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
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