Geoscience Reference
In-Depth Information
in which use has been made of the first of the boundary conditions (10.68), namely
F
=
0
at
x
+
=
0. The two constants
a
and
C
3
will be determined next by imposing the remaining
two boundary conditions (10.68). Imposition of the second of (10.68) requires the derivative
of
F
. Thus, by means of Leibniz's rule (see Appendix) one obtains from (10.79)
F
(2
C
3
−
(2
a
/
3)
F
3
)
1
/
2
dF
dx
+
1
=
(10.80)
At
x
+
=
1 the derivative
dF
/
dx
+
must equal zero, according to the second of (10.68);
moreover, in light of (10.72) at
x
+
=
1 the function
F
must be equal to one, according to
the third of (10.68). Hence, for the left-hand side of (10.80) to be unity, it is necessary that
C
3
=
(
a
/
3). Finally, imposing the third of (10.68), i.e.
F
=
1at
x
+
=
1, also on (10.79),
one obtains
3
2
a
1
/
2
1
y
(1
−
y
3
)
−
1
/
2
dy
1
=
(10.81)
0
By putting
u
=
y
3
, it is easy to show that the integral in Equation (10.81) is equal to
B
(2
/
3
,
1
/
2)
/
3, where the
B
( ) symbol denotes the complete beta function; hence (10.81)
yields the following expression for the constant
a
=
[
B
(2
/
3
,
1
/
2)]
2
/
6. The value of
this beta function can be readily calculated by expressing it in terms of gamma func-
tions (see, for example, 6.2.2 in Abramowitz and Stegun, 1964), to obtain
B
(2
/
3
,
1
/
2)
=
(1
/
2)
(2
/
3)
/
(7
/
6)
=
2
.
587 11, which produces immediately
a
=
1
.
1155. Substitution
of these values of the constants
a
and
C
3
into Equation (10.79) yields the
x
+
-dependent
part of the solution as
F
3
B
(2
/
3
,
1
/
2)
−
y
3
)
−
1
/
2
dy
x
+
=
y
(1
(10.82)
0
or, in a slightly different form,
F
3
1
B
(2
/
3
,
1
/
2)
u
−
1
/
3
(1
−
u
)
−
1
/
2
du
x
+
=
(10.83)
0
In the form of Equation (10.83) the solution is an incomplete beta function for the variable
F
3
. Numerical values of this solution for
F
(
x
+
) have been presented by Aravin and Numerov
(1953); they also indicated that in 1934 L. S. Leibenzon developed the approximation
F
321
x
1
/
2
142
x
3
/
2
179
x
5
/
2
+
); evidently, this expression involves a standard
error of estimate for
F
of 10
−
3
. The function
F
(
x
+
), as given by (10.82) or (10.83), is the
curve for
t
+
=
0 in Figure 10.21. Actually, Figure 10.21 shows the complete solution for the
height of the water table
η
+
, as given by Equation (10.72), for different values of the time
t
+
. It can be seen that, indeed, with (10.72) the water table exhibits self-preservation and
that it maintains the same curvilinear shape throughout the whole drainage process, from
beginning to end. If the aquifer were initially saturated, as required in (10.50) and (10.53),
the water table would only become curvilinear after enough water has drained; therefore,
the solution of (10.30) with (10.65), is only be applicable to describe “long-time” outflow
behavior.
Having obtained the solution ,
F
(
x
+
), one can now determine the value of the constant
b
,
needed for the outflow rate in Equation (10.76). Again, to obtain
b
, as defined in
=
(1
.
+
−
0
.
+
−
0
.
