Environmental Engineering Reference
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the cumulative density function and/or the probability
density function can be estimated (Figure 8.4d). The dis-
tribution is not symmetric, its median is equal to 4.7 m,
which is significantly larger than the deterministic esti-
mate (3.17 m), and it displays a wide range of possible
values between 1.6 m and 15 m. The manager of the con-
struction can use both types of results for risk analysis.
In terms of design for the construction project this opens
the way for two approaches. On the one hand, the man-
ager can include the uncertainty results in the design of
an optimal construction scheme using, for example, loss
function analysis (Srivastava, 1990) to take the decision
that will minimize the expected financial losses. On the
other hand, when the uncertainty is large and, conse-
quently, the risks too high, the manager can decide to
gather additional data to reduce the uncertainty. Again,
the stochastic model can help to locate those additional
measurement points.
As compared to the deterministic method described
earlier, the advantage of the Monte Carlo approach is that
it can be applied for any statistical distribution (including
non-parametric distributions) and make use of any deter-
ministic model. In particular, the deterministic model can
be very complex and does not need to be linear, which is
the main reason why this approach has been used exten-
sively in a wide manifold of applications (for example,
Naff et al ., 1998; Sambridge and Mosegard, 2002). The
limit of the Monte Carlo approach is that it may be
extremely time consuming when complex deterministic
(numerical) models need to be computed many times
(for example, systems that are three-dimensional, or with
nonlinear transients).
In that study, the main source of uncertainty was
on the geology of the coastal aquifer, which is made
of karstic limestone with interbedded conglomerates on
top of a marl deposit. The marine environment causes
partial dissolution of the limestone by interaction with
seawater. Consequently, the host rock presents karstic
cavities as well as a large number of irregularly distributed
small conduits. The hydraulic conductivity is very high
at those karstic features while it drops dramatically at
places where no dissolution occurred. The location of
these karstic features is, indeed, not known a priori .
The work consisted of two main steps. First, the aquifer
was characterized using a stochastic approach. The result-
ing model describes the system in the absence of the target
pumping network. Second, once the model properly rep-
resents the expected patterns of aquifer heterogeneity and
architecture (i.e. when the model is properly inversely
calibrated and honours available data), it is used to fore-
cast the impact of pumping at the new potential wells and
to define the optimum pumping scheme.
The construction of the model followed three main
steps. First, hydraulic conductivity measurements aris-
ing from three pumping tests and geophysical data
(Figure 8.5) were used to build a geostatistical model
of the hydraulic conductivity field k ( x , y ). More precisely,
the field was assumed to be multi-Gaussian and to fol-
low a lognormal univariate distribution. Therefore its
inference requires the definition of just its mean and
covariance from available field data. In addition, a tran-
sient deterministic finite element model representing the
groundwater flow towards the sea was built. This model
allows us to calculate the hydraulic head distribution as a
function of space and time h ( x , y , t ) for a given hydraulic
conductivity field k ( x , y ). Generating many k fields and
solving the deterministic model with them allows us to
estimate the uncertainty on hydraulic head as described
in the previous section.
However, based on this first model only, the uncertainty
would be unrealistically large. Therefore, it is important
to constrain the stochastic model with the available tem-
poral series of hydraulic heads. Indeed, the heads vary in
space and time (Figure 8.6) in response to tidal fluctu-
ations along the northernmost boundary of the aquifer
and in response to the three large-scale pumping tests
conducted in the aquifer for this study. The recorded
signal of hydraulic head variations measured at avail-
able observation wells contains indirect information on
the spatial distribution of hydraulic conductivity. Con-
straining it in such a way that the calculated heads are
similar to the observed heads is a typical inverse problem
8.4 A practical illustration in Oman
This section illustrates the concepts of deterministic and
stochastic modelling introduced previously. To that end,
we use real data gathered in a coastal aquifer in Sur
(Oman), where a desalination facility currently pumps
brackish groundwater at an insufficient rate to satisfy the
growing demand of freshwater in the area. The aim of this
work was to design a new pumping network to increase
the current extraction roughly by a factor of nine (Alcolea
et al ., 2009). To that end, new pumping wells needed to
be sited and their pumping rates defined. The aim was to
achieve the target discharge while minimizing the envi-
ronmental side effects (minimum drop of the hydraulic
head) and the demand for energy for the pumping, thus
minimizing the total cost of the solution.
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