Upscaling of nanoparticle transport in porous media under unfavorable conditions: Pore scale to Darcy scale.

Transport and deposition of nanoparticles in porous media is a multi-scale problem governed by several pore-scale processes, and hence, it is critical to link the processes at pore scale to the Darcy-scale behavior. In this study, using pore network modeling, we develop correlation equations for deposition rate coefficients for nanoparticle transport under unfavorable conditions at the Darcy scale based on pore-scale mechanisms. The upscaling tool is a multi-directional pore-network model consisting of an interconnected network of pores with variable connectivities. Correlation equations describing the pore-averaged deposition rate coefficients under unfavorable conditions in a cylindrical pore, developed in our earlier studies, are employed for each pore element. Pore-network simulations are performed for a wide range of parameter values to obtain the breakthrough curves of nanoparticle concentration. The latter is fitted with macroscopic 1-D advection-dispersion equation with a two-site linear reversible deposition accounting for both equilibrium and kinetic sorption. This leads to the estimation of three Darcy-scale deposition coefficients: distribution coefficient, kinetic rate constant, and the fraction of equilibrium sites. The correlation equations for the Darcy-scale deposition coefficients, under unfavorable conditions, are provided as a function of measurable Darcy-scale parameters, including: porosity, mean pore throat radius, mean pore water velocity, nanoparticle radius, ionic strength, dielectric constant, viscosity, temperature, and surface potentials of the particle and grain surfaces. The correlation equations are found to be consistent with the available experimental results, and in qualitative agreement with Colloid Filtration Theory for all parameters, except for the mean pore water velocity and nanoparticle radius.