Tracers and Modelling in Hydrogeology
(Proceedings of the TraM'2000 Conference
held at Liège, Belgium, May 2000). IAHS Publ. no. 262, 2000.
303
Laboratory experiments for diffusion transport
processes in fractured chalk
K. WITTHÛSER, H. HÔTZL
Applied Geology, University of Karlsruhe, Kaiserstrasse
e-mail: [email protected]
12, D-76128 Karlsruhe,
Germany
B. REICHERT
Geological Institute, University of Bonn, Nufiallee 8, D-53115 Bonn,
Germany
W. STICHLER
GSF-Institute of Hydrogeology,
Ingolstaedter Landstr. 1, D-85764 Neuherberg,
Germany
R. NATIV
Hebrew University of Jerusalem, Seagram Center of Soil and Water, Faculty of Agriculture,
PO Box 12, Rehovol 76100, Israel
2
1
Abstract Effective diffusivities of various artificial tracers of 0.59-2.78 E-6 cm s"
for Maastrichtian chalk (Danish), and of 0.919-6.22 E-7 cm s" for Eocene
(Israeli) chalk samples were obtained with the help of through-diffusion
experiments. The variations can be explained by Archie's law with a distinct
cementation factor for the chalk varieties tested. The Eocene chalk shows
small relative diffusivity resulting in an extremely large m value of 3.3 due to
its narrow and wide range of pore-throat sizes. The rock capacity factors
obtained indicate retardation of pyranine in all chalk samples and batch
experiments performed in parallel confirmed the sorptive behaviour of
pyranine in chalk.
2
1
INTRODUCTION
Diffusive mass transfer between mobile and immobile phases is an important transport
mechanism and a central factor for sorptive uptake and release of solutes in fractured
media. The importance of matrix diffusion as a retardation process for contaminant
transport in fractured media is well known from risk analyses for nuclear and
underground waste deposits. Most of the studies deal with diffusion of radioactive
substances in the crystalline host rocks tested. Little data is available for other
substances and/or other rock types.
In the framework of an international joint project (EC: ENV4-CT97-0441) an
interdisciplinary research group is investigating contaminant transport, monitoring
techniques, and remediation strategies in fractured chalk across Europe. Our work
encompasses the investigation of diffusion processes in different chalk varieties,
compared to other fractured media, by means of laboratory experiments, as well as
with analytical and numerical modelling. This paper deals mainly with throughdiffusion experiments with chalk samples from Denmark and Israel performed with
various substances frequently used as artificial tracers in field investigations. The
tracers being compared are lithium, bromide, uranine, pyranine, sodiumnapthionate,
amidorhodamine G, sulphorhodamine B and deuterium.
304
K. Witthûser
et al.
EXPERIMENTAL SETUP
Chalk samples collected from boreholes in Sigerslev, Denmark (Maastrichtian chalk)
and in the Negev Desert, Israel (Eocene chalk), were saturated and then used for
measuring the diffusion coefficients using diffusion cell devices (e.g. Feenstra et al.,
1984). As far as possible the samples were taken perpendicular to fracture surfaces.
We ensured a complete saturation by evacuating the samples in a desiccator before
leaving them for at least two weeks in degassed water.
BASIC EQUATIONS
The one-dimensional diffusive flux J i of a solute in a porous media due to a gradient in
concentration c can be described by Fick's first law:
c
dc
S dc
(1)
dx
' x dx
where D is the effective diffusion coefficient, D is the aquatic diffusion coefficient,
8 is the constrictivity and x is the tortuosity. Note that the definition of the effective
diffusion coefficient includes the transport porosity s . The change in concentration at a
point in the porous media due to diffusive transport of a sorbing solute is described by
Fick's second law:
-A
aq
2
e
aq
t
/
^ \dc
2
^
dc
(2)
where e is the total porosity, p is the bulk density and Kd is the sorption coefficient for
a linear isotherm. The term in parentheses can be summarized to the rock capacity
factor a. An analytical solution of equation (2) for a porous slab (Carslaw & Jaeger,
1959) is used to describe the observed concentration change (Fig. 1) in the receiving
cell:
C(0
2
e
n
V
2
DE%t
AdC
a
6
n rr,
s
2
a d
(3)
Co denotes the concentration of tracer reservoir cell, d the thickness (1 cm) and A the
surface area (15.5 cm ) of the rock sample. The two fit parameters, the effective
diffusivity D and the rock capacity factor a, were initialized with values determined
from the slope of the linear part of the breakthrough curve (steady-state method, e.g.
Bradbury & Green, 1985) and then estimated with a least-square routine (LevenbergMarquardt-algorithm). This procedure takes into account the transient phase (Fig. 1) of
the breakthrough curve (i.e. the time to develop a linear concentration gradient in the
rock sample) for the determination of the diffusion coefficient and the rock capacity
factor. The resulting parameter and the total porosity (mercury porosimetry) of the
samples are given in Tables 1 and 2.
The determined effective diffusion coefficients were used to calculate the
formation factor F (relative diffusivity D ) and the exponent m (cementation factor) of
Archie's law (Archie, 1942; Klingenberg, 1951), both depending only on the
2
e
Laboratoiy
experiments
for diffusion
transport
processes
in fractured
chalk
305
properties of the rock samples:
1
F = D = D ID
e
aq
= s'"
(4)
The goal of the application of Archie's law is to establish a reasonable relationship for
the investigated chalk varieties, which can be used for the prediction of the diffusion
coefficient for any kind of tracer or substance of interest, thus avoiding timeconsuming diffusion experiments.
T a b l e 1 Results of diffusion cell e x p e r i m e n t s for M a a s t r i c h i a n chalk s a m p l e s (Sigerslev, D e n m a r k ) .
2
1
D (cm s' )
cc(-)
D„ (cm s~')
c
2
2
1
D (cm s" )
s(-)
m
F{-)
m (-)
Uranine
Lithium
Pyranine
Eosin
Amidorhodamine
G
Sulphorhodamine
1.79 E-6
2.625
6.82 E-7
5.90 E-7
1.071
5.51 E-7
6.79 E-7
2.566
2.64 E-7
9.39 E-7
1.196
7.85 E-7
7.57 E-7
2.895
2.61 E-7
4.54 E-6
0.5143
0.395
1.4
6.3 E-6
0.5143
0.094
3.6
3.81 E-6
0.443
0.178
2.1
3.29 E-6
0.4992
0.286
1.8
3.7 E-6
0.5143
0.204
2.4
Bromide
Deuterium
B
Sodiumnaphtionate
8.45 E-7
1.412
5.98 E-7
1.31 E-6
1.587
8.24 E-7
2.78 E-6
10.087
2.76 E-7
4.23 E-6
1.245
3.51 E-6
3.6 E-6
0.4992
0.234
2.1
5.69 E-6
0.443
0.230
1.8
1.19 E-5
0.4992
0.234
2.1
2.15 E-5
0.5143
0.196
2.5
T a b l e 2 Results of diffusion cell e x p e r i m e n t s for E o c e n e chalk s a m p l e s ( N e g e v desert, Israel).
White chalk (24 m depth):
Uranine
Lithium
A , ( c n r V )
<*(-)
2
1
D„ (cm s" )
s(-)
F(-)
m(-)
9.19 E-8
0.386
2.38 E-7
0.352
0.0202
3.73
6.22 E-7
0.432
1.44 E-6
0.389
0.0987
2.45
Pyranine
Grey chalk (113 m depth):
Uranine
Pyranine
1.07 E-7
1.654
6.44 E-8
0.3084
0.0280
3.04
3.46 E-7
0.967
3.58 E-7
0.452
0.07622
3.24
2.78 E-7
2.40
1.16 E-7
0.438
0.0730
3.17
K. Witthûser
306
et al.
RESULTS
Although assumed as comparable non-reactive tracers, remarkable differences have
been observed in the diffusion experiments. A range of effective diffusion coefficients
for Maastrichtian (Danish) chalk samples of 0.59-2.78 E-6 cm s" (Table 1) was
obtained, with the highest values for bromide and uranine and the lowest value for the
lithium cation. The Eocene (Israeli) chalk samples show effective diffusion
coefficients of between 0.919 and 6.22 E-7 cm s" (Table 2), and contrary to the
Danish samples, uranine had the lowest diffusion coefficients and lithium the
highest.
In general our results for Danish and Israeli chalk samples are in accordance with
Hill (1984) and Gooddy et al. (1996), reporting apparent diffusion coefficients in UK
chalk samples, for several anions, in the range 0.28-3.23 E-6 cm s" , and
2.5-8.8 E-6 cm s" respectively. The observed variation of effective diffusion
coefficients can be explained to a large extent by the different aquatic diffusion
coefficients and porosity of the chalk varieties tested (Fig. 2). Figure 3 shows the
relationship between the formation factor and porosity for the different chalk samples.
Porosity variations and differences in the aquatic diffusion coefficients determine the
variation of effective diffusion coefficients significantly. The effective diffusion
coefficient can be estimated according to equation (4) with a cementation factor of m
equal to 2.2 for the Maastrichtian chalk and 3.3 for the Eocene chalk. We used only the
results of the anionic tracers to calculate Archie's law for the different chalk samples
since we can assume different interactions with the matrix for cationic tracers. So far
we have not found a general trend for differences of cationic vs anionic tracer
diffusivities and more experiments related to this question are on the way.
It is obvious from Fig. 3 that no unique relation between porosity and formation
factor for the different chalk samples exists. The lower formation factors and
respectively higher cementation factors of the Israeli chalk samples can be explained
2
2
1
1
2
2
1
1
Laboratory
experiments
for diffusion
transport
processes
in fractured
307
chalk
0,40-4
W h i t e chalk, Israel
m
0,35 4
0,30 4
•
Grey chalk, Israel
A
Danish chalk, Sigerslev
A
Archie's law, m=2.2
Archie's law, m=3.3
A
A
A
..-••A"
I
0,15-j
0,10-4
0,05 H
0,00 |
0,30
.
|
.
i
i
i
.
i
i
i
i
i
I
0,32 0,34 0,36 0,38 0,40 0,42 0,44 0,46 0,48 0,50 0,52 0,54
porosity e [-]
F i g . 3 Plot of f o n n a t i o n factor against porosity for the chalk s a m p l e s .
by the smaller and wider range of pore-throat sizes (Fig. 2) compared to the
Maastrichtian chalk samples.
Adler et al. (1992) found an m value of 1.64 for Fontainbleau sandstone, and
Grathwohl (1998) m values of 1.8-2.4 for several sedimentary rock samples.
Calculating the Archie's exponent for the diffusion coefficients in uncracked UK chalk
samples, Hill (1984) reports values between 2.1 and 2.6. While the exponent of
Archie's law for the Maastrichtian chalk is in the range of the reported values, the m
value of 3.3 of the Israeli chalk tested exceeds values in the literature (e.g. Gooddy et
al, 1996, Grathwohl, 1998, Hill, 1984). The Eocene chalk is generally characterized
by a small relative diffusivity. Therefore it exhibits only limited retardation capacity
with respect to diffusive mass transfer of tracers as well as of inorganic contaminants
compared to the Danish and English chalk. However, diffusion is only one of the
possible retardation factors. As a result of the relative high organic content of the
Grey chalk (1.3%), strong sorption of organic contaminants can be expected and is
currently under investigation.
The rock capacity factor a, calculated from the chalk diffusion experiments
indicates distinct retardation of bromide and pyranine. Both tracers, but in particular
the anion bromide, has been proven to behave as a nearly ideal tracer with respect to its
interaction with the solid matrix, in many tracer studies. Batch experiments performed
in parallel confirmed the sorptive behaviour of pyranine (K<t = 30.09 ml g" ) but not of
bromide (3.4 ml g"'). As usual the distribution coefficients which result are one
magnitude higher due to the larger accessible surface area. More experiments are
necessary to prove the behaviour of bromide in chalk, while pyranine can only be
regarded as a reactive tracer in chalk.
1
Acknowledgements The work was supported by the EC under ENV4-CT97-0441.
Dr J. Bloomfield, British Geological Survey, UK, performed the mercury injection
porosity measurements.
308
K. Witthûser
et al.
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