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

Abstract

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.

Figure 1:

(a) Colloid-surface interaction forces in Newtons. Positive = repulsive (bulk unfavorable), negative = attractive (favorable). (b) Pore scale (Happel cell) particle trajectories emergent from force-torque balance and contact mechanics, showing bulk fluid (yellow) and NSFD (green) trajectories under bulk unfavorable conditions (top) and favorable conditions (bottom). Flow direction is downward. Surface charge heterogeneity (magenta) is attractive attractive under both favorable and unfavorable conditions, whereas bulk surface is attractive under favorable and repulsive under unfavorable conditions. (c) Representative binomial upscaling realizations for 5 colloids (horizontal axis) across 50 pore/grain collectors showing residence in pore bulk fluid (blue), NSFD (magenta), and grain surface (black) based on pore-scale residence times and probabilities. Left side shows upscaled favorable conditions, right side shows upscaled bulk unfavorable conditions. (d) Experimentally observed column-scale profiles of retained colloids for bulk unfavorable conditions (top) and favorable conditions (bottom) as reported in Li and Johnson [2005].

Figure 2:

Predictions from binomial model for a pulse injection of colloids into a 1D network of unit cells at time t = 0. Shown are concentration profiles of colloids in bulk pore water, c^(p), in the NSFD, c⁽ⁿ⁾, and that attached, c^(a). (a) The model parameterization described in the main body of the text results in a non-monotonic retention profile (gray line in the c^(a) plot). (b) Log-linear profile.

Figure 3:

Illustration of the predictions of our simple analytical model for colloid deposition in case of zero reentrainment, in which case attachment only occurs in the 2^nd unit cell or thereafter. (a) Probability of finding a colloid, which was injected into the network at time t = 0, in the NSFD at the time indicated in the legend. (b) Probability that a colloid attaches to a grain surface at the time indicated in the legend. (c) Cumulative probability that a colloid has attached to a grain surface prior or at the time indicated in the legend. In the legends, t_nn and t_na denote the times for colloid transfer from the n state to the n and a states, respectively.

Figure 4:

Comparison between predictions of the binomial model and our new analytical model for the concentration profile of attached colloids and the position of the maximum concentration in case of zero reentrainment. (a) No direct colloid transfer from bulk pore water to primary energy minimum, p_pa = 0. (b) p_pa > 0. Dashed vertical lines indicate the position of the maximum concentration of attached colloids.

Figure 5:

Comparison between the transient retention profiles predicted by new analytical model [Equation (32)] (dashed lines) and the binomial model (solid lines).

Figure 6:

Position of the maximum of the colloid retention profile, i_max, in units of the unit-cell diameter. The reentrainment probability is assumed to be zero, η_r = 0. Other parameters: (a) η₀ = 0.01 and (b) η₀ = 0.001.

Figure 7:

Position of the maximum, i_max, in units of the unit-cell diameter. We varied the reentrainment probability: (a) η_r = 0.1. (b) η_r = 0.5. Other parameters: η₀ = 0.01.

Figure 8:

(a) Measured elution duration series for DA001 bacteria. Binomial modeling for (b) η_r = 0.0, (c) η_r = 0.02 and (d) η_r = 0.05. The experimental data were normalized to match the number of colloids injected (i.e., “1”) into the 1D binomial network model.