Geoscience Reference
In-Depth Information
analog model (Ibrahim and Brutsaert, 1965). Figure 10.14 shows how, as time increases, the
outflows obtained with the one-dimensional hydraulic approximation become practically
the same as with the two-dimensional Laplace equation.
10.3
HYDRAULIC GROUNDWATER THEORY:
A SECOND APPROXIMATION
Free surface representations of flow in unconfined aquifers, as outlined in Section 10.2,
are usually easier to solve than those based on Richards's equation, which include also
flow in the partly saturated zone above the water table. Nevertheless, the implementation
of this simplification for problems in catchment hydrology is rarely straightforward and,
even when obtainable, the resulting solutions can usually not readily be parameterized
for this purpose. Therefore, further simplifications are called for. One very common
approach is based on the observation that unconfined aquifers in natural catchments tend
to be relatively thin compared to their horizontal extent. Thus beside the assumption that
the water table is a true free surface, it is also assumed that under such conditions, the flow
is essentially parallel to the ground surface and/or to the underlying impermeable bed.
Specifically, the first assumption requires that the capillary zone number Ca
=
H
c
/
D
is small, whereas the second requires that the aquifer is shallow, so that
B
+
=
D
is
large. These two assumptions constitute the basis of the hydraulic groundwater theory.
It will become clear below that the hydraulic approach is considerably simpler and more
parsimonious than the more complete formulations described in Sections 10.1 and 10.2;
moreover, in many situations it produces a solution which is quite close to that obtainable
by a more complete formulation. Hence not surprisingly, this approach continues to be the
method of choice in many investigations. The hydraulic approach is usually attributed
to Dupuit (1863). It has also been referred to as the Dupuit-Forchheimer theory, to
acknowledge the fact that Forchheimer (1930) applied it to many different problems.
B
/
10.3.1
General formulation
The governing differential equations for this approach can be derived by combining the
continuity equation with Darcy's law adjusted for the hydraulic assumptions.
Adjustment of Darcy's law
The main assumption is essentially the same as that commonly made for open channel
flow. It is that the curvature of the streamlines is very small, so that the pressure distri-
bution is practically hydrostatic in the direction normal to the impermeable bed. For the
two-dimensional cross section of the aquifer shown in Figure 10.15, Equation (5.5) is
directly applicable and can be rewritten as
∂
p
w
∂
z
+
γ
w
cos
α
=
0
(10.23)
in which
is the slope angle of the underlying impermeable layer, and
z
is the coordinate
normal to that layer. Observe that with a sloping bed
x
and
z
are related with the vertical
α
