A wide variety of analytic solutions have been developed for 1-D 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 upgradient, 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 supporting information.
 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 porous media. A good introduction to currently existing solutions may be found in chapter 6 of Rumynin . The decay of a single species has been modeled in a single fracture by Tang et al.  and in a set of parallel fractures by Sudicky and Frind . A two-species decay chain was also later considered in a system with a single isolated fracture in Sudicky and Frind , modified in a special case by Cormenzana , and reconsidered for longer chains by Sun and Buscheck . 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.)
Table 1. Comparison of Various 1-D Transport Laplace-Domain Analytic Solutions for Two-Component Decay Chains
 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 supporting information.
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 concentration) 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 fractures 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 [1982 and 1984], whose x ran from 0 to , 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):
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.
 In the matrix, advection is not a factor, and diffusion is the dominant mechanism, so the governing equations for the two species are simpler:
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 satisfy the following initial and boundary conditions, which enforce continuity of concentration across the fracture/matrix boundary and also no flux across symmetry boundaries:
 Finally, the sink terms in the fracture (representing flux outward to the porous matrix), as seen in (1) and (2), are given by the following expression:
 In all the above equations, the physical meanings of the variables and parameters are outlined in Table 2.
Table 2. Catalog of Symbols Representing Physical Parameters and Quantities Defining the Transport Problem
Mass concentration of species n [M L−3] in, respectively, the fracture and the matrix
Groundwater velocity in fractures [M T−1]
Upgradient concentration of species n in the fracture (at coordinate z = 0)
First-order decay constant for species n
Effective (tortuosity included) diffusion constant for the porous matrix
Retardation factor for species n in, respectively, the fracture and the matrix
Time since first release of solute (all concentrations are zero at and before t = 0)
Spatial coordinate: distance into matrix from nearest fracture wall
Spatial coordinate: distance along fracture from location where the are specified
Half width of matrix block between two adjacent fractures
Half width of each fracture
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 and are equal.
Table 3. Summary of Solutions in Terms of the Laplace Transform of the Concentration Function for Each Species, in Both Domains
 The functions employed in the solutions summarized in Table 3 are defined below. The location-independent quantities are defined:
 The location-dependent functions are defined:
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 nonlocal 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, 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  and Cormenzana  solution 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 and 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 and identical.
 3. Comparison with Craflush, coded by E. A. Sudicky, which implements the Sudicky and Frind  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.
Units are not used by any of the implementations, but for the sake of concreteness may be thought of as meters and days. Boldface parameter values are distinct from the ones employed in trial 1.
Matrix Diffusion Coefficient D1
Matrix Diffusion Coefficient D2
Fracture Retardation Coefficient R1
Fracture Retardation Coefficient R2
Matrix Retardation Coefficient
Matrix Retardation Coefficient
First-Order Decay Constant
First-Order Decay Constant
 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 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 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 supporting information to this note.
Appendix: Derivation of the Solution in the Laplace Domain
 First consider the governing equation for species 1 in the matrix:
and define . By inspection, the above has the solution
Differentiating and applying the type II BC at x = L yields . Substituting back into , evaluating at x = 0, and applying the type I BC yields
Substituting this all back into (A3) yields
where we may define the function 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
where we may define the function to represent the term in the square brackets. Then, evaluating yields
 Consider now the transform of (1) for the mother species in the fracture. Substituting in the above equation for yields
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
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
 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
where we define the square-bracketed term to be . The above is a second-order linear nonhomogeneous equation, which may be solved by an operator factoring approach. Using operator notation, with (note this is not the Laplacian), we rewrite the problem:
where we again make a simplifying definition, defining Q(z) to represent the square-bracketed term. Since 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:
where the first term represents a particular solution to the ordinary differential equation (ODE), and the remaining terms representing 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 and . For the present, we will assume that they are distinct, and then modify this solution to account for the case when .
A1. The Case of Distinct and
 This case will be most common. Since p is a variable on which each depends, both and for and to be generally equal. If either equation is unsatisfied then there are distinct and , except possibly at isolated points in the p plane. In that case, this double integral in (A15) (which we may term ) may be determined explicitly to be
Applying the boundary conditions, it may be shown that
As an aside, note the similarity of the first term to . 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
Substituting this into the Laplace-transformed governing equation for the daughter species in the fracture yields
where we define M2 to represent the square-bracketed term. Then, substituting in for both and Q(z), we derive
where we may make a final simplification, defining W to represent the square-bracketed term. This leaves us with the following differential equation for :
 To solve the above, first-order, nonhomogeneous ODE is straightforward. In the case when two distinct values of and exist, one is justified in stipulating that , since otherwise it would be necessary that for some constant , for all values of p. It can be shown this is not possible. Thus,
for some γ. Applying the boundary condition when z = 0, (3), yields
 We have thus derived Laplace-transformed solutions for both the mother and daughter species in both the fracture and the matrix.
A2. The Case of Identical and
 Because the solutions derived above contain terms with in their denominator, we cannot directly apply them in the case when . Fortunately, these terms are indeterminate ( , and so L'Hospital's rule can be applied. Let us begin by considering the equation (A18), which was derived assuming and distinct. Defining and , then taking the limit as
yields the following solution in the matrix:
 An identical procedure can be adopted in the fracture, beginning with (A21)
We may make a simplification, defining X to represent the square-bracketed term. This leaves us with the following differential equation for :
It is easy to show that when for all p, for all p. To solve the above first-order, nonhomogeneous ODE is then straightforward. The solution is
for some γ. Applying the boundary condition when z = 0, (3) directly yields
 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 .
 I am extremely grateful to 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.