Environmental Engineering Reference
In-Depth Information
Hydraulic head [m]
5
4
River
Building
3
2
Hill
1
0
0
1000
2000
Distance to river [m]
3000
4000
5000
Time(years)
Time(years)
(a)
(b)
Figure 8.2
Example setup: (a) geometry of the problem, (b) hydraulic head distribution along a flow line computed with the
deterministic solution of Equation (8.11).
This partial differential equation is one typical expres-
sion of a deterministic physical model used in ground-
water hydrology. As explained above, it was derived from
the conservation principle and a phenomenological law
(Darcy's law). It was expressed for a given geometry and
with several simplifying assumptions (compressible fluid,
steady-state, and so forth). Solving for a unique solution
of Equation (8.9) requires some boundary conditions.
In this example, we assume that the hydraulic head is
equal to a given value
h
r
along the river located at
x
analytical expressions cannot be obtained in practice,
either because the geometry is too complex or because
the parameters vary in a complex manner (for example,
hydraulic conductivity changes depending on the geolog-
ical material or precipitation and recharge vary in space
and time). Therefore, deterministic models are usually
solved with numerical techniques such as finite elements
or finite volumes (Huyakorn and Pinder, 1983) (see
Chapters 6, 10 and 11). Overall, the main strength of the
deterministic approach is that it allows us to understand
the influence of certain parameters or processes on the
variables of interest. It also allows us to make forecasts
based on well-established physical principles. In practice,
we have seen that those forecasts can either be obtained
via simple analytical expressions such as Equation (8.11)
or, more generally by solving complex numerical models,
which is often the case in practice.
0
(actually,
h
r
is the elevation of the water table at the river),
and that the flux entering the aquifer along the foothill
(
x
=
=
L
) is equal to
q
. The integration of Equation (8.9)
using the aforementioned boundary conditions yields the
following unique solution:
u
=
x
r
(
L
−
u
)
+
q
k
(
u
)
=
+
h
r
.
h
(
x
)
du
(8.10)
u
−
0
If, in addition one assumes the hydraulic conductivity
k
constant in space, we get:
8.3.3 Stochasticmodels
q
x
The limitation of deterministic models is that they do not
account for uncertainty. The parameters governing the
equations are supposed to be known and the solutions are
therefore unique. This limitation often poses a practical
problem because nature is intrinsically heterogeneous and
the system is only measured at a discrete (and often small)
number of locations. Therefore, even if the physics of the
system is relatively simple and understandable by deter-
ministic equations, it is difficult to trust the solutions
of deterministic models because the input parameters,
r
2
k
x
2
+
rL
h
(
x
)
=
+
+
h
r
.
(8.11)
k
which expresses how the hydraulic head varies within
the domain for any values of the parameters describing
the geometry and properties of the aquifer. For a given
set of parameters, the solution to Equation (8.11) can
be calculated and plotted. Figure 8.2b displays the
solution obtained with k
=
10
−
2
ms
−
1
,
r
=
100 mm a
−
1
,
q
5ma
−
1
,
L
=
=
5 km.
Unfortunately,
such
types
of
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