## 1. Introduction

[2] 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* [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.)

Single Fracture | Parallel Fractures | |
---|---|---|

Single Species | Tang et al. [1981] | Sudicky and Frind [1982] |

Decay Chain | Sudicky and Frind [1984] and Cormenzana [2000] | This paper |

[3] 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

[4] 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.

[5] 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):

and

where *c*_{1} is the concentration of species 1 (the mother in the decay chain) in the fracture, and *c*_{2} 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.

[6] In the matrix, advection is not a factor, and diffusion is the dominant mechanism, so the governing equations for the two species are simpler:

and

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:

[7] 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:

[8] In all the above equations, the physical meanings of the variables and parameters are outlined in Table 2.

Symbol | Meaning |
---|---|

c and _{n} | Mass concentration of species n [M L^{−3}] in, respectively, the fracture and the matrix |

v | 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 | |

D_{n} | Effective (tortuosity included) diffusion constant for the porous matrix |

R and _{n} | Retardation factor for species n in, respectively, the fracture and the matrix |

θ | Matrix porosity |

t | Time since first release of solute (all concentrations are zero at and before t = 0) |

x | Spatial coordinate: distance into matrix from nearest fracture wall |

z | Spatial coordinate: distance along fracture from location where the are specified |

L | Half width of matrix block between two adjacent fractures |

b | Half width of each fracture |