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