A Binomial Modeling Approach for Upscaling Colloid Transport Under Unfavorable Attachment Conditions: Emergent Prediction of Nonmonotonic Retention Profiles

• Published in: Water resources research 2018-01, Vol.54 (1), p.46-60
• Main Authors: Hilpert, Markus, Johnson, William P.
• Format: Article
• Language: English
• Subjects: Attachment; Bacteria; binomial; colloid transport; Colloids; Combinatorial analysis; Design engineering; Diffusion; Dye dispersion; Elution; Exact solutions; Expulsion; Filtration; Grain; Interactions; Mathematical models; Maxima; Modelling; nonmonotonic retention profiles; Pore water; Porous media; Profiles; Reentrainment; Residence time; Residence time distribution; Retention; River banks; Riverbanks; Septic systems; Stagnation; theory; Transport
• ISSN: 0043-1397; 1944-7973
• DOI: 10.1002/2017WR021454

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 nonmonotonic 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 downgradient 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 one‐dimensional (1‐D) 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 nonmonotonic retention profiles occurs when the conditions underlying classical filtration theory are not fulfilled, i.e., when an 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 downgradient 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. Plain Language Summary Transport of colloids in water‐saturated porous media is significantly influenced by their interactions with grain surfaces. In the near‐surface fluid domain, colloids move much slower than in bulk pore water, and they may attach to grain surfaces via strong primary minimum interactions. In previous work, we developed a simple analytical model that predicts how bulk fluid colloids enter the near‐surface fluid domain, reentrain into the bulk fluid via diffusion or expulsion at rear flow stagnation zones, attach to grain surfaces, and move along grain‐to‐grain contacts to the near‐surface fluid domain of an adjacent grain. In th