Colloid filtration theory and the Happel sphere-in-cell model revisited with direct numerical simulation of colloids

The transport of colloids and bacterial cells through saturated porous media is a complex phenomenon involving many interrelated processes that are often treated via application of classical colloid filtration theory (CFT). This paper presents a numerical investigation of CFT from the Lagrangian perspective, to evaluate the role of some of the classical assumptions underlying the theory and to demonstrate a means to include processes relevant to bacterial transport that were inadequately characterized or neglected in the original formulation, including Brownian diffusion and potentially hysteretic potential functions. The methodology is based on conducting a Lagrangian trajectory analysis within Happel's sphere-in-cell porous media model to obtain the collection efficiency (eta), the frequency at which colloids or bacteria make contact with the solid phase of the porous medium. The Lagrangian framework of our model lends itself to mechanistic modeling of the biological processes that may be important in subsurface bacterial transport. The numerical study presented here focuses on the size range of bacterial colloids and smaller (down to 10 nm). Results of our model runs are in good agreement with the deterministic trajectory analysis of Rajagopalan and Tien (when diffusion is neglected) and in excellent agreement with the analytical solution to the Smoluchowski-Levich approximation of the convective-diffusion equation (when external forces and interception are neglected). Simple addition of our result for the deterministic eta to our result for the Smoluchowski-Levich eta matches the overall Rajagopalan and Tien eta to within 5% error or less for all cases studied. When we simulate diffusion and the deterministic forces together, our results diverge from the Rajagopalan and Tien eta as the particle size decreases, with discrepancies as large as 73%. These results suggest that accurate prediction of eta values for bacteria-sized (and all submicrometer) colloids requires simultaneous consideration of the primary transport mechanisms.