Geoscience Reference
In-Depth Information
Equation (10.75), it is necessary to apply Leibniz's rule to (10.82), in the same manner
as done earlier to obtain (10.80); this yields
3
dF
dx
+
2)
(1
−
F
3
)
−
1
/
2
F
1
=
(10.84)
/
,
/
B
(2
3
1
Imposition of the boundary condition at
x
+
=
0, i.e.
F
=
0 to this result, produces imme-
diately
b
=
B
(2
/
3
,
1
/
2)
/
3
=
0
.
862 37.
Outflow rate
With the two constants determined, for future reference it is convenient to rewrite the
outflow rate (10.76) in terms of the original variables as
1
n
e
B
2
t
−
2
bk
0
D
2
B
ak
0
D
q
=−
+
(10.85)
in which the two constants have the following values
a
=
1
.
115
(10.86)
b
=
0
.
862
The applicability of this solution for large values of
t
has been confirmed experimen-
tally by means of a Hele-Shaw model (Ibrahim and Brutsaert, 1965, 1966). Apparently,
equations with the same and similar functional forms as (10.85) were first used by Maillet
(1905; Boussinesq, 1904) in his analysis of drought flows of the Vanne River.
10.4
LINEARIZED HYDRAULIC GROUNDWATER THEORY:
A THIRD APPROXIMATION
A major disadvantage of all formulations to describe unconfined groundwater flows,
discussed so far in this chapter, is that they are nonlinear. Even the simplest, those based
on the hydraulic approach, still suffer from this, and there is no general methodology
available for their solution. It is understandable, therefore, that attempts have been made
in the past to remedy this by linearization. Because the solution to a linear problem
can be represented as a unit response function, it can be generalized and extended
to many different boundary and initial conditions by simple superposition. Moreover,
once obtained, such unit response functions can provide a direct link between the main
underlying physical mechanisms captured by the Boussinesq equation, and the more
abstract mathematical aspects of general linear systems (i.e.unit hydrograph) approaches
at the catchment scale (see Chapter 12).
10.4.1
General formulation
The common way to linearize the Boussinesq equation, in the case of its simplest form,
Equation (10.30), is to expand the second derivative on the right-hand side as follows
∂η
∂
2
∂η
∂
+
η
∂
2
η
k
0
n
e
t
=
(10.87)
x
2
x
∂
