An analytical model for solute transport through a water-saturated single fracture and permeable rock matrix

An analytical model for solute transport through a water-saturated single fracture and permeable rock matrix J. E. Houseworth, 1 D. Asahina, 1 and J. T. Birkholzer 1 The problem of solute transport through a water-saturated single fracture in a permeable rock matrix is examined using an analytical modeling approach. A closed-form analytical solution is obtained that accounts for transverse and longitudinal advective transport in the fracture and matrix and transverse diffusion in the matrix. The solution also accounts for both diffusive and advective solute exchange between the fracture and matrix and a general solute source position in either the fracture or matrix. The novel features are the incorporation of advective transport in the matrix and a general source position into a closed-form solution for the solute-transport problem. Examples of the solution behavior are presented, which demonstrate the effects of matrix advection in combination with advection along the fracture, transverse diffusion in the matrix for solute release in the fracture and matrix. A semianalytical solution in the form of a superposition integral is also derived that includes these transport features, plus independent levels of longitudinal diffusion and dispersion in the matrix and fracture, respectively. Examples are presented that include advective transport in the fracture and matrix, longitudinal and transverse diffusion in the matrix, longitudinal dispersion in the fracture, as well as solute release from the fracture and matrix. An approximate criterion is proposed to evaluate the significance of longitudinal diffusion and dispersion relative to longitudinal spreading caused by fracture- matrix interaction. 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 Pick- ens, 1980]. 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 mod- els 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 cylin- drical macropores and diffusive exchange with a soil ma- trix, 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. 1 Introduction Solute transport in groundwater flow through frac- tured rock is a subject that has been investigated for nuclear waste disposal and other environmental groundwater con- tamination problems [National Research Council, 1996]. Fractures are a common feature of consolidated rock sys- tems 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 unfrac- tured rock or unconsolidated soils [Berkowitz, 2002]. This behavior often makes fracture flow and transport critical characteristics to examine for any geologic site where sol- ute transport is a concern. However, despite their impor- tance, fractures remain a difficult feature to represent accurately in mathematical models for groundwater flow and transport [Matth€ai et al., 2009 ; Wu and Pruess, 2000]. Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA. Corresponding author : J. E. Houseworth, Earth Sciences Division, Law- rence Berkeley National Laboratory, Berkeley, CA 94720, USA. (jehouseworth@lbl.gov)