Infiltration EquationsFundamental Mass Balance Equation:

0

zq

yq

xq

tzyx

Darcy’s Law (z direction):

zhKq zz

Wherezh

Infiltration Equations

h(), K(

Infiltration Equations

h(), K(

Infiltration Equations

h(), K(

Infiltration Equations

)()()( zzz Kz

Kq

Vertical Darcy’s Law is then

Soil water diffusivity is defined

ddKD )()()(

combining

)()( zzz Kz

Dq

Infiltration EquationsIn three dimensions, using K)

zhK

zyhK

yxhK

xt)()()(

defining the specific moisture capacity as C() = d/d

1)()()()(

zK

zyK

yxK

xtC

Infiltration EquationsFor 1-D vertical, saturated flow:

01

zK

z s

Infiltration EquationsBoundary conditions for infiltration at ground surface:

ps ttttz

KzhKr

0),0(),0(1)()(

ps tttht ),0(),0( 0

Infiltration EquationsLower boundary conditions for infiltration

00),( ttL

0)( tKqz

Infiltration EquationsMixed form of Richards Equation

1)(

zK

zt

Infiltration EquationsParameters (van Genuchten, 1980):

0

01)(

s

mnrs

r

2/111)(mm

eles SSKK

where m = 1-1/n for n > 1Se = ( - r)/(s - r)r is the residual volumetric moisture content l, and n are van Genuchten model parameters

Infiltration EquationsParameters: Rosetta by Marcel G. Schaap (1999)

Soil textural classes;Sand, silt and clay percentages;Sand, silt and clay percentages and bulk density;Sand, silt and clay percentages, bulk density and the value of at = 330 cm (33 kPa); andSand, silt and clay percentages, bulk density and the value of at = 330 and 15,000 cm (33 and 1500 kPa).

Infiltration Equations

Infiltration Equations

LLF i )(

Green-Ampt

L

LKL

LhKf )(0

F

FKf

from conceptualization:

Darcy’s Law:

combining:

Infiltration Equations

KttFtF

)(1ln)(

since f = dF/dt

how is this solved?

Infiltration EquationsGreen-Ampt ponding time

)( KiiKt p

for t < tp, how are f and F computed?

Infiltration EquationsGreen-Ampt with a hyetograph

• discrete, steady pulses of rainfall of t duration is used to describe i(t)• F(t + t) = F(t) + i(t)t while f < i• when the surface is ponded throughout the time increment:

tKtF

ttFtFttF

)()(ln)()(

Infiltration EquationsGreen-Ampt with hyetograph continued...

• If f(t + t) < i(t), then ponding occurs during the time interval and using f(t) = i(t) and F(t) = Fp :

KtiKFp

)(

• then tp = t + t' where

)()(

'ti

tFFt p

after Chow et al., 1988

Infiltration EquationsGreen-Ampt parameters, after Chow et al., 1988

Infiltration Equations

Kostiakov

tKf k

Infiltration Equations

Horton

tcoc effff )(

Infiltration Equations

Holtan

ca fSAGIf 4.1

Infiltration Equations

Phillip

sKStf 5.05.0

Infiltration Equations

Smith-Parlange

1)/exp(

1BF

Kf s