CE 394K.2 Hydrology Infiltration Reading AH Sec 5.1 to 5.5 Some of the subsequent slides were prepared by Venkatesh Merwade.

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

Qq

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