Contaminant transport in fractured porous media: Analytical solutions for a system of parallel fractures

TLDR

The assumption of negligible longitudinal dispersion is usually reasonable when advective flux in a fracture is large. The study develops an exact analytical solution for transient contaminant transport in discrete parallel fractures in a porous matrix and compares steady‑state solutions with and without dispersion to establish a criterion for assessing longitudinal dispersion significance. The solution incorporates advective transport, longitudinal and transverse dispersion, molecular diffusion between fracture and matrix, adsorption onto and within the matrix, and radioactive decay, and is expressed as a double integral evaluated by Gauss‑Legendre quadrature, with a simplified form for negligible longitudinal dispersion. Examples show that penetration distances along fractures can be substantially larger through multiple, closely spaced fractures than through a single fracture because the finite matrix has limited solute storage capacity.

Abstract

An exact analytical solution is developed for the problem of transient contaminant transport in discrete parallel fractures situated in a porous rock matrix. The solution takes into account advective transport in the fractures, molecular diffusion and mechanical dispersion along the fracture axes, molecular diffusion from the fracture to the porous matrix, adsorption onto the face of the matrix, adsorption within the matrix, and radioactive decay. The general transient solution is in the form of a double integral that is evaluated using Gauss‐Legendre quadrature. A transient solution is also presented for the simpler problem that assumes negligible longitudinal dispersion along the fracture. This assumption is usually reasonable when the advective flux in a fracture is large. A comparison between two steady state solutions, one with dispersion and one without, permits a criterion to be developed that is useful for assessing the significance of longitudinal dispersion in terms of the overall system response. Examples of the solutions demonstrate that penetration distances along fractures can be substantially larger through multiple, closely spaced fractures than through a single fracture because of the limited capability of the finite matrix to store solute.