Analytical solutions for solute transfer characteristics at continuous fracture junctions

Abstract

Abstract. Simple analytical solutions for the mixing characteristics and transfer probabilities of solute at the continuous fracture junction are proposed in this study. The proposed analytical solution considers the effects of diffusion, and it is simple enough to be applied to the junctions in discrete fracture network. The analytical solutions are compared with the complete-mixing and the streamline-routing models. As the Peclet number increases, the analytical solutions indicate the transition from complete solute mixing to streamline routing at a fracture junction. It is clear from the results that the transfer characteristics at the junctions are a function not only of one dimensionless parameter but of complex combinations of discharge pattern, geometry, and concentration distribution in the related fractures. Comparisons with the existing numerical solutions and possible underestimation of the solute mixing by the analytical solutions are discussed. IntroductionDiscrete and continuum approaches arc two main branches in mod½ling the fluid flow and solute transport in fractured rocks. The conventional continuum mod½l is simple, but it has its limits because of the inability to correctly account for the effect of natural discontinuity geometry and to determine the irregular range and degree of contamination. Thus the simulation using the discrete fracture system or the hybrid discrete continuum system has become increasingly popular they suggested a random walk model at the junction to account for the mixing accelerated by diffusion. In this model the mixing of solute by diffusion across the streamline is explained by the normal probability distribution, and they concluded that the mixing characteristics affect the dispersion to a much further degree than was previously considered. Berkowitz et al.[1994] developed a transport model by using Philip's [1988] analytic solution for fluid flow and particle tracking and a random walk method to evaluate quantitatively the mixing characteristics at discontinuity junction in terms of the Peclet number (Pe), that is, the ratio of the diffusion to the convection. By using this model, they intended to determine the transport conditions under which the complete-mixing and the streamline-routing theories can be applied. Recently, Stockman et al.[1997] suggested a new modeling method to appropriately incorporate the boundary effects by using cellular automata to recognize the importance of the boundary conditions in low-Peclet number transport conditions. As a result, these numerical studies concluded that complete-mixing and streamline-routing theories are valid only in the limited ranges of the Peclet number. Though these experimental and theoretical studies have been performed to date, mixing at the discontinuity junction due to the diffusion is very difficult to incorporate in the analysis of solute transport through the discrete fracture network. The solute transfer characteristics at fracture junctions might be computed numerically, but it requires too much numeri...

Effects of junction transfer characteristics on transport in fracture networks

“…Moreover, the generality of these results, accounting for solute transport patterns in three-dimensional fracture systems, requires investigation. Figure A2) (see Park and Lee [1999] for more detailed discussion). However, these two probabilities are not necessarily identical because of the differences in the discharges in the fracture branches and the apparent mixing behavior (see discussion in sections 2.2.3 and 4.3).…”

“…Resulting mass distributions obtained by assuming these two extreme models were compared to those obtained using analytical solutions [Park and Lee, 1999] (Figure 13). The differences among the concentration distributions when the hydraulic gradients are given by 10-•-10 -7 are relatively minor compared to the differences between the results obtained from the analytical solution and the two extreme models.…”

Implementation of a Time-Domain Random-Walk Method into a Discrete Element Method to Simulate Nuclide Transport in Fractured Rock Masses

“…There are three models: the perfect-mixing model [44], the stream-tube model [45], and the diffusional-mixing model [46]. The focus of this paper is not the mass partition law of solute at fracture intersection.…”

Tracer mixing at fracture intersections

“…Philip (1988) characterizes the mixing process at a fracture intersection in terms of a local Peclet number, representing the interplay between advective and diffusive tracer transfer. Park and Lee (1999) provide simple analytical solutions for the mixing characteristics at the continuous fracture intersections. As the Peclet number increases, the analytical solutions also indicate the transition from complete mixing to streamline routing at a fracture intersection (Park and Lee 1999).…”

“…Park and Lee (1999) provide simple analytical solutions for the mixing characteristics at the continuous fracture intersections. As the Peclet number increases, the analytical solutions also indicate the transition from complete mixing to streamline routing at a fracture intersection (Park and Lee 1999).…”