Geoscience Reference
In-Depth Information
Hence, Boltzmann's transform, i.e.
xt
−
1
/
2
φ
=
(10.54)
can be used here as well. As will be shown next, this greatly simplifies the solution.
In the manner shown in Equations (9.12), the Boltzmann transformation (10.54)
allows the reduction of the Boussinesq equation (10.30) to the following ordinary
differential equation
η
η
+
2
η
k
0
n
e
d
d
d
d
=
0
(10.55)
φ
d
φ
d
φ
The boundary conditions (10.53) now become
η
=
0
φ
=
0
(10.56)
η
=
D
φ
→∞
Regardless of the method used, the solution of (10.55) with (10.56) is of the form
η
=
η
φ
φ
=
φ
η
=
(
)or
(
). The cumulative outflow volume from the aquifer at
x
0is
given by Equation (10.52). Thus, once the solution
φ
=
φ
(
η
) is known, in light of the
Boltzmann transform, this outflow volume becomes
D
t
1
/
2
n
e
∀=
φ
(
η
)
d
η
(10.57)
0
Because the integral in (10.57) is a constant, for conciseness of notation, the outflow
volume can be expressed in terms of the hydraulic desorptivity, defined as
D
De
h
=
n
e
φ
(
η
)
d
η
(10.58)
0
The rate of outflow from the aquifer at
x
=
0, that is
q
=−
d
∀
/
dt
, can now be written
as
1
2
De
h
t
−
1
/
2
=−
q
(10.59)
This can probably serve as a more tangible and practical definition of the desorptivity
than (10.58). Note that the outflow rate
q
can also be obtained by applying the hydraulic
extension of Darcy's law at
x
=
0, namely Equation (10.51); naturally, for a known
solution
η
=
η
(
x
,
t
), the result should be the same as that obtained with (10.59) and
(10.58).
Before any solutions are discussed in detail, some interesting features of the hydraulic
desorptivity
De
h
can be derived from additional similarity considerations. It stands to reason
that the water table height
η
should be normalized with its initial value
D
; insertion of this
normalized depth into Equation (10.55) reveals then immediately the dimensionless form
of
φ
. Thus the problem can be cast in terms of the following scaled variables
η
+
=
(
η/
D
)
(10.60)
φ
+
=
(
n
e
/
k
0
D
)
1
/
2
φ
