Analytical solutions for one-dimensional colloid transport in saturated fractures

Abstract

Closed-form analytical solutions for colloid transport in single rock fractures with and without colloid penetration into the rock matrix are derived for constant concentration as well as constant flux boundary conditions. A single fracture is idealized as two semi-infinite parallel plates. It is assumed that colloidal particles undergo irreversible deposition onto fracture surfaces and may penetrate into the rock matrix, and deposit irreversibly onto rock matrix solid surfaces. The solutions are obtained by taking Laplace transforms to the governing transport equations and boundary conditions with respect to time and space. For the case of no colloid penetration into the rock matrix, the solutions are expressed in terms of exponentials and complimentary error functions; whereas, for the case of colloid penetration into the rock matrix, the solutions are expressed in terms of convolution integrals and modified Bessel functions. The impact of the model parameters on colloid transport is examined. The results from several simulations indicate that liquid-phase as well as deposited colloid concentrations in the fracture are sensitive to the fracture surface deposition coefficient, the fracture aperture, and the Brownian diffusion coefficient for colloidal particles penetrating the rock matrix. Furthermore, it is shown that the differences between the two boundary conditions investigated are minimized at dominant advective transport conditions. The constant concentration condition overestimates liquid-phase colloid concentrations, whereas the constant flux condition leads to conservation of mass.