Exact Upscaling for Transport of Size‐Distributed Colloids

Abstract

The article investigates one-dimensional (1-D) suspension-colloidal transport of size-distributed particles with particle attachment. A population balance approach is presented for computing the particle transport and capture by porous media. The occupied area of each attached particle is particle size-dependent. The main model assumption is the retention rate dependency of the overall vacancy concentration for all particle sizes. For the first time, we derive an exact averaging (upscaling) procedure resulting in a closed system of large-scale equations for average concentrations of suspended and retained particles and of occupied rock surface area. The resulting large-scale 3 × 3 system significantly differs from the traditional 2 × 2 deep bed filtration model. However, the traditional model becomes a particular case that corresponds to an equal occupied area for all particles. The averaging yields the appearance of two empirical suspension and site occupation functions, which govern the kinetics of particle retention and site occupation, respectively. One-dimensional flow problems for the averaged equations are essentially nonlinear. However, they allow for exact solutions. We derive novel exact solutions for three 1-D problems: continuous injection of particulate colloidal suspension, injection of colloidal suspension bank with particle-free chase drive, and fines migration induced by high-rate flows. The analytical model for continuous injection closely matches three series of laboratory tests on nanofluid transport.

Quantitative Linking of Nanoscale Interactions to Continuum-Scale Nanoparticle and Microplastic Transport in Environmental Granular Media

“…A critical characteristic of colloid distribution from the source is that the transition from exponential to nonexponential profiles accompanies the transition from favorable to unfavorable conditions, as demonstrated well for monodisperse colloids. ,,, This critical characteristic highlights the impact of nanoscale colloid–surface interactions on the distribution of colloids down-gradient from the source for monodisperse colloids. While much good work demonstrates that colloid polydispersivity may predominantly drive hyperexponential distribution from the source for polydisperse colloids, − , the underlying role of favorable versus unfavorable nanoscale colloid–surface interactions remains unexplored for polydisperse colloids to the knowledge of the author. An assertion that hyperexponential profiles for stable monodisperse colloids arise from depth-dependent straining, made in an otherwise highly informative review, did not address the absence of straining and the profound impact of favorable versus unfavorable colloid–surface interactions in preceding investigations examining nonexponential distribution of monodisperse colloids from the source. ,, …”

Theory-Guided Targeted Delivery of Nanoparticles in Advective Environmental Porous Media

“…Transport was predicted using a continuum-scale Lagrangian transport model as described in Johnson et al to account for distributions of residence times prior to attachment under unfavorable conditions. Another upscaling option accounts for the impact of colloid size distribution on blocking of colloid attachment by previously attached colloids, which generates transient breakthrough in response to sustained colloid injection. The upscaling utilized here corresponds to a steady-state breakthrough obtained during step injection of the colloids …”

Averaged Boltzmann’s Kinetics for Colloidal Transport in Porous Media

“…Various microscale models have been presented to solve problems related to particle capture, including continuous random walk models (Shapiro, 2007;Yuan & Shapiro, 2010;Yuan et al, 2012), trajectory analysis models (Payatakes et al, 1974), as well as models with distributions of filtration coefficients to model pore-scale heterogeneity (Yuan & Shapiro, 2010). The population balance models to explicitly account for the changes in the particle and pore size distributions have been derived by Bedrikovetsky (2008), Bedrikovetsky et al (2017Bedrikovetsky et al ( , 2019, Shapiro and Yuan (2012), and Yortsos (1987a, 1987b). A similar philosophy is also present in modeling of reactive flows (Kechagia et al, 2002) and biological processes (Knutson et al, 2007) in porous media.…”

“…The population balance models to explicitly account for the changes in the particle and pore size distributions have been derived by Bedrikovetsky (2008), Bedrikovetsky et al. (2017, 2019), Shapiro and Yuan (2012), and Sharma and Yortsos (1987a, 1987b). A similar philosophy is also present in modeling of reactive flows (Kechagia et al., 2002) and biological processes (Knutson et al., 2007) in porous media.…”