advection and dispersion of contaminant in a permeable medium embedding fractures in which advection...

$: Advection and dispersion of contaminant in a permeable medium embedding fractures in which advection velocity is comparatively slow$
Journal of Hydrology

ELSEVIER Journal of Hydrology 199 (1997) 135-162

Advection and dispersion of contaminant in a permeable medium embedding fractures in which advection velocity is

comparatively slow

Hillel Rubin a'*, Dmitry Dveyrin a, Jens Birkh61zer b, Gerhard ROUV6 b

"Department of Civil Engineering, Technion-lsrael Institute of Technology, Haifa, 32000, Israel blnstitut fiir Wasserbau und Wasserwirtschaft der RWTH Aachen, Mies-van-der.Rohe-Str. 1, D-52056 Aachen,

Germany

Received 21 March 1995; revised 7 August 1996, accepted 20 August 1996

Abstract

This paper concerns the advective migration of an aqueous contaminant in a fractured permeable formation. Calculation and analysis refer to fractures in which advection velocity is comparatively small. The slow advective velocity is created by some local cavities or almost vertical orientation of the fractures. It is shown that the advective contaminant transport is characterized by two dimensionless parameters: (1) the ratio between the permeable block and fracture flow rates, termed the 'mobility number'; (2) the ratio between the longitudinal advection velocity of the permeable block flow and that of the fracture flow, termed the 'advection velocity ratio'.

For a sufficiently large domain, continuum parameters can be applied to characterization of the fractured permeable formation. Then the mixing between the permeable block flow and the fracture flow is a mechanism leading to contaminant dispersion. For very small values of the advection velocity ratio, the effective dispersivity is fully determined by the mobility number. Its value decreases with increase of the mobility number. Under such conditions, the equivalent porosity is constant, and its value is identical to that of the permeable blocks. A significant value of the advection velocity ratio increases the equivalent porosity of the fractured permeable formation. With regard to the effective dispersivity of the domain, if the advection velocity ratio is equal to unity then the effective dispersivity vanishes. The larger the deviation of the advection velocity from unity, the larger is the effective dispersivity of the fractured permeable medium.

* Corresponding author.

0022-1694/97/$1%00 © 1997- Elsevier Science B.V. All rights reserved PII S0022-1694(96)03258- I

136

1. Introduction

H. Rubin et aL/Journal of Hydrology 199 (1997) 135-162

Much effort has been invested in various studies concerning the characterization of contaminant transport in tight impermeable fractured formations. In such formations fluid practically flows only through the fracture network, but contaminant advected by the fracture flow may diffuse into the porous impermeable blocks composing the formation. Therefore fractured porous formations are often referred to as double porosity media. Various laboratory experimental studies concern solute transport in single fractures embedded in such tight formations (e.g. Sharp, 1970; Iwai, 1976; Grisak et al., 1980; Moreno et al., 1985; Schrauf and Evans, 1986; Haldeman et al., 1991). Field studies have also been reported (e.g. Novakowski et al., 1985; Abelin, 1986; Raven et al., 1988; Rudolph et al., 1991; Novakovski and Lapcevic, 1994). These studies have provided an understanding of the processes controlling contaminant migration in the fractured domain, such as fracture dispersion, matrix diffusion, and channelling.

Various types of analytical solutions have been developed for contaminant transport through a fracture embedded in an impermeable porous matrix. Neretnieks (1980) developed a solution considering negligible dispersion in the fracture. Tang et al. (1981) have included the fracture flow dispersion in their calculations. Various studies have extended those solutions for cases of instantaneous injection and constant mass flux source of the contaminant (e.g. Maloszewski and Zuber, 1984a, b; Moreno and Rasmuson, 1986). Stochastic analytical solutions have been developed by Rasmuson (1985) and Cvetkovic (1991). Other types of analytical solutions for contaminant advection in the fracture flow associated with diffusion in the porous blocks have been developed by Sudicky and Frind (1982), Sudieky and Frind (1984), Rasmuson and Neretnieks (1986), and Rowe and Booker (1990). In all these studies it is assumed that contaminant transport along the fracture is much more rapid than that within the porous blocks. However, measurements (McKay, 1991; McKay et al., 1993a, b) have shown that sometimes contaminant transport along the fracture is of the order of magnitude of that within the porous blocks. Therefore, Kennedy and Lennox (1995) have developed a control volume model of contaminant transport in a single fracture by which they have evaluated the significance of matrix diffusion in the direction parallel to the fracture axis.

Various studies (e.g. Grisak and Pickens, 1980; Noorishad and Mehran, 1982; Sun and Yeh, 1983) developed finite element numerical models aiming at the simulation of different cases of contaminant transport in fractured porous matrices. Basic issues of modelling of contaminant transport such as scales, algorithms and grid types for assessment of contaminant dispersion in an individual fracture have been considered by various investigators (e.g. Tsang and Tsang, 1987; Moreno et al., 1988; Tsang et al., 1988; Tsang and Tsang, 1989; Preuss and Tsang, 1990; Tsang et al., 1991). Contaminant transport in multifractured porous media has been considered in various studies (e.g. Huyakorn et al., 1983; Smith and Schwartz, 1984; Schwartz and Smith, 1988; Berkowitz et al., 1988, 1994; Germain and Frind, 1989; Sudicky and McLaren, 1992) by the development of different types of numerical modelling techniques and simulation methods.

The combined effect of contaminant advection in the fracture and permeable block flows on dispersion in fractured permeable media has been analysed by Birkhtlzer et al.

H. Rubin et alYJournal of Hydrology 199 (1997) 135-162 137

(1993a), Birkh61zer et al. (1993b), Rubin and Buddemeier (1996) and Rubin et al. (1996). Those studies can be classified as studies of double permeability media, as they refer to cases in which contaminant diffusion is negligible. They address cases of contaminant dispersion originating from advection of the fracture and permeable block flows. The results obtained in those studies may also present some basic information on and explana- tion of the well-documented indication that Fickian-type contaminant behaviour only occurs after an initial advective-dominated period which occurs in any permeable matrix. This phenomenon is termed a scale effect (Fried, 1975; Pickens and Grisak, 1981). Various studies concern the incorporation of the scale-dependent dispersion in modelling approaches (e.g. Matheron and de Marsily, 1980; Hatton and Lightfoot, 1984). The studies of Birkh61zer et al. (1993a, b), Rubin and Buddemeier (1996) and Rubin et al. (1996) have referred to fractured permeable formations in which the advection velocity of the fracture flow is much larger than that of the permeable block flow. They have indicated that, in small domains of fractured permeable medium, contaminant transport is non-Fickian. Furthermore, with regard to contaminant advection and dispersion in fractured permeable formations, the size of the representative elementary volume (REV) depends on the ratio between the fracture flow-rate and the permeable block flow-rate, which is termed the mobility number. In the present study we intend to analyse effects of advection and dispersion in formations composed of permeable blocks embedding fractures in which the advection velocity is of the same order of magnitude as that of the permeable block flow.

2. Basic formulation

Fig. l(a) provides a schematic description of a two-dimensional single fracture embedded in the permeable formation. Fig. l(b) shows an elementary fracture volume and indicates all advective contaminant fluxes entering and leaving this elementary volume. We refer to this elementary fracture volume and use the principle of mass con- servation for the contaminant to obtain (Birkh61zer et al., 1993a)

OC__Cdx , OC , d * Wot , Q f - ~ x +qb(C--Cb) y =0 (1)

where w is the fracture aperture, C is the contaminant (relative) concentration of the fracture flow, t* is the time, x' is a local coordinate extended along the fracture centreline, Qf is the fracture flow rate, qb is the specific discharge of the permeable block flow, Cb is the contaminant (relative) concentration of the permeable block flow which enters the fracture, x* is the longitudinal coordinate, and y* is the transverse coordinate.

Considering a unit length B characterizing the distance between adjacent fracture intersections measured along the fracture and considering a characteristic angle of fracture orientation, O, which is the angle between the fracture centreline and the flow direction, we nondimensionalize the space and time coordinates as follows:

x* y* Vbt * X=BcosO; Y =- - - - -0 ; t = - - (2) Bcos Bcos O

where Vb is the velocity of the permeable block flow.

138 H. Rubin et aldJournal of Hydrology 199 (1997) 135-162

Qb

(a) X'

f ac ~C~d~" ~(c÷ ~d~')

" i d * ~ /C+I~)C dx ' dy*

/ \ Q,c (b)

Fig. 1. Fluxes of fluid and contaminant in the fracture and the permeable block: (a) general description; (b) elementary fracture volume.

H. Rubin et alJJournal of Hydrology 199 (1997) 135-162 139

We also consider the following relationships stemming from Fig. l(b):

dx* dx'= • dy* =dx*tan 0 (3)

~-~s 0'

Introducing Eq. (2) and Eq. (3) into Eq. (1) we obtain

NVR ac + a_C+NM(C--Cb)=O (4) Ot fix

where NVR is the advection velocity ratio and NM is the flow rate ratio or mobility number. The parameters NvR and NM are given as

NM _ qb _ KbBtan 0 (5) qf Mf

VbW V b w 2 NVR = Qfcos---~- Vfcos----O =NM B ~bsin 20 (6)

where qf is the specific discharge induced by the presence of the fracture, K b is the hydraulic conductivity of the permeable block, Mf is the mobility of the single fracture, Vf is the velocity of the fracture flow, and ~b is the porosity of the permeable blocks.

According to Eq. (5) the mobility number represents the ratio between the flow rate of the permeable block flow and the fracture flow rate. According to Eq. (6), the advection velocity ratio represents the ratio between the contaminant longitudinal advection velocity of the permeable block flow and that of the fracture flow. According to Eq. (5) and Eq. (6), if 0 approaches 90 ° then the fracture flow vanishes, and both dimensionless parameters NM and NVR attain very large values. It should, however, be noted that the last expressions of Eq. (5) and Eq. (6) are applicable only for a permeable formation incorporating a uniform fracture network with a constant aperture, w. In such a case, we may assume that the fracture flow is between parallel plates. Then we may apply the plain Poiseuille solution of the viscous flow equations and obtain the following approximation:

Mf =gw3/(12v) (7)

where g is the gravitational acceleration and v is the kinematic viscosity. If the fracture aperture is constant in the entire domain, then we may introduce Eq. (7)

into Eq. (5) and Eq. (6), to obtain

N i = 1 2 kb Btan O (8) w z w

12 k b NVR = ¢#bCO520 W2 (9)

where kb is the permeability of the blocks. Birkh61zer et al. (1993a) have reviewed data obtained in many sandstone aquifers.

These data provide the following ranges of typical physical parameters:

10-4 m < w < 10-3 m; 10-14 m2 < kb < 10-t2 m2; 0.4 < ~b < 0.1 (10)

140 H. Rubin et al./Journal of Hydrology 199 (1997) 135-162

Therefore, if B is of the order of magnitude of 1 m, then maximum values of the mobility number and advection velocity ratio are

N M = O(10tan O); NVR ---- O( 10- 2/COS 20) (11)

These expressions indicate that, for a fracture network of constant aperture, the value of the advection velocity ratio can be significant only when 0 approaches 90 °. For 0 very different from 90 °, the advection velocity ratio can be significant only if the fracture network is rich with cavities. Then the fracture flow rate is determined by the small aperture portions of the fracture network, and the fracture flow velocity is very small in portions of the fracture network in which the aperture is comparatively large. However, reference to all expressions and information given by Eqs. (5)-(11) indicates that the advection velocity ratio value cannot be significantly larger than unity for range of mobility number between 0.1 and 10. Therefore, in the following sections of this paper the maximum considered value of this parameter is three.

The following sections of this paper are devoted to the evaluation of the characteristics of contaminant advection and dispersion in a permeable medium incorporating fractures of slow advection velocity; namely, the mobility number and advection velocity ratio are significant.

3. The conceptual model

We adopt the conceptual model shown in Fig. 2. It is a simplified, idealized and deterministic model. We analyse the migration of a contaminant front which is disposed with uniform concentration at x = 0. The contaminant concentration at that location is Cb=C= 1.

The conceptual model shown in Fig. 2 is highly idealized, but it satisfies the require- ments for the evaluation of phenomena originating from the significant value of the advection velocity ratio. In this model the permeable blocks embed two sets of parallel equidistant fractures. In such a fracture network, there are no dispersion effects stemming from contaminant mixing in fracture intersections (Birkhtlzer et al., 1993a).

This study addresses cases of comparatively large values of the mobility number and the advection velocity ratio. High values of these parameters may be obtained owing to high values of the orientation angle (i.e. O approaches 90°), or can be attributed to the fracture network incorporating cavities, as explained in the following paragraphs.

High values of these parameters can also occur where the orientation angle is about 45 ° . In such cases flow in the fracture network is controlled by portions of the network characterized by narrow apertures. Therefore, the mobility number is fairly high. On the other hand, portions of the fracture network are characterized by comparatively large apertures, namely cavities, as shown in Fig. 2(b). Owing to these cavities the contaminant advection velocity of the fracture flow is small. Therefore, they cause the advection velocity ratio to be large.

The portion of the fracture network bounded between two adjacent fracture intersections is termed a fracture sector. Such a fracture sector may incorporate small and large aperture portions, as shown in Fig. 2(b). Owing to the symmetry of the domain shown in

H. Rubin et alJJoum1 of Hydrology 199 (1997) 135-162

c=cb= 1

60

0 ,...__.__....... .

141

I X+

(b) Fig. 2. The basic conceptual model: (a) macroscale; (b) reference subdomain.

142 H. Rubin et alJJournal of Hydrology 199 (1997) 135-162

Fig. 2(a), it is possible to consider contaminant transport in a subdomain of reference as shown in Fig. 2(b). The number of fracture sectors incorporated in the subdomain is n. Each sector is identified by an i value representing its number. The i sector is located in the following x range:

i - 1 -< x <- i for the i sector (1 ~ i --< n) (12)

For the entire subdomain the following boundary and initial conditions of the contaminant concentration values are applied:

C(0, t) = 1; Cb(0, t) = 1 ( 1 3 )

C(x, 0) =0; Cb(x,0) =0 (14)

However, as the subdomain is divided into n sectors, the boundary conditions represented by Eq. (13) are applied only for the first fracture sector. For this particular sector we apply

C1(0, t)= 1 (15)

Cbl(X,t)=l at 0 ~ x - - < t (16)

Cbl(X,t)=O at t < x <--- 1 (17)

For all other fracture sectors (i > 1) the following boundary conditions are applied:

Ci(i- 1, t)=Ci_l(i- l , t ) (18)

Cbi(X,t)=O at t < 2(x-- i+ 1) (19)

Cbi(X, t) = Ci_ i [2i - x - 2, t - 2(x - i + 1 )] at t > 2(x - i + 1 ) (20)

At the downstream end cross-section of the subdomain incorporating n blocks and n fracture sectors, we obtain

[CbfY, t)]x=. = C.(n- t', t - t') at t > t'

[Cb(Y, t)]x=. =0 at t < t '

where

t' =y (n is even)

t ' = 1 - y (n is odd)

(21)

(22)

(23)

(24)

The value of the flux average contaminant concentration, C~ at the downstream end cross- section of the subdomain incorporating n blocks and n fracture sectors is given as

1 C i

The breakthrough curves represent the variation of Car versus the dimensionless time, t. Analysis of the breakthrough curves provides the basic information concerning the characteristics of the fractured permeable formation, as discussed in the following sections, where we develop analytical, as well as numerical simulation approaches by which the complete information with regard m values of C, Cb, Ca, is obtained.

H. Rubin et alfJournal of Hydrology 199 (1997) 135-162

4. The analytical simulation approach

143

By the use of the Laplace transform, Eq. (4) yields

-~x + (NM + SNvR)C =NMCb

where s is the Laplace transform variable, and

s) = r C(x, t)exp(- st)dt ~(x, Jo

(26)

(27)

f~ ~b(X, s) = J0 Cb(X, t)exp(-- st)dt (28)

The general solution of Eq. (26) is given as

C(x, s) = R(x, s)exp[ - (NM + SNvR)x] (29)

where

R(x, s) = N M J Cbexp[(NM + SNvR)X]dx + F(s) (30)

Values of R(x,s) and F(s) should be determined by the adoption of appropriate boundary conditions.

Eq. (29) and Eq. (30) are applied to each single sector of the fracture network. For the first fracture sector, Eqs. (15)-(17) yield

~l(O,s)= 1 (31) $

~bl(X, $) = 1 at 0 - x <-- t $

Cbl(x,s)=0 at t <-- x < -- 1

For all other fracture sectors (i > 1) Eqs. (18)-(20) yield

Ci( i - 1, s) = ~i_ i ( i - 1, s)

(32)

(33)

(34)

(Tbi(x,s)=O at x > i - 1 +(t/2) (35)

(Tbi(X, S) = C , i -1 [2 i - x - 2, s] at x < i - 1 + (t/2) (36)

Eq. (29) and Eq. (30), subject to the boundary conditions and values of the Laplace transform of Cb, represented by Eqs. (31)-(36), can explicitly provide the complete information on the contaminant concentration distribution in the fracture network of any type of tested subdomain. However, only a limited number of examples based on the analytical approach are given in this paper. The advantage of the analytical approach stems from its provision of valuable basic information about parameters and effects associated with the contaminant migration in the domain. However, for subdomains

144

(a)

H. Rubin et al./Journal of Hydrology 199 (1997) 135-162

x=t

i , 0 cb;o

Ii It '

x=0 x=t/N VR x=t X= 1

NVR >1 Fracture flow zones: Zone 1 C=I Zone 2 0<(2<1 Zone 3 C=0

(b)

x=O

X=t

X=t x=l

NVR= 1

x = t

Cb=l . ~ NVR<I

Fracture flow zones: (C) Zone1 C=1

[,/ Zono3 c-o

x=:O X=t Xmt/NVR X=I

Fig. 3, Zones of contaminant concentration of the fraCttLre flow and regions of contaminant concentnlfion of the permeable block flow in a subdomain incorporating a single fracture.

incorporating a large number of blocks and fracture sectors, the analytical approach becomes very tedious. Therefore, in the present section, we only analyse some simple cases of contaminant migration in a subdomain incorporating a single fracture sector. Additional examples considering contaminant migration in subdomains which incorporate a large number of blocks and fracture sectors will be provided in the next section of this paper by the use of an adequate numerical scheme.

Fig. 3 provides a schematic description of the different zones of contaminant distribution in the first sector of the fracture network. It also shows the various regions of the permeable block flow affected by the mixing between the fracture and the permeable block flows.

H. Rubin et al./Journal of Hydrology 199 (1997) 135-162 145

As indicated by the schematic description of Fig. 3, we may define three basic cases of contaminant advection and dispersion phenomena identified by different ranges of the advection velocity ratio: (a) NVR > I; (b) NVR = 1; (c) NvR < .1. In case (a), contaminant advection velocity of the fracture flow is smaller than that of the permeable block flow. Therefore, in this case we identify three different zones of contaminant concentration distribution in the fracture sector. Contaminant concentration distribution in these zones is obtained by applying the general solution given by Eq. (29) and Eq. (30) and the boundary conditions of Eqs. (31)-(33):

Zone 1 : C = 1 at 0 < x < t/NvR (37)

] Zone 2 : C= 1 -exp - NvR - 1 ( t - x ) at t/NvR < x < t (38)

Z o n e 3 : C = 0 a t t < x < l (39)

Zones 1 and 2 of the fracture sector are fed by the permeable block flow trapezoidal bounded by the fracture sector, the top of the control volume shown in Fig. 3(a), and the vertical lines x = 0 and x = t. The fracture advection velocity is equal to I/NvR. Therefore, the distribution of contaminant concentration of the fracture flow in the first fracture sector is given by Eqs. (37)-(39)). Owing to the mixing between the fracture flow and the permeable block flow, Zone 1 disposes fluid with contaminant concentration equal to unity into the permeable block flow. Zone 2 disposes fluid with contaminant concentration smaller than unity, and Zone 3 disposes fluid with zero concentration into the permeable block flow. The creation of regions in which contaminant concentration varies between zero and unity leads to contaminant dispersion as indicated by calculations given hereafter.

Fig. 3(b) refers to fracture advection velocity identical to that of the permeable block flow (i.e. Nva = 1). This figure indicates that the presence of the fracture sector does not create regions of permeable block flow whose contaminant concentration values vary between zero and unity. Also, in the fracture sector there is a sharp front of contaminant concentration. Therefore, no contaminant dispersion, owing to mixing between the fracture and permeable block flows, is expected when NVR = 1.

Fig. 3(c) refers to fracture advection velocity larger than that of the permeable block flow (i.e. NVR < 1). In this case, owing to the difference between the advection velocity of the fracture and permeable flows, the contaminant distribution of the fracture flow in the first sector of the fracture network is obtained by applying the general solution given by Eq. (29) and Eq. (30) bounded by the boundary conditions of Eqs. (31)-(33):

Z o n e l : C = l a t 0 < x < t (40)

Z o n e 2 : C = e x p [ " N ' - ~ " x-t)] a t t < x < t / N v R I - - N V R J (41)

Zone 3 : C = 0 at t/NvR < x < 1 (42)

As indicated by Fig. 3(c), the difference between the advection velocity of the fracture

146 H. Rubin et al./Jour~l of Hydrology 199 (1997) 135-162

flOW and that of the permeable block flow creates regions of the permeable block flow in which the contaminant concentration varies between zero and unity. Therefore, contaminant dispersion is expected in the domain.

In the following paragraphs we discuss the possible phenomena typical of contaminant advection and dispersion in the fractured permeable formation with regard to cases (a) and (c) of Fig. 3.

When contaminant disposal at the entrance cross-section starts (i.e. at x,t = 0) the abrupt change of contaminant concentration causes the migration of two wave fronts of the contaminant in the downstream direction. One wave front moves with the advection velocity of the permeable block flow; the other migrates with the advection velocity of the fracture flow. The arrival of the fracture flow front at the cross-section x = 1 of Fig. 3(a) or Fig. 3(c) is associated with an abrupt change of the value of Car. The dimensionless arrival time of the fracture flow front depends on the value of the advection velocity ratio, and it is equal to IlNvR. The dimensionless arrival time of the permeable block flow front is t = 1. Therefore, the fracture flow front arrives at x = 1 after a time period shorter than that for the permeable block front, provided that NVR < 1. Where NVR > 1 the permeable block front propagation velocity is larger than that of the fracture flow. The amount of abrupt change in the value of Car at the moment of arrival of the fracture flow front depends on the value of the mobility number. The value of C,v is zero before the arrival of any contaminant front at x = 1. The value of Ca~ starts to grow after the arrival of the first front at x = 1. The build-up of the breakthrough curve is completed when the second front arrives at x = 1.

For NvR > 1 we introduce Eqs. (37)-(39) into Eqs. (21)-(23) and integrate Eq. (25), to obtain

Car = 0 at 0 <- t --< 1 (43)

[ ,1 Car = 10-exp N v S - l - ( t - ~ N M / + 1 +NM NVR_ 1

at 1 <-- t <-- NvR (44)

Fig. 4 shows some breakthrough curves obtained by the use of the analytical solution given by Eq. (43) and Eq. (44). This figure indicates that, in cases of Nva > 1, for all values of the mobility number, following the contaminant disposal at x = 0, the average contaminant concentration at the downstream end cross-section is zero until t = 1. For comparatively large values of the mobility number (e.g. NM = 5) the breakthrough curve incorporates a decaying tail which is extended until the front of C = 1 of the fracture flow arrives at x = 1 (i.e. at t = NvR). The breakthrough curves referring to NM = 5 in Fig. 4 are similar to those of a continuum, and in cases of several fracture sectors the breakthrough curves probably can be reproduced by the use of continuum parameters such as equivalent porosity and dispersivity. This issue will be discussed in the following sections.

For advection velocity ratio larger than unity, the build-up of the breakthrough curves always starts at t = 1, and at t = NvR the build-up of the breakthrough curve is completed. For cases of comparatively small values of the mobility number (e.g. NM = 1, 0.5), large portions of the flow rate are attributed to the fracture flow. Therefore, in such cases the


1.0

0.9

0.8

0.7

C a r 0"6

0.5

0.4

0.3

0.2

O.l

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

t

• -----am NM=0.5, NVR=2. 0 • N M = 0.5, Nvx= 3.0

• -------/t---- NM=I.0 , NvR=2.0 ,t, NM= 1.0 , NVR= 3.0

Nta = 5.0, NvR= 2.0 • N M = 5.0, N VR = 3.0

Fig. 4. Breakthrough curves representing C,, vs. t for the subdomain incorporating a single fracture sector (Nw > 1).

breakthrough curves given in Fig. 4 show a moderate growth of Car until t = NVR; then there is a significant abrupt change in the value of Ca~, and it attains the final value of unity. The abrupt increase of C,v at t = NvR is unaffected by changes of the advection velocity ratio. The smaller the mobility number the more significant is the abrupt increase of the value of C.v.

For cases ofNvR < 1, we introduce Eqs. (40)-(42) into Eqs. (21)-(23) and integrate Eq. (25), t o obtain

Car = 0 at 0 -- t <-- NVR (45)

1 +NM t-NVR

Car= 1 +/eT-NV,,M R e X p l [ " N'-7~" I - t )] atNvs <--t<- - NVR (46)

Fig. 5 shows some breakthrough curves obtained by the use of the analytical solution given by Eq. (45) and Eq. (46). This figure indicates that for NVR < 1, for all values of the mobility number, after the contaminant disposal at x = 0, the flux average contaminant concentration at the downstream end cross-section is zero until t = NVR. At t = NVR there is an abrupt increase in the value of C,v as a result of the arrival of the contaminant front advected by the fracture flow. There is only a single abrupt increase in the value of C,v. The smaller the mobility number the more significant is the abrupt increase of the value of Ca,, as a larger portion of the flow is through the fracture network. For NvR > 1, the breakthrough curves are ended with the abrupt increase o f the Car value. On the other hand, for NvR < 1, they start with such an abrupt increase. None of the breakthrough curves shown in Fig. 5 are similar to those of a continuum. It should be noted that the amount of abrupt increase of Ca, at t = NVR entirely depends on the value of the mobility

148 H. Rubin et al./Journdl of Hydrology 199 (1997) 135-162

1 . 0 , - , - , - , , - , - , , - , -

0.9

0.8

0.7

0.6 ~ N M f f i 0 . 5 ' NvR=0. 0 Cav • NM =0.5 ' Nvlt= 0.5

0.5 .~. N M = 1.0, Nv~= O.O 0.4 ~ NM ffi 1"0, NvRffi 0"5

* NM = 5.0 , NvR= 0.0 0.3 s N M = 5.0, Nvlt~ 0.5

0.2

0.1 - - . ~ J . , ,

0.0 -" 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 l.O

t

Fig. 5. Breakthrough curves representing C,v vs. t for the sutxlomain incorporating a single fractme sector (NvR < I ).

number, e.g. the abrupt increase of Civ is about 0.4 when NM -- 0.5. Therefore, the difference between CIv values referring to zero and finite values of the advection velocity ratio increases with the increase of this parameter. Also, the time period between the start of the build-up of the breakthrough curve (i.e. at t = NVR) and the completion of the breakthrough curve build-up decreases with increase of NVR. This time period vanishes when the advection velocity ratio approaches unity. For cases of comparatively large values of the mobility number (e.g. NM = 5) large portions of the flow rate are attributed to the permeable flow. Therefore, the breakthrough curves given in Fig. 5 and referring to large values of NM show a comparatively small abrupt increase in the value of C,~ at t = NvR followed by the growth of Car until t = 1, and then Cav attains its final value of unity.

5. The numerical simulation a p p r o a c h

The numerical simulation approach applied in the present study uses a finite difference grid for the solution of the differential Eq. (4), and incorporates considerations of boundary conditions and values of Cb entering the particular fracture sector as represented by Eqs. (15)-(20).

The fracture network in the tested subdomain incorporates n sectors. Each sector of the fracture network is divided into k intervals bounded between k + 1 nodes, as shown in Fig. 6. The length of a single interval measured along the x axis is ~x. Values of C and Cb at each nodal point are represented by a three-dimensional matrix referring to the number of the sector, the number of the nodal point in the particular sector, and the number of the time step.

H. Rubin et al./Journal o f Hydrology 199 (1997) 1 3 5 - 1 6 2

(k)_.....~ O) (k- D "-~ i "~'-'-~. (2)

O S : k !

J (1 - i) sector i sector ,._ [

x ,~ i -2 x = i - I x = i

Fig. 6. Intervals and nodes dividing two successive sectors of the fracture network.

149

F_x 1. (4) is a hyperbolic first-order differential equation. We use an explicit implicit finite difference approximation scheme (e.g. Lapidus and Pinder, 1982) of this equation to obtain

+ . - - ' ',j ÷' ÷' : ' ÷ " - " - C~.,/_ l ) + NM (C~,.,j ( 4 7 ) -- ~"bi,j ) -- o x ~ i ' j ~ "

where i is a subscript specifying the sector number (i = 1 ..... n), j is a subscript specifying the number of the nodal point ( j = 0 ..... k), and m is a superscript specifying the number of the time step.

Eq. (47) explicitly provides values of the contaminant concentration distribution in the fracture network for the m + 1 time step, given by

c.m. At -+l NMAt +l C~n~ +1 ',y + ~ i J - 1 + ~"~='-"~bINvR ' j

i,s - ( 48 ) At NMAt 1 + Nv---~-~ + NVR

The numerical scheme represented by Eq. (48) is unconditionally stable. Fig. 6 shows that choice of identical time and space intervals in the dimensionless x,t domain is very convenient. Therefore we choose

Ax = At (49)

For the first fracture sector, the following relationships are applied:

C~b/,/=0 for m < j

C~b/j = 1 for m >--j

(50)

( 5 0


2

k - 1 ~ Ct~1

< n ~'tor ' ~ ~ I C'~

x - - n - I x---ll

Fig. 7. Scheme describing the connection between Cb at a cross-section and C at the adjacent upstream fracture sector.

By applying the scheme of Fig. 6 and Eq. (19) and Eq. (20) we obtain

C~b~,j =0 for m < 2j, i > 1 (52)

-2j C~bi,j=C~-l,k_j for m>--2j, i > 1 (53)

The boundary and initial conditions used for the solution of Eqs. (48)-(53) are given as

C~,o = 1 (54)

C~i,]=O (except for i= 1; j=O) (55)

According to the scheme of Fig. 7, the contaminant concentration of the permeable block flow at the downstream end cross-section of the domain is given as

m ~ -J [C~]x= - C:.,,_j (56)

Using the trapezoidal integration procedure, Eq. (25) and Eq. (56) yield

= C~_ -j (57) j=

In these expressions, whenever a superscript has negative values, the relevant C value should vanish. Eq. (57) is applied to calculate the estimated breakthrough curves at the downstream end cross-section of the subdomain incorporating n fracture sectors.

6. Numerical calculations and results

The numerical calculations are performed in two stages: (1) the numerical scheme


characteristics are determined and its applicability is verified; (2) phenomena taking place in subdomains incorporating several sectors of the fracture network are determined. The numerical scheme characteristics are determined by reference to a tested subdomain incorporating a single fracture sector. Fig. 8 represents the comparison between the analytical and numerical solutions for such a subdomain. The numerical solutions represented in Fig. 8(a) and Fig. 8(b) have been obtained by taking Ax = At = 0.005.

Fig. 8(a) refers to cases of comparatively large value of the mobility number (NM = 5). Fig. 8(b) refers to cases of comparatively small value of the mobility number (NM = 0.5). Both figures show fairly good agreement between the numerical and analytical results. For NVR = 0, the analytical results are completely reproduced by the numerical simulations. However, some phenomena of numerical dispersion have been obtained at moments of abrupt increase of Car values. It should be noted that abrupt changes in Car values are most significant in the case of a subdomain incorporating a single fracture sector. Various methods can be applied to reduce the effect of numerical dispersion (e.g. Lapidus and Piiader, 1982), but we have only performed some tests aiming at the evaluation of the characteristics of the numerical dispersion phenomenon by changing the value of Ax and At. We have found that numerical dispersion in cases of high values of the mobility number (e.g. NM = 5) is very minor provided that NraAx is smaller than 0.1. In cases of low values of the mobility number (NM < 1) we have found that the optimum value of Ax and At is 0.005. We have not applied any special technique leading to suppression of numerical dispersion effects, because our major interest refers to subdomains incorporating several fracture sectors, as shown hereafter. Furthermore, it should be noted that Fig. 8 shows only minor effects of numerical dispersion.

Owing to the steep increase of Ca~ values at t = 1, for NVR > 1, none of the breakthrough curves of Fig. 8(a) referring to such cases can be reproduced by the application of continuum parameters (i.e. effective dispersivity and equivalent porosity). Owing to the absence of decay tail at t values approaching unity, for NVR < 1, none of the breakthrough curves of Fig. 8(a) referring to such cases can be reproduced by the application of continuum parameters.

Fig. 8(b) refers to a comparatively small value of the mobility number (NM = 0.5). Again, there is fairly good agreement between the analytical and numerical results referring to all values of the advection velocity ratio. Only some minor phenomena of numerical dispersion have been obtained at moments of abrupt increase of C,~ values.

Fig. 4, Fig. 5, and Fig. 8 indicate that a subdomain incorporating a single permeable block and a single fracture sector cannot be represented by a continuum. Birkhtilzer et al. (1993b) have shown that, for NVR = 0, the smallest volume of representative continuum, i.e. the representative elementary volume (REV), should incorporate at least four fracture sectors and its length should be significantly larger than the loss of identity length. The loss of identity length is a concept solely associated with NVR = 0. In such a case the fracture flow velocity is considered as infinite compared with the permeable block flow velocity. Then the contaminant concentration decays exponentially along the fracture network at the moment of its disposal at x = 0, and the loss of identity length is defined as the longitudinal distance along which the contaminant concentration of the fracture flow decreases to 1%. In examples of NVR > 0 referred to in the present study, there is no physical reason to apply the loss of identity length concept.

152

1.0

0.9

0.8

0.7

Cav 0.6

0.5

(a)

0.4

0.3

0.2

0.1

0.0

1.0

0.9

0.0

H. Rubin et alJJournal of Hydrology 199 (1997) 135-162

0.2 0.4 0.6 0.8 - I i I " ' " t .

1.0 1.2 1.4 1.6 1.8 2.0

t

* AS, NMffi$.0, Nva=0.0 I NS, NM=5.0, Nvaffi0.0 b AS, NMffiS.0, NvRffi0.5 A INS, NMffiS.0, Nvgffi0.5

.=--.o--- AS, NM=5.0, NvRffi2.0 • NS, NMffiS.0, NVR=2.0

0.8

0.7

0.6 Car

0.5

0.4

0.3

0.2

0.1

0.0

/

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

<b) t

• AS, NMffi0.5, Nw=0.0 N$, NM=0.5, Nwffi0.0

---- / r -- AS, NM=0.5, NvRffi0.5 A N$, NMffi0.5, NVRffi0.5

----o--- AS, N M ffi 0.5, NVR= 2.0 • N$, NMffi0.5, NVRffi2.0

Fig. 8. Comparison between numerical and analytical breakthrough curves of subdomalns incorporating a single fracture sector: (a) NM = 5; (b) NM = 0.5.

Ogata and Banks (1961) have developed the analytical solution to the one-dimensional advection-dispersion equation in a permeable continuum. Their solution, adapted to subdomain of length n, modified into dimensionless parameters and variables, is given by

1 [ n - v t .1 1 r n + v t ] = + ~exp L (58)

L 2(aLvt) l/2J \ a L l

H. Rubin et al./Journal of Hydrology 199 (1997) 135-162 t53

where erfc is the complementary error function, v is the dimensionless advection velocity (normalized with regard to Vb), and aL is the dimensionless dispersivity (normalized with regard to Boos 0).

Eq. (58) is useful for the reproduction of breakthrough curves typical of one-dimensional advection-dispersion of contaminants in a permeable continuum. This equation also provides a basis for the analysis of measured breakthrough curves in laboratory experiments and field tests. In the present study, we apply Eq. (58) for analysis of the possible representation of the fractured permeable formation by a continuum.

If C,v is measured or calculated in a cross-section where

n/aL >> 1 (59)

then the second term of Eq. (58) can be ignored, and the breakthrough curve can provide the value of the advection velocity as

v = n/to. 5 (60)

where t0.5 is the time of arrival of Car = 0.5 at the downstream end cross-section of the subdomain whose length is x = n. When reference is made to the fractured permeable formation, such a subdomain incorporates n fracture sectors and n permeable blocks. The advection velocity is applied for the determination of the equivalent porosity (normalized with regard 4)b) by

1 l + - -

~b = NM (61) v

For fractured permeable medium in which the fracture flow advection velocity is much larger than that of the permeable block flow (i.e. NVR = 0), Birkhtlzer et al. (1993b) have obtained

q)=l; v = I + ( 1 / N M ) (62)

Calculation of the dimensionless dispersivity is based on measurement of the slope of the breakthrough curve at t = t0.s. As implied by Eq. (58),

n

a L = 4~rt02.5S2 (63)

where S is the slope of the dimensionless breakthrough curve at t = t0.s. Eqs. (58)-(63) are applied hereafter to evaluate the possible representation of a frac-

tured permeable formation by a continuum, and what should be the minimum size of the representative elementary volume (REV) of that continuum.

Fig. 9 refers to subdomains incorporating three fracture sectors. Such subdomains cannot be represented by a permeable continuum if the advection velocity ratio vanishes (Birkh~ilzer et al., 1993b). However, some of the breakthrough curves shown in Fig. 9 which refer to NVR > 0 are very similar to those of a continuum, as discussed hereafter. Fig. 9(a) shows that at high values of the mobility number(e.g. NM = 5) and small values of the advection velocity ratio (i.e. N w < 1) breakthrough curves lack the tail of decay which is typical of a continuum. On the other hand, at high values of the advection velocity ratio (e.g. NVR = 2) the breakthrough curve is very similar to that of a permeable continuum.

154 H. Rubin et aL/Journal of Hydrology 199 (1997) 135-162

| o 0

0.9

0.8

0.7

0.6

Cav 0.5

0.3

0.2

0.1

0 . 0 ~'~ -~" -~- , , , t 0

0 1 2 3 4 5 6

(a) t

1.0

0.9

0.8

0.7

0.6

Ca v 0.5

0.3

0.2

0 ! 2 3 4 5 6 7 (b) t

* NvR= 0.0 ----o---- NvR= 0.5

A NvR= 2.0

NvR= 0.0 ----e---- NvR= 0. 5

NvR= 2.0

Fig. 9. Numerical simulation results for subdomains incorporating three fracture sectors: (a) NM = 5, (b) NM = 0.5.

Fig. 9(b) refers to a small value of the mobility number (NM = 0.5). In this case, even at NVR < 1 the breakthrough curves are similar to those of a permeable continuum, but at very low values of the advection velocity ratio, there is a significant abrupt increase of Car value at t = 0. It should be noted that simulation results of Birkh01zer et al. (1993b) have indicated that about ten fracture sectors should be incorporated in the REV when NM = 0,5 and NvR = 0. The incorporation of such a large number of fracture sectors neutralizes the

abrupt increase of Car value at t = 0.

(a)

1.0

0.9

0.8

0.7

Car 0"6

0.5

0.4

0.3

0.2

0.1

0.0


1.0 . . . .

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 . . . . . . . 0 1 2 3 4 5 6 7 8

t

Ca,,

0 1 2 3 4 5 6 7 8 9 10

(b) t

* NVR=0.0 .........n-..- NvRffi 0. 5

--,----b----- NvR= 2.0

--.---,---- NvR= 0. 0 ----n----- N vR= 0. 5

NVR= 2.0

II

155

Fig. 10. Numerical simulation results for subdomains incorporating five fracture sectors: (a) NM - 5; (b) NM ---- 0.5.

Fig. 10 refers to subdomains incorporating five fracture sectors. Fig. 10(a) shows that at high values of the mobility number (e.g. Nu = 5) and all values of the advection velocity ratio, breakthrough curves incorporate the tail of decay which is typical of a continuum. At high values of the advection velocity ratio (e.g. NVR = 2) the breakthrough curve is very similar to that of a permeable continuum. For NVR < 1 the breakthrough curve shows best similarity to a continuum when NvR = 0. Fig. lO(b) refers to a small value of the mobility number (NM = 0.5). Also in this case all breakthrough curves are similar to those of

156

1.6

1.4

1.2

e 1.0

I I

0.8

0.6

0.4

0.2

0.0

(a)

tt. Rubin et aLqournal of Hydrology 199 (1997) 135-162

i | | i | i i .

s I a i a i i

3 4 5 6 7 8 9 10

n

£

V

aLx 100

~- 3

(b)

| ! e | i i |

I I I I • . l I I

2 3 4 5 6 7 8 9 10 n

$ ¥

A aLx I00

Fig. I 1. Variation of continuum parameters characterizing the fractured permeable medium (NvR = 2, n = 2-10): (a) Nm = 5; (b) NM = 0.5.

a permeable continuum, but at NVR -- 0 there is still a minor abrupt increase of Car value at t = 0 .

Fig. 11 concerns the variation of continuum parameters (i.e. cont inuum advection velocity v, equivalent porosity 4, and dispersivity aL) for subdomains incorporating 2 - 1 0 fracture sectors. These parameters have been identified by calculating the value of t0~ and the slope o f the breakthrough curve which have been introduced into Eqs. (60)-(63).


1.0

0.9

0.8

0.7

0.6

0.5 C a r

0.4

0.3

0.2

0.1

0.0 . . . . . . . . . . . . , r - . . . . - . - --

i FM: n= 3.0 - - -o-- - CM: n = 3.0

. FM: n = 4.0 = CM: n = 4.0 N FM: n = 5.0

--.-o---- CM: n = 5.0

157

0 2 4 6 8 10 12

t

Fig. 12. Comparison between breakthrough curves of the fractured medium (FM) and the continuum (CM) model (NM = 0.5, NvR = 2).

These operations have been carded out even in cases of breakthrough curves which were not very similar to those of continua. However, at high values o f the mobility number, as shown in Fig. 1 l(a), as well as low values of this parameter, as shown in Fig. 1 l(b), the only parameter affected by the increase of the size of the subdomain is the dispersivity in cases of high values of the advection velocity ratio (e.g. NVR = 2). Values of continuum parameters shown in Fig. 11 are significantly different from those typical of cases in which NVR = 0 (Birkh61zer et al., 1993b). Values of v and aL shown in Fig. 11 are much smaller than those obtained at Nva = 0. Values of 4~ are significantly larger than those typical of NVR = 0. Variations of continuum parameters owing to the increase of NVR are more significant at low mobility numbers, as shown in Fig. 1 l(b), than at high values of this parameter, as shown in Fig. 1 l(a), because at low mobility numbers larger portions of the flow rate are contributed by the fracture flow.

Fig. 11 indicates good consistency of continuum parameters inferred by the analysis of the data for breakthrough curves. Therefore, we may conclude that, for Nvg > 1, the breakthrough curves, referring to subdomains incorporating even small number of fracture sectors, can be reproduced by the use of continuum domain parameters and Eq. (58). To exemplify the application of this conclusion, we adopt average values for the continuum parameters obtained at n = 5 -10 , Nu = 0.5, NVR = 2, namely

v=0.6; aL=0.047 (q~=5.0) (64)

and reproduce fairly well the breakthrough curves of the fractured permeable medium by applying F.x 1. (58), as shown in Fig. 12.

It should be noted that for NM = 0.5 and NvR = 0, Birkh61zer et al. (1993b) have obtained

v=3.0; aL=0 .96 (~b= 1.0) (65)

158 H. Rubin et al./Journal of Hydrology 199 (1997) 135-162

The difference between values of continuum parameters represented by Eq. (64) and those given by Eq. (65) indicates that cavities present in the fractured permeable formation may lead to extremely large values of the equivalent porosity. They may also lead to significant reduction of the dispersivity. Such effects are more significant for low mobility numbers than for high values of this parameter.

7. Discussion

The present study addresses fractured permeable formations such as sandstone aquifers, in which effects of contaminant diffusion from the fracture network into the permeable blocks are much less significant than effects stemming from contaminant advection and mixing between the fracture and permeable block flows. Dimensionless parameters and their ranges defining the basic reference of the present study have been given by Birkh61zer et al. (1993a). The model applied in this study can be termed a dual permeability model. It considers differences between hydraulic conductivity of the fracture network and the permeable blocks, as well as differences between the contaminant advection velocity of the fracture and the permeable block flows. The model does not consider contaminant diffusion and dispersion in the permeable block and fracture flows. It addresses cases of dispersion originating from mixing between these flows.

It should be noted that the conceptual model consists of advective transport of contaminants within a velocity field which contains two components: (1) constant flow rate, Q f, through the fracture network; (2) constant specific discharge, qb, through the permeable blocks (the matrix). The flux through the permeable blocks is considered to be everywhere parallel to some regional flow and is not affected by its crossing the fracture network. Such a result is obtained as we assume that permeable block flow penetrating the fracture network is immediately mixed with the fracture flow. All these assumptions of the conceptual model simplify the reality. If the fracture network incorporates cavities as shown in Fig. 2(b) then possible short-circuiting out of the fracture and through the permeable block are expected. They would probably be most noticeable near the fracture intersections. Also, in cavities some flow mode can be of three-dimensional nature. How- ever, the present study does not aim to provide all possible details of flow in the permeable block flow. Also, there is no accurate definition of the fracture network domain. We have only studied effects stemming from contaminant advection in a double permeability domain. Our discussion makes it clear that the present study addresses a limited range of phenomena defined by a limited range of dimensionless parameters. For this limited scope, we have developed a simplified mathematical model that has analytical and simple numerical solutions. More sophisticated models which apply the standard advection- dispersion equation in the fracture network and the permeable blocks might provide various types of characteristics which are beyond the scope of the present paper.

8. Summary and conclusions

A basic formulation for transport phenomena in a permeable medium embedding a


fracture network incorporating cavities is developed and presented in dimensionless form. It is shown that the contaminant transport in the domain is governed by two dimensionless parameters, termed the mobility number, NM, and the advection velocity ratio, NvR. A conceptual model representing the domain has been developed. This conceptual model is a deterministic model which simplifies the structure of the domain but incorporates major characteristics which are intended to be studied. The limitations of the model are discussed, and cases of its relevance are presented.

An analytical method for the calculation of transport phenomena in the domain has been developed and applied to a domain incorporating a single fracture sector and a single permeable block. The analytical solution becomes very tedious for domains incorporating a large number of fracture sectors. Therefore, a numerical model has been developed for the simulation of contaminant transport in domains incorporating more than a single fracture sector. Characteristics of the numerical code and verification of its possible use are obtained by comparing analytical results with numerical ones. Numerical simulations have indicated that the presence of cavities in the fractured permeable medium may lead to the creation of breakthrough curves similar to those of a permeable continuum even if the subdomain incorporates a very small number of fracture sectors. The presence of the cavities is represented by high values of the advection velocity ratio. Large values of this parameter may lead to extremely large values of the equivalent porosity, as inferred by the analysis of the breakthrough curve data. If the advection velocity number is large, then analysis of the breakthrough curve data may provide very small values of the dispersivity. Such values are much smaller than those expected if the advection velocity ratio is negligible. The advection velocity ratio can be considered as negligible if its value is smaller than 0.1. For high values of the advection velocity ratio, contaminant dispersion owing to mixing between the fracture and permeable block flows is small. In such cases, contaminant dispersion may be attributed mainly to dispersion effects in the fracture network and the permeable blocks. Aquifers rich in cavities are expected to provide large variability of measured equivalent porosity and dispersivity.

9. Notation

aL

B C

Car Cb Cb F g i J k kb Kb

dimensionless longitudinal dispersivity owing to mixing between the fracture flow and the permeable block flow characteristic length between adjacent fracture intersections (L) fracture flow (relative) concentration Laplace transform of the fracture flow concentration average (relative) concentration in a cross-section permeable block flow (relative) concentration Laplace transform of the permeable block concentration coefficient dependent on s gravitational acceleration (L T -2) number of the considered sector number of the nodal point number of intervals included in a single fracture sector permeability of the permeable blocks (L 2) hydraulic conductivity of the permeable blocks (L T -~)

160

m

Mf n

NM Nvs qb

qf

Qf R REV $

S t to.5 t* V

Vb Vf W

X

X ~

X*

Y y '

At Ax 0 Is


number of the time step mobility of the fracture (L 2 T -t) number of fracture sectors incorporated in the tested subdomaln mobility number advection velocity ratio specific discharge of the permeable block flow (L T -I) specific discharge imposed by the fracture presence (L T -~) discharge flowing through a single fracture (L 2 T -~) coefficient dependent on x and s representative elementary volume Laplace transform variable slope of the dimensionless breakthrough curve at t = t0.s dimensionless time dimensionless time associated with C,v = 0.5 time (7) dimensionless contaminant advection velocity in the fractured permeable medium velocity of the permeable block flow (L T -I) velocity of the fracture flow (L T -I) width of the fracture (L) dimensionless longitudinal coordinate local coordinate extended along the fracture centreline (L) longitudinal coordinate (L) dimensionless vertical or transverse coordinate vertical or transverse coordinate (L) dimensionless time step dimensionless length interval orientation of the fracture sector kinematic viscosity (L 2 T -I) normalized equivalent porosity porosity of the permeable blocks

Acknowledgements

The authors are very grateful to the referees of this paper for many constructive c o m m e n t s .

References

Abelin, H., 1986. Migration in a single fracture: an in situ experiment in a natural fracture, Ph.D. Thesis, Department of Chemical Engineering, Royal Institute of Technology, Stockholm.

Berkowitz, B., Bear, J. and Braester, C., 1988. Continuum models for contaminant transport in fractured porous formations. Water Resour. Res., 24(8): 1225-1236.

Berkowitz, B., Naumann, C. and Smith, L., 1994. Mass transfer at fracture intersections: an evaluation of mixing models. Water Resour. Res., 30(6): 1765-1773.

BirkhiSlzer, J., Rubin, H., Danieis, H. and Rouv6, G., 1993a. Contaminant advection and spreading in a fractured permeable formation 1. Parametric evaluation and analytical solution. J. Hydrol., 144: 1-33.

Birkh61zer, J., Rubin, H., Daniels, H. and Rouv~, G., 1993b. Contaminant advection and spreading in a fractured permeable formation 2. Numerical simulation. J. Hydrol., 144: 35-58.

Cvetkovic, V.D., 1991. Mass arrival of reactive solute in single fractures. Water Rasour. Res., 27(2): 177-183. Fried, JJ., 1975. Groundwater Pollution. Elsevier, New York.


Germain, D. and Frind, E.O., 1989. Modelling contaminant migration in fracture networks: effects of matrix diffusion. In: H.E. Kobus and W. Kinzelbaeh (Editors), Contaminant Transport in Groundwater. A.A. Balkema, Rotterdam, pp. 267-274.

Grisak, G.E. and Pickens, J.F., 1980. Solute transport through fractured media, 1, The effect of matrix diffusion. Water Resour. Res., 16(4): 719-730.

Grisak, G.E., Pickens, J.F. and Cherry, J.A., 1980. Solute transport through fractured media, 2, Column study of fractured till. Water Resour. Res., 16(4): 731-739.

Haldeman, W.R., Chaung, Y., Rasmussen, T.C. and Evans, D.D., 1991. Laboratory analysis of fluid flow and solute transport through a fracture embedded in porous tuff. Water Resour. Res., 27(1): 53-65.

Hatton, T.A. and Lighffoot, E.N., 1984. Dispersion of tracer solutes in flowing groundwater. Water Resour. Res., 20(9): 1253-1290.

Huyakom, P.S., Lester, B.H. and Mercer, J.W., 1983. An efficient finite element technique for modeling transport in fractured porous media, 1, Single species transport. Water Resour. Res., 19(3): 841-854.

Iwai, K., 1976. Fundamental studies of fluid flow through a single fracture. Ph.D. Dissertation, University of California, Berkeley.

Kennedy, C.A. and Lennox, W.C., 1995. A control volume model of solute transport in a single fracture. Water Resour. Res., 31(2): 313-322.

Lapidus, L. and Pinder, G.F., 1982. Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley, New York.

Maloszewski, P. and Zuber, A., 1984a. Interpretation of artificial and environmental tracer in fissured rocks with a porous matrix. In: Isotope Hydrology in Water Resources Development. International Atomic Energy Agency, Vienna.

Maloszewski, P. and Zuber, A., 1984b. On the theory of tracer experiments in fissured rocks with porous matrix. Pep. 1244 AP, Institute of Nuclear Physics, Cracow, Poland.

Matheron, G. and de Marsily, G., 1980. Is transport in porous media always diffusive? A counterexample. Water Resour. Res., 16(5): 901-917.

McKay, L.D., 1991. Groundwater flow and contaminant transport in a fractured clay till. Ph.D. Thesis, University of Waterloo, Waterloo, Ont.

McKay, L.D., Cherry, J.A. and Gillham, R.W., 1993a. Field experiments in a fractured clay till, 1, Hydraulic conductivity and fracture aperture. Water Resour. Res., 29(4): 1149-1162.

McKay, L.D., Gillham, R.W. and Cherry, J.A., 1993b. Field experiments in a fractured clay till, 2, Solid and colloid transport. Water Resour. Res., 29(12): 3789-3890.

Moreno, L. and Rasmuson, A., 1986. Contaminant transport through a fractured porous rock: impact of the inlet boundary condition on the concentration profile in the rock matrix. Water Resour. Res., 22(12): 1728-1730.

Moreno, L., Neretnieks, I. and Eriksen, T., 1985. Analysis of some laboratory tracer runs in natural fissures. Water Resour. Res., 21(7): 951-958.

Moreno, L., Tsang, Y.W., Tsang, C.F., Tsang, Y.V., Hall, F.V. and Neretnieks, I., 1988. Flow and tracer transport in a single fracture: a stochastic model and its relation to some field observations. Water Resour. Res., 24(12): 2033-2048.

Neretnieks, I., 1980. Diffusion in the rock matrix: an important factor in radionuclide retardation? J. C-eophys. Res., 85(B8): 4379-4397.

Noorishad, J. and Mehran, M., 1982. An upstream finite element method for solution of transient transport equation in fractured porous media. Water Resour. Res., 18(3): 588-596.

Novakovski, K.S. and Lapcevic, P.A., 1994. Field measurement of radial solute transport in fractured rock. Water Resour. Res., 30(1): 37-44.

Novakowski, K.S., Evans, G.V., Lever, D.A. and Raven, K., 1985. A field example of measuring hydrodynamic dispersion in a single fracture. Water Resour. Res., 21(8): 1165-1174.

Ogata, A. and Banks, R.B., 1961. A solution of the differential equation of longitudinal dispersion in porous media. US Geol. Surv. Prof. Pap. 411-A.

Pickens, J.F. and Grisak, G.E., 1981. Modeling of scale dependent dispersion in hydrologic systems. Water Resour. Res., 17(6): 1701-1711.

Preuss, K. and Tsang, Y.W., 1990. On two-phase relative permeability and capillary pressure at rough-wailed rock fractures. Water Resour. Res., 26(9): 1915-1926.


Rasmuson, A., 1985. Analysis of hydrodynamic dispersion in discrete fracture networks using the method of moments. Water Resour. Res., 21(! I): 1677-1683.

Rasmuson, A. and Neretnieks, I., 1986. Radionuclide transport in fast channels in crystalline rock. Water Resour. Res., 22(8): 1247-1256.

Raven, K.G., Novakowski, K.8. and Lapcevic, P.A., 1988. Interpretation of field tracer tests of a single fracture using a transient solute storage model. Water Resour. Res., 24(12): 2019-2032.

Rowe, R.K. and Booker, J.R., 1990. Contaminant migration through fractured till into an underlying aquifer. Can. G-eotech. J., 27(4): 484-495.

Rubin, H. and Buddemeier, R.W., 1996. Transverse dispersion of contaminants in fractured permeable formations. J. Hydrol., 176: 133-151.

Rubin, H., Soliman, A.M., Birkhtlzer, J. and Rouv~, G., 1996. Transport of a tracer slug in a fractured permeable formation. J. Hydrol., 176: 153-180.

Rudolph, D.L., Cherry, J.A. and Farvolden, R.N., 1991. Groundwater flow and solute transport in fractured lacustrine clay near Mexico City. Water Resour. Res., 27(9): 2187-2201.

Schrauf, T.W. and Evans, D.D., 1986. Laboratory studies of gas flow through a single natural fracture. Water Resour. Res., 22(7): 1038-1050.

Schwartz, F.W. and Smith, L., 1988. A continuum approach for modelling mass transport in fractured media. Water Resour. Res., 24(8): 1360-1372.

Sharp, J.C., 1970. Fluid flow through fissured media. Ph.D. Dissertation, Imperial College, London. Smith, L. and Schwartz, F.W., 1984. An analysis of the influence of fracture geometry on mass transport in

fractured porous media. Water Resour. Res., 20(9): 1241-1252. Sudicky, E.A. and Frind, E.O., 1982. Contaminant transport in fractured porous media--analytical solution for a

system of parallel fractures. Water Resour. Res., 18(6): 1634-1642. Sudicky, E.A. and Frind, E.O., 1984. Contaminant transport in fractured porous media: analytical solution for a

two member decay chain in a single fracture. Water Resour. Res., 20(7): 1021-1029. Sudicky, E.A. and McLaren, R.G., 1992. The Laplace transform Galerkin technique for large scale simulation of

mass transport in discretely fractured porous formation. Water Resour. Res., 28(2): 499-514. Sun, N.Z. and Yeh, W.W.G., 1983. A proposed upstream weigh numerical method for simulating pollutant

transport in groundwater. Water Resour. Res., 19(6): 1489-1500. Tang, D.H., Frind, E.O. and 8udicky, E.A., 1981. Contaminant transport in fractured porous media: analytical

solution for a single fracture. Water Resour. Res., 17(3): 555-564. Tsang, C.F. and Tsang, Y.W., 1989. Flow channeling in a single fracture as a two-dimensional strongly hetero-

geneous permeable medium. Water Resour. Res., 25(9): 2076-2080. Tsang, C.F., Tsang, Y.W. and Hale, F.V., 1991. Tracer transport in fractures: analysis of field data based on a

variable aperture channel model. Water Resour. Res.~ 27(12): 3095-3106. Tsang, Y.W. and Tsang, C.F., 1987. Channel model of flow through fractured media. Water Resour. Res., 23(3):

467 -479. Tsang, Y.W., Tsang, C.F., Neretuieks, I. and Moreno, L., 1988. Flow and tracer transport in fractured media: a

variable aperture channel model and its properties. Water Resour. Res., 24(12): 2049-2060.

advection and dispersion of contaminant in a permeable medium embedding fractures in which advection...

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