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.

This paper presents several exact and approximate analytical solutions of the equations describing convective-dispersive solute transport through large cylindrical macropores with simultaneous radial diffusion from the larger pores into the surrounding soil matrix. Adsorption effects were included through the introduction of linear isotherms for both the macropore region and the soil bulk matrix. In one formulation the macropores are surrounded by cylindrical soil mantles of finite thickness. Another formulation considers diffusion from a single cylindrical macropore into a radially infinite soil system. A relatively simple but very accurate approximate solution that ignores dispersion in the macropore region is also derived. The various analytical solutions in this paper can be used to calculate temporal and spatial concentration distributions in the macropore system. In addition, approximate solutions are presented for the radial concentration distribution within the adjacent soil matrix. By means of an example, it is demonstrated that at early times, little accuracy is lost when the radially finite soil mantle is replaced by an infinite system.

A complete description of the convective‐dispersive transport of a contaminant from an injection well requires consideration of a velocity dependent dispersion coefficient. An analytical solution of this equation for steady fluid flow conditions can be obtained. The solution is significantly different from existing approximate analytical solutions. A numerical solution to this problem converges to the new analytical solution upon refinement of the finite difference net.

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