Geoscience Reference
In-Depth Information
where
c
and
b
are constants. Integrating (9.33) with the first of conditions (9.14) (provided
b
≤
0), one finds
D
w
dS
n
d
φ
cS
−
b
n
+
=
0
(9.34)
A second integration of (9.33) with the second of (9.14) yields
1
φ
=
c
−
1
y
b
D
w
dy
(9.35)
S
n
The constants
b
and
c
remain to be determined. As shown elsewhere (Brutsaert, 1976),
c
can
be determined in several ways. But the more accurate form, for the purpose of infiltration
calculations, can be obtained by means of the integral condition (9.31). Substitution of
Equation (9.34) for
S
n
=
1 into the first term and (9.35) into the second term of (9.31)
produces upon integration by parts by means of Leibniz's rule (see Equation (A1))
2
c
+
c
−
1
⎡
⎛
⎝
⎞
⎠
⎤
⎦
1
1
1
⎣
S
n
−
y
b
D
w
(
y
)
dy
+
c
−
1
S
1
+
b
n
D
w
(
S
n
)
dS
n
=
0
(9.36)
S
n
0
0
Because the second term of (9.36) is zero, this results in
⎡
⎣
⎤
⎦
1
/
2
1
S
1
+
b
n
c
=
D
w
(
S
n
)
dS
n
/
2
(9.37)
0
The solution (9.35) can therefore be written as
⎛
⎞
⎝
2
1
1
/
2
1
D
w
S
1
+
b
n
⎠
D
w
(
y
)
y
b
dy
φ
=
dS
n
(9.38)
0
S
n
Comparison with the more general form (9.32) shows that this method to determine
c
produces
a
=
1
+
b
.
Some comments on this approximate method of solution are in order. Richards's equation
in the form of Equation (9.13) is derived from the equation of continuity and Darcy's law,
and thus it embodies the principles of mass and momentum conservation. Because (9.33)
is only an approximation of (9.13), it may no longer satisfy these conservation principles.
However, by constraining the solution of (9.33) with (9.31) in the determination of
b
and
c
, one ensures that this solution satisfies them at least in an integral or average sense.
Optimal value of the exponent b
The procedure to derive an optimal value of
b
can be understood by recalling that Equation
(9.27) represents an exact solution for a possibly approximate diffusivity function (9.26),
whereas Equation (9.38) represents an approximate solution for an unspecified but presum-
ably exact diffusivity function. Therefore, that value of
b
can be adopted, which makes
(9.38) come closest to the exact result. Because infiltration is the phenomenon of interest,
the sorptivity should be used for this purpose.
