Maria @ PR

5.81 sq mi

40.4 sq mi Ppt Stg Qcfs

Fetter, 2001

Freeze & Cherry, 1978

Criss 2003

-1

0

1

2

3

4

5

-1 0 1 2 3 4

Log

Q cfs

Log A m i2

Mean Flowsslope 1:1

Peak Flowsslope 0.57:1

Missouri

updated after Criss 2003

STREAM GAGING: Establish link between Stage S & Discharge Q 1)   THEORETICAL EQUATIONS 2) SEMI-QUANTITATIVE EQUATIONS 3) WEIRS 4)   VELOCITY-AREA METHOD

STREAM GAGING: Establish link between Stage S & Discharge Q 1)   THEORETICAL EQUATIONS 2) SEMI-QUANTITATIVE EQUATIONS 3) WEIRS 4)   VELOCITY-AREA METHOD

THEORY of STEADY LAMINAR FLOW of Newtonian Fluid !

!Channel Flow (slot) !u = (G/2µ)(a2-y2) !

!uavg = Ga2/3µ !

! !Q ~ g s W a3/3ν cm3/sec

!Pipe Flow !u = (G/4µ)(a2-r2) !

! !uavg = Ga2/8µ

! !Q = g s π a4/8ν cm3/sec •  where G= “pressure gradient”, s=slope, 2a = slot depth or tube radius;

W=width µ viscosity; kinematic viscosity ν =µ/ρ cm2/sec!

LAMINAR SLOT FLOW

0

a

a

a

a

0

LAMINAR PIPE FLOW

u =G(a2-y2)/2µ

uavg = Ga2/3µ

u = uavg @ a/√3 = 0.577 down

u =G(a2-r2)/4µ

uavg = Ga2/8µ u = uavg @ a/√2 = 0.707 down

LINEAR RESERVOIR (Chow, 14-27) Storage ∝ Outflow => S = Q/k Also, - dS/dt = Q (material balance requirement) Total flow = Base Flow:

!

Vol = Qdt =Qo0

"

# /k!

Q =Qoe"kt

where Qo is (peak) discharge @ t ≡ 0 For complete depletion, the "Total Potential GW Discharge" is,

0

2

4

6

8

10

0 1 2 3 4 5

Linear Reservoir

Q

Time

Qo =10 k=1

!

Q =Qoe"kt

0

2

4

6

8

10

0 1 2 3 4 5

Linear Reservoir

Q

Time

Qo =10 k=1

!

Q =Qoe"kt

!

Vol = Qdt =Qo0

"

# /k

Note: not linear, but Concave Up

Note: not linear, but Concave Up

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5

Linear Reservoir

-3.2

-2.4

-1.6

-0.8

0

0.8

1.6

2.4

3.2

LogQ

Time

LnQ

!

Q =Qoe"kt

0

1

2

3

4

5

6

7

8 0

0.5

1

54 56 58 60 62 64

Bluegrass Spring

Disc

harg

e (c

fs)

Tyson PPT (in/hr)

YearDay 2001

observed

Q =7.07*Exp{-1.25*(t-tpk)}

Q =1.2*Exp{-0.2083*(t-tpk)}

observed

0

1

2

3

4

5

6

7

8 0

0.5

1

1.5

2

2.5

354 56 58 60 62 64

Bluegrass SpringDi

scha

rge

(cfs

)Tyson PPT (in/hr)

Year Day 2001

QBGS = 7.07* Q (0.35, 56.167, 1)

observed

2) SEMI-QUANTITATIVE EQUATIONS!!

!a. Chezy Equn (1769) U = C Sqrt [RS]! !

!where !! ! !“U” = water velocity !! ! !“C” = discharge coeff. !! ! “R” = hydraulic radius = A/P = cross sectional area/wetted perimeter!! ! !“S” = energy gradient (slope of H2O sfc.)!! ! !!! !! ! ! ! !!

!!

2) SEMI-QUANTITATIVE EQUATIONS!!

!a. Chezy Equn (1769) U = C Sqrt [RS]! !

!where !! ! !“U” = water velocity !! ! !“C” = discharge coeff. !! ! “R” = hydraulic radius = A/P = cross sectional area/wetted perimeter!! ! !“S” = energy gradient (slope of H2O sfc.)!! ! !!! !! ! ! ! !!

!! Units ?

2) SEMI-QUANTITATIVE EQUATIONS!!

!a. Chezy Equn (1769) U = C Sqrt [RS]! !

!where !! ! !“U” = water velocity ft/s or m/s!! ! !“C” = discharge coeff., in units of √g. !! ! “R” = hydraulic radius = A/P = cross sectional area/wetted perimeter (in ft) !! ! !“S” = energy gradient (slope of H2O sfc, dimensionless, e.g. ft/ft)!! ! ! !!! !!! ! ! ! !!

!! Units ? U vs Q ?

2) SEMI-QUANTITATIVE EQUATIONS!!

!a. Chezy Equn (1769) U = C Sqrt [RS]! !

!where !! ! !“C” = discharge coeff., in units of √g. !! ! “R” = hydraulic radius = A/P = cross sectional area/wetted perimeter (in ft) !! ! !“S” = energy gradient (slope of H2O sfc, dimensionless, e.g. ft/ft)!! !! ! ! ! !!

! b. Manning (1889) Equn Uavg = Q/A = (1/n) R2/3 S1/2 m/s note: units!!! where:! !

! “n” = Manning roughness coeff. “n” , in units of sec/m1/3 !! ! ! !n= 0.012 (concrete) !! ! ! !n= 0.05 (rocky mountain stream)! ! Note: !1) 1/n => 1.49/n if use ft, cfs (English units) !!! ! ! !2) Manning eq is not compatible w/ Poiseuille flow!! ! ! ! !as these have different proportionalities!!!! ! ! !3) Manning Eq. is asserted to be the “same” as Chezy Equn! !! ! ! ! with n=3R1/6/2C where C=Chezy coeff. impossible unless n or C depends on scale!!

STREAM GAGING: Establish link between Stage S & Discharge Q 1)   THEORETICAL EQUATIONS 2) SEMI-QUANTITATIVE EQUATIONS 3) WEIRS 4)   VELOCITY-AREA METHOD !!

0

a

a

Laminar Channel Flow (slot) !!

u = (G/2µ)(a2-y2) uavg = Ga2/3µ !

H

WEIRS

3) WEIRS !!Rectangular: !Qcfs = 3.333 ( L - H/5) H3/2 !!!90° V-Notch: !Qcfs = 2.5 H5/2!!

! ! ! ! where H, L in ft. Fetter p. 58 !! !!!!!!!Culvert: !See Chow 15-33; USGS Circ. 376)!!

H H

3) WEIRS !!Rectangular: !Qm3/s = 1.84 ( L - H/5) H3/2 !!!90° V-Notch: !Qm3/s = 1.379 H5/2!!

! ! ! ! where H, L in m. Fetter p. 58 !! !!!!!!!Culvert: !See Chow 15-33; USGS Circ. 376)!!

H

3) WEIRS !!Rectangular: !Qm3/s = 1.84 ( L - H/5) H3/2 !!!90° V-Notch: !Qm3/s = 1.379 H5/2!!

! ! ! ! where H, L in m. Fetter p. 58 !! !

“Note that equations…. are empirical and ! not subject to dimensional analysis” Fetter p. 58!!!!Culvert: !See Chow 15-33; USGS Circ. 376)!!

3) WEIRS !!Rectangular: !Qcfs = 3.333 ( L - H/5) H3/2 !!90° V-Notch: !Qcfs = 2.5 H5/2 Units!!! V-Notch: ! ! ! ! ! !!

! ! ! ! ! ! !Cd= “discharge coeff”;!! ! ! ! ! ! ! Chow 7-46!

!!

!Culvert: !See Chow 15-33; USGS Circ. 376)!!!

!

Qcfs =815Cd 2g tan "

2# $ % & ' ( H 5 /2

H

http://www.hubbardbrook.org

V-Notch Weir http://www.hubbardbrook.org

4) AREA-VELOCITY METHOD!!

Current Meter !! ! !Divide current into 15-30 segments !! ! !Measure velocity @ 0.6*depth of segment (60% down) !

or, if channel is deep, take average v @ 0.8 and 0.2 times the depth.!!

! !Q = Vavg*A !!

! ! !Q = Σqi = Σ vi di wi !!

! ! ! !where: !! ! ! ! !vi = segment velocity !! ! ! ! !di = segment depth !! ! ! ! !wi = segment width !!!

!Rating Curve: Graph of Discharge (cfs) vs. Stage (ft) !! !Use entire river channel as a weir !! !Need to revise curve if channel changes!

!

! !Qcfs = Sa or Qcfs = (S - So)a where So = stage @ zero flow ! !

! !Make polynomial fit !

USGS

0

2 105

4 105

6 105

8 105

1 106

1.2 106

-10 0 10 20 30 40 50

Mississippi River @ St. Louis1980-2006 9,820 pts.Q, cfs

Stage, ft.

Rating Curve:Q = 79340 + 6517.6 S + 241.52 S2

Bob Criss Washington University

THEORETICAL HYDROGRAPH

and its

APPLICATIONS

Fetter, 2001

Freeze & Cherry, 1978

Criss 2003

Meramec River

May 2000 Stage 27.8 ft. Discharge 56,000 cfs

Oct. 2000 Stage 1.8 ft. Discharge 500 cfs

Rockwoods Spring

Criss

March 1996

May 2000

/

Q

0

L

x

/ 0

L

x

Q

P

b. Real Watershed

a. Hypothetical Watershed (eq. 5)

Darcy’s Law

!

QA

= "K#h#x

!

"h"t

= D"2h"x2

Diffusion Eq.

!

h = B+C"Dt

e#x 2

4Dt

Plane Source Solution

Criss’ Dimensionless Theoretical Hydrograph

!

QQmax

=2eb3t

"

# $

%

& '

32e(bt

!

Q = A KCx2D "D# $ %

& ' ( 1

t# $ % & ' ( 3/2

e)x 2 /4Dt

!

Qmax = A KCx2D "D# $ %

& ' ( 6D

x2# $ %

& ' ( 3/2

e)3 /2

!

at tmax = x 2 /6D

/

Q

0

L

x

!

QQmax

=2eb3t

"

# $

%

& '

32e(bt

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Q/Qmax

Time, t

Lag time = 2b/3

b = 1

/

Q

0

L

x

!

QQmax

=2eb3t

"

# $

%

& '

32e(bt

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Q/Qmax

Time, t

Lag time = 2b/3

b = 1

!

Vol = Qdt = bQp "2e3

# $ %

& ' ( 0

)*3/2

+ 4.324 b Qp

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Q/Q

p

Time, days

0.25

0.5

1.0

b=2.0

SYNTHETIC HYDROGRAPHtp = 2b/3

after Criss (2003)

Bluegrass Spring Criss

0

2

4

6

8 0

1

54 56 58 60 62 64

Bluegrass Spring

Area = 0.4 mi2

Q, c

fsP

recipitation, in/hr

Day Number, 2001

Criss & Winston 2003

0

2

4

6

8 0

1

54 56 58 60 62 64

Bluegrass Spring

Area = 0.4 mi2

Q, c

fsP

recipitation, in/hr

Day Number, 2001

b = 0.4 days

Criss & Winston 2003

0

1

2

3

4

5

6

7

8 0

1

55 56 57 58 59 60 61

Bluegrass Springb = 0.3 days

DischargeQ calcExp Fit

Tyson PPT (in)

Day Number, 2000

Exp

Precipitation, in/hr Q

, cfs

0

5

10

15

20 0

1

176 178 180 182 184

Bluegrass Spring

b = 0.27 days

Day Number, 2000

Precipitation, in/hr Q

, cfs

Criss & Winston 2003

1PPT

(cm

/hr)

Temp (°C)Temp Model (b=3.4 days)

!18O (‰)!18O Model (b=1.6 days)

12.8

13.2

13.6

-8.0

-7.5

-7.0

‰

0

50

100

400

600

800

1000

74 75 76 77 78 79 80

Discharge (l/s)

Baseflow Model (b=0.5 days)

Baseflow Separation (l/s)

Event Water Model (b=0.4 days)

Event Water Separation (l/s)

SpC Model (b=3.4 days)

SpC (!S/cm)

Dis

char

ge (l

/s)

Specific Conductivity (!S/cm

)

Year Day, 2000

Model Pulse D

ate (74.89)

Winston & Criss 2004