Geoscience Reference
In-Depth Information
In terms of these dimensionless variables, the desorptivity (10.58) can now be written as
De
h
=
a
(
k
0
n
e
)
1
/
2
D
3
/
2
(10.61)
In Equation (10.61)
a
represents the definite integral
1
a
=
φ
+
d
η
+
(10.62)
0
which is a dimensionless constant, and thus simply a number whose value depends on the
solution.
To summarize, this brief analysis has shown how, except for a constant
a
, the exact
functional form of the rate of outflow from the aquifer, that is (10.59) with (10.61), can be
obtained by using similarity, that is almost by inspection, without actually deriving the solu-
tion. Two types of similarity were invoked here. The first, Boltzmann's transform, which
results from the nature of the boundary conditions, involves a combination of the indepen-
dent variables; it states that the dependency of
η
on
x
is similar to its dependency on
t
−
1
/
2
.
The second type of similarity involves the scaling of the variables to make the formulation
dimensionless, and thus universally applicable to any aquifer, with any dimensions and
consisting of any type of porous material.
Solutions
Several solutions of this problem have been derived. Polubarinova-Kochina (1952, p. 507)
was able to obtain a solution, by transforming Boussinesq's Equation (10.30) to the Blasius
equation for the viscous boundary layer. From her result it can be shown that
a
=
0
.
664 12
(10.63)
but the details of the derivation are beyond the present scope. A similar but slightly more
accurate procedure was later used by Hogarth and Parlange (1999). There are also several
approximate solutions available, that while less accurate, still yield values of
a
close to
(10.63). One such solution, based on an approximation of Equation (10.30) by successive
steady states, was proposed in 1886 by K. E. Lembke (cited by Polubarinova-Kochina,
1952, p. 573); this assumption can be shown to lead to
a
=
(1
/
3)
1
/
2
, which is within 13% of
Equation (10.63). Incidentally, the assumption of successive steady states is equivalent with
the quasi-steady approach, which was used in the solution of the horizontal infiltration prob-
lem by Parlange (1971). A second approximate solution can be obtained by linearization;
in 1947 J. H. Edelman (cited by Kraijenhoff, 1966) proposed its use to describe free surface
groundwater flow; this solution yields
a
=
(4
p
/π
)
1
/
2
, in which
p
is a parameter used to
compensate for the approximation due to the linearization and discussed further in Section
10.4. Comparison with Equation (10.63) shows that the linear solution can produce the
same result as the exact outflow rate, provided
p
=
0.3465.
Outflow rate
The rate of outflow from the aquifer into the adjacent stream or some other type of open
water body is the main item of interest in catchment hydrology; on the basis of simple
similarity, it was already shown to be given by Equation (10.59), in which
De
h
was
defined as a constant but unspecified hydraulic desorptivity. By combining (10.59) with
