Applying the boundary layer concept to model transport of dissolved chemicals in preferential flow paths

Abstract

The exchange of dissolved chemicals between preferential flow paths and the surrounding matrix has a major effect on the concentration at the flow path outlet and, on a larger scale, on contaminant transport in structured soils and fractured rock systems. A model for dissolved chemical transport in a discrete fracture accompanied by simultaneous exchange with the surrounding porous matrix is presented and solved analytically. The fracture transport is modeled by a kinematic wave equation which ignores longitudinal dispersion, and the lateral solute transport in the matrix, where fluid is stagnant, is modeled by a diffusion equation. The exchange of dissolved chemicals between the fracture and matrix is assumed to take place through a boundary layer along the fracture‐matrix interface. The analytical solution to the equations is a convergent series, even for large values of the characteristic dimensionless parameter ϵ = kb2/(Dθ2), (where k is mass transfer coefficient, b is half the fracture width, and D and θ are the matrix diffusion coefficient and porosity, respectively) the column length, and long time periods. The number of terms in the series depends on the parameter values and desired accuracy. The model was successfully verified by fitting its output to breakthrough curves measured for highly fractured clay‐till soils. The role of the boundary layer (also known as film resistance) was studied by comparing breakthrough curves predicted by the current model (boundary layer (BL) model) and by a model that assumes a uniform concentration across the fracture‐matrix interface (local equilibrium (LE) model). Analyses of the model equations and simulations revealed that solute displacement from a fracture surrounded by a matrix with immobile solute can be divided into two stages with respect to the boundary layer's effect on breakthrough curve shape. Soon after the initiation of displacement, the boundary layer controls the solute flux to the matrix and the rate of concentration increase at the fracture outlet. The BL model predicts lower fluxes into the matrix than the LE model, and use of the latter for this stage underestimates the displaced solute mass at the fracture outlet. The duration of this stage and the deviation between breakthrough curves predicted by the current and LE models depend on ϵ, flow velocity, and column length. During the following stage, the boundary layer's effect on the rate‐limited exchange steadily diminishes, and the solute flux is controlled by matrix diffusion. At this latter stage, both models predict similar breakthrough curves at the fracture outlet.