Geoscience Reference
In-Depth Information
The basic assumption in the linearization is that position
η
of the free surface is never
very different from an unperturbed average value, say
η
0
. Thus, because
η
remains close
to constant, in (10.87) the first term on the right becomes negligible, and
η
in the second
term can be replaced by
η
0
. Equation (10.87) then becomes
∂η
∂
k
0
η
0
n
e
∂
2
η
t
=
(10.88)
∂
x
2
Equation (10.88) is in the form of the standard diffusion equation, with a constant
hydraulic (groundwater) diffusivity
k
0
η
0
n
e
D
h
=
(10.89)
In a similar way, the more general form (Equation (10.31)) of the Boussinesq equation can
be linearized to produce
cos
∂
y
2
∂η
∂
t
=
k
0
η
0
n
e
α
∂
η
∂
x
2
2
+
∂
2
η
k
0
sin
α
n
e
∂η
∂
x
+
I
n
e
+
(10.90)
in which, as before,
α
is the slope of the underlying impermeable bed.
A second but less common way of linearizing the Boussinesq equation consists of
multiplying both sides by
η
, and then bringing it inside the first derivatives or replacing it
by
η
0
, whichever appears more appropriate. For instance, in the case of Equation (10.30)
this yields
∂η
2
∂
t
=
k
0
η
0
n
e
∂
2
∂
x
2
2
η
(10.91)
2
. This approach was probably first used by N. A. Bagrov and later
by N. N. Verigin (Polubarinova-Kochina, 1952; Aravin and Numerov, 1953). A theoretical
advantage of Equation (10.91) over (10.88) is that for steady conditions it reduces to (10.39)
as it should; this means that it accords better with the hydraulic assumptions on which the
Boussinesq equation is based. Nevertheless, the few studies on this have not been conclusive
(Polubarinova-Kochina, 1952, p. 501; Brutsaert and Ibrahim, 1966) as to which of the two
linearizations is preferable; but for some practical applications (see below) this may be
immaterial.
which is linear in
η
A few comments are in order on the optimal value of
η
0
to be used in the linearization.
It stands to reason that the optimal value of
η
0
should never be very different from the
average water table height, namely
B
η
=
η
dx
/
B
(10.92)
0
The difficulty with this is that
is unknown. Nevertheless, two known solutions for
special situations may give some indication. One occurs when
D
and
D
c
in the boundary
conditions (10.50) have nearly the same value. Inspection of the Dupuit formula (10.43)
for steady flow shows that it can be considered in some way as a finite difference form of
η