18
Journal of Contaminant Hydrology, 6 (1990) 85-102 85 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK: ANALYTICAL SOLUTIONS FOR DELTA-TYPE SOURCE CONDITIONS* YOKO FUJIKAWA and MASAMI FUKUI Research Reactor Institute, Kyoto University, Kumatori-Cho, Sennan-Gun, Osaka 590-04 (Japan) (Received April 24, 1989; revised and accepted December 12, 1989) ABSTRACT Fujikawa, Y. and Fukui, M., 1990. Adsorptive solute transport in fractured rock: Analytical solutions for delta-type source conditions. J. Contam. Hydrol., 6: 85-102. Solutions for adsorptive solute transport equations in a single fracture-rock system were derived under two different delta-type source conditions. One was the delta-type, flux injection condition. The correspondent solution can be used for the analysis of experimental column breakthrough curves obtained by injecting the tracer into the rock fracture. The other was the delta-type, resident fluid injection condition. The solution can be used for the crude estimation of pollutant migration from the underground radioactive waste repository. In formalizing the resident fluid injection, an initial distribution of solute between solution and solid phase was assumed in order to satisfy the mass balance between injected and detected solute. Each of the two solutions was also expressed in two ways reflecting the flux and the resident fluid detection. Solutions expressed in terms of the flux detection correspond to the effluent concentration measured experimentally by a fraction collector system. On the other hand, solutions expressed in terms of the resident fluid detection describe spatial distribution of the solute in the fluid. Since considerably long tailings of breakthrough curves are often observed in column tracer experiments using fractured rocks, effects of some parameter values on the tailing were also discussed. It was shown that the adsorption of solute to rock matrix caused longer tailing. It was also shown that the resident fluid injection caused longer tailing of breakthrough curves than the flux injection condition. INTRODUCTION There are a number of studies concerning analytical solutions of mass-trans- port models for rock-single fracture systems (Neretnieks, 1980; Rasmuson and Neretnieks, 1981; Tang et al., 1981; Matoszewski and Zuber, 1985; Chen, 1986). Most of these authors except Matoszewski and Zuber (1985) have focused on continuous injection of pollutants which is represented as: c(x,t) = Co or c(x,t) = Co exp (-At) reflecting the effect of radioactive decay. These source conditions describe the * Contribution to the oral presentation of the International Conference on Chemistry and Migra- tion Behavior of Actinides and Fission Products in the Geosphere, held in Munich, F.R.G., 1987. 0169-7722/90/$03.50 © 1990 Elsevier Science Publishers B.V.

Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

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Page 1: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

Journal of Contaminant Hydrology, 6 (1990) 85-102 85 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

A D S O R P T I V E SOLUTE T R A N S P O R T IN F R A C T U R E D ROCK: ANALYTICAL S O L U T I O N S FOR D E L T A - T Y P E SOURCE CONDITIONS*

YOKO FUJIKAWA and MASAMI FUKUI

Research Reactor Institute, Kyoto University, Kumatori-Cho, Sennan-Gun, Osaka 590-04 (Japan)

(Received April 24, 1989; revised and accepted December 12, 1989)

ABSTRACT

Fujikawa, Y. and Fukui, M., 1990. Adsorptive solute transport in fractured rock: Analytical solutions for delta-type source conditions. J. Contam. Hydrol., 6: 85-102.

Solutions for adsorptive solute transport equations in a single fracture-rock system were derived under two different delta-type source conditions. One was the delta-type, flux injection condition. The correspondent solution can be used for the analysis of experimental column breakthrough curves obtained by injecting the tracer into the rock fracture. The other was the delta-type, resident fluid injection condition. The solution can be used for the crude estimation of pollutant migration from the underground radioactive waste repository.

In formalizing the resident fluid injection, an initial distribution of solute between solution and solid phase was assumed in order to satisfy the mass balance between injected and detected solute.

Each of the two solutions was also expressed in two ways reflecting the flux and the resident fluid detection. Solutions expressed in terms of the flux detection correspond to the effluent concentration measured experimentally by a fraction collector system. On the other hand, solutions expressed in terms of the resident fluid detection describe spatial distribution of the solute in the fluid.

Since considerably long tailings of breakthrough curves are often observed in column tracer experiments using fractured rocks, effects of some parameter values on the tailing were also discussed. It was shown that the adsorption of solute to rock matrix caused longer tailing. It was also shown that the resident fluid injection caused longer tailing of breakthrough curves than the flux injection condition.

INTRODUCTION

There are a number of studies concerning analytical solutions of mass-trans- port models for rock-single fracture systems (Neretnieks, 1980; Rasmuson and Neretnieks, 1981; Tang et al., 1981; Matoszewski and Zuber, 1985; Chen, 1986). Most of these authors except Matoszewski and Zuber (1985) have focused on continuous injection of pollutants which is represented as:

c ( x , t ) = Co or c ( x , t ) = Co exp (-At)

reflecting the effect of radioactive decay. These source conditions describe the

* Contribution to the oral presentation of the International Conference on Chemistry and Migra- tion Behavior of Actinides and Fission Products in the Geosphere, held in Munich, F.R.G., 1987.

0169-7722/90/$03.50 © 1990 Elsevier Science Publishers B.V.

Page 2: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

8 6 Y. FUJIKAWA AND M. FUKUI

release of nuclides at constant concentrations (Rasmuson, 1984). However, other conditions such as finite or very short duration of injection, variations of source strength and the finite volume of source are equally important and likely to occur in actual sites and laboratory experiments. Therefore, it is more convenient to obtain solutions for delta-type (instantaneous or point) injections. Application of the concept of superposition to the solution of delta- type injections would enable us to cope with the above problems. In this study, we are dealing with this delta-type injection.

On the other hand, many authors have pointed out the necessity to distin- guish between two modes of injection or of detection in convection-dispersion equations (Kreft and Zuber, 1978; Kreft and Zuber, 1979; Parker and van Genuchten, 1984; van Genuchten and Parker, 1984).

The two modes of injection, the flux injection and the resident fluid injection, represent the case where solutes are added to the fluid that enters the system at the inlet boundary and solutes are placed in the system at a given moment, respectively. The former may correspond to column experiment conditions where tracers are injected into the fluid at the inlet boundary, and the latter may correspond to source conditions of underground waste disposal repositories from where pollutants are leaching.

The two modes of detection, the resident fluid mode of detection and the flux mode of detection, are the measurement of resident fluid concentrations cr and flux concentrations cf, respectively. The two kinds of concentration, cr and cf, are introduced by Kreft and Zuber (1978). cr is defined as "the mass of solute per unit volume of fluid contained in an elementary volume of the system at a given instant", and cf as ~the mass of solute per unit volume of fluid passing through a given cross-section at an elementary time interval". Namely, c r represents the amount of solute in a unit volume of fluid at each point of the space and is used in the usual continuum approach to solute transport in porous media (Parker and van Genuchten, 1984). On the other hand, cf represents the effluent con- centration at each point of the space which can be measured experimentally by fraction collector systems. Consequently, Cr can be used to describe the spatial distribution of solutes in the fluid, and cf to describe breakthrough curves (designated BTC's hereafter) of solutes at a given point in the system. The relation between Cr and cf in one-dimensional systems is generally expressed as follows (Kreft and Zuber, 1978):

UCf = UC r -- D~cr/~x (1)

Taking into consideration that Cr(X,t) is finite at x --* 0o, the following equation can be derived from eq. 1 (Carslaw and Jaeger, 1959):

Cr(X,t) = aID ~ exp(- u~/D)cf(x + ~,t)d~ (2) 0

Since conventional solutions have sometimes been applied to the interpreta- tion of experimental results without distinguishing cr from cf, some problems have occurred (Parker and van Genuchten, 1984). For example, the use of Cr

Page 3: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 87

instead of cf in analyzing BTC's is known to induce an overestimation of pore velocity (Nkedi-Kizza et al., 1983). Solutions for resident fluid injection are often used to interpret experimental results obtained in the situation interpret- able as flux injection, which will also lead to erroneous inference. It is apparent that these two modes of injection and detection should be distinguished carefully in solving contaminant transport problems.

Limiting the considerations to injections of the delta type, combination of two injection modes and two detection modes yields four kinds of concentra- tions to be considered. Using the notation introduced by Kreft and Zuber (1978), these can be expressed as follows: Cir~ = point injection in resident fluid, and detection in resident fluid; cirf = point injection in resident fluid, and detection in flux; Clfr = instantaneous injection in flux, and detection in resident fluid; and ci~ = instantaneous injection in flux, and detection in flux.

As shown in the following part of the paper, four kinds of solutions of adsorptive solute transport equations for a rock-fracture system, i.e. ci~r, c~rf, c~, and ci,, were obtained. Effects of the adsorption reaction on source conditions were also considered and the mass balance between the injected and observed solute was mathematically checked.

MODEL

The physical model considered was essentially the same as that of Tang et al. (1981) who considered: (1) advective transport in the fracture; (2) longitudi- nal mechanical dispersion and longitudinal molecular diffusion in the fracture; (3) molecular diffusion from the fracture into the matrix; (4) linear adsorption onto the fracture surface (interface between the fracture and the rock matrix) and within the rock matrix; and (5) radioactive decay, in a single fracture-rock system. Mixing in the fracture cross-section was assumed to be complete and diffusion in rock matrix parallel to the fracture axis was neglected.

Though radioactive decay term is neglected in this paper for simplicity, the effect can be included by multiplying the obtained solution by exp ( - ~t) in case of delta-type injection, where ~ is radioactive decay constant and t is the elapsed time since the delta-type injection occurred. We also obtained solutions for irreversible adsorption combined with a linear adsorption isotherm, which will be referred to in a forthcoming paper.

Matoszewski and Zuber (1985) have also obtained a solution for a delta-type injection in a similar rock-fracture system. The main differences between their study and ours are that they considered the ci~ case only and did not include into the model solute adsorption onto the fracture wall. The solution by Tang et al. (1981) is for continuous type injection for which the boundary conditions were improperly posed according to Parker and van Genuchten (1984).

The rock-fracture system considered in our study is shown in Fig. 1. Model equations for mass transport in the fracture are as follows:

1 ~s 0cr 02Cr ~C~ q - - - - + -- D - u - - - ( 3 ) b ~)t ~t 0z 2 ~z b

Page 4: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

88

X-(OXiS /]/~/ Rock rnotrix / ~

2b ,

/ /

Y. FUJIKAWA AND M. FUKUI

F ig . 1. R o c k f r a c t u r e s y s t e m .

s = g a c r

q = - 0 D ' (?c'r/~Xlx-b

Equa t ions for mass t r anspor t in the rock mat r ix are as follows:

t~OC'~ D,t~20c'~ ~t t3x 2

@ b Q

Ot

with

Q =

(4)

(5)

(6)

Kdc'~ (7)

where t = t ime [T]; z = coord ina te a long the f rac tu re [L]; x = coord ina te pe rpend icu la r to z-axis [L]; Cr = res ident fluid co n cen t r a t i o n of solute in the f r ac tu re (Cr = Cr (z,t)) [M/L3]; c~ = res ident fluid co n cen t r a t i o n of solute in the fluid in the rock ma t r ix (Cr = C'~(Z,X,t)) [M/L3]; s = mass of solute adsorbed per un i t length of f r ac tu re surface [M/L2]; Q = mass of solute adsorbed per un i t mass of solid in the rock mat r ix (Q = Q(z , x , t ) ) [M/M]; 2b = f rac tu re width [L]; u = flow ve loc i ty in the f r ac tu re [L/T]; D = hydrodynamic dispers ion coeffi- c ient of the flow th rough the f rac tu re [L2/T] (with D = Dm + ~u, where Dm is molecu la r diffusion coefficient [L2/T] and ~ is dispers ivi ty [L]); 0 = poros i ty of the rock matr ix; D' = effect ive diffusion coefficient in the rock mat r ix [L2/T]; Ka = d is t r ibu t ion coefficient of the solute adsorp t ion to f r ac tu re surface [L]; and Kd = d is t r ibu t ion coefficient of the solute adsorp t ion to rock mat r ix [L3/M].

Combining eqs. 3-5 and mak ing use of the r e t a rda t i on cons tan t R, the final equa t ion for the solute t r anspo r t in the f r ac tu re is ob ta ined as follows:

~Cr D ~2Cr U ~Cr OD' OCr' x=b (8) ~t - R ~z 2 R ~z + b--R- ~--x-

with

R = l + K a / b

Similarly, combin ing eqs. 6 and 7 and mak ing use of the r e t a rd a t i o n cons tan t R', we have as final equa t ion for the solute t r anspo r t in the rock matr ix:

Page 5: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 89

0Cr D' ~2C' r - ( 9 )

8t R' 8x 2

with

R" = 1 + flbKd/O

As for further development, we also tried to incorporate surface diffusion, diffusion in the adsorbed phase, proposed by Skagius and Neretnieks (1982, 1985). Defining the surface diffusion coefficient for the fracture surface and the rock matrix as D~ and D's, respectively, the incorporation can be achieved by using the diffusion coefficients D + DsK a and D' + KdD'~/O instead of D and D', in eqs. 8 and 9, respectively.

SOURCE CONDITIONS

In this section, we will derive the source conditions for the two modes of delta-type adsorptive solute injections.

The initial and boundary conditions (IBC) of the two modes of delta-type injections for the following non-adsorptive one-dimensional solute transport equation (Kreft and Zuber, 1978):

~nCr-- D ~2ncr ~nCr (10) " Ot ~z 2 - u OZ

are as follows:

(1) The IBC of the resident fluid injection into an infinite area are:

C r ( z , O ) : M6(z)/(nn')

lira Cr (z,t) = 0

(2) The IBC of the flux injection into a semi-infinite area are:

cr(0,t) Du ~cr(z,t)~z ~=0+ - --nA'uM 6(t)

Cr(Z,0) = 0 for Z > 0

lim Cr(Z,t) = 0

Here, n is the porosity of the one-dimensional porous medium where solutes are transported; A' is cross-section area [L2]; M is the total mass [M] injected into the system; and 6(z) and 6(t) are Dirac delta functions [1/L and l/T], respective- ly.

We expanded the IBC for one-dimensional non-adsorptive solute transport to obtain the IBC for adsorptive solute transport in the two-dimensional, permeable rock-fracture system shown in Fig. 1. Here the porous medium where one-dimensional solute transport takes place is replaced by the rock fracture. Since the existence of fracture-filling materials is not assumed in our

Page 6: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

90 Y. FUJIKAWA AND M. FUKUI

study, the value of porosity n is set to 1. Since we deal with adsorptive solutes instead of non-adsorptive solutes, the effect of instantaneous adsorption to the fracture wall has to be included in the conditions. The development of the IBC is shown in Appendix A. The obtained IBC's are as follows.

(1) IBC's of the delta-type resident fluid injection into an infinite medium are:

Cr(Z,O ) = M6(z)/(2ab R) (11)

cr (+ ~, t ) = 0 (12)

C'r(b,z,t ) = Cr(Z,t ) (13)

Cr(~,z,t) = 0 (14)

G(x,z,O) = 0 (15)

where a is the width of the rock (see Fig. 1). Here we considered that the solute is previously distributed between solution and solid phase. Thus, under the assumption of instantaneous sorption of solute, the solute adsorbs to the solid phase at the same moment of resident fluid injection and therefore, from the beginning, the solute exists in both the solution phase and the solid phase. The initial condition (11) implicitly means that s(z,0), i.e. the initial distribution of solute adsorbed per unit length of fracture surface, equals KaM6(z)/(2abR) and thus the total amount of mass in the system is M. If the initial distribution of solute in the fluid, c(z,0), is M6(z)/(2ab), then the total mass existent in the system becomes RM. In order to make M the total mass in the system, c(z,0) should be M6(z)/(2abR). The concept of the previous distribution can be seen in Lindstrom and Boersma (1971).

(2) IBC's of the delta-type flux injection into a semi-infinite medium are:

D ~C r z=O+ (16) Cr(0,t) = M6(t) + u ~z

Cr(Z,O ) : 0 (17)

C r ( ~ , t ) = 0 (18)

~G(~,t)/Oz = 0 (19)

cr(b,z,t) = cr(z,t) (20)

c~(~,z, t) = 0 (21)

Cr(X,z,O) = 0 (22)

Now we will discuss to what cases these two kinds of injection conditions can be applied.

The delta-type resident fluid injection into a rock fracture means that at t = 0 a certain amount of solute exists at one point of fracture fluid and wall. We may think of underground repositories which are in their early stage of

Page 7: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 91

destruction. Most of the leachate stays in the nearest neighborhood of the repositories and therefore can be regarded as a point source of contamination. Par t of the leachate is adsorbed to the geological medium. The major pathway of leachate migrat ion is the fracture running beside the repository. Our solutions for resident fluid injection may be used for a crude estimation of leachate migration from such repositories.

The delta-type flux injection into a rock fracture means that a certain amount of mass is instantaneously injected at the entrance of the rock-f rac ture system. Such si tuation may correspond to the laboratory column experiment using a rock core with a single fracture where the t racer material is injected at one side of the core together with solution and the condition of constant flow rate is maintained throughout the experiment. The solutions for the flux injection, if expressed in flux-mode of detection form, may be used for analyses of experimental BTC's.

SOLUTION

Analytical solutions of the rock-f rac ture system were obtained by applying the Laplace transform to the governing eqs. 8 and 9. The development of the solutions is summarized briefly in Appendix B. The solutions are listed in Table 1 for the four combinations of different inject ion-detect ion modes together with the IBC used. Since flux concentrat ions in the rock matrix have no relevance to physically measurable values, solutions c]rf and c~ are not listed in the table.

Eqs. 23 and 24 are the mass-balance equations that solutions of flux injection and those of resident fluid injection should satisfy, respectively:

L L :c t

2abR f cifr dz + 2aOR" f f C]frdxdz + 2ab f uciffdt 0 o b 0

= 2abRicifrdz + 2aOR'iiC~frdzdx 0 0 b

= M

L

2abR f Cir r dz + 2aOR'

L

= 2abR f ci~dz

= M

(23)

ci~dxdz + 2ab UCirfdt ~z b 0

+ 2aOR' i ?C~rrdzdx :c b

(24)

Here both first lines of eqs. 23 and 24 are the sum of mass in the region z < L and the mass outflow from the point z = L. The second lines are the integration of the spatial distribution of solute concentrat ion. The third lines are the total mass of solute injected into the system, M. Solutions (1-3), (1-4), (1-5), (1-6), (1-7) and (1-8) in Table 1 are integrable and mathematical ly proved to satisfy the mass-balance equations (23) and (24). This indicates tha t the source conditions

Page 8: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

TA

BL

E 1

Sol

utio

ns f

or i

nsta

ntan

eous

in

ject

ions

int

o th

e ro

ck-f

ract

ure

syst

em

Init

ial

and

boun

dary

con

diti

ons

Sol

utio

ns

Inje

ctio

n in

res

iden

t fl

uid

(i

nfin

ite

med

ia):

M

cr(z,

O)

= ~

~(z

)

cr(

+ ~

,t)

=

0

c~(b

,z,t

) =

C

r(Z,

t )

G(~

,z,t

) =

0

G(X

,z,O

) =

0

(1-1

)

eirr

Cirf C~

rr --

M

2a

bD

~ D

t/(4

R)

G71

/2 e

xp

z2

u27

2T 3/

2 2-

D

167

D 2

~

d7

) [

J __

(6

7) 2 7

M

Dz

G7~

/2 ex

p u

z z 2

u2

7 d~

2a

bD

n

o +

~

2T3/

2 2

D

167

D 2

4T

Dt/

(4R

)

2a b

Drc

o

ex [U C

ry +

(R

'/D

')ll

2(x

- b)

2 T

312

z ~

u27

[G~'

+

(R

'/D

')U

2(x

- b)

]27

167

D 2

4T

]

d7

(1-3

)

(1-4

)

(1-5)

Page 9: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

Inje

ctio

n in

flu

x (s

emi-

infi

nite

med

ia):

D ~

c~ ~

=o.

e~(O

,t)

= M

S(t)

+

u ~z

cr(z

,O)

= 0

cr(o

o,t)

= 0

~c---z

(~,t)

=

0 O

z

c'~(

b,z,t

) =

c~(z

,t)

e'~(

oo,z,

t) =

0 G(

x,z,O

) -

0

(1-2

)

Cifr

Ciff

~ciDt/

(fR)

I (Z

-}-

?~)2

u2

7 (G

7)2

u(z-

q)]

M

G(z

+

tl)e

xp

D 2

--

+

--

dT

dt /

(1-6

) 2a

b D

~ 8(

Ta7

)u 2

167

4T

2D

o o

Dt/

(4R

) I

J M

f

Gz

z 2

U27

(6

7) 2

UZ

7

2abu

~ o

8(T

3Y)l

/2ex

p 16

7 D

2

4T

+~-

~ d7

(1

-7)

c~f,

co

Dt/

(4R

)

M

f f

[GT

+(R

'/D

')l/

2(x-

b)](

z+q)

2a

bDn

8(Ta

7)1/

2 0

o (z +

q)

2 u2

7 [G

7 +

(R'/D

')II2

(x

- b)

] 2

u(z

- tl)

1 x

exp

167

/92

4T

+ -2

D

dTdr

/ (1

-8)

> ©

> z © ~8

>

G =

40

(D'R

')I/

e/(D

b) a

nd T

=

t 4R

T/D

in

the

abov

e eq

uati

ons.

©

C3

Page 10: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

TA

BL

E 2

Sol

utio

ns f

or c

onti

nuou

s in

ject

ions

int

o th

e ro

ck-f

ract

ure

syst

em

Init

ial

and

boun

dary

con

diti

ons

Sol

utio

ns

Inje

ctio

n in

res

iden

t fl

uid

(inf

init

e m

edia

):

c o fo

r z

< 0

c~(z

,O)

= 0/

2 fo

r z

= 0

Ccrr

for

Z>

0

S(z,

O)

=

Kac

r(z,

O )

c~(-

~,t

) -

c o

c'~(

b,z,

t) =

c~(z

,t)

c'A

~,z

,t)

=

0 c'

~(x,

z,O

) =

0

Dt/(

4R)

Co

Dn

o

2G7

~-~

exp

[-

(GT

/2)2

/T]

erf¢

[z/

(4) '1

/2)

- u~

,t/2/

D]

d),

Dt/(

4R)

(2-1

) c o

f

2G7

o

D

exp

[-

(G~/

/2)'~

/T]

erfc

[Z/

(471

/2)

--

u71/

2/D

] +

-

-

4~ U

2 U

(2-4

)

3 ex

p [

z/(47

1/'2

) u}

,lJ2/

De]

ld 7

x

(2-5

)

Inje

ctio

n in

flu

x (s

emi-

infi

nite

med

ia):

cr(z

,O)

= 0

Df°r

~cr

z >

0 (2

-2)

Cr(

0,t )

=

C O +

--

--

u t?

z

cr(b

,z,t)

=

cr(z

,t)

c'~(

=~,

z,t)

= 0

c'~(

x,z,O

) =

0

i t

t ̧

Ccfr

= ~

{ ~0

4~(~

- ~

)~/2

ex

p

c o

zAR

1/2

cc

~ =

--

ex

p ~o

4[D

(t"

- fl)

3fl]~

j2

o

Solu

tion

by

Tan

g et

al.

(198

1) (

sem

i-in

fini

te m

edia

):

cr(O

,t )

= co

~

Dt/(

4R)

cr(~

,t)

= 0

cr(z

,O)

= 0

c'~(

b,z,

t) =

c~(z

,t)

(2-3

) c~

,(~,

z,t)

=

0 c'

~(x,

z,O

) =

0

(Aft

) 2

u(z

- rl)

~

dfld

t'dq

--

+

-2b

J I

(Z +

~/

)2R

u2

fl

4Dfl

4D

R

4(t'

- fl

)

z2R

u2

fl (A

ft) 2

uz

7 4D

fl

4DR

4(

t' -

fl)

+ -~

J df

ldt"

Cr

CO

f 4~

U 2

0 D

t/(4R

)

= c

o f

Cf

7~ "I/2

Z -~

exp

[uz/

(2D

) z2

/(16

~)

- (v

/D2)

ql

erfc

[G

~/(2

T)]

d~

[z/8

-

D/(

4u)

+ D

z2/(

32uq

)]

r/~'2

exp

[uz/

(2D

) z~

/(16

q)

(v/D

fiql

er

fc [

Gq/

(2T

)]d~

/

(2-6

)

(2-7

)

(2-8

)

(2-9

)

T =

t -

4RT/

D,

G

= 40

(DtR

')~'~

/(D

b)

and

A

= O

(D'R

')I"~

/(Rb

) in

the

abo

ve e

quat

ions

.

Page 11: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 95

a r e f o r m u l a t e d p r o p e r l y so t h a t t h e t o t a l m a s s in t h e s y s t e m e q u a l s M. O n e s h o u l d n o t e t h a t t h e r e t a r d a t i o n f a c t o r R s h o u l d be i n c l u d e d in t h e s o u r c e c o n d i t i o n s for t h e r e s i d e n t f lu id i n j e c t i o n (eqs. 11-15). O t h e r w i s e , s i g n i f i c a n t m a s s i m b a l a n c e w o u l d o c c u r in t h e s o l u t i o n .

S o l u t i o n s fo r c o n t i n u o u s i n j e c t i o n o f s o l u t e w e r e o b t a i n e d b y u t i l i z i n g t h e t r a n s f o r m a t i o n f o r m u l a g i v e n b y K r e f t a n d Z u b e r (1978). T h e I B C ' s (2-1) a n d (2-2) a n d t h e s o l u t i o n s (2-4), (2-5), (2-6) a n d (2-7) a r e l i s t e d in T a b l e 2 t o g e t h e r w i t h t h o s e b y T a n g e t al . (1981) (eqs. 2-3, 2-8 a n d 2-9) w h i c h w i l l be r e f e r r e d on p. 98.

RESULTS AND DISCUSSION

I n t e g r a l t e r m s o f t h e s o l u t i o n s fo r t h e r o c k - f r a c t u r e s y s t e m w e r e n m n e r i - c a l l y c a l c u l a t e d to d i s c u s s e f fec t s o f i n j e c t i o n m o d e s , d e t e c t i o n m o d e s , a n d s o m e p a r a m e t e r v a l u e s on B T C ' s o f t h e s o l u t e . P a r a m e t e r v a l u e s fo r e a c h r u n a r e l i s t e d in T a b l e 3.

(1) E f f e c t s o f i n j e c t i o n m o d e s w e r e c o n s i d e r e d . I n F ig . 2, B T C ' s fo r d e l t a - t y p e r e s i d e n t f lu id i n j e c t i o n a n d f lux i n j e c t i o n a r e s h o w n . T h e f lux c o n c e n t r a t i o n fo r

TABLE 3

Parameters used

Figure Injection mode Detection mode ~ K. Kd (× 10 -2 ) (×10 2) (× 10 3)

Fig. 2 delta-type flux detection 0.1 0.0 0.0 (1) resident fluid 1.0

injection 10.0 (2) flux injection

(c~rf, ci~)

Fig. 3 delta-type (1) resident fluid 0.1 0.0 0.0 flux injection detection 1.0

(2) flux detection 10.0

Fig. 4 delta-type flux detection 0.1 0.0 0.0 flux injection 0.02

0.08

Fig. 5 delta-type flux detection 0.1 0.0 0.0 flux injection 0.5

5.0

Fig. 6 continuous flux detection 0.1 0.0 0.0 (1) resident fluid 1.0

injection 10.0 (2) flux injection (3) Tang et al. (1981) type

injection

The other parameter values are fixed to b = 0.1.10 _2 , u = 0.015-10 2 Dm _ 0.0009.10 4, Pb - 2.65 • 103, 0 = 0.01, D' = Din. Here m, kg, s are used as units.

Page 12: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

96 Y. FUJIKAWA AND M. FUKUI

J

o

c

c~

m

3"0 / T Cif f injection in flux

OC=lO.Ox 10 -2 Cir f injection in resident 2.0- fluid

Ot =0.1 x 10 -2

0.0 ' ~ . . . . I I 0 .0 " ,!O 2.~00 3.0 41.0

Dimensionless Effluent Volume u t / l

Fig . 2. B r e a k t h r o u g h c u r v e a t z = L f o r t w o d i f f e r e n t m o d e s o f i n j e c t i o n ,

resident fluid injection (ci~) is the solution for an infinite medium and is susceptible to larger back-diffusion. Therefore BTC's of ci,f have lower peak value and longer tailing compared with those of ci~, the flux concentration for flux injection. The difference between ci~ and ci~ becomes more significant as dispersivity constant values become larger.

(2) Effects of detection modes were considered. In Fig. 3, two concentrations of different detection modes for the flux injection observed at a fixed point of the system, namely ci~ and cifr, are shown. The resident fluid concentration Cifr shows the lower peak and longer tailing than the flux concentration cir. The area under the curve in Fig. 3 corresponds to ~ 2abuci~dt and ~ 2abucifflt for

3 . 0 - -

e . J v

c

3

c~

2.0

I.O-

0.0 0.0

- - C i f f detection in flux

C(= IO.Ox IO -2 - - - Cif r detection in resident fluid

II a=0 .1x l0 -2

-'/ i . . . . . . . . I 1.0 2.0 3.0 4.0

Dimensionless Effluent Volume u t / /

F ig . 3. B r e a k t h r o u g h c u r v e a t z = L f o r t w o d i f f e r e n t m o d e s o f d e t e c t i o n .

Page 13: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 97

g

g

c o

L

3.0-

2.0

1.0

K a = 0.0 x IC) z

o.o I 0.0 4.0

/ • K O = 0 . 0 2 x I 0 - 2

\ \ /,,"\ < :oo xo / / \ ~ S - - , , / °

,.o "\'--Td--- Dimensionless Effluent Volume u i / L

F i g . 4. S e n s i t i v i t y o f b r e a k t h r o u g h c u r v e a t z = L t o p a r a m e t e r K s.

the concentrat ion of flux detection and of the resident fluid detection, respec- tively. Although the integral of the flux concentration ~121 2abuci~dt expresses the mass of solute transported through a given cross-section for a given time interval, the integral of the resident fluid concentration ~tt21 2abucifrdt lacks this property (Kreft and Zuber, 1979). Namely, while S~ 2abuq~dt equals M (the total mass injected into the system), S~ 2abucifrdt does not. Since the integral of resident fluid concentrat ion ~ 2abucifrdt did not satisfy the mass-balance law, it is apparent that the resident fluid concentration should not be used as effluent concentrat ion to analyze experimental BTC's.

(3) Sensitivity analyses of BTC's were conducted for two parameters, Ka and

_J

o =

"E

x

u_

m E

3 .0 -

2 .0 -

1.0-

0.0

0.0

K d = 0.0

//, 1.0 2.0 3.0

Dimensionless Effluent Volume u t /L

~ 6 s

: 5 . 0 \ \ \ . . x 10-3\ "--~

4.0

Fig . 5. S e n s i t i v i t y o f b r e a k t h r o u g h c u r v e a t z = L t o p a r a m e t e r K d.

Page 14: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

98 Y. FUJIKAWA AND M. FUKUI

Kd. A column tracer experiment where tracer material is introduced into the fluid instantaneously at the inlet boundary was assumed and therefore ci~ was used to describe the effluent concentration. Fig. 4 shows that the adsorption of solute to fracture wall causes lower and retarded peaks in BTC's. The tailing effect is rather insignificant here. Fig. 5 shows that the adsorption of solute to rock matrix causes lower and retarded peaks and also longer tailings in BTC's. As can be seen from eq. 10, larger R' means a smaller apparent diffusion coefficient D ' / R ' . Therefore, rocks with finer mineral structure (which means smaller effective diffusion coefficient) and larger adsorption capacity may produce longer tailings in BTC's.

(4) BTC's were compared for three modes of continuous injections (see Fig. 6). Solutions used to calculate the BTC's were Ccr~ (eq. 2-5) for resident fluid injection, cc~ (eq. 2-7) for flux injection, and the transformed solution of Tang et al. (1981) (eq. 2-9) in Table 2. Here eq. 2-9 was obtained by converting the original solution by Tang et al. (1981), which is expressed as resident fluid concentration form, into flux concentration form. All these solutions, with radioactive decay constant set to zero, are listed in Table 2.

As shown in Fig. 6, the transformed solution of Tang et al. (1981) yields an earlier point of breakthrough and greater mass inflow into the system than the other two. The difference between them is greater for higher dispersivity constants. The fact that the solute flux at the inlet boundary ucr(O,t) - D~cr(O,t)/~z is UCo - D~cr(O,t)/~z under IBC (2-3) in Table 2 indicates that rates in the solutions by Tang et al. (1981) at which solutes are supplied to the system depend on dispersion coefficients and the concentration gradient

_J

, j -

g o

c

g £.)

=* E

;=,

1.5 Ccf f injection in f lux

. . . . Ccr f injection in resident fluid - - - s o l u t i o n by Tong et ol.

1.0 ~- 1.5 c .,-~-~ 0~: OA x I0 -2

o.5 ~- t ' r.s-T . . . . . . . . ct: t.o × Io -z

I / I J;E - - - - . . . . . . o.o

0.0 3.0 d. = IO.O x I 0 -2

0.0 2 .0

0 . 0 'v I I P 0.0 1.0 2.0 3.0

Dimensionless Effluent Volume u t / L

Fig. 6. Compar ison of b r eak th rough curves at z = L for three kinds of con t inuous injection modes.

Page 15: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 99

at the inlet boundary . Therefore, the appl ica t ion of flux in jec t ion condi t ions (2-2) is preferable when the solute is supplied to the system at a cons tan t rate.

CONCLUSIONS

The model equat ions (8) and (9) for mass t r anspor t in the r o c k - f r a c t u r e system shown in Fig. 1 were solved for two different del ta- type in ject ion and detec t ion modes. The obta ined solut ions are listed in Table 1. The IBC's (1-1) and (1-2) were formula ted proper ly so tha t the integral of the solut ion equals the amoun t of solute injected into the system.

The effect of different in ject ion and detec t ion modes became more signifi- can t for h igher dispersivi ty cons tan t values. BTC's of res ident fluid in ject ion yielded lower peak and longer ta i l ing compared to those of flux injection. BTC's obta ined by regard ing res ident fluid concen t r a t i on as effluent concent ra- t ion yielded lower peak and longer ta i l ing compared to those obta ined by regard ing flux concen t r a t i on as effluent concent ra t ion . It is discussed tha t the use of res ident fluid concen t r a t i on as effluent concen t r a t i on should be avoided because the res ident fluid concen t r a t i on does not satisfy the mass-balance law tha t effluent concen t r a t i on should satisfy. La rge r values of the f rac ture surface adsorp t ion coefficient Ka produced lower and re ta rded peaks in BTC's. La rge r values of the rock mat r ix adsorp t ion coefficient K d produced lower and re ta rded peaks and also longer ta i l ings in BTC's.

Solut ions for con t inuous solute in ject ion (eqs. 2-4, 2-5, 2-6 and 2-7 in Table 2) were obta ined t h rough t r ans fo rma t ion of solut ions for delta-type injection. Compar ing the obta ined solut ions with those of Tang et al. (1981), it was shown tha t the IBC of cons tan t concen t r a t i on bounda ry (eq. 2-3) used by Tang et al. (1981) should not be applied to the case where solutes were supplied to the system at a cons tan t rate.

APPENDIX A -- DEVELOPMENT OF INITIAL AND BOUNDARY CONDITIONS FOR ADSORPTIVE SOLUTE TRANSPORT

(1) Conditions of the delta-type resident fluid injection

Source conditions (initial conditions for resident fluid injection) should be set so that the total mass of solute that initially exists in the system equals the sum of mass in the liquid phase and adsorbed phase at t = 0. Since adsorption of the solute to the rock matrix is controlled by diffusion and requires a finite period of time, adsorption to the rock matrix is not likely to occur at t - 0. On the other hand, adsorption to the fracture wall which is always in contact with the solution phase is expected to occur at the same moment of solute injection into the fracture fluid. Therefore, initial distribution of solute between solution in the fracture and the fracture wall was assumed. Let the initial values of resident fluid concentration c r and adsorbed solute per unit length of fracture surface s be:

c~(z,O) - Ml~(z ) (A4)

s(z,O) = M25(z ) (A-2)

Page 16: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

100 Y. FUJIKAWA AND M. FUKUI

whe re M 1 a n d M 2 are u n k n o w n cons t an t s . In order to sa t i s fy m a s s - b a l a n c e cr i ter ia , cr(z,0) and s(z,0) shou ld sa t i s fy t he fo l lowing e q u a t i o n (here M is to ta l m a s s in jec ted in to the system):

2ab ? Cr(Z,O)dz + 2a i s(z,O)dz = M (A-3)

S u b s t i t u t i o n of eqs. A-1 and A-2 in to eq. A-3 gives:

2abM 1 + 2aM 2 = M (A-4)

Since the so lu te is d i s t r ibu ted by l i nea r adso rp t ion i s o t h e r m (4), t he fo l lowing r e l a t i onsh ip be tween M 1 and M s is obta ined:

M,~ - KaM1 (A-5)

So lv ing eqs. A-4 and A-5, the u n k n o w n c o n s t a n t s M, and M 2 a re de t e rmined as follows:

MI = M/(2abR)

M 2 = KaM/(2abR )

Thus , t he sou rce condi t ions :

cr(z,0) - MS(z)/(2abR)

s(z,O) - KaMg(z)/(2abR)

were obta ined . O t h e r cond i t i ons were d e t e r m i n e d af te r T a n g et al. (1981) and Matoszewsk i and Zuber (1985) as l i s ted in Table 1.

(2) Conditions of the delta-type flux injection

Source cond i t i ons (bounda ry cond i t i ons for flux in jec t ion) shou ld be set so t h a t the to ta l m a s s of so lu te in jec ted a t t he in le t b o u n d a r y equa l s M. Let the flux c o n c e n t r a t i o n cf(0,t) be:

cf(O,t) = M~5(t) (A-6)

whe re M 3 is an u n k n o w n cons t an t , cf(0,t) shou ld sa t i s fy t he fo l lowing m a s s - b a l a n c e equat ion:

2abu icf(O't) = M (A-7)

0

S u b s t i t u t i n g eq. A-6 in to eq. A-7, M 3 is de t e rmi ned as follows:

M~ = M/(2abu)

Since eq. A.6 def ines t he a m o u n t of m a s s inflow in to t he sys tem, t he effect of adso rp t ion phase need no t be cons ide red as in t he r e s iden t fluid in jec t ion case. Thus , t he sou rce condi t ions :

cf(O,t) = M~(t)/(2abu)

s(z,0) = 0

were obta ined . O the r cond i t i ons were de t e rmi ned af te r T a n g et al. (1981) and Maloszewsk i and Zuber (1985) as l i s ted in Table 1.

A P P E N D I X B - - S O L U T I O N OF T HE M O D E L E Q U A T I O N S

The who le so lu t ion p rocedure is s imi la r to t h a t of T a n g et al. (1981). In the case of r e s iden t fluid in jec t ion , the Lap lace t r a n s f o r m was appl ied to eqs. 8 and 9 wi th

Page 17: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

ADSORPTIVE SOLUTE TRANSPORT IN FRACTURED ROCK 101

respec t to t. The ob ta ined e q u a t i o n for 5r, t he Lap lace t r a n s f o r m of c,., was solved u n d e r IBC's (11) (15) as follows:

M { i A1/z OF - 2abD exp [u(z - ~)/(2D) - (~ - z)/2] 5(~)/A~/~d~ z

+ i exp [u(z - ~)/(2D) - A1/'2(z - ~)/215(~)/A~;zd~

U s i n g the fol lowing fo rmula (Tang et al., 1981):

( 2X2Xlj,2)/X]/2 = 1/7r~/~ ~ e x p ( - X 1 7 - X~/?)d7 (B-l) exp 71/2

5r could be simplif ied as follows:

Cr exp [uz/(2D) - A 7 - z2/(167)l/71/2d'/ (B-2) 2ab Dn v2

o

where

A (u /D) 2 + 4 R p / D + 40(D'R'p)~/2/(Db)

The Lap lace i nve r s i on of eq. B-2 is C~r r (eq. 1-3 in Table 1). In order to ob ta in t he flux concen t ra - t ion C,r f (eq. 1-4 in Table 1), eq. 1 was appl ied to eq. 1-3.

In the case of flux in jec t ion , model e q u a t i o n s (8) and (9) and the IBC (1-2) in Table 1 could be r ewr i t t en in t e rms of c r as follows:

~?cr D c2c~ R Ocf + "-b-R"~xOD" Oc'f ~ b (~t R cTz 2

~?c~ D' ~2c~

(?t R ' ~x 2

cf(O,t) = MS( t ) / (2abu)

cf(z,O) 0

c~(,~,t) 0

c'f(b,z,t) - cf(z,t)

c'f( o~ ,z,t) 0

c'~(x,z,O) 0

The Lap lace t r a n s f o r m was applied to t he above equa t i ons and ~f, the Laplace t r a n s f o r m of cf, could be ob ta ined as follows:

M cf = 2abu exp [uz/(2D) - A1/2/2] (B-3)

where

A (u/D) 2 + 4 R p / D + 40(D'R'p)l lZ/(Db)

The Lap lace i nve r s i on of eq. B-3 is ci~. (eq. 1-7 in Table 1). App ly ing eq. 2 to eq. 1-7, t h e flux c o n c e n t r a t i o n ci~ was t r a n s f o r m e d in to the r e s iden t fluid c o n c e n t r a t i o n c~e r (eq. 1-6 in Table 1).

Page 18: Adsorptive solute transport in fractured rock: analytical solutions for delta-type source conditions

102 Y. FUJIKAWA AND M. FUKUI

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