Modeling of colloid and colloid-facilitated contaminant transport in a two-dimensional fracture with spatially variable aperture

Mathematical models are developed for two-dimensional transient transport of colloids, and cotransport of contaminant/colloids in a fracture-rock matrix system with spatially variable fracture aperture. The aperture in the fracture plane is considered as a lognormally distributed random variable with spatial fluctuations described by an exponential autocovariance function. Colloids are envisioned to irreversibly deposit onto fracture surfaces without penetrating the rock matrix; whereas, the contaminant is assumed to decay, sorb onto fracture surfaces and onto colloidal particles, as well as to diffuse into the rock matrix. The governing stochastic transport equations are solved numerically for each realization of the aperture fluctuations by a fully implicit finite difference scheme. Emphasis is given on the effects of variable aperture on colloid and colloid-facilitated contaminant transport. Simulated breakthrough curves of ensemble averages of several realizations show enhanced colloid transport and more pronounced fingering when colloids are subject to size exclusion from regions of small aperture size. Moreover, it is shown that an increase in the fracture aperture fluctuations leads to faster transport and increases dispersion. For the case of contaminant/colloids cotransport it is shown, for the conditions considered in this work, that colloids enhance contaminant mobility and increase contaminant dispersion.Nomenclature b fracture aperture, L - c contaminant concentration in the fracture, M/L³ - c _m contaminant concentration in the rock matrix, M/L³ - c _o source contaminant concentration, M/L³ - c ^* contaminant concentration adsorbed onto fracture surfaces, M/L² - c _m ^* contaminant concentration adsorbed inside the rock matrix, M/M - d _p colloidal particle diameter, L - D hydrodynamic dispersion coefficient dyadic, L²/t - D Brownian diffusion coefficient for colloids and molecular diffusion coefficient for contaminants, L²/t - D _m effective diffusion coefficient in the rock matrix, L²/t - h total head potential in the fracture, L - K _f partition coefficient for contaminant sorption onto fracture surfaces, L - K _m contaminant partition coefficient in the rock matrix, L³/M - K _n partition coefficient for contaminant sorption onto suspended colloids, L - K _n* partition coefficient for contaminant sorption onto deposited colloids, L³/M - _x fracture length in thex-direction, L - _y fracture length in they-direction, L - n colloid concentration in the liquid phase, M/L³ - n _o source colloid concentration, M/L³ - n ^* colloid concentration adsorbed onto fracture surfaces, M/L² - n _max ^* maximum deposited colloid concentration on fracture surfaces, M/L² - N ^* number of deposited colloidal particles per unit surface area of the fracture, 1/L² - N _max ^* maximum number of deposited colloidal particles per unit surface area of the fracture, 1/L² - q ^* diffusive mass flux into the rock matrix, M/L²t - R retardation factor in the fracture - R _m retardation factor in the rock matrix - s contaminant concentration adsorbed on colloids in the liquid phase, M/M - s _o source solid-phase contaminant concentration onto suspended colloids, M/M - s ^* contaminant concentration adsorbed on deposited colloids, M/M - t time, t - U interstitial velocity vector, L/t - x coordinate along the fracture length, L - y coordinate along the fracture width, L - z coordinate perpendicular to the fracture plane, L - area blocked by a deposited colloidal particle, L² - _L longitudinal dispersivity, L - _T transversal dispersivity, L - fluid specific weight, M/L²t² - fraction of the fracture surface physically covered by colloids - gz dummy integration variable - porosity of the rock matrix - colloid deposition coefficient, L - first-order decay coefficient, 1/t - fluid dynamic viscosity, M/Lt - defined in (18) - _b bulk density of the rock matrix, M/L³ - _p colloidal particle density, M/L³ - standard deviation of the lognormally distributed fluctuations of the fracture aperture