Maria @ PR
5.81 sq mi
Ppt Stg Qcfs 4.4 sq mi
Fetter, 21 Freeze & Cherry, 1978 Criss 23
5 4 Missouri Peak Flows slope.57:1 Log Q cfs 3 2 1 Mean Flows slope 1:1-1 -1 1 2 3 4 Log A mi 2 updated after Criss 23
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)(a 2 -y 2 ) u avg = Ga 2 /3µ Q ~ g s W a 3 /3ν cm 3 /sec Pipe Flow u = (G/4µ)(a 2 -r 2 ) u avg = Ga 2 /8µ Q = g s π a 4 /8ν cm 3 /sec where G= pressure gradient, s=slope, 2a = slot depth or tube radius; W=width µ viscosity; kinematic viscosity ν =µ/ρ cm 2 /sec
LAMINAR SLOT FLOW u =G(a 2 -y 2 )/2µ u avg = Ga 2 /3µ u = u avg @ a/ 3 =.577 down a a LAMINAR PIPE FLOW u =G(a 2 -r 2 )/4µ u avg = Ga 2 /8µ u = u avg @ a/ 2 =.77 down a a
LINEAR RESERVOIR (Chow, 14-27) Storage Outflow => S = Q/k Also, - ds/dt = Q (material balance requirement) Total flow = Base Flow: Q = Q o e "kt where Qo is (peak) discharge @ t For complete depletion, the "Total Potential GW Discharge" is, Vol = " # Qdt = Q /k o
1 8 Qo =1 k=1 Linear Reservoir Q 6 4 2 Q = Q o e "kt 1 2 3 4 5 Time
1 Qo =1 Linear Reservoir Q 8 6 k=1 Vol = " # Qdt = Q o /k 4 2 Q = Q o e "kt 1 2 3 4 5 Time
Note: not linear, but Concave Up
Note: not linear, but Concave Up
LogQ 1.5 1.5 Linear Reservoir 3.2 2.4 1.6.8 LnQ -.5-1 Q = Q o e "kt -.8-1.6-2.4-1.5 1 2 3 4 5 Time -3.2
Discharge (cfs) 8 7 6 5 4 3 2 Q =7.7*Exp{-1.25*(t-t pk )} observed Bluegrass Spring.5 1 Tyson PPT (in/hr) 1 Q =1.2*Exp{-.283*(t-t pk )} 54 56 58 6 62 64 YearDay 21
Discharge (cfs) 8 7 6 5 4 Bluegrass Spring Q BGS = 7.7* Q (.35, 56.167, 1) observed 3 2.5 1 1.5 Tyson PPT (in/hr) 2 1 2.5 54 56 58 6 62 64 Year Day 21 3
2) SEMI-QUANTITATIVE EQUATIONS a. Chezy Equn (1769) where U = water velocity U = C Sqrt [RS] 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) where U = water velocity U = C Sqrt [RS] 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) where U = C Sqrt [RS] 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) where U = C Sqrt [RS] 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) R 2/3 S 1/2 m/s note: units where: n = Manning roughness coeff. n, in units of sec/m 1/3 n=.12 (concrete) n=.5 (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=3r 1/6 /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 a H a WEIRS Laminar Channel Flow (slot) u = (G/2µ)(a 2 -y 2 ) u avg = Ga 2 /3µ
3) WEIRS Rectangular: Qcfs = 3.333 ( L - H/5) H 3/2 9 V-Notch: Qcfs = 2.5 H 5/2 where H, L in ft. Fetter p. 58 H Culvert: See Chow 15-33; USGS Circ. 376)
3) WEIRS Rectangular: Qm 3 /s = 1.84 ( L - H/5) H 3/2 9 V-Notch: Qm 3 /s = 1.379 H 5/2 where H, L in m. Fetter p. 58 Culvert: See Chow 15-33; USGS Circ. 376) H
3) WEIRS Rectangular: Qm 3 /s = 1.84 ( L - H/5) H 3/2 9 V-Notch: Qm 3 /s = 1.379 H 5/2 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) H 3/2 9 V-Notch: Qcfs = 2.5 H 5/2 Units V-Notch: # Q cfs = 8 % & ( H 5 /2 C d = discharge coeff ; 15 C d 2g tan " $ 2' Chow 7-46 Culvert: See Chow 15-33; USGS Circ. 376) H
http://www.hubbardbrook.org
V-Notch Weir http://www.hubbardbrook.org
4) AREA-VELOCITY METHOD Current Meter Divide current into 15-3 segments Measure velocity @.6*depth of segment (6% down) or, if channel is deep, take average v @.8 and.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 = S a or Qcfs = (S - S o ) a where S o = stage @ zero flow Make polynomial fit USGS
1.2 1 6 1 1 6 Q, cfs 8 1 5 6 1 5 Rating Curve: Q = 7934 + 6517.6 S + 241.52 S 2 Mississippi River @ St. Louis 198-26 9,82 pts. 4 1 5 2 1 5-1 1 2 3 4 5 Stage, ft.
THEORETICAL HYDROGRAPH and its APPLICATIONS Bob Criss Washington University
Fetter, 21 Freeze & Cherry, 1978 Criss 23
Meramec River Oct. 2 Stage 1.8 ft. Discharge 5 cfs May 2 Stage 27.8 ft. Discharge 56, cfs
March 1996 Rockwoods Spring May 2 Criss
/ x L Q
a. Hypothetical Watershed (eq. 5) / x P Q L b. Real Watershed
Darcy s Law Q A = "K#h #x "h Diffusion Eq. 2 "t = D" h "x 2 Plane Source Solution h = B+ C "Dt e#x 2 4Dt
Q = A # % $ KCx 2D "D & ' ( # % 1& $ t ' 3/2 ( e )x 2 / 4 Dt Q max = A # % $ KCx 2D "D & ( ' # % $ 6D x 2 3/2 & ( e )3/2 at t ' max = x 2 /6D Q = Q max " $ # 2eb % ' 3t & 3 2 e (b t Criss Dimensionless Theoretical Hydrograph
/ Q = Q max " $ # 2eb % ' 3t & 3 2 e (b t x 1.8 Lag time = 2b/3 Q/Qmax.6 L.4.2 b = 1 Q 1 2 3 4 5 Time, t
/ Q = Q max " $ # 2eb % ' 3t & 3 2 e (b t x 1.8 Lag time = 2b/3 Q/Qmax.6 Q L.4.2 ) # Vol = Qdt = bq p " 2e & * % ( $ 3 ' + 4.324 b Q p 1 2 3 4 5 Time, t 3/2 b = 1
1 SYNTHETIC HYDROGRAPH t p = 2b/3.8 Q/Qp.6.4 1. b=2..5.2.25 1 2 3 4 5 Time, days after Criss (23)
Bluegrass Spring Criss
Q, cfs 8 Bluegrass Spring Area =.4 mi 2 1 6 4 Precipitation, in/hr 2 54 56 58 6 62 64 Day Number, 21 Criss & Winston 23
Q, cfs 8 Bluegrass Spring Area =.4 mi 2 1 6 4 Precipitation, in/hr 2 b =.4 days 54 56 58 6 62 64 Day Number, 21 Criss & Winston 23
8 7 6 Bluegrass Spring b =.3 days 1 Precipitation, in/hr 5 Q, cfs 4 3 Discharge Q calc Exp Fit 2 1 Exp 55 56 57 58 59 6 61 Day Number, 2
2 Bluegrass Spring b =.27 days 15 1 Precipitation, in/hr Q, cfs 1 5 176 178 18 182 184 Day Number, 2 Criss & Winston 23
PPT (cm/hr) 1 13.6 13.2 12.8 SpC Model (b=3.4 days) SpC (S/cm) Temp ( C) Temp Model (b=3.4 days) 1 8 6 4-8. Specific Conductivity (S/cm) 18 O ( ) -7.5 Discharge (l/s) 1 5 Model Pulse Date (74.89) 18 O Model (b=1.6 days) Discharge (l/s) Baseflow Model (b=.5 days) Baseflow Separation (l/s) Event Water Model (b=.4 days) Event Water Separation (l/s) -7. 74 75 76 77 78 79 8 Year Day, 2 Winston & Criss 24