Geoscience Reference
In-Depth Information
of Darcy's equation and the principle of mass conservation, any changes to it result in a
violation of these physical principles. Therefore, in some cases it has been found useful to
consider the integral of (9.13), with both of (9.14), namely,
θ
0
2 (
D
w
d
θ/
d
φ
)
θ
=
θ
0
+
φ
d
θ
=
0
θ
i
or
,
in terms of
S
n
(9.31)
0
φ
dS
n
=
0
1
2 (
D
w
dS
n
/
d
φ
)
S
n
=
1
+
and use it as a constraint on the approximate solution. An example of the application of this
approach is found in the next section.
9.2.3
A nearly exact solution for strongly nonlinear soils
In the hydrologic literature many numerical solutions of (9.13) with (9.14) have been
published starting with those of Klute (1952) and Philip (1955). Such solutions can be quite
accurate, but their implementation in practical simulations, especially over larger areas,
is still a cumbersome task. Therefore, it is often useful to describe the phenomenon with
parameterizations that satisfy the dual requirement of physical realism and computational
simplicity. Over the years a number of simple analytical solutions have been formulated,
which satisfy this requirement. Although these solutions are not exact, their accuracy is
still reasonable, and they involve much smaller mathematical error than those generated
by the uncertainty of the parameter functions
k
=
k
(
θ
) and
H
=
H
(
θ
). Moreover, they are
closed form and concise so that they are easy to apply. It can be shown (Brutsaert, 1976)
that several of these solutions are special cases of the following general form
⎛
⎞
⎠
⎝
2
1
1
/
2
1
D
w
S
n
dS
n
D
w
(
y
)
y
b
dy
φ
=
(9.32)
0
S
n
where
a
and
b
are constants. The specific values of
a
and
b
depend on the nature of the
approximation used in the solution. For example, in the quasi-steady state solution (Landahl,
1953; Macey, 1959; Parlange, 1971),
a
=
1 and
b
=
0; in a second approximation of the
quasi-steady state solution (Parlange, 1973),
a
=
0 and
b
=−
1; in the sharp-front solution
(Brutsaert, 1974),
a
=
b
=−
1; finally, in a first weighting solution
a
=−
b
=
1
/
2 (Parlange,
1975), and in a second weighting solution
a
=
b
=−
3
/
2. By comparing them with the
exact solution (9.28), it was found (Brutsaert, 1976) that the error involved in all these
solutions for
φ
is at most of the order of 3% or 4%; however, as will be shown next, the first
weighting solution with
a
=−
b
=
1
/
2 is accurate within a few thousandths.
Derivation of general form
The general form of Equation (9.32) can be obtained by direct integration of (9.13), with
the assumption that in its second term on the left,
φ
can be replaced by a power function of
S
n
. With this approximation (9.13) becomes
d
d
φ
D
w
dS
n
d
φ
cS
−
n
=
d
d
φ
+
0
(9.33)
