Contaminant transport in fractured porous media: Analytical solution for a single fracture

Abstract

A general analytical solution is developed for the problem of contaminant transport along a discrete fracture in a porous rock matrix. The solution takes into account advective transport in the fracture, longitudinal mechanical dispersion in the fracture, molecular diffusion in the fracture fluid along the fracture axis, molecular diffusion from the fracture into the matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. Certain assumptions are made which allow the problem to be formulated as two coupled, one-dimensional partial differential equations: one for the fracture and one for the porous matrix in a direction perpendicular to the fracture. The solution takes the form of an integral which is evaluated by Gaussian quadrature for each point in space and time. The general solution is compared to a simpler solution which assumes negligible longitudinal dispersion in the fracture. The comparison shows that in the lower ranges of groundwater velocities this assumption may lead to considerable error. Another comparison between the general solution and a numerical solution shows excellent agreement under conditions of large diffusive loss. Since these are also the conditions under which the formulation of the general solution in two orthogonal directions is most subject to question, the results are strongly supportive of the validity of the formulation. tant attenuation mechanism exists in the form of molecular diffusion into the solid matrix [Golubev and Garibyants, 1971]. This mechanism acts to reduce contaminant concentrations in the fracture and thereby delays the migration of the contaminant.

An upstream finite element method for solution of transient transport equation in fractured porous media

“…In comparing the results with analytic solution of Tang et al [1981], we also assumed that concentration gradient is perpendicular to the fracture.…”

“…In situations where the transport is purely convective or convective et al [1981] and the numeric solution of shows that in the middle ranges of diffusion coefficients (1o-8 to 1o-9 cm2/s) considerable amount of discrepancy exists as shown by Tang et al [1981].…”

“…Recent advances in analytical solution of the simplified transport phenomena problems [Tang et al, 1981] bring valuable contributions to the understanding of solute transport through fractures and accompanied diffusion into the host rock. However, in the more realistic problems, one needs to resort to numerica 1 approaches.…”

“…The analytic solutions, although important to the understanding of the fundamental phenomena in solute transport and also necessary for verification of numerical methods, suffer from the usual limitations of initial and boundary conditions plus the restrictions imposed by multidimensionality of transport [Neretnieks, · 1980;Rasmuson and Neretnieks, 1981;Tang et al, 1981]. The numerical solution of on the other hand.models the fractures by the two-dimensional continuum elements with different material properties.…”

Monte Carlo simulation of radionuclide migration in fractured rock for the performance assessment of radioactive waste repositories

“…Within the simplifying setting of a single, longitudinal fracture, both analytical [12][13] and numerical [14] solutions have been successfully compared to the experimental results of [15]. The interested reader may also refer to [16] for a review of the most important existing approaches.…”