Exact upscaling for transport of size distributed colloids
The article investigates one‐dimensional (1D) 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 3x3 system significantly differs from the traditional 2x2 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. 1D flow problems for the averaged equations are essentially non‐linear. However, they allow for exact solutions. We derive novel exact solutions for three 1D 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 nano‐fluid transport.