.0
flow flow
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Open-Channel Flow: Open-Channel Flow:
• Types of FlowTypes of Flow• States of FlowStates of Flow
• Regimes of FlowRegimes of Flow• Basic equationsBasic equations
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Regimes of Flow: Regimes of Flow:
• Combined effect of viscosity and gravity Combined effect of viscosity and gravity 4 regimes of flow4 regimes of flow 1) Sub-critical – laminar: F < 1; R 1) Sub-critical – laminar: F < 1; R << 500 500
2) Super-critical – laminar: F > 1; R 2) Super-critical – laminar: F > 1; R << 500 500
3) Sub-critical – turbulent: F < 1; R 3) Sub-critical – turbulent: F < 1; R >> 2000 2000
4) Super-critical – turbulent: F > 1; R 4) Super-critical – turbulent: F > 1; R >> 2000 2000
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Regimes of Flow: Regimes of Flow:
From Chow, 1959
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Upstream Natural ControlUpstream Natural Control
Upstream control- Flow past gage is supercritical
Upstream view
Downstream view
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Rating and controls, San Francisquito Cr.Rating and controls, San Francisquito Cr.
Offset = 0.0 Low Section Control
0.01
0.1
1
10
100
0.001 0.01 0.1 1 10 100 1000 10000
Discharge
Ga
ge
He
igh
t
PZF = 0.07
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Rating and controls, San Francisquito Cr. Rating and controls, San Francisquito Cr. (cont.)(cont.)
Offset = 1.0 High Section Control
0.1
1
10
100
1 10 100 1000 10000
Discharge
Gag
e H
eig
ht
- o
ffs
et
Measurement at moderate flowG.H. = 5.4
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Rating and Controls, San Francisquito Cr. Rating and Controls, San Francisquito Cr. (cont.)(cont.)
Offset = 3.0 Channel Control
1
10
100
100 1000 10000
Discharge
Gag
e H
eigh
t -
offs
et Overbank flow?
Channel Controlbeginning to dominateat this stage (6.25 feet)
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Rating and controls, San Francisquito Cr. Rating and controls, San Francisquito Cr. (cont.)(cont.)
Offset = 3.0 Channel Control
1
10
100
100 1000 10000
Discharge
Gag
e H
eigh
t -
offs
et Overbank flow?
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Shifting ControlsShifting Controls
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Shifting ControlShifting Control
The non-cohesive streambed in this photo is subject to The non-cohesive streambed in this photo is subject to scour and fill, as well as changing vegetation conditions. scour and fill, as well as changing vegetation conditions.
Unstable Unstable channelchannel
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ShiftsShifts
• Shift is a “temporary rating”Shift is a “temporary rating”
• Generally used for a Generally used for a temporarytemporary change change in the controlin the control– Case 1: Assumes control will move back to Case 1: Assumes control will move back to
the ratingthe rating– Case 2: Control changes so frequently, Case 2: Control changes so frequently,
shifts applied to avoid always making a shifts applied to avoid always making a new rating new rating
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Shift CorrectionsShift Corrections
• Change the shape and/or position of the rating Change the shape and/or position of the rating curvecurve
• Creates a “temporary rating”Creates a “temporary rating”• By timeBy time
– SimpleSimple
• By stageBy stage– Variable shift or V-shift diagramsVariable shift or V-shift diagrams– A better reflection of what actually happens in A better reflection of what actually happens in
streamstream
• Combination of bothCombination of both
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Template for Content SlideTemplate for Content Slide
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Rating Curve Shifts
Discharge
Sta
ge
Positive shift
Plus shift
+
Negative shift
Minus shift
-
Fill or deposition on control,Temporary dams (natural or human-made),
Seasonal vegetative or algal growthDebris piled on control
Base Rating
Shift to the Left
Shift to the Right
Scour at the control,Gravel mining,
Change in channel geometry, (human-induced)Clearing of of debris from control (by flood event or humans)
Often will prorate to shiftfrom the start of a riseto the peak.
Often will prorate to shift on a recession.
Possible causes of shift:
Possible causes of shift:
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Shift Curve Shapes and RatingsShift Curve Shapes and Ratings
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looking at rating for shiftlooking at rating for shift
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0
1
2
3
4
5
6
0 2 4 6 8 10
ADAPS uses up to 3-point “V-diagrams” to document shifts to ADAPS uses up to 3-point “V-diagrams” to document shifts to ratingsratings
Shift Adjustment
0
1
2
3
4
5
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Gag
e H
eigh
tG
age
Hei
ght
Discharge
d
d
c
c
b
a
b
a
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How many shifts do you see?How many shifts do you see?