An analytical model for solute transport in unsaturated flow through a single fracture and porous rock matrix

Abstract

An exact analytical solution is presented for solute transport in an unsaturated, vertical fracture and porous rock matrix. The problem includes advective transport in the fracture and rock matrix as well as advective and diffusive fracture‐matrix exchange. Linear sorption and radioactive decay are also treated. The solution is derived under the assumptions that longitudinal diffusion and dispersion are negligible and that the fracture and rock properties are homogeneous. The water flux and saturation within the matrix are assumed to be steady and spatially uniform. The fracture flux, saturation, and wetted fracture‐matrix interface area are steady but are allowed to be spatially variable. The problem is first solved in terms of solute concentrations that result from an instantaneous point source in the fracture. An integrated form of the solution is also derived for cumulative solute mass flux at a fixed downstream position. The closed‐form analytical solution is expressed in terms of algebraic functions, exponentials, and error functions. Analyses indicate limited sensitivity to fracture porosity and fracture retardation under typical conditions but strong sensitivity to matrix diffusion, matrix retardation, and matrix imbibition.