Infiltration Equations
Fundamental Mass Balance Equation:
 q x q y q z



0
t
x
y
z
Darcy’s Law (z direction):
h
qz   K z
z
Where
h   z
Infiltration Equations
h(), (), K()
Infiltration Equations
h(), (), K()
Infiltration Equations
h(), (), K()
Infiltration Equations
Vertical Darcy’s Law is then
 ( )


qz   K z ( )
 K z ( ) 
z


Soil water diffusivity is defined
d ( )
D( )  K ( )
d
combining



qz   Dz ( )
 K z ( ) 
z


Infiltration Equations
In three dimensions, using K():

 
h   
h   
h 

K
(

)

K
(

)

K
(

)
t x 
x  y 
y  z 
z 
defining the specific moisture capacity as
C() = d/d
C ( )

 
   
   
 


K
(

)

K
(

)

K
(

)

1







t
x 
x  y 
y  z 
 z

Infiltration Equations
For 1-D vertical, saturated flow:
   

Ks 
 1  0

z   z

Infiltration Equations
Boundary conditions for infiltration at ground surface:
r  K ( )
h
 

 K ( )
 1  (0, t )   s  (0, t )  0 t  t p
z
 z

 (0, t )  h0  (0, t )   s
t  tp
Infiltration Equations
Lower boundary conditions for infiltration
 ( L, t )  0 t  0
q z   K ( ) t  0
Infiltration Equations
Mixed form of Richards Equation

 
 

  K ( )
 1
t z 
 z

Infiltration Equations
Parameters (van Genuchten, 1980):
s  r

 r 
n
 ( )  
1  

 s
(

)
K ( )  K s S 1  (1  S
l
e
 0
m
 0
)

2
1/ m m
e
where m = 1-1/n for n > 1
Se = ( - r)/(s - r)
r is the residual volumetric moisture content
l,  and n are van Genuchten model parameters
Infiltration Equations
Parameters: 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); and
Sand, silt and clay percentages, bulk density
and the value of  at  = 330 and 15,000 cm
(33 and 1500 kPa).
Infiltration Equations
Infiltration Equations
Green-Ampt
from conceptualization:
Darcy’s Law:
combining:
F  L(   i )  L
 h0  (  L) 
  L 
f  K
 K

L
 L 


  F 
f  K

 F
Infiltration Equations
since f = dF/dt

F (t ) 
F (t )   ln 1 
  Kt
  
how is this solved?
Infiltration Equations
Green-Ampt ponding time
tp 
K
i (i  K )
for t < tp, how are f and F computed?
Infiltration Equations
Green-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:
 F (t  t )   
F (t  t )  F (t )   ln 
 Kt

 F (t )   
Infiltration Equations
Green-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 :
K
Fp 
i (t )  K
• then tp = t + t' where
t ' 
Fp  F ( t )
i (t )
after Chow et
al., 1988
Infiltration Equations
Green-Ampt parameters, after Chow et al., 1988
Infiltration Equations
Kostiakov
f  Kk t 
Infiltration Equations
Horton
f  f c  ( f o  f c )e  t
Infiltration Equations
Holtan
f  GI A Sa1.4  f c
Infiltration Equations
Phillip
f  0.5St
0.5
 Ks
Infiltration Equations
Smith-Parlange



f  K s 1 

exp(

F
/
B
)

1

