Modeling transport in a single crack by the dual-porosity concept with a boundary layer at the interface

Abstract

A model for contaminant transport along a discrete fracture in a porous rock matrix is developed and solved analytically. The focus here is on the dynamics of solute exchange between the matrix and a single crack and its effect on solute transport within the crack and the chemical distribution with time at any depth. A laminar boundary layer of near stagnant fluid exists at the interface between the fracture and the matrix through which a rate-limited mass transfer takes place. The flow and the dissolved chemical concentration normal to the flow direction are essentially uniform in the crack except for the boundary layer. Chemical concentration in the immobile porosity varies with time and space in the lateral direction. The driving force for the dissolved chemical exchange through this layer is the difference between the fluid concentration in the crack and the matrix concentration at the interface, both varying with time and space. The dissolved chemical concentration in the crack is modeled by the kinematic wave equation and the diffusion equation is used to model the dissolved chemical transport in the matrix. Exact analytical solutions for the matrix and crack equations are obtained in the Laplace domain and an approximate solution in the time domain for the case where ζε/ s is sufficiently small. ζ is the dimensionless distance along the crack, ε is a dimensionless parameter obtained when dimensionless variables were introduced into the differential equations and boundary conditions, and s is the Laplace transformation variable. The deviations between the approximate and exact solutions for different values of ε and other variables, calculated in the Laplace domain, enable us to evaluate the cases where the approximate solution can be satisfactorily used. The solution obtained by the current model was verified by its comparison with BTCs measured by Neretnieks et al. (Neretnieks, I., Eriksen, T., Tahtinen, P., 1982. Tracer movement in a single fissure in granitic rock: some experimental results and their interpretation. Water Resour. Res. 18, 4, pp. 849–858) for a granite core with a natural fissure parallel to its axis. A very good agreement was obtained between the measured and predicted BTCs. The physical meaning of the fitted parameters has been discussed. The relative role of the two rate-limited processes, namely film transfer and diffusion in the stagnant matrix solute, on the overall chemical exchange and BTC shape was analyzed by the non dimensional version of the mass balance equations. The displacement duration has been divided into two stages. Soon after its initiation, the chemical transfer through the stagnant film controls the chemical exchange between the preferential path and matrix. The duration of this stage depends on different properties of the system. Subsequently, the matrix–diffusion controls the chemical exchange between the two domains and the model can then be simplified by replacing the rate-limited transfer by a local equilibrium. For cases for which the current study is dealing with, the preferential flow is very fast and the rate-limited transfer through the stagnant film dominates the BTC shape.