ln T at a very fine scale over the entire field. From this almost continuous ln T field, two sets of ln T values are
required for the cokriging equations. Firstly, the ln T values at the points to be estimated need to be selected.
These points, which normally constitute a regular grid, will be denoted by the coordinate vector x
i
¼ x
i
, y
i
. Sec-
ondly, the ln T value associated to each point x
a
must be
obtained where x
a
¼ x
a
,y
a
represents the location of the experimental transmissivity data. This involves regrouping
the simulated punctual ln T values around x
a
to form a value that is representative of the zone of influence of the given
well test. This point, the regularized nature of the transmis- sivity data, will be discussed in more detail in Section 4.1.
2.2 Numerical hydraulic head calculation steps 2 and 3
The hydraulic head field corresponding to a given transmis- sivity simulation is obtained via a numerical flow simulator
that, in our case, is based on a finite differences algorithm. A slight modification
19
to the standard finite differences algorithm, based on the direct entry of interface transmis-
sivities into the finite differences flow model, is used and leads to the resulting hydraulic head fields being globally
unbiased
20
. That is, the flux generated by the resulting head field is equal to that of the equivalent homogeneous aquifer,
the equivalent transmissivity being equal to the geometric mean in our case. Each interface transmissivity is calculated
as the local equivalent transmissivity from one grid mesh centre to the next. The numerical flow simulator then pro-
duces a punctual hydraulic head H value assigned at the centre of each discrete grid mesh.
Given that our particular case study lends itself to be modelled in two-dimensional steady-state flow, the diffu-
sion equation solved by the numerical flow simulator is: div{T
x ,
y grad H
x ,
y } þ Q
x ,
y ¼
where Q is the effective recharge assumed to be known. From this equation, we propose to use the variations,
denoted by f, of the hydraulic head about its mean h as the second variable of interest. These local variations best
reflect the influence of local variations of the log-transmis- sivity. The value of the hydraulic head itself essentially
provides the global form of the flow that somewhat masks the influence of the smaller-scale head variations
which result from the local variations of the log-transmis- sivity ln T.
The mean hydraulic head, hx ¼ EHx, can be calcu-
lated at each finite differences grid node as the average head value over the series of independent head simulations.
The value of the head variation f is then obtained as the difference between the hydraulic head and its mean: fx ¼
H
x ¹ hx. Variographic studies of the head perturba-
tion
12,8,21
show it to vary in a much smoother fashion than ln T. This smoothness justifies calculating f at the experi-
mental hydraulic head data points x
b
by a simple linear interpolation of the four nearest grid node f values.
3 THE NUMERICALLY OBTAINED COKRIGING SYSTEM
3.1 Calculating numerical covariances step 4 Values of ln Tx
