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Tracer tests in natural fissures performed in the laboratory are analyzed by means of fitting two different models. In the experiments, sorbing and nonsorbing tracers were injected into a natural fissure running parallel to the axis of a drill core.

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WATER RESOURCES RESEARCH, VOL. 21, NO. 7, PAGES 951-958, JULY 1985
Analysis of Some Laboratory Tracer Runs in Natural Fissures
LuIs MORENO AND IVARS NERETNIEKS
Department of Chemical Engineering, oyal Institute of Technology, tockholm, weden
TRYGGVE ERIKSEN
Department of Nuclear Chemistry, Royal Institute of Technology, tockholm, weden Tracer tests n natural fissures performed n the laboratory are analyzed by means of fitting two
different models. n the experiments, orbing and nonsorbing racers were injected nto a natural fissure running parallel to the axis of a drill core. The models ake into account advection, dispersion, diffusion
into the rock matrix, and sorption onto the surface of the fissure and on the microfissures nside the
matrix. For the second mechanism, one of the models considers hydrodynamic dispersion, while the
other model assumes hanneling dispersion. The models take into account time delays n the inlet and
outlet channels. The dispersion characteristics nd water residence ime were determined from the
experiments with nonsorbing racers. Surface and volume sorption coefficients nd data on diffusion nto
the rock matrix were determined or the sorbing tracers. The results are compared with values ndepen-
dently determined n the laboratory. Good agreement was obtained using either model. When these models are used or prediction of tracer transport over larger distances, he results will depend on the model. The model with channeling dispersion will show a greater dispersion han the model with hydrodynamic ispersion, ssuming onstant dispersivity.
INTRODUCTION
The final disposal of radioactive wastes rom nuclear power
plants has been studied n many countries. n Sweden, crys-
talline rock has been selected as the most suitable bedrock in
which to build a repository. f a canister s broken, adio-
nuclides will be carried by the water flowing in the bedrock.
The radionuclides may interact with the rock by mea ns of
sorption onto the surface of the fissures nd by diffusion nto
•he rock matrix and sorption onto the surfaces f the inner
microfissures.
The sorption of radionuclides on the rock and tl•e diffusion into it have been studied n laboratory experiments. he trans-
port through fissures n the rock has been studied both in the laboratory and in in situ experiments Neretnieks et al., 1982; Abelin et al., 1984].
The aim of this study s to test the capability of two models to predict the transport of Strontium hrough a single issure using data from experiments carried out in the labo•ratory.
Velocity variations are modeled by hydrodynamic dispersion
in one model and by channeling ispersion n the other model.
The influence of the selected model, when the results from
these ests are used o predict breakthrough curves or longer
migration distances, s also studied.
EXPERIMENTAL
Two granitic rill cores aken rom the Stripa mine were
used n the experiments. ach drill core had a natural issure
running parallel to the axis. The dimensions f drill cores were core A, 18.5 cm in height and 10.0 cm in diameter, and core B,
27.0 crn n height and 10.0 cm in diameter.
Figure 1 shows he experimental setup. Tracers were inject-
ed into the upper channel in core A and into the lower
channel in core B. The outlet channel was simultaneously flushed with water to reduce he delay time due to the channel
volume. The water residence time in the inlet channel as well
Copyright 1985 by the American Geophysical Union.
Paper number 5W0275.
0043-1397/85/005W-0275505.00
as n the outlet channel s important ompared ith the resi-
dence ime in the fissure proper. The channel used o distrib-
ute he racer long he issure nlet had a volume f 1.1 mL
and the volume of the outlet channel was about the same. The
experimental echnique s described n some detail by Neret-
nieks t al. [1982]. The tritiated water, odide, romide, nd
lignosulphonate ons were selected as nonsorbing racers.
Strontium was used as a sorbing on. The injection flow rates
were between .22 and 1.62 mL/min for the nonsorbing
tracers. For the tests with sorbing tracer the flow rates were between 0.75 and 1.25 mL/min. The experiments were per-
formed t the department f Nuclear Chemistry y Eriksen
and coworkers.
The porosity of Stripa granite and diffusivity of the iodide
tracer and the tritiated water have been determined by Ska-
gius nd Neretnieks 1982, 1983]. A piece f granite -10 mm
in thickness was fixed between two chambers. One chamber
contained a solution with a high concentration of the non-
sorbing ion and the other chamber with a very low con-
centration of the ion. The diffusivity s determined rom the
rate at which the ion is transported hrough the piece of gran-
ite. The porosity was determined by weighing dry and wet
samples.
The sorption of strontium on crushed granite was deter-
mined by Skagius t al. [1982]. Different article ize ractions
(from 0. D to 5.0 mm) were Used. he particles were equilibrat-
ed or more han 1 year with raced olution. he arger arti-
cles had an equilibrium ime of about 1 year. From these data
surface and volume distribution coefficients nd diffusivity of
strontium were determined. The volume distribution coef-
ficient and diffusivity of strontium for pieces of granite have also been determined Skagius nd Neretnieks, 983], Pieces of granite 5 mm in thickness were used.
CONCEPTS oF THE MODELS
The transport of a species carried by a fluid flowing in a
fissure n rock is influenced by (1) molecular diffusion n the liquid, (2) variations n fluid velocity n the fissure, 3) velocity
variations between channels n the fissure, and (4) chemical or
951
952 MORENO ET AL.' TRACER TaTS IN NATURAL FISSURES
.••endp•ece _•
utlet hssure
flushing J
water J
e••jper•stalt•
J pump
n m ' - ..... __•solubon
frachonal collector
Fig. 1. Experimental setup.
physical nteractions with the solid material. Two models are
used to describe he transport through a thin fissure. Both models describe he tracer transport as taking place through a
parallel-walled fissure. The tracers penetrate the matrix by molecular diffusion and they may be sorbed onto the fissure
walls and onto microfissure surfaces within the rock matrix.
The transversal dispersion n the plane of the fissure s as- sumed to bc negligible. The tracer movement n the fissure
proper is studied considering wo different models: (1) the
hydrodynamic dispersion-diffusion model and the (2)
channeling dispersion-diffusion odel.
The first model assumes hat tracers flow in a parallel-
walled fissure, and the spreading of the tracer in the fissure s
taken into account by means of hydrodynamic dispersion
which s modeled as Fieklan dispersion. n the second model t
is assumed hat the fissure consists of parallel unconnected channels n the plane of the fissure, he channel widths having a lognormal distribution. This causes he tracer to bc carried
different distances n a given time due to the velocity vari-
ations between the channels. The velocity variations are duc
to the differences n channel width and/or flow resistance. he
hydrodynamic dispersion n each channel s assumed o be
negligible; .e., n this case he spreading s entirely due to the
variation in flow velocity hrough the different channels. he
models are shown graphically n Figure 2.
The water residence ime in the inlet channel depends on
the flow of tracer injection. The ratio of the flow of flushing water to the flow through the fissure determines he water
residence time in the outlet channel. If the overall flow
through the fissure s thought to be the sum of the flows
following several distinct pathways, each pathway will have a
different time delay in the inlet channel and in the outlet channel. The pathway closest o the inlet will have the least time delay in the inlet and the greatest n the outlet. Disper- sion for each pathway in the inlet and outlet channels s as- sumed to be negligible. Continuous flushing of the outlet
channel makes the residence time there less than in the inlet
channel. The flow through he inlet channel, issure, nd outlet channel s divided into various pathways as shown n Figure 3. Each pathway has a different ime delay.
GOVERNING EQUATIONS
H ydrodynamic ispersion-Diffusion odel
The model considers he transport of contaminants in a
fluid that flows through a thin fracture in a water-saturated , ROCK
Fissure
I
Water flow
Same velocity and hydrodynamic
dispersion in all channels
Fig. 2a. Hydrodynamic dispersion-diffusion odel.
porous rock. The tracers may be sorbed on the fracture sur-
face by an instantaneous eversible eaction. The tracers may also penetrate he rock matrix by molecular diffusion and may
be sorbcd onto the microfracture surfaces within the rock
matrix. The following processes ill be considered: 1) advec-
tive transport along the fracture, 2) longitudinal mechanical dispersion n the fracture, 3) sorption onto the surface of the
fracture, 4) molecular diffusion rom the fracture nto the rock matrix, and (5) sorption within the rock matrix. By assuming a linear isotherm for the surface sorption, the
differential equation for the transport of a tracer n the fissure
may be written n the ollowing way:
3C•. •. 2C•. •. ?C•. D CpJ
•t Ro •x 4 Ro c•x 6 R. c•z = = O (1)
where Ro s the surface etardation coefficient efined as
2
Ro=I +•Ko (2)
Other coefficients are defined in the notation.
The differential equation or the porous matrix is
c•Cp De c•ZCP 0 (3)
c•t Kdp c•z
Kd is the bulk distribution coefficient; t is based on the mass
of microfissured solid and includes the nuclide which is on the
solid as well as in the water in the microfissures. a is related
to the distribution coefficient based on the mass of the solid
proper, Ka', by
K•pp = ep + K•'pp (4)
Kdpp s usually he entity determined n sorption measure-
ments. he difference etween a'pp nd K•pp s negligible or
sorbing tracers on low-porosity materials such as crystalline
rock; K•'pp >> p or such ystems.
The system of (1) and (3) with the initial and boundary conditions existing n the tracer tests has an analytical solu- tion [Tang et al., 1981-1. he initial and boundary conditions
• ..... _•_ /"'ROCK •//
water tlOW -
Different velocities in different
pathways. No dispersion.
FiS. 2•. Channelins dis•rsion-di•usion m•el.
MORENO ET AL.' TRACER TESTS N NATURAL FISSURES 953
are zero initial concentration, constant concentration at the
inlet during the tracer injection, and a matrix and fissure of
very large extension. The tracer concentration n the outlet
may be written as
C(t)/C = (2/x/•) exp Pe/2) exp _•2 _ pe2/16•2)
( (Pe o/8A•2)
rfc t- (Pe o/4•2)} /2' • (5)
where
l = (Pe to/4t) M (6)
Pe = U x/D•. (7)
t o = Ro x/Uf = Rot• (8)
A = 6Rff2(DeKdpp) M (9)
Channeling ispersion-Diffusion odel
In this model the dispersion hat occurs n the direction of the flow is accounted or by means of channeling dispersion. The velocity differences n the channels will carry a tracer
different distances ver a given ime.
The transport of the tracers akes place through a fracture
in which parallel channels with different widths exist. This is
shown in Figure 2b. It is assumed hat the fissure aperture
widths have a lognormal distribution and the interconnection
between he different channels s negligible. he hydrodynamic dispersion n each single channel s also assumed o be negligi-
ble compared o the effects of channeling. The model includes
the following mechanisms: 1) advective ransport along the fissure, 2) channeling ispersion, 3) sorption onto the surface
of the channels, 4) diffusion into the rock matrix, and (5) sorption within the rock matrix.
For a tracer flowing through a fissure with negligible ongi- tudinal dispersion, he equation for the concentration n the
fissure is
c3Cf f 3Cf D 3Ct, = 0 (10)
3t R c3x 6 R c3z =o
The equation or the diffusion nto the rock matrix is given
as before by (3). The solution for (10) and (3) is found in the literature [Carslaw and Jaeger, 1959]'
where
Bt,• (11)
_ erfc 6(t - -•)l/2•
o
t o = Rat w (12)
B = (DeKctpt,) /2 (13)
If, in each pathway separate channels exist with different
fissure widths, 6, the fluid will have different velocities in the
different channels when flowing through the fissure. n this
case Ra will be different or the different channels. he con-
stant entity is Ka, the surface distribution coefficient. f the
breakthrough curve for each channel n the actual pathway is
given as Cf(6, t) then he concentration f the mixed effluent
from all the channels n the pathway is [Neretnieks et al.,
1982]:
©
(t) f(6)Q(6)C i6, ) a6
- (14)
Cø 6)Q(6) 6
Outlet
I I I I I I
Injection
Different locations of
the tracer due to delay
in injection channel
Flushing
Fig. 3. Residence ime distributions n inlet and outlet channels are
accounted for in the model.
In a parallel-walled hannel of width 6, the flow rate for
laminar flow is proportional to the fissure width cubed. Snow [1970], studying he fissure requencies or consolidated ock,
found that the fissure widths have a lognormal distribution. The density unction has the form
1 1 ([ln(6/#)]2.)
(6) o.(2•r)1/2 exp (15)
a 2
FIT OF THE EXPERIMENTAL DATA
Breakthrough Curve or the Effluent
To account for the influence of the finite volumes of the
inlet and outlet channels, ime delays must be considered. n each pathway the time delay is determined by the distance o
the tracer inlet. A dimensionless istance between he position
of the respective pathway and the inlet is defined: W (see
Figure 3). The time delay for a pathway may be written as
t n = tn•(• , q) + tn2(•, q, g)
(16)
where q is the water flow rate through the fissure, and g is the
ratio of the flushing low to the flow q. The breakthrough
curve at the outlet or a given pathway s
C = C'(t - tD)= C(W, t)
(17)
and the concentration of the mixed effluent rom all the path-
ways s
1
(t) = C(W, t) dW (i 8) Determination of the Parameters
The concentration of the effluent rom all the pathways for the hydrodynamic dispersion-diffusion model may be written
as
C(t)
-f(Pe, t,•, Ka, A, t) (19)
Co
For the channeling dispersion-diffusion model the con-
centration of the effluent becomes
C(t)
-f (a, œw, ,,, B, t) (20)
Co
The determination of the parameters was done by means of a
nonlinear least squares fitting. The runs with nonsorbing
tracers were used for the determination of the hydraulic
954 MORENO ET AL.' TRACER TESTS IN NATURAL FISSURES
TABLE 1. Experiments with Nonsorbing Substances Fitted With
the Dispersion-Diffusion Model, Core A
Run Tracer
Water Standard
Peclet Residence Fissure Deviation
Number Time Width of Fit
Pe t,•, rain 6, mm s/Co
A1 NaLS
A2 NaLS A3 NaLS
A4 NaLS
A5 NaLS A6 NaLS
A7 NaLS
A8 NaLS
A9 NaLS
A10 NaLS
All 3-H A12 3-H
A13 3-H
A14 I
A15 Br
15.7 1.22 0.13 0.01 11.2 1.79 0.14 0.01
14.2 2.68 0.14 0.04
9.5 3.52 0.14 0.03 91.5 5.26 0.14 0.02
36.8 10.0 0.15 0.02
59.0 5.00 0.14 0.01
26.2 5.44 0.15 0.03
14.4 10.1 0.15 0.02 15.5 5.07 0.14 0.01
109.0 2.99 0.16 0.03
43.0 5.94 0.16 0.03
85.2 11.9 0.17 0.04 15.0 2.42 0.13 0.02 29.6 2.61 0.14 0.01
D e = 0.1 x 10 •2 m2/s.
properties. These properties Pe and tw for the hydrodynamic
dispersion odel and a and •w or the channeling odel) re
used n the runs with the sorbing racer to determine he other
parameters. When the water residence ime tw (or equivalent
fissure width) is known from the nonsorbing racer runs, the
surface quilibrium oefficient, a, and the product DeKap
are obtained from the sorbing racer fits. The modelling considers he existence of various pathways
with different time delays. These time delays are calculated
considering hat the inlets for the tracer and the flushing water are on the same side of the respective channels and that the
outlet is on the opposite side of the inlet; these ocations are
shown n Figure 3.
First, the experimental data was fitted using the hy-
drodynamic dispersion-diffusion model. For the tracer tests
with nonsorbing ubstances, he determination f the parame-
ter which takes into account the interaction with the rock
matrix, the A parameter, cannot be determined with any accu-
racy. The reason for this is the short tracer residence ime,
which results in a small interaction with the matrix for a
nonsorbing tracer. The water residence imes were in the
range 2-10 min. In the runs with nonsorbing tracer the A parameter was calculated assuming a value of De=0.1
'FABLE 2. Experiments with Nonsorbing Substances itted With
the Dispersion-Diffusion Model, Core B
Run Tracer
Water Standard Peclet Residence Fissure Deviation Number Time Width of Fit
Pe t,•, min 6, mm s/Co
B1 NaLS
B2 NaLS B3 NaLS
B4 NaLS B5 NaLS B6 NaLS
B7 3-H B8 3-H
B9 3-H
B10 I
Bll Br
18.6 1.94 0.14 0.01
9.9 2.24 0.12 0.02
14.9 3.67 0.12 0.02
80.2 4.56 0.12 0.01
14.4 7.14 0.13 0.01
13.2 15.3 0.15 0.02
15.8 4.55 0.15 0.01
48.2 8.97 0.16 0.02
40.0 18.1 0.18 0.03
9.5 4.07 0.13 0.03
9.6 4.30 0.14 0.03
D e -' 0.1 x 10 •2 m2/s.
Fig. 4.
o.o 5.0 o.o
TIME, MIN
Curves fitted with the hydrodynamic dispersion model for nonsorbing racers runs A1, A2, and A3).
x 10-12 m2/s. This value of the effective iffusivity as a very
small impact on the breakthrough curve compared to when
De- 0; i.e., the diffusion into the matrix for a nonsorbing
tracer with short residence ime is negligible for a value of
De- 0.1 x 10 12 m2/s and less. Diffusion values or iodide
and tritiated water were determined by Skagius nd Neretnieks
[1982] for pieces of granite. These values are in the range
0.07-0.18 x 10 12 m2/s. The effective iffusion oefficient or
the lignosulfonate on is assumed o be of the same order of magnitude or less han the iodide and tritiated water because
this molecule s very large. A large molecule would have low
access o the micropores of the rock matrix. The surface sorp-
tion coefficient s Ka = 0 for a nonsorbing racer.
The other parameters Pe and tw) are determined by means of a data fitting. From these values the dispersion coefficient and the average issure width are calculated. The fissure width
is directly btained rom he measured low rate and he resi-
dence ime tw. The results are shown n Tables 1 and 2. Some fitted curves are shown in Figures 4 and 5. For core A the
average Peclet number was about 20 and the fissure width was
0.14 mm; for core B these values were 15.0 and 0.13 mm,
respectively.
For the runs with sorbing racer the values of the Peclet
number and fissure width determined in the runs with the
nonsorbing tracers are used. The tracer residence ime to-- Ratw and the parameter A which takes nto account he inter- action with the rock matrix are determined by a data fitting.
c•. + ß ß ..x. x
o- x x .-x ..............
+ 1•,,
0.0 lO.O 20.0
TIME, MIN
Fig. 5. Curves fitted with the hydrodynamic dispersion model for
nonsorbing racers runs B 1, B2, and B3).
MOPENO T AL.' TRACER •STS N' NATURAL FIssug• 955
TABLE 3. Experiments With the Sorbing Substance Strontium TABLE 4. Experiments with Nonsorbing Substances itted With
Fitted With the Hydrodynamic ispersion-Diffusion odel the Channeling ispersion Model, Core A
Run*
Tracer Surface Standard Standard Water Mean Standard
Residence Sorption Deviation Deviation Residence Fissure Deviation
Time A Coefficient, DeKdpr, of Fit in Lognormal Time Width of Fit t o, min Parameter K a x l0 s m 1012 m 2 s/C Run Distribution t• min $, mm s/C
A16 12.7 33.5 27 101 0.05
A17 14.3 65.1 31 34 0.07 A18 8.4 44.2 27 58 0.06
B12 13.2 70.5 16 10 0.02 B13 11.7 43.7 13 20 0.02 B14 6.6 42.0 12 19 0.02
*The Peclet numbers are 20 for core A and 15 for core B.
$D•Kdpr etermined n the laboratory s 3.5 x 10 12 m2/s or
crushed ranite particles nd 2.4 x 10 x2 m2/s or sawn pieces f
granite.
The inlet concentration s obtained directly from the experi-
mental data (concentration f the solution, njection low and flushing low). The results are shown n Table 3. The break-
through curve of a run with strontium s shown n Figure 6.
The surface orption oefficient as about 28.3 x 10 s m
for core A and about 13.7 x 10 -s m for core B. The value
determined in the laboratory [Skagius et al., 1982] is 6.6
x 10 • m. From the value of the A parameter t is only
possible o determine he product DeKaPv. n core A it was
about 6.4 x 10 • m2/s, while t was about 1.6 x 10 • m2/s
in core B. The values determined n the laboratory were 2.4
x 10 •2 m2/s for sawn pieces of granite and 3.5 x 10 •2
m2/s when hese alues were determined sing rushed ranite
particles 5 mm in diameter. The sawn and crushed pieces had
no coating material.
The channeling ispersion-diffusion odel was used only to evaluate he experiments n core A both for the nonsorbing substance NaLS) and sorbing substance strontium). n the
runs with the nonsorbing racer NaLS the B parameter was
calculated sing value of 0.1 x 10-x2 m2/s or the effective
diffusivity of the lignosulphonate on. The other parameters
were calculated y means f a fitting process a and •). The
results for the nonsorbing substance are shown in Table 4.
The mean fissure width was 0.14 mm. This same value was
obtained when the hydrodynamic dispersion model was used
to fit these data. The mean standard deviation for the loga-
rithm of the fissure width was a - 0.155 or these experiments.
To compare he values f a and Pe determined rom he it
A1 0.176 1.27 0.133 0.01 A2 0.206 1.87 0.151 0.01 A3 0.186 2.79 0.149 0.03 A4 0.220 3.71 0.150 0.03 A5 0.055 5.32 0.130 0.02 A6 0.119 10.2 0.148 0.02 A7 0.096 5.08 0.124 0.02 A8 0.138 5.57 0.151 0.03
A9 0.180 10.5 0.152 0.02
A10 0.181 5.30 0.130 0.01
D• = 0.1 x 10-x2 m2/s; NaLS racer. a theoretical elationship s determined. A breakthrough curve is created by the channeling model for different values of a. The curves so created are then used as if they were experi-
mental results. The Peclet number is determined from the first
and second moments of the breakthrough curve as ELev-
enspiel, 1972]
2/Pe = ttt2/t 2 (21)
where or a step unction njection
c = o (oo) c(t)
(•) at (22)
at = 2 t [C(c•)- (t)] t,
C(c•) dt (23)
It is possible o find an analytical expression etween Pe and
a 2 [Neretnieks, 983]'
2
• = exp 4a 2) -- 1 (24)
Pe
This can be used o give he relation between e and a 2 and,
of course, gives the same results as the method described
above. n the curve itting method, however, other information
can also be obtained such as higher moments. The relation so
obtained is shown by the full line in Figure 7. The values
0.0 50.0 100.0
TIME, MIN
Fig. 6. Curves itted with the hydrodynamic ispersion model solid
line) and channeling model (dashed ine) for strontium run A16).
o.o' g.o so.o A.o ,oo.o
PECLET NUMBER
Fig. 7. Relationship etween Peclet number and standard deviation
of the logarithm of the fissure width.

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