The simulation grid and finite differences grid are aligned to the new axes after this rotation.
It is the flexibility of the methodology based on joint flow simulations, as opposed to analytical models, that allows
these complex physical features of the aquifer to be taken into account. The irregular boundary conditions, the effec-
tive recharge, and the impermeability of overlying geologi- cal layers are all used as supplementary information by the
numerical flow simulator during the calculation of each hydraulic head field. The subject of numerous geological
and hydrogeological studies, this aquifer has been very well sampled. The position of the 41 transmissivity values
and the 153 hydraulic head measurements is also shown in Fig. 1. The transmissivity ranges from 2 3 10
¹ 6
to 1.3 3 10
¹ 3
m
2
s with an average value of 2.1 3 10
¹ 4
m
2
s.
4.1 Variographic analysis of the log-transmissivity
The variographic analysis of the transmissivity needs to take two factors into account: the error component associated
with each experimental transmissivity value and the regu- larized nature of this data, that is, the fact that they do not
represent punctual i.e. point-support values. Each trans- missivity value is representative of the area around the
well over which the influence of the pumping test is felt by a modification to the pre-pumping aquifer level. This
area, defined by the so-called radius of action of the well, i.e. the distance from the well beyond which the drawdown
caused by the pumping test is negligible, is the support over which the transmissivity measurement has been taken.
Under certain simplifying hypotheses
23
the radius of action can be approximated as R
a
¼ 1:5
Tt=S
p where t is
the pumping time and S the storage coefficient. Each trans- missivity measurement is therefore assumed to be represen-
tative of the average transmissivity within the circle of radius R
a
centred on the well. The distribution of the experimental R
a
is presented in Fig. 2, where the size of each open circle is proportional to
the value of R
a
. The lack of an accompanying piezometer prevents us from calculating the radius of action for the five
wells marked by a dot. Two distinct populations can be identified: the group of wells to the northwest of the aquifer
having an average R
a
of about 120 m, and the rest for which the average R
a
is about 20 m. The first group is due to the presence of recent superficial alluvial deposits only found in
that part of the aquifer. To account for these differing sup- port sizes, the transmissivity data is divided into two groups,
with the mean R
a
of each group being assigned to all wells within that group as the support size associated with the
transmissivity values. Two series of experimental variograms are calculated.
The first series is calculated from only the southern population 34 data, for which the average circular
support has a radius of about 20 m. The second series is then calculated from all the data. In this case the average
support size more than doubles. The regularization of a point variogram model allows us to correctly model the
changing behaviour of the experimental variogram accord- ing to the size of the support considered. The link between
the point variogram g and its regularized counterpart g
R
is
Fig. 2. Proportional representation of the experimental radius of action: five missing values X.
Combining geostatistics and flow simulators to identify transmissivity 559
given by: g
R
h ¼
measurement error variance þ
1 V
R
a
V
R
a
p
Z
V
Ra
x
du
Z
V
Ra
p
x þ h
g u ¹ v
dv
where V
R
a
and V
R
a
p are the areas of the circular supports whose radii are R
a
and R
a
, respectively. These circles are
centred on two points separated by the distance vector h. The measurement error variance, due to the incertitude
associated with the transmissivity data, is discussed in the following paragraph. The sill and range of the point vario-
gram model are chosen so that the resulting g
R
h correctly
fits the experimental variograms for different support sizes. As an example Fig. 3 shows the omnidirectional variogram
for all data fitted by a spherical variogram model: the con- tinuous line represents the regularized variogram model
g
R
h for which the support of the data is taken into
account, and the dotted line is its point support equivalent g
h. Anisotropic behaviour was seen from the individual
