A binomial modeling approach for upscaling colloid transport under unfavorable attachment conditions: Emergent prediction of non-monotonic retention profiles.

We used a recently developed simple mathematical network model to upscale pore-scale colloid transport information determined under unfavorable attachment conditions. Classical log-linear and non-monotonic retention profiles, both well-reported under favorable and unfavorable attachment conditions, respectively, emerged from our upscaling. The primary attribute of the network is colloid transfer between bulk pore fluid, the near surface fluid domain (NSFD), and attachment (treated as irreversible). The network model accounts for colloid transfer to the NSFD of down-gradient grains and for reentrainment to bulk pore fluid via diffusion or via expulsion at rear flow stagnation zones (RFSZs). The model describes colloid transport by a sequence of random trials in a 1D network of Happel cells, which contain a grain and a pore. Using combinatorial analysis that capitalizes on the binomial coefficient, we derived from the pore-scale information the theoretical residence time distribution of colloids in the network. The transition from log-linear to non-monotonic retention profiles occurs when the conditions underlying classical filtration theory are not fulfilled, i.e., when a NSFD colloid population is maintained. Then, nonmonotonic retention profiles result, potentially both for attached and NSFD colloids. The concentration maxima shift downgradient depending on specific parameter choice. The concentration maxima were also shown to shift downgradient temporally (with continued elution) under conditions where attachment is negligible, explaining experimentally-observed down-gradient transport of retained concentration maxima of adhesion-deficient bacteria. For the case of zero reentrainment, we develop closed form, analytical expressions for the shape and the maximum of the colloid retention profile.