Geotechnical News • June 2013
33
GROUNDWATER
because the variances are small for
this fairly homogenous sand aquifer.
The Blainville test (Quebec) was
carried out after reaching steady-state
pumping conditions in an unconfined
stratified aquifer (
K
ave
= 1.2 x 10
-4
m/s), using a spike of lithium chloride
in the screen of a monitoring well. The
test gave
n
e HEHA
= 5%, whereas
n
≈
40% for each layer. Trenches revealed
1- to 5-cm thick nearly horizontal
layers, varying from pea gravel to silt,
each soil being uniform, and
K
vary-
ing from 10
-7
to 10
-3
m/s. A lognormal
K
distribution, with
μ
ln
K
= -11.107
and
σ
ln
K
= 2.039 (Fig. 4), covers the
K
range: it yields
K
ave
= 1.2 x 10
-4
m/s and
n
e HEHA
=
5%. For com-
parison, a normal
K
distribution
with the same
K
ave
strongly
differs from
the lognormal
distribution (Fig.
4), and predicts
n
e HEHA
= 0.40
as for individual
layers, a value
much higher than
the field value of
5%.
The data of those
field tracer tests,
for which all
needed informa-
tion could be
gathered, appear in Table 2 and are
plotted in Fig. 1: eq. (1) predicts cor-
rectly the experimental
n
e HEHA
values.
Discussion
Despite academic progress, there is
no reliable method to predict
n
e
. The
missing information about
n
e
is regret-
table for all specialists who need to
predict the fate of contaminants and
protect drinking water supplies. The
objective of this paper was to verify a
new analytical solution for the field
n
e
in stratified aquifers having a lognor-
mal
K
distribution, under plane flow
(Chapuis 2015).
The approach was to use stratified
aquifers with no local dispersion, in
order to highlight the role of heteroge-
neity in velocity fields. This approach
may seem outdated for those currently
involved in research on tracer tests. In
fact, the equations of this paper could
have been developed in the 1970s
or 1980s, but they were not before
2015. Recent theories on tracer tests
have increased complexity, number
of parameters, and yet they cannot
predict effective porosity
n
e
and have
limited predictive capacity for longi-
tudinal dispersivity
α
L
. The existing
1D, 2D or 3D methods are only fitting
methods, with many fitting param-
eters, which obscure physics and are
too vague for practitioners.
As a result, this paper is the first one
with a predictive equation for the
effective porosity of the hydrauli-
cally equivalent homogenous aquifer
(HEHA). This new equation is sup-
ported by field data.
The recently proposed analytical equa-
tions of Chapuis (2015) had shown
that a lognormal
K
distribution can
fully explain: (i) the early arrival of
the tracer in field tests, using an equa-
tion providing the
n
e HEHA
value; (ii)
the increase of
α
L
with distance and
also with the variance of the lognor-
mal
K
distribution; and (iii) the long
thick tail of field breakthrough curves.
This paper has added an experimental
verification of the predictive equation
for
n
e
.
Conclusion
Current bylaws require to delineate the
total area from which groundwater is
captured (the catchment), and a few
capture zones which are used to define
protection perimeters. However, only
a few national bylaws require field
tracer tests converging towards the
pumping well. As a result, groundwa-
ter professionals need to predict the
values of effective porosity,
n
e
, and
longitudinal dispersivity,
α
L
. However,
despite academic progress, there is
still no reliable method to predict
n
e
.
Figure 4. Blainville field tracer test (steady-state pump-
ing): the normal and lognormal K distributions which fit
K
ave
are quite different. The lognormal K distribution is the
only one that also fits the K range and the n
e HEHA
of 5%
given by the tracer test.
Table 2 – Collected data for
n
and
n
e HEHA
,
to assess eq. (1), as shown in Fig. 1.
No.
site
n
n
e, HEHA
σ
ln
K
n
e, HEHA
/
n
e
1
2
3A
3B
4A
4B
5A
5B
6
7
CapeCod
Tucson
Scheldt
Scheldt
Hanford
Hanford
Borden
Borden
Lachenaie
Blainville
0.39
0.315
0.39
0.40
0.35
0.40
0.332
0.332
0.40
0.40
0.39
0.17
0.38
0.38
0.10
0.10
0.30
0.30
0.33
0.05
0.37-0.49
0.96
0.15
0.20
1.5
1.6
0.54
0.70
0.66
2.0
confused
0.540
0.974
0.950
0.286
0.250
0.904
0.904
0.825
0.125
