## 1 Introduction

[2] Solute transport in groundwater flow through fractured rock is a subject that has been investigated for nuclear waste disposal and other environmental groundwater contamination problems [*National Research Council*, 1996]. Fractures are a common feature of consolidated rock systems and typically present much higher permeability than unfractured rock matrix, such that flow through fractures often dominates overall flow behavior. Matrix, on the other hand, typically dominates the overall pore volume of a fractured rock. These attributes often lead to much higher solute-transport velocities through fractures than in unfractured rock or unconsolidated soils [*Berkowitz*, 2002]. This behavior often makes fracture flow and transport critical characteristics to examine for any geologic site where solute transport is a concern. However, despite their importance, fractures remain a difficult feature to represent accurately in mathematical models for groundwater flow and transport [*Matthäi et al*., 2009; *Wu and Pruess*, 2000]. Flow through fractures displays highly heterogeneous and complex flow patterns controlled by small-scale fracture features. Furthermore, transport processes at small scales that govern fracture-matrix exchange can dramatically influence larger-scale transport behavior [*Grisak and Pickens*, 1980].

[3] Transport through saturated fractured rock was first investigated in an analytical model by *Neretnieks* [1980]. *Tang et al*. [1981] extended this work to include the effects of longitudinal dispersion in the fracture, and *Sudicky and Frind* [1982] and *Maloszewski and Zuber* [1985] further generalized the transport problem from a single fracture to a system of parallel fractures with interceding rock matrix. *Sharifi Haddad et al*. [2012] developed a semianalytical model of solute transport in a system of parallel fractures for a radially symmetric flow field associated with well injection. *Maloszewski and Zuber* [1990] considered the effects of linear kinetic interactions between solute and the rock for a single fracture and rock matrix. All of these models were restricted to advective transport in the fracture, transverse diffusion in the matrix, and diffusive fracture-matrix solute exchange. A review of these modeling approaches among others is provided by *Bodin et al*. [2003a, 2003b]. *Cihan and Tyner* [2011] developed exact analytical solutions for advective transport through cylindrical macropores and diffusive exchange with a soil matrix, for an instantaneous release of solute into a macropore, a constant concentration of solute at the top of a macropore, and a pulse release of solute into a macropore. They also compared their analytical solutions with numerical simulations that included longitudinal and transverse matrix advection and longitudinal dispersion in the macropore. *Roubinet et al*. [2012] added the effects of transverse dispersion in a fracture and longitudinal diffusion in the matrix to these mechanisms, and found that transverse dispersion in fractures had little effect on solute transport, but that longitudinal diffusion in the matrix becomes important at low Peclet numbers. The solution provided by *Roubinet et al*. [2012] also is capable of treating spatially varying and time-dependent source conditions.

[4] Other modeling efforts have focused on advective-dominated systems. *Birkhölzer et al*. [1993a, 1993b] present an analytical model for transport through a two-dimensional fracture network and permeable rock matrix under conditions in which fracture-matrix exchange is dominated by advective processes, such that diffusion could be neglected. A diffusion-advection number was also developed to help ascertain conditions for which diffusive fracture-matrix exchange is negligible compared with advective fracture-matrix exchange. The methodology developed by *Birkhölzer et al*. [1993a, 1993b] was used by *Rubin et al*. [1996] to investigate a tracer slug injection in a fractured rock with permeable rock matrix, and by *Rubin et al*. [1997] to investigate transport for cases with slow advective velocities in the fracture. *Odling and Roden* [1997] used a numerical model to study transport in fracture networks embedded in a permeable rock matrix, focusing on the role the permeable rock matrix plays when fracture networks have limited connectivity, including fractures that are disconnected from the network.

[5] *Cortis and Birkholzer* [2008] and *Geiger et al*. [2010] have utilized a continuous-time random-walk numerical method to investigate the effects of diffusion and advection in fractured, permeable rock. Transport calculations for a two-dimensional fracture network having a range of fracture and matrix permeabilities were used to identify the parameter ranges over which matrix advection has a significant role or may be neglected.

[6] *Houseworth* [2006] extended analytical modeling approaches for unsaturated flow, and included longitudinal and transverse flow and advective transport in the matrix, as well as advective and diffusive transport between the fracture and rock matrix. Longitudinal diffusion/dispersion was not included for the fracture or matrix, because this greatly simplifies the analytical model and is a reasonable approximation in some cases. For an unsaturated fractured rock, local differences in capillary conditions between the fracture and the rock matrix tend to dominate fracture-matrix flow exchange, such that flow exchange is more likely to result either in fracture discharge into the matrix or convergent matrix flow into the fracture.

[7] An analogous problem was analyzed by *Zhan et al*. [2009] for transport through a saturated system consisting of an aquifer confined above and below by aquitards of infinite extent. The analogy is that the aquifer corresponds to the fracture and the aquitards correspond to the rock matrix surrounding a fracture. Longitudinal advection and longitudinal and transverse dispersion were included as transport mechanisms in the aquifer. Transport in the aquitards was limited to the transverse direction but included both advection and diffusion. The general solution led to analytical results in the Laplace domain that required numerical inversion. Transverse dispersion in the aquifer and a more general treatment of transverse advection were new features introduced into the analyses. *Zhan et al*. [2009] compared their solution with previously derived solutions in which solute concentration was assumed to be transversely well mixed in the aquifer. They concluded that accounting for transverse solute gradients and transverse solute-transport processes in the aquifer had a significant impact on the results. In addition, the total solute mass entering the aquitard from the aquifer was found to be sensitive to the Peclet number for advective and diffusive transport in the aquitards.

[8] The analytical model presented here is for solute transport during steady state saturated flow occurring in a single fracture and a porous, permeable rock matrix. This model goes beyond existing analytical models by including the combination of lateral matrix diffusion and flow through the matrix in any direction relative to the orientation of the fracture axis, as well as flow through the fracture. Thus, the flow direction in the matrix and fracture may have components both parallel to and orthogonal to the fracture axis, leading to fracture-matrix exchange through cross-flow [*Birkhölzer and Rouve*, 1994]. Diffusive fracture-matrix exchange and general diffusive transport orthogonal to the fracture axis in the rock matrix is also included. The location of solute release as an instantaneous point source is generalized for an arbitrary point within the model domain. Closed-form analytical solutions for transport are obtained for these conditions neglecting longitudinal diffusion and dispersion. Solutions are also developed including simultaneous dispersive transport along the fracture and longitudinal diffusion in the matrix. These solutions take the form of superposition integrals of the closed-form results.

### 2. Flow Model

[9] Transport processes are considered for a two-dimensional, saturated flow system with a single fracture embedded in a permeable rock matrix. Specification of the flow field is a necessary first step to define the transport problem. The permeability of the rock matrix is taken to be homogeneous and isotropic and the fracture is assigned a different, higher, permeability than the matrix along the fracture axis. It also has permeability equal to that of the matrix in the direction transverse to the fracture axis. This anisotropy in the fracture simplifies the flow problem. In general, the transverse permeability of the fracture is generally not too significant because of the narrow transverse dimension of the fracture, unless this permeability is much lower than the matrix and represents a flow barrier. Flow is also restricted to steady state conditions, implying steady state pressure boundary conditions. For simplicity, the flow process will be discussed for a situation in which flow is driven exclusively by pressure differences.

[10] The flow system investigated is a generalization of the flow driven by a simple uniform pressure gradient aligned with the fracture axis. To understand the steady state, two-dimensional flow field, the flow system is diagrammed in Figure 1 with the fracture axis oriented in the same direction as the *z* axis. The corners of the rectangular flow domain are the origin, ; ; ; and , moving around the rectangle in counterclockwise order. The red arrows at the corners of the domain are axes displaying pressure. The pressures at the corners are set so that pressure drops across the domain in the longitudinal and transverse directions are uniform. From this configuration, it is clear that there will be a uniform transverse flow within the domain with the water flux rates . Longitudinal water flux rates within the matrix and the fracture are uniform within their respective domains; however, . The total pressure gradient is also shown and is uniform across the entire flow domain.

[11] Rotating the flow system in Figure 1 such that the rotated *z* coordinate, , aligns with the total pressure gradient results in the flow diagram given in Figure 2. This flow system is equivalent to the system shown in Figure 1. Because the coordinate is parallel to the total pressure gradient, there is no flow along the constant boundaries (upper and lower boundaries in Figure 2) and the constant boundaries (left and right boundaries in Figure 2) coincide with contours of constant pressure. The flow field (relative to the fracture) remains the same as in Figure 1. Through this rotation, the flow field in Figure 1 is shown to be equivalent to a flow field resulting from a uniform pressure gradient that is not (necessarily) aligned with the fracture axis.

[12] The transport problem to be solved is assumed to be sufficiently far from the pressure boundaries that any effects of these boundaries on transport are negligible. The configuration as shown in Figure 1 is used to compute transport processes, with the longitudinal direction defined to be the same direction as the axis of the fracture and the transverse direction is defined to be orthogonal to the fracture.