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CE 394K.2 Hydrology. Infiltration Reading AH Sec 5.1 to 5.5 Some of the subsequent slides were prepared by Venkatesh Merwade. Darcy’s Law. K = hydraulic conductivity q = specific discharge V = q/n = average velocity through the area. Richard’s Equation. Recall Darcy’s Law - PowerPoint PPT Presentation
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CE 394K.2 Hydrology
Infiltration
Reading AH Sec 5.1 to 5.5
Some of the subsequent slides were prepared by Venkatesh Merwade
Darcy’s Law
• K = hydraulic conductivity
• q = specific discharge
• V = q/n = average velocity through the area
L
hKAQ
z
hKqz
L
hhK
A
updown
Richard’s Equation
• Recall – Darcy’s Law– Total head
• So Darcy becomes
• Continuity becomes
z
hKqz
Kz
D
Kz
K
z
zKqz
KD
Soil water diffusivity
Kz
Dzz
q
t
Kz
Kqz
Infiltration
• General– Process of water
penetrating from ground into soil
– Factors affecting• Condition of soil surface,
vegetative cover, soil properties, hydraulic conductivity, antecedent soil moisture
– Four zones• Saturated, transmission,
wetting, and wetting front
depth
Wetting Zone
TransmissionZone
Transition ZoneSaturation Zone
Wetting Front
Philips Equation• Recall Richard’s
Equation– Assume K and D are
functions of , not z
• Solution– Two terms represent
effects of • Suction head• Gravity head
• S – Sorptivity– Function of soil suction
potential– Found from experiment
Kz
Dzt
KtSttF 2/1)(
KSttf 2/1
2
1)(
Infiltration into a horizontal soil column
x0
Θ = Θn for t = 0, x > 0
Θ = Θo for x = 0, t > 0
zD
xt
Equation:
Boundary conditions
Measurement of Diffusivity by Evaporation from Soil Cores
Air stream
q = soil water flux = evaporation rate
xDq
q
x
http://www.regional.org.au/au/asa/2006/poster/water/4521_deeryd.htm
Measurement of Diffusivity by Evaporation from Soil Cores
Numerical Solution of Richard’s Equation
(Ernest To)
http://www.ce.utexas.edu/prof/maidment/GradHydro2007/Ex4/Ex4Soln.doc
Implicit Numerical Solution of Richard’s Equation
x (i)
t (j)
i-1 i i+1
jj -1
Implicit Numerical Solution of Richard’s Equation
Matrix solution of the equations
Θ
f
F
Definitions
solid
Pore withair
Pore withwater
Element of soil, V(Saturated)
Element of soil, V(Unsaturated)n0content;moisturenS
V
V
S0;saturationV
VS
porosityV
Vn
waterofvolumeV
solidsofvolumeV
poresofvolumeV
elementofvolumegrossV
w
v
w
v
w
s
v
1
Infiltration
• Infiltration rate– Rate at which water enters the soil at the surface
(in/hr or cm/hr)
• Cumulative infiltration– Accumulated depth of water infiltrating during given
time period
t
dftF0
)()(
)(tf
dt
tdFtf
)()(
Infiltration Methods
• Horton and Phillips – Infiltration models developed as approximate
solutions of an exact theory (Richard’s Equation)
• Green – Ampt– Infiltration model developed from an
approximate theory to an exact solution
Hortonian Infiltration
• Recall Richard’s Equation– Assume K and D are
constants, not a function of or z
• Solve for moisture diffusion at surface
Kz
Dzt
z
K
zD
t
2
2
02
2
zD
t
ktcc effftf )()( 0
f0 initial infiltration rate, fc is constant rate and k is decay constant
Hortonian Infiltration
0
0.5
1
1.5
2
2.5
3
3.5
0 0.5 1 1.5 2
Time
Infi
ltra
tio
n r
ate,
f
k1
k3
k2
k1 < k2 < k3
fc
f0
Philips Equation• Recall Richard’s
Equation– Assume K and D are
functions of , not z
• Solution– Two terms represent
effects of • Suction head• Gravity head
• S – Sorptivity– Function of soil suction
potential– Found from experiment
Kz
Dzt
KtSttF 2/1)(
KSttf 2/1
2
1)(
Green – Ampt Infiltration
Wetted Zone
Wetting Front
Ponded Water
Ground Surface
Dry Soil
0h
L
n
i
z
LLtF i )()(
dt
dL
dt
dFf
Kz
Kf
fz
hKqz
MoistureSoilInitial
Front WettingtoDepth
i
L
Green – Ampt Infiltration (Cont.)
• Apply finite difference to the derivative, between – Ground surface– Wetting front
Kz
Kf
Wetted Zone
Wetting Front
Ground Surface
Dry Soil
L
i
z0,0 z
fLz ,
KL
KKz
KKz
Kff
0
0
F
L
LtF )(
1
FKf
f
Kz
Kf
1
LK
dt
dL f
1
FKf
f
dt
dLf
Green – Ampt Infiltration (Cont.)
LtF )(
Wetted Zone
Wetting Front
Ground Surface
Dry Soil
L
i
z
L
dLdLdt
K
f
f
CLLtK
ff
)ln(
Integrate
Evaluate the constant of integration
)ln( ffC
0@0 tL
)ln(L
LKtf
ff
Green – Ampt Infiltration (Cont.)
)ln(L
LKtf
ff
)1ln(f
fF
KtF
1
FKf
f
Wetted Zone
Wetting Front
Ground Surface
Dry Soil
L
i
z
See: http://www.ce.utexas.edu/prof/mckinney/ce311k/Lab/Lab8/Lab8.html
Nonlinear equation, requiring iterative solution.
Soil Parameters
• Green-Ampt model requires – Hydraulic conductivity, Porosity, Wetting Front
Suction Head– Brooks and Corey
Soil Class Porosity Effective Porosity
Wetting Front
Suction Head
Hydraulic Conductivity
n e K (cm) (cm/h) Sand 0.437 0.417 4.95 11.78 Loam 0.463 0.434 9.89 0.34 Clay 0.475 0.385 31.63 0.03
re n
ees )1(
e
res
Effective saturation
Effective porosity
Ponding time
• Elapsed time between the time rainfall begins and the time water begins to pond on the soil surface (tp)
Ponding Time
• Up to the time of ponding, all rainfall has infiltrated (i = rainfall rate)
if ptiF *
1
FKf
f
1
* p
f
tiKi
)( KiiKt
fp
Potential Infiltration
Actual Infiltration
Rainfall
Accumulated Rainfall
Infiltration
Time
Time
Infi
ltra
tion
rate
, f
Cu
mu
lati
ve
Infi
ltra
tion
, F
i
pt
pp tiF *
Example
• Silty-Loam soil, 30% effective saturation, rainfall 5 cm/hr intensity 30.0
/65.0
7.16
486.0
e
e
s
hrcmK
cm
340.0)486.0)(3.01()1( ees
340.0*7.16
hr17.0))(65.00.5(0.5
68.565.0
)(
KiKiiKt
fp
