Semi-analytic solution for transport of a two-member decay

chain in discrete parallel fractures

Scott K. Hansen, Department of Environmental Science and Energy Research, Weizmann

Institute of Science, Rehovot 76100 Israel (scott.hansen@weizmann.ac.il)

This article has been accepted for publication and undergone full peer review but has not beenthrough the copyediting, typesetting, pagination and proofreading process which may lead todifferences between this version and the Version of Record. Please cite this article as an‘Accepted Article’, doi: 10.1002/wrcr.20451

Key points

• Solution derived for transport of 1st-order decay chains in parallel fractures

• This generalizes two existing solutions and is verified against them

Abstract

A wide variety of analytic solutions have been developed for 1D contaminant transport, but to

date the author is aware of none modeling a decay chain in parallel discrete fractures in porous

media. In this note, the derivation is presented for a two-species first-order decay chain in such an

environment, with an arbitrary concentration history specified up-gradient, fracture advection, and

diffusion into the porous matrix. The solution is presented in brief, followed by corroboration of its

numerical implementation against two different existing numerical codes. An appendix contains a

detailed derivation of the solution, and a Mathematica notebook that implements it and may be

used by practitioners is enclosed as supplementary material.

Index terms

1009 - Geochemical modeling

1832 - Groundwater transport

3299 - Mathematical geophysics (general or miscellaneous)

1 Introduction

Previously, there has been interest in analytic solutions for transport of decaying contaminants in

fractured porous media, and solutions have been derived in a number of simplified geometries. In the

literature, those solutions have been widely cited by authors working in the transport of radionuclide

chains in fractured rock, as well as by authors verifying various numerical codes. These solutions

are also applicable to transport and first-order biological decay of organic compounds in fractured

2

porous media. A good introduction to currently existing solutions may be found in Chapter 6 of

Rumynin (2011). The decay of a single species has been modeled in a single fracture by Tang et al.

(1981), and in a set of parallel fractures by Sudicky and Frind (1982). A two-species decay chain

was also later considered in a system with a single isolated fracture in Sudicky and Frind (1984),

modified in a special case by Cormenzana (2000), and reconsidered for longer chains by Sun and

Buscheck (2003). However, no solution the author is aware of exists which accommodates both a

decay chain and a set of parallel discrete fractures, despite zones of multiple fractures being common

in reality. The work presented here fills that lacuna. (See Table 1 for a comparison of approaches.)

In this section, the exact problem formulation is provided. In Section 2, the Laplace-domain

solution is presented (without derivation), along with discussion of its form. In Section 3, computer

implementation of the given solution is discussed, and results of trials of a computer implementation

of the new solution against existing codes are given. Full derivation of the new solution is presented

in Appendix A, and a computer implementation is enclosed as supplementary material.

1.1 Exact problem formulation

This document presents a Laplace-domain expression for the concentration histories of two chemical

species comprising a straight, mother-daughter, decay chain in an infinite set of parallel fractures,

under uniform groundwater flow down the fractures, given an arbitrary Type I (specified concen-

tration) source at the origin. Assumptions made are that: (1) groundwater flow is known, constant,

and is longitudinal along the fracture direction, (2) groundwater flow is sufficient for transport in

the fractures to be advection dominated, (3) in the porous matrix, diffusion is the only transport

process, and it occurs only orthogonal to the fracture planes, (4) geometry is uniform, with frac-

tures of uniform width, evenly spaced, and of infinite extent, (5) all decay processes in the chain

can be modeled as first-order or pseudo-first order, and 6) sorbed and free solute concentrations

are proportional, allowing use of constant retardation factors. In Figure 1, the geometry of the

fracture system is shown. Note that the origin for the x coordinate is at the edge of the fracture,

rather than the midpoint. This is slightly different from the convention used by Sudicky and Frind,

3

whose x ran from 0 to (L+ b), rather than 0 to L. This change was made in order to simplify some

algebra when deriving the solution.

In the fracture, the governing equations for the various species are the standard advection equations

with first-order decay, plus a sink term corresponding to matrix diffusion in and out of the fracture

walls. The governing equations for the two species are (variables are defined below):

∂c1∂t

+v

R1

∂c1∂z

+ λ1c1 +q1R1b

= 0 (1)

and∂c2∂t

+v

R2

∂c2∂z

+ λ2c2 +q2R2b

=R1λ1

R2c1 (2)

where c1 is the concentration of species 1 (the mother in the decay chain) in the fracture, and c2

is the concentration of species 2 (the daughter). These are solved subject to the following initial

and boundary value equations, which define fractures initially devoid of solute, with arbitrary type

1 (specified concentration history) boundary conditions specified at the upgradient boundary for

both the mother and daughter.

c1(0, t) = f1(t)

c2(0, t) = f2(t)

c1(z, 0) = c2(z, 0) = 0

(3)

In the matrix, advection is not a factor, and diffusion is the dominant mechanism, so the governing

equations for the two species are simpler:

∂c′1∂t

+D1

R′1

∂2c′1∂z2

+ λ1c′1 = 0 (4)

and∂c′2∂t

+D2

R′2

∂2c′2∂z2

+ λ2c2 =R′

1λ1

R′2

c′1 (5)

where the prime (i.e. ′) symbol indicates a concentration in the matrix, and the numeric subscripts

1 and 2 again identify the mother and daughter species, respectively. The matrix solutions must

4

satisfy the following initial and boundary conditions, which enforce continuity of concentration

across the fracture/matrix boundary, and also no flux across symmetry boundaries:

c′1(0, z, t) = c1(z, t)

c′2(0, z, t) = c2(z, t)

∂c′1∂x (L, z, t) = 0

∂c′2∂x (L, z, t) = 0

c′1(x, z, 0) = c′2(x, z, 0) = 0

(6)

Finally, the sink terms in the fracture (representing flux outwards to the porous matrix), as seen in

(1) and (2) are given by the following expression:

qn = −θDn∂c′n∂x

∣∣∣∣x=0

(7)

In all the above equations, the physical meanings of the variables and parameters are outlined in

Table 2.

2 Summary of the solutions

The solutions in the Laplace domain can be expressed in terms of the Laplace transforms of the

boundary conditions, the spatial variables, and a number of simplifying auxiliary functions (which

are defined below). The solutions are summarized in Table 3. As in the work of Cormenzana, two

different forms are given for the concentration of species 2 in the fracture, on account of potential

zero division when the auxiliary functions κ1 and κ2 are equal.

The functions employed in the solutions summarized in Table 3 are defined below. The location-

independent quantities are defined:

κn ≡√

R′n

Dn(p+ λn) (8)

5

Mn ≡ Rn

v

(p+ λn − θDnhp(κn, 0)

Rnb

)(9)

W ≡ λ1f1v

(R1 −R′

1

θ

b

(hp(κ1, 0)− hp(κ2, 0)

κ21 − κ2

2

))(10)

X =λ1f1v

(R1 −R′

1

θ

b

(1− e4κL − 4e2κLκL

2κ (1 + e2κL)2

))(11)

The location-dependent functions are defined:

P (x) ≡ −e−κx

(2e2κL

(e2κx − 1

)L+

(e2κL + 1

) (e2κL − e2κx

)x)

(e2κL + 1)2 (12)

Q(z) ≡ −R′1

D2λ1e

−M1z f1 (13)

h(k, x) ≡ eκ(L−x) + e−κ(L−x)

eκL + e−κL(14)

hp(κ, x) ≡∂h

∂x(κ, x) = k

[e−κ(L−x) − eκ(L−x)

eκL + e−kL

](15)

3 Numerical results and solution corroboration

The analytic solution presented briefly above (and completely in the appendix) is in the Laplace

domain, where time has been transformed to a non-local variable, p. In order to use the solutions to

determine the concentration history at a given spatial location, it is necessary to invert the Laplace

transform. While analytical inversions are sometimes possible, using a computer for evaluation of

the solution is inevitable given the symbolic complexity here. Thus, it is simpler to numerically

invert the Laplace transforms directly. A computer code (Parfrac) that does this inversion to

calculate concentration profiles downgradient from the source, in the fractures, was written. The

computer implementation was slightly simplified, assuming only the mother species is present at

the source. This is in keeping with computer implementations of prior solutions, though there is

no technical reason why the program could not cover the full range of solutions outlined above,

6

if needed. Three trials were run to test the code implementing the new solution against two

previously implemented solutions in order to compare fracture concentration profiles generated.

Since the fracture solutions are derived from the matrix solution, this testing has the potential to

indirectly corroborate them, also. The trials were:

1. Comparison with Chainf, an existing code written by C. Neville which implements Sudicky

and Frind (1984); Cormenzana (2000) for a two-species decay chain in a single fracture. This

was run for against Parfrac for a very large fracture spacing (so the parallel and single fracture

approaches converge) for κ1 and κ2 both distinct. Concentration upgradient was 1.0 for the

first 100 days and 0.5 thereafter.

2. A repetition of the above with intrinsic parameters altered to make κ1 and κ2 identical.

3. Comparison with Craflush, coded by E.A. Sudicky, which implements the Sudicky and Frind

(1982) solution for decay of a single species in parallel fractures. This was compared, for a

narrow fracture spacing, with the output of Parfrac for the mother species (species 1) only.

Concentration upgradient was 1.0 for all time.

The hydrogeologic parameters used in the simulations are given in Table 4. Results for the three

trials are shown in Figures 2 and 3, respectively. As is apparent, correspondence between output

from Parfrac and the existing codes is excellent in all cases.

4 Summary

A Laplace-domain analytical solution to the problem of a straight two-species decay chain in a set

of parallel fractures was derived. This solution allows for a broader range of calculations than are

possible using of existing analytic solutions. It may be of use for making engineering estimates for

natural attenuation problems, particularly concerning the transport of radionuclides, and also for

verifying numerical models. A computer implementation of the model was generated for the case of

a two species decay chain, with only the parent species present at the upgradient source. Both the

7

analysis and its computer implementation have been corroborated by empirical comparison with

existing solutions using a few different sets of fracture and solute properties. The implementation

has been included as supplementary material to this note.

A Derivation of the solution in the Laplace domain

First consider the governing equation for species 1 in the matrix:

pc′1 − c′1(x, z, 0)−D1

R′1

∂2c′1∂x2

+ λc′1 = 0 (16)

∂2c′1∂x2

− R′1

D1(p+ λ)c′1 = 0 (17)

and define κ21 ≡ (R′

1/D1)(p+ λ) . By inspection, the above has the solution

c′1 = αe−κ1x + βeκ1x (18)

Differentiating, and applying the type II BC at x = L yields α = βe2κ1L. Substituting back into

c′1, evaluating at x = 0, and applying the type I BC yields

β =c1

1 + e2κ1L(19)

So,

α =c1

1 + e−2κ1L(20)

Substituting this all back into (18) yields

c′1 = c1

[eκ1(L−x) + e−κ1(L−x)

eκ1L + e−κ1L

](21)

8

where we may define the function h(κ1, x) to represent the term in square brackets. Now we have

an expression for the Laplace-transformed concentration in the matrix of species 1 in terms of its

concentration in the fracture. To determine the concentration in the fracture, we need to determine

the flux into the matrix, to determine the sink term there, and then solve. We determine this by

differentiating spatially, yielding

∂c′1∂x

= c1

[κ1

(e−κ1(L−x) − eκ1(L−x)

)eκ1L + e−κ1L

](22)

where we may define the function hp(κ1, x) to represent the term in the square brackets. Then

evaluating hp(κ1, 0) yields

q1 = −θD1c1hp(κ1, 0) (23)

Consider now the transform of(1), for the mother species in the fracture. Substituting in the above

equation for q1 yieldsdc1dz

+

[R1

v1

(p+ λ1 −

θD1h(κ1, 0)

R1b

)]c1 = 0 (24)

where we define M1 to be equal to the square-bracketed quantity. Since the above is a homogeneous

first-order equation, employing the boundary condition (3) yields

c1 = f1e−M1z (25)

which is the Laplace transform of the solution for species 1 (the mother) in the fracture. To complete

the analysis, this can be plugged back into the expression for species 1 in the matrix, yielding

c′1 = f1e−M1zh(κ1, x) (26)

Now, we can do a (more complicated) analog of the above analysis, for the daughter species.

Beginning again in the matrix, Laplace transforming (5) and rearranging yields

d2c′2dx2

−[R′

2

D2(p+ λ2)

]c′2 = −R′

1

D2λ1c′1 (27)

9

where we define the square-bracketed term to be κ22. The above is a second-order linear nonhomoge-

neous equation, which may be solved by an operator factoring approach. Using operator notation,

with ∆ ≡ ddx (note this is not the Laplacian), we re-write the problem:

(∆− κ2) (∆ + κ2) c′2 =

[−R′

1

D2λ1f1e

−M1z

]h(κ1, x) (28)

where we again make a simplifying definition, defining Q(z) to represent the square-bracketed term.

Since (∆ + a)−1 has an explicit integral form for any real value a, we may write down the following

expression, using the indefinite integral form and arbitrary constants of integration b1 and b2:

c′2 = eκ2x

[ˆe−κ2x

[e−κ2x

[ˆeκ2xQ(z)h(κ1, x)dx+ b1

]]dx+ b2

](29)

Rearranging yields:

c′2 = Q(z)eκ2x

[ˆe−2κ2x

[ˆeκ2xh(κ1, x)dx

]dx

]+ α2(z)e

κ2x + β2(z)e−κ2x (30)

where the first term represents a particular solution to the ODE, and the remaining terms repre-

senting the solution to the associated homogeneous equation. It is possible to evaluate the integral

explicitly, but two separate cases must be considered, depending on the relative values of κ1 and

κ2. For the present, we will assume that they are distinct, and then modify this solution to account

for the case when κ1 = κ2.

A.1 The case of distinct κ1 and κ2

This case will be most common. Since p is a variable on which each κn depends, both R′1/D1 =

R′2/D2 and λ1 = λ2 for κ1 and κ2 to be generally equal. If either equation is unsatisfied then there

are distinct κ1 and κ2, except possibly at isolated points in the p-plane. In that case, this double

10

integral in (30) (which we may term ˆc′2,p) may be determined explicitly to be

ˆc′2,p =Q(z)

κ21 − κ2

2

h(κ1, x) (31)

Consequently,

c′2 = α2eκ2x + β2e

−κ2x +Q(z)

κ21 − κ2

2

h(κ1, x) (32)

Applying the boundary conditions, it may be shown that

c′2 = c2h(κ2, x) +Q(z)

κ21 − κ2

2

[h(κ1, x)− h(κ2, x)] (33)

(As an aside, note the similarity of the first term to c′1. Essentially, the solution has been divided

into two additive terms, the first representing the behavior of the daughter species on its own, and

the second representing a source due to the decay of the mother species into the daughter species.)

Finally, we may use this expression in the matrix to determine the sink term for the daughter

species in the fracture. We may write

q2 = −θD2

[c2hp(κ2, 0) +

Q(z)

κ21 − κ2

2

[hp(κ1, 0)− hp(κ2, 0)]

](34)

Substituting this into the Laplace-transformed governing equation for the daughter species in the

fracture yields

[R2

v

(p+ λ2 −

θD2hp(κ2, 0)

R2b

)]c2 +

ˆdc2dz

=R1

vλ1c1 +

θD2Q(z) (hp(κ1, 0)− hp(κ2, 0))

vb (κ21 − κ2

2)(35)

where we define M2 to represent the square-bracketed term. Then, substituting in for both c1 and

Q(z), we derive

dc2dz

+M2c2 =

[λ1f1v

(R1 −R′

1

θ

b

(hp(κ1, 0)− hp(κ2, 0)

κ21 − κ2

2

))]e−M1z (36)

11

where we may make a final simplification, defining W to represent the square-bracketed term. This

leaves us with the following differential equation for c2:

dc2dz

+M2c2 = We−M1z (37)

To solve the above, first-order, non-homogeneous ODE is straightforward. In the case when two

distinct values of κ1 and κ2 exist, one is justified in stipulating that M1 = M2 , since otherwise it

would be necessary that hp(κ1, 0) = rhp(κ2, 0) for some constant r = 1, for all values of p. It can

be shown this is not possible. Thus,

c2 = e−M2z

(W

M2 −M1e(M2−M1)z + γ

)(38)

for some γ. Applying the boundary condition when z = 0, (3), yields

c2 = f2e−M2z +

W

M2 −M1

(e−M1z − e−M2z

)(39)

We have thus derived Laplace transformed solutions for both the mother and daughter species in

both the fracture and the matrix.

A.2 The case of identical κ1 and κ2

Because the solutions derived above contain terms with κ21 − κ2

2 in their denominator, we cannot

directly apply them in the case when κ1 = κ2. Fortunately, these terms are indeterminate (0/0),

and so L’Hospital’s rule can be applied. Let us begin by considering the equation (33), which was

derived assuming κ1 and κ2 distinct. Defining κ1 ≡ κ and κ2 ≡ κ + δ, then taking the limit as

δ → 0

c′2 = c2h(κ2, x) +Q(z)

limδ→0

∂∂δ [h(κ, x)− h(κ+ δ, x)]

∂∂δ

[κ2 − (κ+ δ)

2]

(40)

12

yields the following solution in the matrix:

c′2 = c2h(κ, x) +Q(z)

2κ

(e−κx

(−2e2κL

(−1 + e2κx

)L−

(1 + e2κL

) (e2κL − e2κx

)x)

(1 + e2κL)2

)(41)

An identical procedure can be adopted in the fracture, beginning with (36)

dc2dz

+M2c2 =

λ1f1v

R1 −R′1

θ

b

limδ→0

∂∂δ [hp(κ, 0)− hp(κ+ δ, 0)]

∂∂δ

[κ2 − (κ+ δ)

2]

e−M1z (42)

leading todc2dz

+M2c2 =

[λ1f1v

(R1 −R′

1

θ

b

(e4κL + 4e2κLκL− 1

2κ (1 + e2κL)2

))]e−M1z (43)

We may make a simplification, defining X to represent the square-bracketed term. This leaves us

with the following differential equation for c2:

dc2dz

+M2c2 = Xe−M1z (44)

It is easy to show that when κ1 = κ2 for all p, M1 = M2 for all p. To solve the above first-order,

non-homogeneous ODE is then straightforward. The solution is

c2 = e−M2z (Xz + γ) (45)

for some γ. Applying the boundary condition when z = 0 (3) directly yields

c2 =(f2 +Xz

)e−M2z (46)

Again, we have derived Laplace transformed solutions for both the mother and daughter species in

both the fracture and the matrix, this time for the special case where κ21 − κ2

2.

13

Acknowledgment

I am extremely grateful to Mr. Christopher Neville of S.S. Papadopulos and Associates, who

suggested this work and also provided his implementation of Cormenzana’s single fracture solution

in source code form. I would not have undertaken this project without his impetus.

References

Cormenzana, J. (2000), Transport of a two-member decay chain in a single fracture: Simplified

analytical solution for two radionuclides with the same transport properties, Water Resources

Research, 36(5), 1339–1346.

Rumynin, V. G. (2011), Subsurface Solute Transport Models and Case Histories, Theory and Ap-

plications of Transport in Porous Media, vol. 25, Springer Science+Business Media.

Sudicky, E. A., and E. O. Frind (1982), Contaminant transport in fractured porous media: Ana-

lytical solutions for a system of parallel fractures, Water Resources Research, 18(6), 1634–1642.

Sudicky, E. A., and E. O. Frind (1984), Contaminant transport in fractured porous media: Analyt-

ical solution for a two-member chain decay in a single fracture, Water Resources Research, 20(7),

1021–1029.

Sun, Y., and T. Buscheck (2003), Analytical solutions for reactive transport of n-member radionu-

clide chains in a single fracture, Journal of Contaminant Hydrology, 62, 695–712.

Tang, D. H., E. O. Frind, and E. A. Sudicky (1981), Contaminant transport in fractured porous

media: Analytical solution for a single fracture, Water Resources Research, 17(3), 555–564.

Table 1: Comparison of various 1D transport Laplace domain analytic solutions for two-componentdecay chains.

Single fracture Parallel fractures

Single species Tang et al. (1981) Sudicky and Frind (1982)Decay chain Sudicky and Frind (1984); Cormenzana (2000) This paper

14

Table 2: Catalog of symbols representing physical parameters and quantities defining the transportproblem

Symbol Meaningcn and c′n Mass concentration of species n [M L-3] in, respectively, the fracture

and the matrixv Groundwater velocity in fractures [M T-1]

fn(t) Upgradient concentration of species n in the fracture (at coordinatez = 0)

λn First-order decay constant for species nDn Effective (tortuosity included) diffusion constant for the porous

matrixRn and R′

n Retardation factor for species n in, respectively, the fracture and thematrix

θ Matrix porosityt Time since first release of solute (all concentrations are zero at and

before t = 0)x Spatial coordinate: distance into matrix from nearest fracture wallz Spatial coordinate: distance along fracture from location where the

fn(t) are specifiedL Half-width of matrix block between two adjacent fracturesb Half-width of each fracture

Table 3: Summary of solutions in the Laplace transform of the concentration function for eachspecies, in both domains.

Species 1 Species 2

Fracture c1 = f1e−M1z c2 =

{f2e

−M2z +Xze−M2z if κ1 = κ2

f2e−M2z + W

M2−M1

[e−M1z − e−M2z

]if κ1 = κ2

Matrix c′1 = c1h(κ1, x) c′2 =

{c2h(κ2, x) +

P (x)Q(z)2κ if κ1 = κ2

c2h(κ2, x) +Q(z)κ21−κ2

2[h(κ1, x)− h(κ2, x)] if κ1 = κ2

15

Table 4: Parameters for simulation trials. Units are not used by any of the implementations, butfor the sake of concreteness may be thought of as meters and days. Boldface parameter values aredistinct from the ones employed in trial 1.

Trial 1 Trial 2 Trial 3Velocity v 0.1 0.1 1

Fracture width 2B 1.0e-4 1.0e-4 1.0e-4Fracture spacing 2L 10 10 0.1

Porosity θ 0.01 0.01 0.01Matrix diffusion coefficient D1 8.64e-6 8.64e-6 8.64e-6Matrix diffusion coefficient D2 4.68e-6 8.64e-6

Fracture retardation coefficient R1 1 1 1Fracture retardation coefficient R2 1 1Matrix retardation coefficient R′

1 1 1 1Matrix retardation coefficient R′

2 1 1First-order decay constant λ1 3.798e-4 4.220e-4 3.798e-4First-order decay constant λ2 4.220e-6 4.220e-4

Figure 1: Schematic diagram of fractured rock, with coordinate system overlaid. Only two fracturesare shown, but the system is taken to be of infinite extent in the x-direction, so that lines ofsymmetry exist in the middle of each porous matrix block. Note that the fractures are shownhorizontal, but the solution derived here is valid regardless of the orientation of the x-axis relativeto gravity.

Figure 2: Comparison of concentration profile predictions from Parfrac (solid lines) and Chainf(hollow markers). Square, cross, and circle markers respectively indicate profiles at 1000, 10,000,and 100,000 days. Black curves and markers are for species 1, grey for species 2. The top axes showoutput of trial 1 (distinct κ1 and κ2), the bottom show output of trial 2 (identical κ1 and κ2)

16

Figure 3: Comparison of concentration profiles at three different times, generated for the motherspecies (species 1) for trial 3. Lines represent output from Parfrac and markers represent outputfrom Craflush. Square, cross, and circle markers respectively indicate profiles at 1000, 10,000, and100,000 days

17

Auxiliary material for 

Semi‐analytic solution for transport of a two‐member decay chain in discrete parallel fractures 

Scott K. Hansen, Weizmann Institute of Science 

Water Resources Research 

Introduction 

The enclosed Mathematica notebook implements the solution derived here, using a well known algorithm for numerically inverting the Laplace transform. 

 

Use of “Mathematica workbook generating profiles.nb” 

To use it, you simply need to: 

1) Open the notebook in Mathematica (it has been tested on version 8). 

2) Fill in the appropriate subsurface parameters in the section "USER SPECIFIED: intrinsic parameters and boundary condition". 

3) Set "bc[p_] :=" in the same section to the Laplace transform of the Type I upgradient boundary condition on the mother species in the fracture. In this line, "p" represents the Laplace variable that replaces time. The daughter species is taken to have an upgradient boundary condition of 0 for all time. 

4) Sequentially press [Enter] or [Shift]+[Return] in each of the execution groups in the notebook. The second last execution group will plot two sets of concentration profiles: the first for the mother species, and the second for the daughter species. The last execution group exports those curves so that they may be graphed in more sophisticated tools. 

Good luck! Note that you may need to adjust the PlotRange parameter in the last execution group, depending on the value of Type I boundary condition that you apply.