Geoscience Reference
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occurs when soil is saturated from below. This may occur at relatively low rates of
rainfall in humid climates, when downward low is limited by subsoil layers with low
permeability or bedrock, or in seepage situations. The irst type might be denoted as
'surface soil control', the second type as 'subsurface soil control'.
Numerical models that solve Richards' soil water low equation with the proper
boundary conditions at the top (e.g., rainfall and potential evapotranspiration rates)
and bottom (e.g., relations between lux and pressure head) are able to calculate
runoff amounts of both types. An important condition is that the time and space
steps of the numerical model near the soil surface are ine enough, as discussed in
Section 9.1.2
. Although we may meet this condition for one-dimensional models
at the plot level, in general multidimensional models based on Richards' equation
require too much computation time to simulate runoff in a reliable way at ield
or larger spatial level. Therefore these models require simpliied, semi-empirical
methods to approximate runoff amounts. An extensive overview of these methods
has been given by Smith (
2002
). Here we discuss two semi-empirical methods:
Horton and Green-Ampt. Both methods refer to the runoff type with surface soil
control.
4.8.1 Horton Iniltration Model
Horton (
1933
,
1939
) was one of the pioneers in the study of iniltration in the ield.
Horton anticipated that the reduction in iniltration rate with time after the initia-
tion of iniltration was controlled largely by factors at the soil surface. These fac-
tors included swelling of soil colloids and the closing of small cracks that progres-
sively sealed the soil surface. Compaction of the soil surface by raindrop action
was also considered important where it was not prevented by vegetation cover.
Horton's ield data, similar to those of many other workers, indicated a decreasing
iniltration rate for 2 or 3 hours after the initiation of the storm runoff. The iniltra-
tion rate eventually approached a constant value that was often somewhat smaller
than the saturated hydraulic conductivity of the soil. Air entrapment and incom-
plete saturation of the soil were assumed to be responsible for this latter inding.
Horton used an exponential function to describe the decreasing iniltration rate
(Jury et al.,
1991
):
=+ −
( )
−
β
t
II
I
I
e
(4.26)
f
0
f
where
I
is the iniltration rate (m d
-1
),
I
0
is the initial iniltration rate (m d
-1
) at
t
= 0,
I
f
is the inal constant iniltration rate (m d
-1
) that is reached after a long time, and
β
(d
-1
)
is a soil parameter that describes the rate of decrease of iniltration. The cumulative
iniltration
I
cum
(m) follows from integration of Eq. (
4.26
):
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