Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet granite, Manitoba, Canada

$Page 1: Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet granite, Manitoba, Canada$
Journal of Contaminant Hydrology, 14 (1993) 207-232 207 Elsevier Science Publishers B.V., Amsterdam

Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet

granite, Manitoba, Canada

Y o k o Fuj ikawa a, Masami Fukui a, D o u g J. Drew b and Tjalle T. Vandergraaf b

aResearch Reactor Institute, Kyoto University, Kumatori-cho, Sennan-gun, Osaka, 590-04, Japan

bWhiteshell Nuclear Research Establishment, Atomic Energy of Canada Limited, Pinawa, Man. ROE 1LO, Canada

(Received February 20, 1992 revised and accepted April 21, 1993)

ABSTRACT

Fujikawa, Y., Fukui, M., Drew, D.J. and Vandergraaf, T.T., 1993. Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet granite, Manitoba, Canada. J. Contam. Hydrol., 14: 207-232.

Breakthrough curves of 137Cs and tritiated water injected instantaneously into artificial fractures in Lac du Bonnet granite were analyzed using the analytical solution for a single rock-fracture system and assuming the linear sorption isotherm of the solute. Parameters of nuclide diffusion and sorption in rock matrices, obtained by fitting, varied depending on the flow velocity in the fractures. According to theoretical calculations, different fracture flow velocities lead to different diffusion distances of nuclides in matrices at the same injection volume. As microscopic inhomogeneity is considered to exist in the rock matrix, the average diffusion sorption characteristics of the matrix within the diffusion distance may have varied with the fracture flow velocity. Surface sorption was marked in fractures that had relatively high matrix sorption diffusion capacities. The phenomenon was interpreted using the theoretical relationships developed between the surface sorption, matrix sorption and pore diffusion coefficient, and the porosity of matrices.

The effect of the nonlinear sorption of solute was examined by numerically solving model equations that incorporate the nonlinear isotherm. This incorporation may contribute to the reduction of deviations between theoretical and experimental BTC's.

INTRODUCTION

Disposal in mined cavities of plutonic rock has been the most common way to manage spent nuclear fuel in most countries that utilize nuclear energy. The safety of disposal depends on the isolation ability of artificial and natural barriers. Natural rocks contain fractures with spacings ranging from 1.5 to

0169-7722/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

2 0 8 Y. FLIJIKAWA ET AL,

40 m and openings of up to 300 #m (e.g., Snow, 1968). Transport of radionuclides through these water-bearing fractures after the corrosion of waste containers is expected to be the principal pathway for radionuclide migration to the biosphere. Because migration is known to be retarded significantly by the interaction of nuclides with rock, a number of studies have been conducted, in particular on the diffusion-sorption of nuclides to rock.

Rocks consist of complicated structures represented by pores, the grain boundary of minerals, inhomogeneous mineral distribution, and weathered outer and inner surfaces. These characteristics differentiate "rock" from a mere assortment of minerals. Torstenfelt et al. (1982) compared the sorption coefficient, Kd, of rocks with those of rock-forming minerals and found that the weighted mean total of the Kd of each mineral was lower than that of the whole rock. This indicates that the effect of the complicated structures of rocks should be taken into account when studying sorption phenomena.

In experimental investigations of radionuclide sorption to rocks, various techniques have been used to distinguish the transient effect of matrix diffusion from sorption. One is to enhance the rate of diffusion by crushing the rocks or to force solutions into rock matrices under high hydraulic pressure. Berry et al. (1988) conducted a batch experiment with crushed rock and a high-pressure convection experiment in which solution was trans- ported through intact rock cores under a high hydraulic gradient. They obtained relatively comparable Ka-values from the two experiments. The shortcomings of these procedures are: (1) overestimation of the Kd-values may occur as the inner surfaces of dead-pores, which are irrelevant to sorption under normal conditions, may be involved because of the high hydraulic gradient or the crushing of the rocks; and (2) spatial distribution of Kd-values in rock matrices cannot be measured by this method. Another way of separating the kinetic effect of matrix diffusion is to "wait" until diffusion ends (Fujikawa and Fukui, 1991a). Although overestimation of Kd-values can be avoided with this method, the spatial distribution of Kd remains unknown.

Skagius et al. (1982), and Fujikawa and Fukui (1991a) separated "surface" sorption from "matrix" sorption in their analyses of radionuclide sorption to rocks in batch experimental systems. Here, surface sorption can be interpreted as the sorption to that part of the solid matrix easily accessible to the radionuclides. The boundary between the "surface" and the "matrix", however, is considered to be arbitrary rather than absolute and to vary depending on the time scale of interest (Fujikawa and Fukui, 1991 a).

Although the laboratory experiment has various constraints when simu- lating field conditions, it is the only means by which to study long-lived radionuclide migration under well-defined conditions. A number of laboratory tracer-migration experiments on fractured rocks have been con-

MIGRATION OF INSTANTANEOUSLY INJECTED CESIUM IN ARTIFICIAL FRACTURES 209

ducted. Grisak et al. (1980) and Vandergraaf et al. (1988) conducted experiments under the continuous (step) input of tracer materials. Neretnieks et al. (1982) made the finite step injection of solutes into a fractured rock core and observed the channeling between fractures. Failor et al. (1982) conducted pulsed-injection experiments. Generally, pulsed injections give more information on the hydrology and sorption characteristics of the system than do continuous injections (Kool et al., 1989). In experiments using radioactive tracers with long half-lives, the amount of tracer and radioactive waste to be disposed of after the experiment can be greatly reduced by using a pulsed rather than a continuous injection. Migration experiments in rock fractures conducted under well-controlled pulsed-injection conditions and analyzed by convection-dispersion-type models are, however, relatively rare.

In the study reported here, results of tracer experiments in artificial fractures of Lac du Bonnet granite were analyzed using the analytical solution developed by Fujikawa and Fukui (1990). Effects of the flow velocity on the parameter values obtained were examined. A theoretical relationship was developed between the sorption coefficient defined in terms of the fracture surface and the diffusion-sorption parameters in the matrix. Effects of sorption nonlinearity on breakthrough-curve shape were investigated by comparing results of a numerical simulation. The possibility of obtaining a better fit of experimental breakthrough curves (called BTC's hereafter) by the incorporation of a nonlinear sorption isotherm also was examined.

MATERIALS AND METHODS

Rocks were obtained from the Lac du Bonnet batholith, Manitoba, Canada. The mineral and chemical compositions of the rocks are given in Walton et al. (1984). A core with the diameter 2.54 cm and length 6 cm having one artificial fracture with an opening of 1.2 mm (core 1), and three geo- metrically identical cores with the diameter 2.54 cm and length 9 cm, each having one artificial fracture with an opening of 0.2 mm (cores 2 4), were made from the Lac du Bonnet rocks. The fracture in core 1, created by inducing stress on the core by chiseling along the length of the sample, had rough surfaces. The fractures in cores 2-4, created by sawing each core lengthwise, had relatively smooth surfaces. Standard synthetic granitic groundwater like that used in AECL (Walton et al., 1984) was used in this experiment. The composition of this water is given in Table 1.

A schematic diagram of the high-pressure radionuclide migration apparatus (HPRM) developed in the AECL in order to conduct tracer experiments on the rock cores is given in Fig. 1 (Drew and Vandergraaf, 1989). A solution tagged with carrier-flee 137Cs and tritiated water was instantaneously introduced through the core by loading it into the injection valve of the HPRM

210

TABLE 1 Granitic groundwater compositions

Y. F U J I K A W A ET AL.

Composition Concentrat ion Composition Concentrat ion (g m -3) (g m -3)

Na ÷ 8.3 HCO~ (*) K + 3.5 CI- 5.0 Mg 2~ 3.9 SO42- 8.6 Ca 2+ 13 NO3 0.62 Sr 2+ - F - 0.19

Fet Si pH 6.5 ± 0.5

- below detection limit. * Not determined. In equilibrium with the CO2 in the atmosphere.

then turning the handle. Cores were placed under a confining pressure of ~20 MPa. The effluent from the core was passed through an NaI(T1) detector to measure the 137Cs then gathered in a fraction collector, after which the tritiated water was measured with a liquid scintillator counter. The effect of 137Cs o n the 3H count was found to be negligible if t h e 3H/137Cs counting ratio was > 10. A detailed description of this H P R M apparatus is given in Drew and Vandergraaf (1989).

Consecutive injections to the same artificial fracture of core 1 were made at different flow velocities. The fracture was flushed with groundwater prior to the next injection. Injections also were made to the fractures of cores 2-4 at different flow velocities. The experimental conditions are summarized in Table 2.

Air Water

. . . . . . . . . . ~ ~[~ Confining Core Sample . . . . Pressure Rupture Pump,~

Radionuclide Injection Valve sparer Pressure Relief Valve / NaI(TI) Detector [ / /

I / / Tran ~ ~' Solution I Reservoir

Pressure I Vessel Heaters -----__

..... ~Confining Pressure Cavity

~) Active Drain

Fig. 1. Diagram of HPRM (Drew and Vandergraaf, 1989).

MIGRATION OF INSTANTANEOUSLY INJECTED CESIUM IN ARTIFICIAL FRACTURES

TABLE 2

Parameter values

211

Figure No. Core Nuclide u (*') c~ (.2) /(d Dp 0 (.2) (×10 -5) (×10 2) (×10-13)

6 1 3H 16.4 0.8 K' d = 0 7a 1 137Cs 16.4 0.8 2.0 7b 1 137Cs 0.45 0.8 0.25 7c 1 137Cs 0.233 0.8 0.70 8a 2 137Cs 12.8 0.8 12 8b 3 137Cs 2.56 0.8 3.5 8c 4 137Cs 0.61 0.8 4.5

Figure No. K~a 2) Error .3 M (.4) V~ .2) K~d (* 5) (cpm) (× 10 -6)

6 0.0 +4.2 60,720 2.07 0.0 7a 0.0 +3.2 76,863 2.07 2.0 7b 0.0 +6.1 45,466 2.07 0.25 7c 0.0 ±6.5 58,476 2.07 0.7 8a 1.0.10 -3 ±1.7 1,012,991 0.45 10.0 8b 8.0.10 3 +6.9 856,620 0.45 3.5 8c 2.0.10 -3 +3.5 1,136,829 0.45 4.5

Units are in kg, m and s, unless otherwise stated, , i Experimental condition. ,2 Fitted to the BTC of tritiated water in core 1 at u = 1.64.10 -4 m s i. This value was used for cores 2-4. ,3 Normalized with respect to peak height. ,4 Total amount of tracer injected. ,5 K~ was estimated from the fitted values of K~dOpOp on the assumption that Op = 5.0.10 -11 m 2 s -1 and 0p = 2.0.10 -3 .

L I N E A R S O R P T I O N I S O T H E R M M O D E L O F T R A C E R M I G R A T I O N IN A R O C K - F R A C T U R E SYSTEM

Linearization of the nonlinear sorption isotherm

Skagius et al. (1982) found that the sorption of cesium to Finnsj6n and Stripa granites, Sweden, can be expressed as a nonlinear Freundlich isotherm. Vandergraaf et al. (1986) also found a nonlinear sorption isotherm of cesium in Lac du Bonnet granite. The isotherm obtained for the Lac du Bonnet granite in a static experiment within the cesium concentration range of 1.10 3-1 kg m -3 was:

. . . . 0.5627 (1) qeq = U ' ~ Z C e q

212 Y. FUJIKAWA ET AL.

where qeq is the radionuclide sorbed per unit mass of rock at equilibrium (unit kg-1); and Ceq the radionuclide concentration in the solution at equilibrium (unit m 3). The correlation coefficient between log10 qeq and log10 Ceq is 0.9995.

Because transport models that incorporate nonlinear sorption isotherms cannot be solved analytically, nonlinear isotherms often are linearized to derive analytical solutions. In contrast, van Genuchten et al. (1977) accounted for the nonlinear isotherm numerically in order to predict solute migration in a soil column. They found that the difference between the BTC's calculated for the linear and nonlinear isotherms was not large under the finite input of solutes. The effect of sorption nonlinearity on the shape of the BTC in a single rock-fracture system is discussed later (p.218).

The linearization method proposed by van Genuchten et al. (1977) is:

J Cmax / / [ Cmax ) Kdl = Kfcndc cdc

Cmin / \ dCmin (2)

where Kdl is the linearized sorption coefficient (m 3 kg-1); Kf the coefficient of the Freundlich isotherm (unit 1-n m 3n kg 1); n the order of the Freund- lich isotherm; and £min and Cmax are the minimum and maximum values (unit m -3) of the equilibrium concentration range of interest, respectively [Cmi n = 0 in van Genuchten et al. (1977)]. For example, the sorption coefficient of the Lac du Bonnet granite obtained in the static experiment can be linearized by combining Eqs. 1 and 2, and assuming Cmi n ~ O:

,.~ ~- ,-c, -0.4373 Kdl = l . z ~ O i S C m a x (3)

In conducting the series of parameter-fitting calculations, we assume that the matrix sorption coefficient of cesium can be linearized. The effect of the incorporation of a nonlinear isotherm also is discussed.

Model equations and the solution

The analytical solution for a single rock-fracture system (Fujikawa and Fukui, 1990) was used in the analysis of the experimental data. The model equations are:

Ocf D 02 Cf lg Ocf OpD ~ Oc~

Ot Ra Oz 2 Ra Oz ~- bR a Ox x=b

Oc~ D' 02eff

Ot R ~ Ox 2

(4)

(5)

M I G R A T I O N OF INSTANTANEOUSLY INJECTED CESIUM IN ARTIFICIAL F R A C T U R E S 213

where

R a = - 1 + Ka/b (6) /

R ' = 1 + pbKd/Op (7)

In the equations above, t is the time (s); z the coordinate along the fracture (m); x the coordinate perpendicular to the z-axis (m); cf the flux concentration of the solute in the fracture (unit m-3); c~ the flux concentration of the solute in the fluid in the rock matrix; 2b the fracture opening (m); u the flow velocity in the fracture (m s-l); D the hydrodynamic dispersion coefficient of the flow through the fracture (m 2 s-l), where D = D m ÷ au [D m the molecular diffusion coefficient (m 2 s 1), c~ the dispersivity (m)]; Pb the bulk density of the matrix (kg m 3); 0p the porosity of the rock matrix; D' the pore diffusion coefficient in the rock matrix (m 2 s -1); K a the coefficient of the solute sorption to the fracture surface (m); and K~ the coefficient of the solute sorption to the rock matrix (m 3 kg l).

The initial and boundary conditions for a delta-type flux injection are:

M of(o, t) = 6( t) (8)

cf(z, 0) = 0 for z > 0 (9)

lim cr(z, t) = 0 (10) z - ~ ÷ o c

c (b, z, t) : f(z, t) (11)

c~(ec,z, t) = 0 (12)

c (x, z, 0) = 0 (13)

where ~(t) is a Dirac delta function [T-l]; n the porosity in the fracture (n = 1 in this study); A' the cross-sectional area of the fracture (m2); and M the total mass or radioactivity of the solute injected into the system (unit).

The flux concentration cf corresponds to the breakthrough concentration as discussed in Fujikawa and Fukui (1990). The concentration c~ has been introduced formally for the convenience of defining the boundary condition (Eq. 11), but it does not correspond to any of the physically measurable values. The concentration in the matrix is expressed generally by the resident fluid concentration C~r as derived from c~:

! ! cr = u / D exp( -ur l /D)c f ( z+r l , x, t)drl (14)

jo

214 Y. F U J I K A W A ET AL.

The above equation is derived from the definition of c~:

, , O OCtr e f = c r - - - - ( 1 5 ) u Oz after Carslaw and Jaeger (1959, p.70). The final solutions for the breakthrough concentration cf and concentration in the rock matrix dr from Fuji- kawa and Fukui (1990) are:

M [Dt/(4R") Gz ( z2 u27 (G7)2 uz ) C f - n ~ u l r J 0 8(T37) l/2exp 167 D 2 4T + ~ d7 (16)

!

C r -- M IG- + b/J(z +

nA~DTr Jo J0/ 8(T373)1/2

167 D 2 4 T +- dTdr/

(17)

where

G = 40p (R'O')l/2/(Db) (18)

T= t - 4Raf/D (19)

For highly sorptive tracers such as cesium, it can be assumed that:

4( D' OpPbK~) l /2 (20) G ~ Db

because for large values of K~:

R t = 1 + pbKtd/Op ~, PdKtd/Op (21)

Eq. 20 implies that the breakthrough concentration for highly sorptive tracers is a function of K a, u, o~ and the product D'K'dOp. Therefore, even if the parameters D ~, K~ and 0p are unknown, fitting can be conducted for the product D~KtdOp, and the number of parameters to be fitted can be reduced.

Sensitivity analysis

An investigation of the geometrical characteristics of BTC's through sensitivity analysis provides useful information for conducting the fitting of experimental data. The responses of theoretical BTC's to different c~, Ka and K~-

MIGRATION OF INSTANTANEOUSLY INJECTED CESIUM IN ARTIFICIAL FRACTURES 2 ] 5

values under pulsed input of solute has been examined by Maloszewski and Zuber (1985), and by Fujikawa and Fukui (1990). For a larger dispersivity oz, the peak point of the BTC shifts to the left (appears earlier), the entire shape becoming more asymmetrical, and there is longer tailing. For a larger matrix sorption coefficient K~, longer tailing is seen in BTC's; whereas, the shift of the peak point location is minor. For a larger surface sorption coefficient Ka, the peak shifts to the right (appears later), the whole shape becoming broader. If the K~-value were large, longer tailing also would occur for larger Ka-values.

The effects of the flow velocity in the fracture on the theoretical BTC's are shown in Fig. 2. Here, the fracture geometry is the same as in core 1, and the velocity is varied within the range of the experimental velocity variations given in Table 2. Fo r sorptive tracers (K~ = 2.0m 3 kg -1, Ka = 0), the fracture flow velocity has a critical effect on BTC's. Namely, when the sorption coefficient is large, smaller flow velocities (which means longer solute residence time) increase the amount of solute diffused and sorbed to the rock matrix; therefore, longer tailings are produced in BTC's. Although not shown in the figure, BTC's for different flow velocities were identical for non-sorptive tracers (K~ -- 0 m 3 kg l, Ka _-- 0). We infer from these results that under the experimental conditions used the effect of diffusion in the matrix is negligible for a non-sorptive tracer like tritiated water.

Concentration distributions of the sorptive solute (K~ = 2.0 m 3 kg -1) in the matrix are shown in Fig. 3a and b for different fracture flow velocities (u) and surface sorption coefficients (Ka). Calculations were made with Eq. 17 at time t = L / ( u / R a ) , the average time for a solute to travel from the inlet to the outlet of a core with the length L. In Fig. 3a, u = 1.64.10 -4 m s -1 and Ka = 0, and in Fig. 3b, u = 6.1.10 -6 m s 1 and Ka = 0. The fracture geome-

x 10 4

~_. 2 '5 t

2 0 I t', u = 1.64 x 10- 4 ,., : , f

~> 1.5 L I : 6 6 7 X 10 -5 o | :f/i

1.0

-6 /I \ o: ,o x,o

O 0 , , , , , - ~ - 0.0 I 0 2.0 3'.0

Effluent Volume (m 3)

4:0 x 10 -5

Fig . 2. S e n s i t i v i t y a n a l y s i s o f B T C ' s f o r d i f f e r e n t f r a c t u r e f l o w ve loc i t i e s , u = 1 . 6 4 . 1 0 - 4 , 6 . 6 7 - 1 0 -5 a n d

6 . 1 0 . 1 0 -6 m s 1 f o r t h e dotted, broken a n d solid lines, r e s p e c t i v e l y ; a = 0 . 8 0 c m K a = 2 .0 m 3 k g - l ; Ka = 0 m ; D ' = 5 . 0 . 1 0 - u m 2 s l; Op = 2 . 0 - 1 0 3; M = 6 0 , 7 2 0 cp rn .

216 v. FUJIKAWA lET AL.

x i o" (a) ~ I o" 1.23 -4

~ 0.8] t: L/(u/Ro) ~ ] U: 6'0x'O

o~o o o ~ ~ ~ 5.0

o

i n t o , o "'.i ~-- '<"~ ~ o D,$iaoce along D i ~ i o ~ e ~ . o \ ~ / 4.° 20 Distance a m a t ~ o ~ " ~ ~ ~ u]s^~u,[[~, u,-z (Xi65mint° matrix) I O'X~x'x~'~-~ ~ ' ~ 0 .0~0 long

(XlO-Srn) o.o fracture (XlO m) fracture (X 10-2rn

F i g . 3. S e n s i t i v i t y a n a l y s i s o f t he s o l u t e d i s t r i b u t i o n in the r o c k m a t r i x , a = 0 .80 c m ; K ~ = 2 . 0 m 3 k g - l ;

D ~ = 5 . 0 . 1 0 11 m 2 s i ; M = 60 ,720 c p m ; 0p = 2 . 0 . 1 0 -3 .

a. u = 1 .64 .10 4 m s - J ; Ka = 0 .0 m a t t i m e t = L/(u/Ra). b. u = 6 . 1 - 1 0 -6 m s - l ; Ka = 0 .0 m a t t i m e t = L/(u/R,).

try was assumed to be the same as that in core 1, and the flow velocity used in the calculation was the same as that adopted in the experiment.

The distance of solute diffusion is < 30 #m from the matrix-fracture interface at the high velocity of 1.64.10 4 m s 1 (Fig. 3a); whereas, at the lower flow velocity of 6.10- 10 -6 m s -1 , the solute is diffused > 50 #m into the matrix (Fig. 3b). A comparison between solute distributions in the matrix, as calculated for different Ka-values, shows that diffusion into rock matrices is deeper for a larger Ka (result of the calculation not shown). A long solute residence time induced by a small flow velocity or large surface sorption therefore enhances solute diffusion into the matrices.

The plots in Fig. 3a and b were calculated under the same conditions as the respective dotted- and solid-line BTC's in Fig. 2. Because the solid-line BTC shows longer tailing as compared to the dotted-line BTC, we know that the microscopic difference in the diffusion distance of the sorptive solute in the matrix (on the order of a 10 #m length) causes the considerable difference in the tailings of the BTC's.

Fitting strategy

The porosity value of 2.0.10 .3 reported by Noronha et al. (1990) for Lac du Bonnet granites was used in the fitting procedure. As the longitudinal molecular diffusion coefficient Dm usually is small in comparison to the longitudinal hydrodynamic dispersion coefficient au, D m w a s approximated as zero for both tritiated water and 137Cs.

In the tracer experiments on core 1, experimental BTC's were obtained for both tritiated water and 137Cs at the fracture flow velocity of u = 1.64.10 - 4

m s -1 . Because through the sensitivity analysis the effect of matrix diffusion on

M I G R A T I O N OF I N S T A N T A N E O U S L Y I N J E C T E D C E S I U M IN A R T I F I C I A L F R A C T U R E S 217

BTC's has been shown to be negligible for non-sorbing tracer, the value of the matrix diffusion coefficient D' of tritiated water was fixed as the literature value of 5.0.10 -12 m 2 s -1 (Skagius and Neretnieks, 1988) for convenience. The longitudinal dispersivity c~ and dead volume V 0 were estimated by fitting the BTC of tritiated water at u = 1.64-10 -4 m s -1. The surface sorption coefficient Ka and the product K~dD'Op were estimated by fitting the BTC's of 137Cs at different velocities in core 1 using the c~- and V0-values estimated for tritiated water.

In the tracer experiments on cores 2-4, data on the 137Cs effluent were available for analysis; but, for tritiated water only the location of a peak was identified at one pore volume of effluent. For simplicity, V 0 was set equal to the fracture volume. Its value is considered to have a minor effect on the final results of fitting because a very large retardation in the BTC peak point, in comparison to the expected value of V0, was found. Because the experimental setups were identical, the oz-value that was used in the fitting of the BTC's of core 1 was utilized. The values of K a and K~dD'Op for 137Cs were estimated by fitting each of the experimental BTC's for cores 2-4.

The error between the experimental data and the fitted theoretical curve was calculated as:

1 N ~ ~-'~ ICk - Ek I (22)

F27

where Ck, Ek and N are the theoretical effluent concentration, the experimental effluent concentrat ion and the number of experimental data points, respectively. Opt imum parameter values were chosen by repeating fitting and calculation of the error.

As the efficiencies of the radioactivity detectors used to measure the injected and effluent radioactivity were not estimated, no values for the total injected radioactivity expressed in the same units as the effluent radioactivity were available. The amount of injected tritiated water therefore was calculated by numerical integration of the experimental effluent data. For ~3VCs, numerical integration was impossible because the measurement of effluent radioactivity was not continued until the effluent concentration became zero. Therefore, both the experimental data for 137Cs and the theoretical curves were normalized with respect to the peak height before fitting so that the geometrical similarity between the two curves could be compared. The total injected radioactivity was calculated from the model after fitting was com- pleted. Normalizat ion may cause loss of information concerning the recovery of the mass initially injected, as Maloszewski and Zuber (1990) have pointed out; but, mechanisms that cause the net loss of mass (e.g., irreversible sorption) are less probable in our experiments because granitic rocks do not

218 Y. FUJIKAWA ET AL.

contain clay minerals which are known to specifically adsorb cesium (Tamura and Jacobs, 1960) and because the experiment was conducted within too short time for irreversible sorption to occur (Walton et al., 1984; Fujikawa and Fukui, 1991b). The approach used in our study therefore is considered justified.

THE EFFECT OF NONLINEAR FREUNDL1CH-TYPE SORPTION

Mode l equations and the solution

The model equations of solute transport in a single rock-fracture system incorporating a Freundlich isotherm are:

10S 0¢ r 02¢r -- U OCr q- OpDt Octr x=b -bOt + - - ~ = D ~ z 2 Oz b Ox + F (23)

2 ! OpbQ OOpOc~r -- D' 00pC r (24)

Ot ~- Ot Ox 2

where

s = ke n

Q = k, ctr n'

r = (M/A' ) I~(z - Zo)~5(t - to)

(25)

(26)

(27)

In the above equations, Cr is the resident fluid concentration in the fracture (unit m-3); C'r the resident fluid concentration in the fluid in the rock matrix (unit m-3); s the amount of solute sorbed per unit length of fracture surface (unit m-2); Q the amount of solute sorbed per unit mass of rock matrix (unit kg-1); F the delta-type source term of radionuclides (unit m -3 s-l); k, k ' the coefficients of the Freundlich isotherm on the fracture surface and in the rock matrix (unit 1-n m 3n kg-1), (unit l-n' m 3n' kg-1), respectively; and n, n' the orders of the Freundlich isotherm on the fracture surfaces and in the rock matrix, respectively. The other parameters are the same as in Eqs. 4 and 5.

The concentration c~ in Eqs. 23-27 hereafter is designated Cirr, subscripts of which mean instantaneous (delta-type) resident fluid injection-resident fluid detection concentration (Fujikawa and Fukui, 1990). The numerical solution of ¢irr (designated dirr) is obtained by solving the model equations by combining the finite-element formulation of Huyakorn et al. (1983) and a substitu- tion-iteration scheme, which gives the solution for the nonlinear sorption term (Huyakorn and Pinder, 1983, pp. 156-158). Because the effluent con-


centration which corresponds to the flux detection concentration principally is dealt with in this study, the flux concentration dirf (instantaneous resident fluid injection-flux fluid detection concentration) was obtained by numerically converting the resident fluid concentration Cir r using the relationship:

D 0Cir r Cirr = Cirr U OZ (28)

Details of the solution scheme will be provided by Fujikawa and Fukui (1993). Because the Dirac delta function, defined as:

6(0) = oc (29)

6(0 = 0 (t ¢ 0) (30)

I ~ 6(0 = 1 (31)

is extremely steep, simulation of delta-type source conditions by a discrete numerical approach is difficult. Spatial integration of Eq. 23, done in the course of finite element formulation, however, helped to simplify the delta- type source term:

F = M 6 ( z - Zo)6(t - to ) /A '

into

J Fdz = M 6 ( t - to)/A' R

Assuming linear sorption of the solute (n = n ' = l) and approximating 6(t - to) as l / A t ( A t is the time step size of simulation), the numerical solutions Cirr and Cirf w e r e obtained with less than a few percent error when compared with the analytical solutions of Cir r and Cirf in Fujikawa and Fukui (1990).

Sensi t iv i ty analysis

A comparison of the flux concentration for resident fluid injection and that for flux injection showed that the difference could not be ignored for solutes governed by a linear sorption isotherm (Fujikawa and Fukui, 1990). Assum- ing that this also is the case for solutes governed by a nonlinear sorption isotherm, the fitting of experimental effluent data obtained by the flux injection of a solute with the numerical solution for resident fluid injection mode (Oir0 cannot be justified.

220

0.05

0.04

0.03 ..J "4

0.02

0.01

0.0025

O. 002

--I - , ~ 0 . 0 0 1 5

f f

0.001

0.0005

0.3

~ 0 . 2 \

0.1

l i n e a r

non I i n e a r

-8 5 xlO

-7 1 xlO

v o l u m e ~m~

-6 2.5×10

-6 5 x lO

v o l u m e ( m 3)

(c)

il , nol~linear~ -8 -8 -8 -7

2.5'10 5 'i0 7.5xi0 ix i0

volume (rn3~

(a)

-7 1.5xlO

(b)

-6 7.5×10

Y. FUJIKAWA ET A


0.05

0.04

~0.03

'~o.o2 u

0.01

(d)

inear

-8 -7 -7 5 x l O 1 x l O 1 . 5 x l O

v o l u m e / m 3 !

0.6

0.5 °°°°°°'°'%°°°

u:2 .56~10 -4 ~; 0 . 4 • "* . ° . . / (e}

~1 %. - ~ o . 3 • u = 2 . 5 6 , 1 0 ~ ........

o . 2

0.1 "'~ J

- 8 - 8 -8 - 8 2 x 10 4 ~ 10 6 x 3.0 8 x 10

v o l u m e m 3 !

Fig. 4. C o m p a r i s o n o f CirfAIL/M ( l inear s o r p t i o n i so the rm) a n d ~irfAIL/M ( non l i nea r s o r p t i o n i so the rm) p lo t t ed a g a i n s t effluent vo lume . L = 1 cm; M = 856,600 cpm; u = 2 .56-10 -5 m s - l ; D m = 5-10 - t l m 2 s - l ;

c~ - 0.80 cm; b = 100/~m; a = 2.54 cm.

a. C o m p a r i s o n w h e n s o r p t i o n o n the f r a c t u r e su r face is n o n l i n e a r a n d n < 1. s - 2.0c °7 (uni t m -2) (solid line, nonl inear ) ; K~ - 0.45 m m (black dots, l inear) a n d Ka - 0.50 m m (gray dots, l inear); K~ = 3 .5m 3 k g -1

for all.

b. C o m p a r i s o n w h e n s o r p t i o n o n the f r ac tu re sur face is n o n l i n e a r a n d n > 1. s - 0 .0002c 12 (uni t m 2) (solid line, nonl inear ) ; K a - 3.06 c m a n d K~ - 0 .005m 3 kg l (solid small dots, l inear); K a = 3.06 cm a n d

K~ = 0.05 m 3 kg - t (solid dots, l inear); Ka = 3.06 c m a n d K~ = 0 . 3 m 3 k g ] (gray dots, l inear).

c. C o m p a r i s o n w h e n s o r p t i o n to the rock m a t r i x is n o n l i n e a r a n d n ' < 1. q - 3500c/07 (uni t kg -1) (solidline, nonl inear ) ; K~ - 0 .525m 3 k g - ] (solid dots, l inear); K~ - 0 .5775m 3 k g ] (gray dots, l inear); K~ - 0 .6125m 3

k g t (solid small dots, l inear) .

d. C o m p a r i s o n w h e n s o r p t i o n to the rock m a t r i x is n o n l i n e a r a n d n' > 1. q = 0 .035c 12 (uni t kg 1) a n d

K, - 0 m (solidline, nonl inear ) ; K~ = 7 .0m 3 kg i a n d Ka = 71 # m (solid dots, l inear); K~ -- 7.5 m 3 k g l

a n d Ka = 71 izm (solid small dots, l inear); K~ 8.0 m 3 k g ] a n d Ka = 71 /~m (gray dots, l inear).

e. Veloci ty d e p e n d e n c e o f B T C ' s when s o r p t i o n to the ma t r i x is g o v e r n e d b y a n o n l i n e a r i so therm. q 3500c '07 (uni t k g 1) a n d K ~ - - 0 m; u - 2 . 5 6 . 1 0 -4 m s I (solid dots, nonl inear ) ; u - 2 . 5 6 . t 0 5

m s ] (gray dots, nonl inear ) ; u 2 .56-10 6 m s I (solid line, non l inea r )

Instead of conducting direct fitting, we examined the effect of sorption nonlinearity on the geometrical shape of BTC's. Breakthrough curves of £'irf (numerical solution for the nonlinear isotherm) and Cir f (analytical solution for the linear isotherm), which have similar peak heights and locations, are compared in Fig. 4a-d. The Cirf c u r v e s for different flow velocities

222 Y. F U J I K A W A ET ~L.

are shown in Fig. 4e. Here breakthrough concentrat ions are normalized by crAWL/M, with respect to the amoun t of injected solute, where cf is the flux concentra t ion of the solute (Cir f o r Cirf); A~ the cross-sectional area of the fracture; and L the distance of a breakthrough point f rom the source.

In Fig. 4a and b, Cirf curves are compared with ~irf curves for n = 0.7 (n < 1) and n = 1.2 (n > 1), calculated on the assumpt ion of linear sorpt ion in the matrix and nonlinear solute sorpt ion on the fracture surfaces. Fig. 4a shows that BTC's with a nonlinear isotherm have delayed breakthrough points and sharper b reak through fronts than BTC's with linear isotherms when n < 1. In contrast , Fig. 4b shows that when n > 1, BTC's with a nonlinear isotherm have an earlier breakthrough point and a more spreading breakthrough front than BTC's with linear isotherms. In Fig. 4c and d, cirf curves are compared with ~irf curves fo rn t = 0.7 (n I < 1) and n ~ = 1.2 (n ~ > 1), on the assumption of linear sorpt ion on the fracture surfaces and nonlinear solute sorpt ion in the matrix. When n' < 1, the shape of BTC's with nonlinear sorpt ion is sharper a round the peak than for a linear isotherm (Fig. 4c); whereas, it is broader when n' > 1.

Characteristics of the parameter values of a linear isotherm which yields BTC's similar to those calculated assuming a nonlinear isotherm can be elucidated by compar ing the parameter values of linear and nonlinear BTC's which have similar peak heights and locations, as in Fig. 4a d. We have used the subscripts nlin and lin to distinguish between the parameters used to calculate BTC's with nonlinear isotherms and those used to calculate comparable BTC's with linear isotherms. For nonlinear sorpt ion on the fracture surface, Ka(lin) < k(nlin) a n d K~l(lin ) = K~t(nlin) for n < 1 (Fig. 4a), whereas Ka(lin) > k(nlin) and K~(lin) < K~(nlin ) for n > 1 (Fig. 4b). For nonlinear sorp- t ion in rock-matrices, Ka(lin) = Ka(nlin ) a n d K~(lin) ~ k inlin) for n' < 1 (Fig. 4c), whereas Ka0in) > Ka(nlin) and K~t(lin ) > k~nlin) for n > 1 (Fig. 4d).

In Fig. 4e, Cirf c u r v e s with nonlinear isotherms calculated for different flow velocities are indicated. They have longer tailings and lower peaks for smaller flow velocities, which is essentially the same as for the BTC's calculated assuming a linear isotherm.

RESULTS AND DISCUSSION

Experimental results

Fig. 5a shows BTC's of 137Cs obtained in core 1 under different flow velocities. The BTC of tritiated water also is shown. Fig. 5b shows BTC's of t37Cs obtained in cores 2-4, each under different velocities. For tritiated

M I G R A T I O N OF INSTANTANEOUSLY INJECTED CESIUM IN ARTIFICIAL F R A C T U R E S 223

water, only the peak point of the BTC is indicated in the figure because detailed effluent concentrations were not obtained.

Experimental BTC's of 137Cs obtained for cores 1-4 showed longer tailing at a slower flow velocity, typical of matrix diffusion sorption governed systems as shown in the sensitivity analysis in Fig. 2. The observed features of the experimental BTC's indicate the possibility of obtaining a good-fit using the analytical solution. Considerable retardation in the elution peak of 137Cs as compared with that of tritiated water was found in cores 2 4 (Fig. 5b); but there was no such retardation in core 1 (Fig. 5a). Therefore Ka should be negligible in core 1 and large in cores 2-4.

7.0 ~103

N

E O. U

×

× x / 3 H u=1"64×104 ×

x

13?,"~ ° o / ~ s u4.64×104 (a) o o

x x

o ~

o o 13? C xo / Cs u=4.50×10

x ° O o

#,z~ 0 0 e o " ~ % ~2XXxxO o ,3?Cs u=2.33×!00 ~ ~OOo[3DD (3 DD~Of~3[~D~

2.0 9.0 ~I I) G vo l u rne(m 3) ×I lj6

~H i0 4 - /~

oo

o o

o o u_J.28xlO'-4 %/ (b)

~ 137Cs ,,, o

E ~- 56x10 -~

A

~ u=6.10×10 -6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 ' ' ' 2',0 v o l u m e (m 3] x lO-4

Fig. 5. Experimental BTC's. a. Experimental BTC's obtained for core 1 at different velocities. b. Experimental BTC's for cores 2 4 obtained at different velocities.

224 v FUJIKAWA ET AL.

Results of fitting

The parameter values used and fitted are given in Table 2 (p.211), together with the mean errors normalized with respect to maximum breakthrough concentration. The results of fitting also are shown in Figs. 6, 7a -c and 8a-c.

Fig. 6 shows the fitting of the BTC of tritiated water obtained in core 1 under u = 1.64-10 -4 m s -1. The best-fit c~ was 0.80 cm, the dead-volume V 0 being 2.07.10 -6 m 3. Results of calculations of the +10% values of the opti- m u m c~ also are shown. Here V 0 corresponds to the sum of the volumes of the fracture, tubes and column end-fittings. V 0 therefore is expected to be larger than the fracture volume. Because the measured fracture volume in core 1 was 1.92.10 - 6 m 3, the obtained V0-value appears to be valid.

Results of fitting the 137Cs BTC's obtained for core 1 under three velocities, 1.64.10 -4 , 4.50.10 -6 and 2.33.10 -6 m s 1, are given in Fig. 7a, b and c, respectively. The opt imum surface sorption coefficient Ka was 0, the opt imum K~D'Op-values being 2.0-10 -13, 2.5.10 -14 and 7.0.10 -14 m 5 kg -1 s 1. BTC's calculated for +10%-values of the opt imum K~ also are shown.

The respective results of fitting 137Cs BTC's obtained for core 2 at u - - 1.28.10 - 4 m s -1, for core 3 at u = 2.56.10 -5 m s -1, and for core 4 at 6.1.10 -6 m s -1 are shown in Fig. 8a, b and c, respectively. The opt imum Ka-values were 1.0, 8.0 and 2.0 mm, and the optimum K~dD~Op-values 1.0.10-12, 3.5.10 -13 and 4.5.10 -13 m 5 kg -1 s -1 for cores 2, 3 and 4, respectively.

The fits between the experimental and theoretical BTC's were generally good, except for Fig. 6 (tritiated water in core 1), Fig. 8b (137Cs in core 3), and Fig. 8c (137Cs in core 4). As for the BTC of tritiated water in Fig. 6, the error between the theoretical and the experimental BTC's is large in the tailing of the BTC's where the radioactivity count of 137Cs exceeds that of the

× 10 3

7.0 ~,'~ d, = 0.72 × [(~2

60 ~P~'~ ~ = 0 8 xtO ~2 ~ ~'=0 88 xld2

~_ 5.0

, q ( ?

E u = 164 x I0 4

~ o ~ 2.()

1,02

0.0 e TM , , , , ~"'~ , 0.0 0.2 0 4 0.6 08 1.0

× j0 -5 Ef f luent Volume (m 3)

Fig. 6. F i t t ing o f the B T C o f t r i t i a t e d water in core 1. u = 1.64.10 4 m s I; D p - 5 . 0 . 1 0 II m 2 s i

V 0 = 2 . 0 7 . 1 0 - 6 m 3 ; K a = 0 m ; K ~ - - 0 m 3kg i ; 0 p _ 2 . 0 . 1 0 3; o = 0.22, 0.80 and 0.88 cm fo r the broken,

dotted (bes t fit) a n d chain lines, r e spec t ive ly .


x I0 3

4 0-

~ 3 0- ~6

E

U = 164 x I() 4 /•o• k u = 450 x IO e

",,,:

/ . . ~:~., ,K~:o2z5 c~ Kd :0.225 ~'~..~

.K'd=2 2

*"~..~ K'd = 18 ~-'<" "~'-"~-'-~- -~ (a)

~o ~io 7:o 8o 9~o V o l u m e (m 3) xlO-6

× 10 3 1.4

1.2

1.0

"~ O8

0.6 o

~ 0,4

0.2

0 .0 2.0

0 O. ' J - (b) i E

2 0 3 0 4'0 6.0 I0.0 14'.0

E f f l uen ! E f f l u e n t Vo lume (m 3) x 10 .6

xIO 2

6.0-

=~ 3 _ _ 4 0 a>

c~ 0 "-- 2 ,0 E-_

o

u :2 .33 xlO -6

~'~,X,,, / K'd " 0 7

~,',, ..K~ =0 7 7 '~.,,f.

(c) 0.0~--

o o ,o 2o 3o 4;Io_ 5

Ef f l uen t Vo lume (m 3)

Fig . 7. F i t t i n g o f the B T C o f 137Cs in c o r e l . ct - 0 . 8 0 c m ; Ka = 0 m ; D ~ - 5 .0 . l 0 I1 m 2 s - l 0p = 2 . 0 . 1 0 3.

a. u - 1 . 6 4 . 1 0 4 m s 1; K ~ = 1.8, 2 .0 a n d 2 .2 m 3 k g - I (K~D'Op - 1 . 8 . 1 0 13, 2 . 0 . 1 0 - 1 3 a n d 2 . 2 . 1 0 13m5

k g I s - I ) f o r t h e broken, dotted (bes t fit) and chain lines, respectively. b. u 4 . 5 . 1 0 -6 m s 1; K ~ = 0 . 2 2 5 , 0 .25 a n d 0 . 2 7 5 m 3 k g -1 (KrdffOp=2.25.10 -14, 2 . 5 . 1 0 14 and 2 . 7 5 . 1 0 H m 5 k g - I s I) f o r t h e broken, dotted ( b e s t fit) and chain lines, respectively. c. u - 2 . 3 3 - 1 0 -6 m s ]; K ~ = 0 .63 , 0 .7 a n d 0 .77 m 3 k g - l (K~dDtOp = 6 . 3 - 1 0 14, 7 . 0 . 1 0 14 a n d 7 . 7 - 1 0 i4 m 5

k g - I s I) f o r t h e broken, dotted (bes t fit) and chain lines, respectively.

tritiated water (Fig. 5a). Because the measurement of tritiated water in a mixed 137Cs solution may not be accurate, especially if the radioactivity of 137Cs is high, deviation might have been caused by the experimental setup rather than by an unknown phenomena related to the behavior of tritiated water. As for the BTC's of 137Cs in Fig. 8b and c, sorption nonlinearity, which was ignored in the course of fitting, may be relevant. The discussion is given below (p.229).

Comparison of K'dD'Op-values

Considerable variation in K'dD'Op-values was found in the BTC's obtained

226 Y. FUJIKAWA ET At..

x I0 3 io.o[

~ 8 0-

E c~, 4, 0 -

E X2.Q

~ ' u : 1.28 x 164

'~, K'd :10 0 ~ < ~ , . o

0,0 i J , i i , i i

0.0 0.4 0.8 12 16

Effluent Volume (mS)

(a) i

2.0 xlO -4

x I03

4 . 0 -

m

~ 3 0

~ 2 . 0 -

~ 1 . 0 -

0 0- 0.0

/ ? ~ . x u = 2.56 × 10 -5

/K~°30 K d = 3.5

K' 40

X ooo o

(b)

0.4 08 12 1.6×10_ 4

E f f l u e n t Volume (m 3)

x 10 3 1.6

~ [.2

E O . 8

~ 0 . 4

/ , " / ~ q.Y @o o i,./" , \ .-°b-~ /,,2' K ~, 4 5 ~ \ k-.oto-

/ / ,' ~ . ~ • K~=40

I~ !'' u ~ 6.1o, t0-~

o. o j . . . . . . . . (c) o o 0.4 08 L2 1.6

X 10 -4 Effluent Volume (rn 3)

Fig. 8. F i t t ing of the BTC's of 137Cs in cores 2-4. c~ 0.80 cm; D' = 3.0.10 -5 m 2 s 1 0p -- 2.0.10 -3.

a. F i t t ing in core 2. u = 1.28.10 _4 m s - I ; K ~ - - 1 . 0 . 1 0 -3 m; K ~ - 9 . 0 , 10.0 and 11.0 m 3 kg -1 (K'oD'Op - 9 . 0 . 1 0 13, 1.0.10-12 and 1.1.10 t2 rn 5 kg t s - I ) for the broken, dotted (best fit) and chain lines, respectively. b. F i t t ing in core 3. u = 2,56.10 5 m s ~; K~ = 8.0.10 3 m; K ~ - 3.0, 3.5 and 4.0 m 3 kg (K~dD~Op - 3 . 0 . 1 0 J3, 3.5.10 13 and 4.0.10 13 m 5 kg i s i ) for the broken, dotted (best fit) and chain lines, respectively. c. F i t t ing in core 4. u - 6 . 1 . 1 0 -6 m s t; K a = 2 . 0 . 1 0 3 m; K ~ = 4 . 0 , 4.5 and 5.0 m 3 kg 1 (K~D'Op --4.0.10 -13, 4.5.10 -t3 and 5.0.10 13 rn 5 kg i s i) for the broken, dotted (best fit) and chain lines, respectively.

when different flow velocities were applied to core 1. The value was ahnost one order higher at the flow velocity of 1.64.10 4 m s -1 than at 4.5.10 .6 and 2.33.10 -6 m s -1 . As shown in Fig. 3a and b, the distance of nuclide diffusion into the matrices varies with the flow velocity in the fracture. As microscopic inhomogeneity is expected to exist with respect to the spatial distribution of sorption and the diffusion coefficients as well as the porosity of rock matrices,


the variation in ~ KdD 0p is considered to be caused by variation in the matrix sorption-diffusion capacity averaged over the different diffusion distances.

Values of K~dDtOp also varied for cores 2-4. The variations are considered partly to be caused by the inherent differences in the characteristics of the cores, which is inevitable in samples from natural inhomogeneous rocks. Different fracture flow velocities, which give different diffusion distances into the matrix, may also have caused variation.

The fitted values of K~dD~Op were larger in cores 2-4 than in core 1. Varia- tions in the lots, as well as the difference in the method used to introduce the fractures into cores, may have affected the results, i.e. the sawing of cores 2-4 might have produced more microfissures than the chiseling of core 1.

Comparison of Ka-values

Theoretically, Ka can be related to matrix diffusion-sorption characteristics as follows: Neretnieks (1980), as well as Rasmuson and Neretnieks (1986), considered the contact-time-dependent penetration thickness of solute into the rock matrix. Here we consider ~i, the depth of solute penetration attained within a very short time (ts (s)), compared with the solute residence time. According to Neretnieks (1980), ~i should satisfy the equation:

gdpC~i KdpCtd ~ (32)

where c (unit m 3) is the solute concentration at the fracture-matrix interface; c' (unit m 3) the solute concentration in the matrix; Kdp (m 3 kg 1) the local sorption coefficient in the matrix to which the solute diffused; and r]~ (m) the depth of the diffusion front of the solute in the rock matrix where c/c o = ~ ~ 0 (c is infinitesimal amount). The penetration thickness ~i that satisfies Eq. 32 is approximated as:

~i ~ ~/o.5 (33)

where r/0.5 is the penetration depth to the point where c'/c equals 0.5 at time t~. The amount of solute sorbed to the matrix within the very short time ti is converted to the amount sorbed per unit surface area as follows:

S = ~iPp_localKdpC ~ 170.5Pp_localKdpC (34)

where Pp-local (kg m 3) is the local bulk-density of the matrix to which the solute diffused. As the surface sorption coefficient Ka is defined by the relationship:

s = K ~ c (35)

228 y. FUJIKAWA ET AL.

the coefficient Ka can be represented as follows by compar ing the r ight-hand sides of Eqs. 34 and 35:

K a ~ 770.5Pp_localKdp (36)

Rasmuson and Neretnieks (1986) derived the penetrat ion thickness r/h (m) to the point at which c' /c = h, assuming that the matr ix- f rac ture interface is mainta ined at a constant concentrat ion. During the very short period of time ti, the concentrat ion in the fracture (c) can be regarded as constant, and q0.5 is derived as follows using the equat ion they developed:

r/0.5 = 2 erfc-l(0.5)(Dtti/Rtp) 1/2 (37)

Rip = 1 + pp localKdp/Op_local (38)

where R~p, Dp local ( m2 s- l ) and 0p local are the retardation constant, pore- diffusion coefficient and porosity, respectively, defined for the rock matrix near fracture surfaces. Here erfc-l is the inverse of the complementary error function and erfc -l (0.5) = 0.48. F r o m Eqs. 36 and 37:

Ka 2erfc-I (0.5) , 1/2 = (Op_ loca l t i /Rp ) Pp_localKdp (39) !

If we consider a highly sorptive solute so that Rp ~ pp localKdp/Op local, Ka is expressed as:

Ka -- 2erfc-l (0.5)(tiPp_local Op-localOp-localKdp) 0"5 (40)

In order to calculate the values ofTi (~ r/0.5), we assumed that the time ti in Eq. 40 can be defined using the coefficient r as:

ti = r L / u (41)

r << 1 (42)

where L (m) is the spatial scale of interest. Assuming r = 0.01, L = 0.09 m (length of cores 2-4) , Kdp = 2.0 m 3 kg - l , Dp ---- 10 l l -10 7 m 2 s - I and using the velocity values applied in the experiments, the ~i-values were calculated as a few #m at most.

The op t imum surface sorption coefficient Ka, obtained by fitting the BTC's of 137Cs breakthrough, was zero for core 1 and 1.0 8.0 m m for cores 2-4 . Variations in the fitted Ka-values between the cores are considered to reflect the difference in the local sorpt ion-di f fus ion characteristics of the matrix to which the solute was diffused. According to theoretical equat ion (39), Ka becomes smaller for the rock matrix with smaller local sorpt ion and/or diffusion parameters. As discussed in the sensitivity analysis (p.216), the term "local" as used here means a diffusion depth of the order of 10 # m , which is far smaller than the usual scale under which we measure K~-values. It is


possible that the local sorption-diffusion capacity is extremely small in core 1, which in effect would reduce the Ka-values to zero. As indicated in Table 2, the

t fitted values of parameter KdDpO p w e r e smaller in core 1 than in the others. If the rock matrix in each core is almost homogeneous and KdpO p local is equal

t t o KdDpOp, the fact that the Ka-value in core 1 was smaller than in the other cores is qualitatively in accordance with the theoretical relationship in Eq. 40.

As stated above (p.228), the coefficient K a probably is governed by the microscopic characteristics of the fracture surfaces and the near-fracture- surface matrices. Although the fractures in our experiment were artificial, the various phenomena that govern K a through the microscopic characteristics of the rock also should exist in natural fractures. As for the fracture surfaces, the existence of coating minerals, such as clay minerals of high adsorption capacity, in natural fractures has been reported (Tammemagi et al., 1980; Allard et al., 1985; Suksi et al., 1991). The effects of these coating minerals on the diffusion sorption of nuclides to rocks have been studied in static batch and diffusion experiments (Torstenfelt et al., 1982; Walton et al., 1984; Allard et al., 1985; Skagius and Neretnieks, 1986a, b). According to Eqs. 39 and 40, which indicate that the near-surface diffusion sorption characteristics of rocks govern K a, surface minerals also should govern Ka-values. Alteration zones often occur from natural fracture surfaces to certain depths (Neretnieks et al., 1982; Vandergraaf et al., 1988). Neretnieks et al. (1982) suggested that an altered zone may have an sorption-diffusion capacity that differs from that of the non-altered zone. The existence of such zones also may affect the Ka-values.

Because K~ is an important parameter that governs the solute residence time, as well as the the solute elution profiles, its fundamental characteristics needs to be studied using natural fractures.

The effect o f a nonlinear sorption isotherm on fit t ing

Theoretical BTC's calculated for solutes governed by a linear sorption isotherm simulated various features of experimental BTC's (i.e. breakthrough fronts, location, height and sharpness of peak, and tailing) very well. In cores 3 and 4, however, deviations in the shape of the fitted and experimental BTC's were conspicuous. The experimental breakthrough front was broader than that of the fitted curve in core 3 (Fig. 8b), and in core 4 (Fig. 8c) the shape of the peak of the fitted curve was broader than that of the experimental one.

Direct fitting of the experimental data using the numerical solution for a nonlinear isotherm is not possible because the numerical solution ciff could not be obtained for the delta-type flux fluid injection conducted in the present experiment. The sensitivity analysis of the nonlinear isotherm conducted for Oirl curves, however, indicates that the breakthrough front becomes broader if

230 V F U J I K A W A ET AL.

solute sorption to the fracture surface is nonlinear and n > 1 (Fig. 4b). More- over, the shape of the peak is known to become broader if solute sorption to the rock matrix is nonlinear and n' > 1 (Fig. 4d). Assuming that the same is the case for the ~iff curves, it is probable that the incorporation of a nonlinear sorption isotherm with n > 1 for the fracture surface and with n' > 1 for the matrix improves the fit in cores 3 (Fig. 8b) and 4 (Fig. 8c). The following contradictions and drawbacks, however, apply to the incorporation of the nonlinear isotherm: (1) because the nonlinear isotherm obtained for the Lac du Bonnet granite has the coefficient n < 1 (n' < 1), as indicated in Eq. l, the introduction of a nonlinear isotherm with n > 1 (n / > 1) is not justified; and (2) the number of sorption diffusion parameters to be fitted is increased from two (Ka, K~dD'Op) to six (k, n, k i, n/, D t, 0p) by the incorporation. It is natural that the increase in the number of parameters to be fitted yields better fits, but it does not always mean that the newly introduced parameters are intrinsic. To verify the existence of nonlinear sorption, intrinsic parameters should be selected by analyzing various effluent concentration data and by comparing the results with those of batch sorption experiments.

Implication of the present findings for safety assessment

Estimation of the rate of underground nuclide migration from a nuclear waste repository to the biosphere is an important step in the safety assessment of underground radioactive waste repositories. Both , / KdDOp and Ka are expected to govern this rate. The variability of these parameters seen in our study indicates that in conducting a safety assessment of an underground repository, statistical distributions of the K~dD10p - and Ka-values need to be assumed rather than a single representative value being used for each of them.

CONCLUSIONS

BTC's obtained under pulsed injections of 137Cs into artificial rock fracture systems of Lac du Bonnet granite were analyzed. Long tailing typical of tracer migration governed by matrix diffusion and sorption was present in experimental BTC's of 137Cs. Using the analytical solution for solute transport in a single rock-fracture system and assuming a linear sorption isotherm, good agreement between the experimental and theoretical BTC's was obtained.

For highly sorptive t racers , K~dDlOp, the product of the rock matrix sorption coefficient, pore diffusion coefficient and the porosity, was shown to govern theoretical BTC's. Therefore, instead of optimizing the three parameters, the product K~dD'Op was fitted to the BTC's of ~37Cs. The value varied even in the same fracture depending on the fracture flow velocity. This is


attributed to the velocity-dependent diffusion distance the solute travels in the rock matrix and the microscopic inhomogeneity of the matrix.

The surface sorption coefficient Ka, obtained by fitting the BTC's of 137Cs was almost zero for core 1, the matrix of which had relatively small diffusion- sorption characteristics, whereas the other cores had greater diffusion-sorption characteristics for the matrix and larger Ka-values. This tendency was qualitatively in accordance with the theoretical equation for Ka, which indicates that Ka is larger for a matrix with larger local sorption-diffusion parameters.

The effect of a nonlinear Freundlich sorption isotherm of solutes on the geometrical shape of BTC's also was examined using a sensitivity analysis of numerical solutions. Improvement of the fit between the experimental and theoretical BTC's was shown to be possible by the incorporation of a nonlinear isotherm.

REFERENCES

Allard, B., |ttner, T. and Torstenfelt, B., 1985. Migration of trace elements into water-exposed natural fissure surfaces of granitic rock. In: Y. Kitano (Guest-Editor), Water Rock Inter- action. Chem. Geol., 49:31 42 (special issue).

Berry, J.A., Bourke, P.J., Coates, H.A., Green, A., Jefferies, N.L. and Littleboy, A.K., 1988. Sorption of radionuclides on sandstones and mudstones. Radiochim. Acta, 44/45:135 141.

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Documents

Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet granite, Manitoba, Canada