Geoscience Reference
In-Depth Information
C
(
) and
K
(
) may not be simple single valued functions of
or
but may be subject to
hysteresis
, that is varying with the history of wetting and drying at a point. Various models
of hysteretic soil characteristics have been proposed (see Jaynes, 1990) but they are not often
used in rainfall-runoff modelling. For saturated soil and in the saturated zone of an aquifer,
K
(
) approaches the saturated conductivity
K
s
and
C
(
) takes on a very small value due to the
compressibility of the soil or aquifer.
An equivalent form of the Richards equation may be written with soil moisture content
as
the dependent variable as:
K
(
)
d
∂
∂t
=∇
∂K
z
(
)
∂z
d
∇
+
−
E
T
(
x,y,z,t
)
(B5.1.8)
The product
K
(
)
d
d
is known as the
diffusivity
of the soil and is often written as
D
(
).
The Richards equation applies to both saturated and unsaturated flow through a porous
medium. For near surface flows, it is normally assumed that the water is of constant density
and that the soil is also incompressible. For deep aquifers, such an assumption may not be
valid and it may be necessary to take account of the compressibility of both water and rock. A
typical parameterisation takes the form:
n
(
P
)
=
s
(1
+
bP
)
(B5.1.9)
where
n
(
P
) is the porosity of the soil at pore pressure
P
, for
P>
0,
s
is the porosity at atmo-
spheric pressure, and
b
is a compressibility coefficient that varies with the nature of the soil or
rock.
The temperature dependence of hydraulic conductivity due may also be important and may
be expressed in the form:
k
s
k
r
(
)
g
K
(
)
=
(B5.1.10)
where
k
s
is called the intrinsic permeability of the soil at saturation which should be a char-
acteristic only of the porous medium,
k
r
(
) is a relative conductivity (0
<k
r
(
)
<
1) varying
with capillary potential
, and
is the dynamic viscosity of the soil. Both
and
vary with
temperature.
The Richards equation does not have an analytical solution for most cases of interest and
solutions must be obtained by approximate numerical calculations (see Box 5.3).
The following assumptions are made in developing this form of the Richards equation:
A1 For both saturated and unsaturated flow, flow velocity may be assumed to be a linear
function of the gradient of hydraulic potential in accordance with Darcy's law.
A2 Functional relationships can be specified for the soil moisture characteristic curves to
relate moisture content, capillary potential and hydraulic conductivity of the soil.
A3 Fluxes of water in vapour form can be neglected.
Additional assumptions that are often made in applying the Richards equation are:
A4 The soil moisture characteristics are non-hysteretic.
A5 The hydraulic conductivity tensor is isotropic.
A6 The porous medium is incompressible.
A7 The water is of constant temperature and density.
Assumptions A4-A7 reduce the number of parameters that must be specified before the
model can be run.
