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

TLDR

The authors develop a general analytical solution for contaminant transport along a discrete fracture in a porous rock matrix and compare it to a simplified model. The solution incorporates fracture advection, longitudinal dispersion, fracture and matrix diffusion, adsorption on and within the matrix, and radioactive decay, formulated as two coupled one‑dimensional PDEs and evaluated via Gaussian quadrature. Comparisons reveal that neglecting longitudinal dispersion causes large errors at low groundwater velocities, while the general solution matches numerical results under high diffusive loss, confirming its validity.

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.