Geoscience Reference
In-Depth Information
To derive the general soil water low equation for variably saturated soils, we com-
bine Eq. (
4.16
) and (
4.21
):
∂
∂
+
h
z
∂
kh
()
1
∂
∂
=
θ
t
∂
∂
=
h
t
Ch
()
−
Sz
()
(4.22)
∂
z
where
C
(= ∂
θ
/∂
h
) is the differential soil moisture capacity (m
-1
). Equation (
4.22
) is
called
Richards
'
equation
, and is generally used to solve soil water low problems in
the vadose zone. It is written in pressure head rather than water content, as pressure
head is continuous with depth at soil layer transitions. To solve Richards' equation for
an arbitrary situation, we should know:
1. The so-called soil hydraulic functions that relate
θ, h
, and
k
2. The actual root water extraction rate
S
3. The top and bottom boundary condition
4. The initial soil moisture amounts.
Under strict assumptions some analytical solutions of Richards' equation can be
derived. In general the soil hydraulic functions are strongly nonlinear and the ield
boundary conditions are highly dynamic. In that case numerical solutions are the only
feasible way to solve Richards' equation.
4.7 Soil Hydraulic Functions
The soil hydraulic functions relate
h
with
θ
(retention function) and
k
with either
θ
or
h
(conductivity function). Although tabular forms of
θ
(
h
) and
k
(
θ
) have been used for many
years, currently analytical expressions are preferred for a number of reasons. Analytical
expressions are more convenient as model input and a rapid comparison between hori-
zons is possible by comparing parameter sets. Various concepts for modelling hysteresis
of the retention function, require analytical soil hydraulic functions. Also scaling, which
is used to describe spatial variability of
θ
(
h
) and
k
(
θ
), requires an analytical expression of
the soil hydraulic functions. Another reason is that extrapolation of the functions beyond
the measured data range is possible. Last but not least, analytical functions allow for cal-
ibration and estimation of the soil hydraulic functions by inverse modelling.
Important requirements for analytical expressions of
θ
(
h
) and
k
(
θ
) are that they are
lexible in order to describe the wide variability among soils, contain only a few param-
eters in order to facilitate unique calibration, and that these parameters have some phys-
ical meaning, such that they can be related to soil texture, organic matter content and
soil bulk density. Van Genuchten (
1980
) proposed an analytical expression for
θ
(
h
) that
met the above requirements and has become widespread among soil scientists:
−
+
( )
−
θθ
θ θ
()
h
=+
s
r
(4.23)
r
n
1
h
n
n
1
α
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