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

Ratings developed by making discharge Ratings developed by making discharge measurementsmeasurements

1

10

100

1 10 100 1000 10000 100000Discharge (cfs)

Sta

ge

(ft)

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

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

Stage-Discharge Relations for Stage-Discharge Relations for Artificial ControlsArtificial Controls

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

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

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

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

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

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)

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)

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

Stage-Discharge Relations for Stage-Discharge Relations for Natural ControlsNatural Controls

Section ControlsSection Controls

Section ControlsSection Controls

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)

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

Channel ControlsChannel Controls

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)

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.)

Different Controls, Same SiteDifferent Controls, Same Site

Channel control orpartial channel control

Section control

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.)

Open-Channel Flow: Open-Channel Flow:

• Types of FlowTypes of Flow

• States of FlowStates of Flow

• Regimes of FlowRegimes of Flow

Open-Channel Flow: Open-Channel Flow:

• Types of FlowTypes of Flow• States of FlowStates of Flow• Regimes of FlowRegimes of Flow• Basic equationsBasic equations

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

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

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

Gradually Varied FlowGradually Varied Flow

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

Open-Channel Flow: Open-Channel Flow:

• Types of FlowTypes of Flow

• States of FlowStates of Flow• Regimes of FlowRegimes of Flow

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

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

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

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

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

Open-Channel Flow: Open-Channel Flow:

• Types of FlowTypes of Flow• States of FlowStates of Flow

• Regimes of FlowRegimes of Flow• Basic equationsBasic equations

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

Regimes of Flow: Regimes of Flow:

From Chow, 1959

Upstream Natural ControlUpstream Natural Control

Upstream control- Flow past gage is supercritical

Upstream view

Downstream view

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

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

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)

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?

Shifting ControlsShifting Controls

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

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

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

Template for Content SlideTemplate for Content Slide

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:

Shift Curve Shapes and RatingsShift Curve Shapes and Ratings

looking at rating for shiftlooking at rating for shift

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

How many shifts do you see?How many shifts do you see?