Water Resources Final Equation Sheet

Santa Clara University Department of Civil Engineering

CENG 140 Water Resources Engineering Spring 2013

1

Random Supplemental Data 1 acre = 43,560 ft2

1 mi2 = 640 acres = 2.78784 x 107ft2

1 m = 3.28 ft

1 m3 = 1000 liters

1 km2 = 106 m2

1 m = 1000 mm

1 hectare=2.47 acres

1 inch = 2.54 cm = 25.4 mm

Surface area of a sphere = 4πr2

π=3.14159265358979323846

Diameter of the earth = 12,756 kilometers (at the equator)

Relative humidity, Rh=e/es ; e = actual vapor pressure of the air. e≈es at dew point temperature

Weibull plotting position (prob. of exceedence, P): m/(N+1), m=rank; N=total number

P(either A or B) = P(A) + P(B) (if A and B are independent and mutually exclusive)

P(both A and B) = P(A) x P(B) (if A and B are independent)

P(not A) = 1 - P(A) or (Prob. of exceedence) = 1 – (Prob. of non-exceedence)

P(no occurrences of A in any of N time periods) = (1 – P)N where P is the probability of any one

event occurring (probability of exceedence)

P(one or more occurrence in N time periods) = 1-(1-P)N

Return Period T: FP

T−

==1

11 where P=prob. of exceedence; F=prob. of non-exceedence

Residence time = Volume/Σ(inflows)

Standard Normal Deviate, z = (x-µ)/σ

Mass (Volume) Balance: I – O = dS/dt ≈ ∆S/∆t I=inflow O=outflow Standard flood frequency analysis equation: KSXX +=

Rational Formula Q=CiA (i in inches/hr, A in acres, Q in cfs; or i in m/s, A in m2, Q in m3/s)

Bed Shear Stress oo ySγτ = where γ=specific weight of water, y=flow depth and So=bed slope

Reservoir Storage Required=max(ΣI-ΣO) (maximum between peak and subsequent low value)

Infiltration Trench Design: Vin=Qt; Vout=KHLt; Vstorage=WHLn

Infiltration/runoff Equations: Horton NRCS

( ) ktccp effff −−+= 0 ( )

SPSPQ

8.02.0 2

+−

= when P> 0.2S P, Q in length units

Ia=0.2S

−= 101000

CNS α

α=1 for S in inches, α=25.4 for S in mm.



2

Equations for time of concentration (length L in m, slope S0 is unitless, time units as noted): Kirpich

385.00

77.0

019.0(min)SLtc =

Kerby

467.0

0

44.1(min)

=

SLrtc

NRCS

Sheet Flow: ( )4.0

05.0

2

8.0

0288.0(hrs)SP

nLts =

Overland/Concentrated Flow: sc

sc VLt =

where 5.00kSVsc =

Kinematic Wave

6.0

04.0

99.6(min)

=

SnL

itc where i is in mm/h

solved for i: 5.1

05.2

129

=

SnL

ti

c



3

Note: F(-z) = 1 – F(z)

Values in table are F(z), vertical axis is z to one decimal, columns are for the second decimal place.



4

Unit peak discharge for SCS rainfall distribution type I, used in TR-55 graphical peak flow formula.



5

NRCS TR-55 equations: SI a 2.0=

α=1 for S in inches, α=25.4 for S in mm. NRCS TR-55 graphical peak flow equation: Qp=quAQF where A is basin area, F is pond/swamp factor, and qu unit peak flow. For Central/Southern California, assume a Type 1 rain distribution, so qu can be determined from the figure above. Muskingum method: ( )[ ]111 1 OXXIKS −+= = K[weighted flow] 1211202 OCICICO ++=

( )( ) ( )K

tX

XKt

C∆+−

−∆=

12

20

( )( ) ( )K

tX

XKt

C∆+−

+∆=

12

21

( ) ( )( ) ( )K

tXK

tXC

∆+−

∆−−=

12

122

Storage indication equation: ( )

−

∆++=+

∆ −−

− 11

122

tt

tttt O

tS

IIOt

S

111

11 2

22−−

−−

− −

+

∆=−

∆ ttt

tt OO

tS

Ot

S

Darcy’s Law and groundwater hydraulic head

LhKV

LhKAQ ∆

=∆

= or Hydraulic head, h ≈ EGL elevation = γPz +

Actual velocity Vactual = V/n where V=“Darcian velocity,” determined from Darcy's law. n=porosity. Reservoir Trap Efficiency

−= 101000

CNS α

( ) 8.02.0 2

SPSPQ

+−

=when P> 0.2S (P, S, Q have same units)



6

NRCS Curve Number Adjustment

Shields Diagram:

x-axis: R*=U*ds/ν where ds=mean particle diameter (d50, the diameter for which 50% is finer by weight) , ν=kinematic viscosity of water. U*=(τo/ρ)½ where ρ is the density of water. y-axis: τ*=τo/[(γs-γ)ds] where γs=specific weight of the sediment material (for sand: γs =2.65γ) .

Water Resources Final Equation Sheet

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