Geoscience Reference
In-Depth Information
p
w
<0
WT
p
w
>0
D
z
D
c
x=B
x
Fig. 10.2 Schematic representation of the cross section of an unconfined riparian aquifer, lying on a horizontal
impermeable layer. The origin of the coordinates is taken at the stream (
x
=
0) and at the
impermeable layer (
z
=
0);
D
c
is the water depth in the adjoining stream,
D
is the aquifer thickness,
and
B
is the aquifer breadth, that is the distance from the stream to the divide. The water table (WT),
is the locus where the water pressure
p
w
is atmospheric, or
p
w
=
0. Above the WT the soil is partly
saturated, and the water pressure smaller than atmospheric; below the WT the water pressure is
larger than atmospheric.
assumption that it is possible to define (or imagine) a spatially uniform model aquifer,
with effective parameter functions
k
) which, upon solution of
Equation (10.1) with appropriate boundary conditions, produces the same flow charac-
teristics of interest, as the real spatially variable prototype aquifer. A second simplifi-
cation is based on the observation that the vertical dimensions of unconfined riparian
aquifers tend to be much smaller than their horizontal extent. This has led to the assump-
tion that the boundary conditions of real aquifers are rarely very different from those
of a two-dimensional model aquifer of rectangular cross section; for purposes of flow
analysis, the aquifer depicted in Figure 10.1 can thus be represented schematically as
shown in Figure 10.2. These two simplifications have allowed some standardization of
the groundwater outflow problem, while maintaining its main characteristics.
=
k
(
θ
) and
H
=
H
(
θ
10.1.2
Some common approximate formulations
Even with the simplifications just mentioned, the governing Richards equation (10.1)
remains highly nonlinear and most problems involving combined saturated and partially
saturated flow must be analyzed by numerical methods. With the availability of high
speed digital computers, at present there is no dearth of efficient numerical codes for this
purpose, and rapid advances continue to be made in this field. One drawback of such
exact solutions of (10.1), however, is that they cannot be easily parameterized in practical
terms for incorporation in basin-scale analyses. Thus further simplifications, which may
be valid under special conditions and for which solutions may be more readily obtained,
are often called for.
In a first approximation the flow in the zone above the water table, where
p
w
<
0, is
neglected, and the water table is treated as a true free surface; with the assumption of
an effective hydraulic conductivity and porosity, the governing equation (10.1) reduces
then to Laplace's. This case is discussed in Section 10.2. In a second approximate for-
mulation, beside the free surface assumption, the distribution of the water pressure in the
general direction normal to the flow is assumed to be hydrostatic; these two assumptions,
also called the Dupuit assumptions, are the basis of the hydraulic groundwater theory,
