Analytical solutions for reactive transport of N-member radionuclide chains in a single fracture

$Page 1: Analytical solutions for reactive transport of N-member radionuclide chains in a single fracture$
Analytical solutions for reactive transport of

N-member radionuclide chains in a single fracture

Yunwei Sun*, Thomas A. Buscheck

Lawrence Livermore National Laboratory, P.O. Box 808, L-646, Livermore, CA 94550, USA

Abstract

Several numerical codes have been used to simulate radionuclide transport in fractured rock

systems. The validation of such numerical codes can be accomplished by comparison of numerical

simulations against appropriate analytical solutions. In this paper, we present analytical solutions for

the reactive transport of N-member radionuclide chains (i.e., multiple species of radionuclides and

their daughter species) through a discrete fracture in a porous rock matrix applying a system

decomposition approach. We consider the transport of N-member radionuclide chains in a single-

fracture–matrix system as a starting point to simulate more realistic and complex systems. The

processes considered are advection along the fracture, lateral diffusion in the matrix, radioactive

decay of multiple radionuclides, and adsorption in both the fracture and matrix. Different retardation

factors can be specified for the fracture and matrix. However, all species are assumed to share the

same retardation factors for the fracture and matrix, respectively. Although a daughter species may

penetrate farther along the fracture than its parent species when a constant-concentration boundary

condition is applied, our results indicate that all species retain the same transport speed in the fracture

if a pulse of the first species is released into the fracture. This solution scheme provides a way to

validate numerical computer codes of radionuclide transport in fractured rock, such as those being

used to assess the performance of a potential nuclear-waste repository at Yucca Mountain.

D 2002 Elsevier Science B.V. All rights reserved.

Keywords: Analytical solution; Multi-species; Radionuclide; Reactive transport; Fracture; Decay

1. Introduction

During the last few decades, much effort has gone into improving our understanding of

flow and transport in fractured rock (Nitao and Buscheck, 1991). Much of the motivation

for this paper comes from the need to understand radionuclide transport processes in

0169-7722/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0169-7722(02)00181-X

* Corresponding author. Tel.: +1-925-422-1587; fax: +1-925-423-1997.

E-mail address: [email protected] (Y. Sun).

www.elsevier.com/locate/jconhyd

Journal of Contaminant Hydrology 62–63 (2003) 695–712

fractured rock and to validate numerical codes used to assess the performance of the

potential geologic repository for high-level nuclear waste at Yucca Mountain. The

potential repository site at Yucca Mountain, much of which is comprised of a fractured

volcanic tuff, has led to an increased interest in the behavior of radionuclide transport in

fractures. The Yucca Mountain Project utilizes complex numerical flow and transport

codes to simulate radionuclide transport through fractured rock. One of the difficulties in

relying on such complex codes is code verification and validation. An important

component of such validation efforts is comparing the output of numerical codes with

analytical solutions. However, appropriate analytical solutions for multiple-species trans-

port are lacking, so numerical codes are often validated by comparing with single-species

solutions (Steefel and Lichtner, 1998; Nitao, 1998; Xu et al., 1998). The behavior of

daughter species is not addressed in such validation exercises.

Determination of analytical solutions of radionuclide transport usually involves

complex mathematical manipulation in order to derive closed forms of inverse Laplace

transforms (Bear, 1979; Pigford et al., 1980; van Genuchten, 1985). For this reason,

analytical solutions that use Laplace transforms are limited to a small number of species

(van Genuchten, 1985). Sun et al. (1999) developed a simple linear transform method

and extended analytical solutions to N-member chains in homogeneous systems. In this

paper, we provide analytical solutions for reactive transport of N-member radionuclide

chains (i.e., multiple species of radionuclides and their daughter species) through a

single fracture adjacent to a matrix that is impermeable to fluid flow but allows diffusive

transport. We consider a single-fracture–matrix system as a starting point for more

realistic and complex systems. This work expands previous studies by Tang et al.

(1981), Sudicky and Frind (1984), and Cormenzana (2000). Tang et al. (1981)

developed an analytical solution to a mathematical model and investigated the funda-

mental dynamics of the transport of a single radionuclide along a single fracture.

Sudicky and Frind (1982) extended the solution to multiple parallel fractures. Further,

Sudicky and Frind (1984) derived an analytical solution for the reactive transport of a

two-member decay chain in a single fracture. More recently, Cormenzana (2000)

provided a simplified form of the Sudicky and Frind (1984) solution.

Because of the difficulties involved in inverse Laplace transforms, analytical solutions

for transport in fractured systems are limited to only one or two species. To avoid the

difficulties noted by Sudicky and Frind (1984) in extending their approach to multiple

species, we use the transform of Sun et al. (1999) to decompose the partial differential

equations, which are coupled by reaction terms, into independent subsystems. Then, the

solutions for single-species transport in the single fractured system can be applied to derive

solutions for N-member decay chains.

2. Conceptual model

A contaminant source with specified concentrations of N-species is assumed to be at

the origin of the vertical fracture in Fig. 1. The groundwater velocity in the fracture is

assumed to be constant. The aperture of the fracture is much smaller than the transport

distance, so the fluid composition within the fracture can be assumed to be well mixed at

Y. Sun, T.A. Buscheck / Journal of Contaminant Hydrology 62–63 (2003) 695–712696

all times. The permeability of the porous rock matrix is low enough that advection in the

matrix can be neglected. Therefore, transport in the matrix only results from molecular

diffusion. The transport along the fracture is much faster than that within the matrix.

Different retardation factors can be specified for the fracture and matrix. However, all

species are assumed to share the same retardation factors for the fracture and matrix,

respectively. The conceptual system is summarized by the last column in Table 1.

3. Mathematical model

The stoichiometry of radionuclide reactions can be described by the sequential first-order

reaction scheme:

C1 Zk1C2 Z

k2 : : :Ci: : : Z

kN�1

CN ZkNCNþ1 ð1Þ

where Ci is the product of reaction i� 1 and a reactant of reaction i; ki [T� 1] is the reaction

constant (decay rate) in reaction i (ki = ln 2/ti,1/2; ti,1/2 is the half-life of species i); N is the

number of species.

Table 1

Conceptual model definitions

Tang et al.

(1981)

Sudicky and

Frind (1984)

Cormenzana

(2000)

This paper

Number of species 1 2 2 N

Advection along fracture yes yes yes yes

Dispersion along fracture yes/no no no yes/no

Advection in matrix no no no no

Diffusion in matrix yes yes yes yes

Same retardation no yes yes

Reaction 1st order 1st order 1st order 1st order

Fig. 1. The physical model.

Y. Sun, T.A. Buscheck / Journal of Contaminant Hydrology 62–63 (2003) 695–712 697

The mass-action rate equation of Eq. (1), in terms of species concentrations, is

dci=dt ¼ �kici þ yi�1ki�1ci�1; bi ¼ 1; 2;: : :;N ð2Þ

where ci [ML� 3] is the concentration of species Ci and yi (with yo = 0) is the stoichio-

metrical yield factor which is calculated as the concentration ratio of ci + 1 to ci in reaction

i. For radionuclide decay problems, yi = 1, bi = 1,. . .,N. Note that the concentration of the

end product, cN + 1, is not considered.

The transport processes of sequentially decaying species in the fracture–matrix

system can be expressed using two sets of partial differential equations, one for the

fracture and one for the porous matrix, with reaction coupling among the species within

each subsystem (fracture or matrix). Molecular diffusion provides continuity of concen-

tration for each species along the fracture–matrix interface. The system of equations

describing transport along the fracture, and the system of differential equations describ-

ing the diffusive transport of first-order decay chains in the direction perpendicular to the

fracture, are expressed by mass balance equations (Bear, 1979; Sudicky and Frind,

1984):

Fðc1Þ ¼ �k1c1

Fðc2Þ ¼ �k2c2 þ y1k1c1

]

FðciÞ ¼ �kici þ yi�1ki�1ci�1

]

FðcN Þ ¼ �kNcN þ yN�1kN�1cN�1

;

Mðc1VÞ ¼ �k1c1V

Mðc2VÞ ¼ �k2c2Vþ y1k1cV1

]

MðciVÞ ¼ �kiciVþ yi�1ki�1ci�1V

]

MðcNVÞ ¼ �kNcNVþ yN�1kN�1cN�1V

;

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð3Þ

where F represents the transport processes in the constant flow along the fracture

and M represents the diffusive process in the matrix. F can be explicitly written

as

FðciÞ ¼Bci

Bt� D

R

B2ci

Bz2þ v

R

Bci

Bzþ qi

Rb; 0VzVl; bi ¼ 1; 2;: : :;N ð4Þ

where R is the retardation factor in the fracture, which is assumed to be the same for all

species, b [L] is the half aperture of the fracture, v is the constant flow velocity [LT� 1], and

D represents the hydrodynamic dispersion coefficient [L2T� 1] along the fracture.D is given

by (Bear, 1979):

D ¼ aLvþD* ð5Þ


where aL is the longitudinal dispersivity [L] along the fracture, D* is the molecular

diffusion coefficient [L2T� 1] in water, and qi [MT� 1L� 2] is the mass loss in the fracture

due to the diffusion crossing the fracture–matrix interface, which can be expressed by

Fick’s first law as

qi ¼ �/DVBciV

Bx

��x¼b

; bi ¼ 1; 2;: : :;N ð6Þ

where / is the matrix porosity, DV= sD* is the diffusion coefficient in the matrix, s is the

matrix tortuosity, and ciV[ML� 3] is the concentration of species i in the matrix. M can be

explicitly written as

MðciVÞ ¼BciV

Bt� DV

RV

B2ciV

Bx2ð7Þ

and RVis the retardation factor in the matrix, which is assumed to be the same for all

species. Following Tang et al. (1981) and Sudicky and Frind (1984), we define two sets of

initial and boundary conditions in Table 2.

4. Solution development

The transform of Sun et al. (1999) is used to solve the system in Eq. (3). The terms that

couple the parent-species concentrations in the partial differential equations can be

eliminated in the transformed domain. The transport system of each species becomes

independent with an identical mathematical format in the transformed domain. The

analytical solution derived for a single species by Tang et al. (1981) for the ‘‘no-

dispersion-along-the-fracture’’ case is applicable to the ‘‘constant-concentration-boun-

dary’’ case, and the analytical solution derived by Sudicky and Frind (1984) and by

Cormenzana (2000) is applicable to the pulse injection case.

The transform is defined as a linear function of species concentrations as (Sun et al.,

1999):

ai ¼ ci þXi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �cj; aiV¼ ciVþ

Xi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �cjV: ð8Þ

Table 2

Initial and boundary conditions

Initial/boundary type Reference Fracture Matrix

Constant source Tang et al. (1981) ci (0, t) = cio ciV(b, z, t) = ci (z, t)

ci (l, t) = 0 ciV(l, z, t) = 0

ci (z, 0) = 0 ciV(x, z, 0) = 0Pulse injection Sudicky and Frind (1984) ci (0, t) = ci

od(t� 0) ciV(b, z, t) = ci (z, t)ci (l, t) = 0 ciV(l, z, t) = 0

ci (z, 0) = 0 ciV(x, z, 0) = 0


Applying the transform to Eq. (3) for species i,

FðaiÞ ¼ FðciÞ þXi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �FðcjÞ

¼ yi�1ki�1ci�1 � kici þXi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �½yj�1kj�1cj�1 � kjcj�: ð9Þ

Combining terms with common factors of concentrations,

FðaiÞ ¼ �ki ci þXi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �cj

( )¼ �kiai: ð10Þ

The detailed derivation for Eq. (10) can be found in Sun et al. (1999).

Therefore, the transformed system equations of Eq. (3) become

Fða1Þ ¼ �k1a1

Fða2Þ ¼ �k2a2

]

FðaiÞ ¼ �kiai

]

FðaN Þ ¼ �kNaN

;

Mða1VÞ ¼ �k1a1V

Mða2VÞ ¼ �k2a2V

]

MðaiVÞ ¼ �kiaiV

]

MðaNVÞ ¼ �kNaNV

:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð11Þ

The solutions of Eq. (11) are available in previous publications (Tang et al., 1981;

Sudicky and Frind, 1984; Cormenzana, 2000) when the initial and boundary conditions in

Table 2 are applied,

ai ¼ Sðaoi ; kiÞ; aiV¼ SVðaoi ; kiÞ; bi ¼ 1; 2;: : :;N ð12Þ

where S and SVsymbolically represent analytical solutions, and aio is the initial or

boundary condition in the transformed domain, which is given by:

aoi ¼ coi þXi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �coj : ð13Þ


4.1. Pulse injection: Sudicky and Frind (1984), Cormenzana (2000)

For the pulse injection case, when D = 0, the analytical solution for multiple species

transport in the single fracture system can be expressed in the transformed domain as:

ai ¼ aoicz

2ffiffiffip

pT3

exp � c2z2

4T2� kit

� ; T 2 > 0

ai ¼ 0; T2V0 ð14Þ

aiV¼ aoiczþ Bðx� bÞ

2ffiffiffip

pT 3

exp � ½czþ Bðx� bÞ�2

4T 2� kit

!; T2 > 0; xzb

aiV¼ 0; T2V0 ð15Þwhere

c ¼ /ffiffiffiffiffiffiffiffiffiffiRVDV

p

bv; B ¼

ffiffiffiffiffiffiffiffiffiffiffiRV=DV

p; T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffit � Rz

v

r:

For coding convenience, the solutions of ai and aiV can be combined and expressed in a

general solution form:

ai ¼ aoiczþ Bðx� bÞH

2ffiffiffip

pT3

exp � ½czþ Bðx� bÞH�2

4T2� kit

!; T 2 > 0

ai ¼ 0; T2V0; ð16Þ

where H is the Heaviside step function, with H(x>b) = 1 and H(xV b) = 0.

4.2. Constant concentration boundary: Tang et al. (1981)

For the constant concentration boundary case, when D = 0, the solution for multiple

species transport in the single fracture system can be written in the transformed domain

as:

ai ¼aoi2exp � kiRz

v

� exp � k1=2i Rz

vA

!erfc

Rz

2vAT� k1=2i T

� þ exp

k1=2i Rz

vA

!"

erfcRz

2vATþ k1=2i T

� #; T 2 > 0

ai ¼ 0; T2V0 ð17Þ


aiV¼aoi2exp � kiRz

v

� expð�k1=2i W Þerfc W

2T� k1=2i T

� þ expðk1=2i W Þ

�

erfcW

2Tþ k1=2i T

� �; T2 > 0; xzb

aiV¼ 0; T2V0 ð18Þ

where

A ¼ bR

/ffiffiffiffiffiffiffiffiffiffiRVDV

p ; W ¼ Rz

vAþ Bðx� bÞ:

Similarly to Eq. (16), Eqs. (17) and (18) can be combined as:

ai ¼aoi2exp � kiRz

v

� expð�k1=2i WÞerfc W

2T� k1=2i T

� þ expðk1=2i WÞ

�

erfcW

2Tþ k1=2i T

� �; T 2 > 0

ai ¼ 0; T 2V0 ð19Þ

where

W ¼ Rz

vAþ Bðx� bÞH:

When the dispersion along the fracture is considered, Eqs. (35) and (38) of Tang et al.

(1981) should be applied instead of Eq. (19).

4.3. Inverse transform to real concentration domain

In this section, we illustrate how to derive a solution in terms of the real

concentrations based on the solution of concentrations in the transformed domain.

Taking the pulse injection case as an example, we present the derivation process of a

four-member decay chain in the fracture–matrix system based on the analytical solution


of Sudicky and Frind (1984). Using the analytical solutions (Eq. (16)) of ai in the

transformed domain, the solutions of ci can be expressed as:

ci ¼ ai �Xi�1

j¼1

ji�1

l¼j

ylklkl � ki

� �cj; bi ¼ 1; 2;: : :;N : ð20Þ

For simplicity, we use the most compact solution (Eq. (16)) to generate a solution for a

four-member decay chain and assume T2>0, cio = 0, bi = 2,3,4. Then, the transformed

boundary condition is written:

ao1 ¼ co1

ao2 ¼y1k1

k1 � k2co1

ao3 ¼y1y2k1k2

ðk1 � k3Þðk2 � k3Þco1

ao4 ¼y1y2y3k1k2k3

ðk1 � k4Þðk2 � k4Þðk3 � k4Þco1

Since c1 = a1, the solution for the first species remains the same form as Eqs. (23) and

(24) of Sudicky and Frind (1984):

c1 ¼ co1czþ Bðx� bÞH

2ffiffiffip

pT3


4T2� k1t

!: ð21Þ

For the second species,

a2 ¼y1k1co1k1 � k2|fflfflfflffl{zfflfflfflffl}

ao2

czþ Bðx� bÞH2ffiffiffip

pT 3


4T 2� k2t

!ð22Þ

c2 ¼y1k1co1k1 � k2


pT3


4T 2� k2t

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

a2

� y1k1k1 � k2

co1czþ Bðx� bÞH

2ffiffiffip

pT3


4T2� k1t

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c1

¼ y1k1co1k1 � k2


pT3


4T 2

!

½expð�k2tÞ � expð�k1tÞ�: ð23Þ


When y1 = 1, Eq. (23) is identical to Eqs. (24) and (26) of Cormenzana (2000) derived

from inverse Laplace transforms, when H = 0 and when H = 1, respectively.

For the third species,

c3 ¼ a3 �y2k2

ðk2 � k3Þc2 �

y1y2k1k2ðk1 � k3Þðk2 � k3Þ

c1

¼ y1y2k1k2ðk1 � k3Þðk2 � k3Þ

co1

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ao3


pT3


4T2� k3t

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

a3

� y2k2k2 � k3

y1k1co1k1 � k2


pT3


4T2

!½expð�k2tÞ � expð�k1tÞ�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c2

� y1y2k1k2ðk1 � k3Þðk2 � k3Þ

co1czþ Bðx� bÞH

2ffiffiffip

pT3


4T2� k1t

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

c1

¼ y1y2k1k2co1


pT3


4T2

!

expð�k3tÞ � expð�k1tÞðk1 � k3Þðk2 � k3Þ

� expð�k2tÞ � expð�k1tÞðk1 � k2Þðk2 � k3Þ

� �:

ð24Þ

Finally, for the fourth species,

c4 ¼ y1y2y3k1k2k3co1


pT3


4T2

!

expð�k4tÞ � expð�k1tÞðk1 � k4Þðk2 � k4Þðk3 � k4Þ

� expð�k3tÞ � expð�k1tÞðk3 � k4Þðk1 � k3Þðk2 � k3Þ

�

þ expð�k2tÞ � expð�k1tÞðk2 � k4Þðk2 � k3Þðk1 � k2Þ

�: ð25Þ

The solution for additional species can be derived successively in the same way.

However, for computational purposes, it is unnecessary to explicitly write out the

mathematical expressions of solutions if the number of species is large. Since the basic

solution for a single-species transport with first-order decay in fracture–matrix systems

(Tang et al., 1981; Sudicky and Frind, 1982; Sudicky and Frind, 1984; Cormenzana, 2000)

can be coded as standard classes or subroutines, it is straightforward to implement the

transform processes to extend those solutions to N-species systems. Similarly, the closed-

form solution of four species with a constant boundary condition in the given system,

when T2>0, cio = 0, bi= 2,3,4, D = 0, is written as:

c1 ¼ co1f1; c2 ¼ co1b2

f2 � f1

k1 � k2ð26Þ


c3 ¼ co1b3

f3 � f1

ðk1 � k3Þðk2 � k3Þ� f2 � f1

ðk1 � k2Þðk2 � k3Þ

� �ð27Þ

c4 ¼ co1b4

f4 � f1

ðk1 � k4Þðk2 � k4Þðk3 � k4Þ� f3 � f1

ðk1 � k3Þðk2 � k3Þðk3 � k4Þ

�

� f2 � f1


�ð28Þ

where

fi ¼1

2exp � kiRz

v

� expð�k1=2i WÞerfc W

2T� k1=2i T

� þ expðk1=2i WÞ

�

erfcW

2Tþ k1=2i T

� �

i ¼ 1; 2; 3; 4

b2 ¼ y1k1; b3 ¼ y1y2k1k2; b4 ¼ y1y2y3k1k2k3:

5. Applications

5.1. Four-member radionuclide decay chain with pulse injection boundary

We simply extend the first example of Sudicky and Frind (1984) from two species to

four species. Except for the decay rates of the third and fourth members, other parameters

remain the same. The parameter values are listed in Table 3.

Table 3

System parameters for a four-member radioactive decay chain with pulse injection

2b= 100 Am /= 0.01

v= 0.1 m/day s= 0.1D*= 8.64 10� 5 m2/day

t1,1/2 = 5 years c1o = 1.0

t2,1/2 = 450 years c2o = 0.0

t3,1/2 = 900 years c3o = 0.0

t4,1/2 = 2000 years c4o = 0.0


Fig. 2 shows the concentration profile of four species along the fracture for four

different times. The results for the first two species are identical to Fig. 2 of Cormenzana

(2000). When all radionuclides have the same retardation factor in the fracture, the same

retardation factor in the matrix, and the same diffusion coefficient, they advance in the

same speed along the fracture if the pulse injection boundary is assumed. (Note: the

daughter species will advance farther than its parent species if a constant concentration

boundary is used and the dispersion along the fracture is not neglected.) To prove that

all species advance in the same speed in the fracture (H= 0), we use the first spatial

moment,

ziðtÞ ¼1

Mi

Z l

0

cizdz; i ¼ 1; 2; 3; 4; Mi ¼Z l

0

cidz: ð29Þ

We recall that analytical solutions of ci, i= 1,2,3,4, Eqs. (21) and (23)–(25), are products

of two factors; one is a distance-dependent and species-independent function and the

Fig. 2. Concentration profile along the fracture for a four-member decay chain with a pulse injection boundary

condition.


other is a function of decay rates. The solution of species concentrations along the

fracture (xV b) can be further written as

ci ¼ PQi bi ¼ 1; 2; 3; 4

P ¼ co1cz2ffiffiffip

pT3

exp � c2z2

4T2

� ð30Þ

Q1 ¼ expð�k1tÞ ð31Þ

Q2 ¼ y1k1expð�k2tÞ � expð�k1tÞ

k1 � k2

� �ð32Þ

Q3 ¼ y1y2k1k2expð�k3tÞ � expð�k1tÞ

ðk1 � k3Þðk2 � k3Þ� expð�k2tÞ � expð�k1tÞ

ðk1 � k2Þðk2 � k3Þ

� �ð33Þ

Q4 ¼ y1y2y3k1k2k3expð�k4tÞ � expð�k1tÞ

ðk1 � k4Þðk2 � k4Þðk3 � k4Þ� expð�k3tÞ � expð�k1tÞ


�

þ expð�k2tÞ � expð�k1tÞðk2 � k4Þðk2 � k3Þðk1 � k2Þ

�: ð34Þ

Therefore,

ziðtÞ ¼

Z l

0

PzdzZ l

0

Pdz

; bi ¼ 1; 2; 3; 4: ð35Þ

The first moment is independent of the species index i. Similarly, the location of the

peak concentration along the fracture for all species is found to be the same after solving

Bci

Bz¼ BPQi

Bz¼ Qi

BP

Bz¼ 0Z

BP

Bz¼ 0: ð36Þ

This demonstrates that BP/Bz is species-independent.

5.2. Four-member radionuclide decay chain with a constant boundary concentration

When a constant concentration boundary is applied at the origin (see Fig. 1), Eq. (19)

(also see Tang et al., 1981) becomes the base solution to derive the solution of multiple

decay radionuclides in the fracture–matrix system. Using the same system parameters in

Table 3, the concentration profiles of four radionuclides along the fracture are given in

Fig. 3 for four different times.

As indicated by Tang et al. (1981), the solution of Eq. (17) can reach steady state

when the simulation time is sufficiently large:

ai ¼ aoi exp � R

vki þ

ffiffiffiffiki

p

A

� z

� �: ð37Þ


Fig. 3. Concentrations of radionuclides along the fracture with a constant-concentration boundary condition.

Fig. 4. Steady-state concentrations along the fracture when the constant-concentration boundary is applied.


The semilog plot of the steady-state concentrations along the fracture is shown in Fig. 4.

It shows that the peak concentration of the first species is at the origin, while that of its

daughter product is located downstream.

5.3. Four-member reactions with cio p 0 and yi p 1

Applying the analytical solutions of Tang et al. (1981), Sudicky and Frind (1984), and

Cormenzana (2000) for more general reactive transport purposes (beyond radionuclide

migration) poses two challenges. First, the daughter species may not necessarily maintain

zero concentration at the boundary. Second, one unit of parent species may not produce

one unit of daughter species. As in Eq. (8), we use yield coefficients, yi, i = 1,2,. . .,N� 1,

in the system decomposition.

Table 4

System parameters for a four-member biodegradation chain

b= 0.01 m / = 0.01

v= 1.0 m/day s= 0.1D*= 1.0 10� 5 m2/day R = 1.0 RV= 1.0k1 = 0.005 1/day y1 = 0.79 c1

o = 100

k2 = 0.003 1/day y2 = 0.74 c2o = 20

k3 = 0.0005 1/day y3 = 0.64 c3o = 10

k4 = 0.0002 1/day c4o = 2

Fig. 5. Concentration profile of a four-member decay chain with pulse injection in the fracture–matrix system

after 1000 days. c1o = 100, c2

o = 20, c3o = 10, c4

o = 2.


Fig. 6. Distance from peak concentration to fracture–matrix interface.

Fig. 7. Concentration profile of a four-member decay chain with constant-concentration boundary conditions in

the fracture–matrix system after 1000 days. t= 1000 days, c1o = 100, c2

o = 20, c3o = 10, c4

o = 2.


Four-species sequential reactions as defined by Sun et al. (1999) are used to

demonstrate the solution in a single fractured matrix system. The velocity in the fracture

and the diffusion coefficient in the matrix are assumed to be constant. System parameters

are given in Table 4.

Fig. 5 shows the concentration distributions in both the fracture and matrix when a

pulse injection boundary is assumed. The peak concentrations of all species are

concentric, but not located at the fracture–matrix interface. The distance from the

highest concentration to the interface is

x* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DV

RVt � Rz

v

� s� /DV

bv; x*z0: ð38Þ

Fig. 6 shows the distance as a function of time and z distance for a given R, RV, DV, v, and b.

Fig. 7 also shows the concentration distributions of four species in both the fracture and

matrix domains when a constant concentration boundary is applied. The highest concen-

tration of the first species is at the source in the fracture. Although the boundary

concentrations of daughter species are assumed (20, 10, 2), the peak concentrations of

daughter species are located downstream from their parent species.

6. Conclusions

Analytical solutions have been developed for the reactive transport of N-member decay

chains in a single fracture–matrix system based on the solutions to single-species transport

problems derived by Tang et al. (1981), Sudicky and Frind (1984), and Cormenzana

(2000). Different retardation factors and transport properties can be specified for the

fracture and matrix. However, all species are assumed to share the same retardation factors

and transport properties for the fracture and matrix, respectively. The reactions are

assumed to be sequential, first-order, and irreversible.

A closed-form solution for a four-species transport problem in a single fractured porous

medium, as defined by Sudicky and Frind (1984), is explicitly provided. It is mathemati-

cally demonstrated that parent and daughter species concentrations advance at the same

speed along the fracture if a pulse injection boundary is assumed and dispersion is

neglected in the fracture. The solution with the pulse injection boundary condition also

indicates that the peak concentrations of all species in the matrix may be located inside the

matrix rather than at the fracture–matrix interface. The solutions developed in this paper

not only provide a framework for studying the processes of radionuclide transport in

fractured porous media but also apply to non-unimolar reactive transport, such as

biodegradation of chlorinated solvents and denitrification.

Acknowledgements

The authors wish to thank Nina D. Rosenberg, Kayyum Mansoor, and anonymous

reviewers for careful review and helpful comments. This work was performed under the


auspices of the U.S. Department of Energy by the University of California, Lawrence

Livermore National Laboratory under Contract No. W-7405-Eng-48.

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Analytical solutions for reactive transport of N-member radionuclide chains in a single fracture