Colloid Filtration Theory and the Happel Sphere-in-Cell Model Revisited with Direct Numerical Simulation of Colloids Kirk E. Nelson* and Timothy R. Ginn Department of Civil and Environmental Engineering, University of California at Davis, One Shields Avenue, Davis, California 95616 Received June 28, 2004. In Final Form: December 22, 2004 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 (η), 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 η to our result for the Smoluchowski-Levich η matches the overall Rajagopalan and Tien η 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 η as the particle size decreases, with discrepancies as large as 73%. These results suggest that accurate prediction of η values for bacteria-sized (and all submicrometer) colloids requires simultaneous consideration of the primary transport mechanisms. I. Introduction Understanding processes of colloidal (including biocol- loidal) transport in saturated porous media is an important goal relevant to the maintenance and restoration of clean drinking water supplies. Colloidal particles are intricately involved in the problems associated with speciation, fate, and transport of chemical contaminants in aquatic systems. 1 In natural soils and subsurface materials, as well as in industrial and wastewater streams, and in drinking water supplies, contaminants and other impor- tant species dynamically transfer between colloidal and other phases by weathering, erosion, coagulation, frag- mentation, adsorption, and desorption. Biotic colloids such as bacteria may operate as either harmful or beneficial agents: the former when they are pathogens originating from sources such as septic tanks or agricultural waste, the latter when they are used for in situ bioremediation of toxic chemicals in groundwater. In any case, it would be very useful to have capable tools for predicting the behavior of the colloidal particle and its filtration during transport through porous media. A. Colloid Filtration Theory (CFT). Understanding and predicting the transport of colloids in the natural subsurface is a challenging scientific problem, because the controlling processes include physical and chemical factors, acting at solid-aqueous interfaces. The complexity increases for the case of biotic colloids such as bacteria. One of the theoretical frameworks commonly used to predict colloid transport through porous media is the CFT. CFT has enjoyed widespread use since its inception in the 1970s for the modeling of colloidal particle deposition in general and more recently (since the late 1980s) for the modeling of subsurface microbial fate and transport. CFT describes the deposition of suspended particles onto porous media grains as a two-step process. The first step, transport to the grain surface, is quantified by the collection efficiency (η), the frequency at which particles in the aqueous phase come into contact with the solid phase (i.e., the “collector”). The second step, attachment to the grain surface, is quantified by the sticking efficiency (R), the frequency at which the particles coming into contact with the collector actually attach to it. The conventional approach for the application of CFT to subsurface bacterial transport has been to calculate η a priori from the properties of the bacteria, the flowing water, and the porous media grains. Then R is calibrated with the use of data from column transport experiments. Researchers have also calibrated R on the basis of field data. 2 There currently exists no complete theory for either the transport step or the attachment step when the particles in question are living cells. However, the transport step of CFT is reasonably well understood for nonliving colloidal particles. The physicochemical theory underlying the calculation of η applies to bacterial cells insofar as they share the general properties of all colloidal particles. However, the transport of these living colloids * To whom correspondence should be addressed at U.S. Bureau of Reclamation, MP 740, 2800 Cottage Way, Sacramento, CA 95825. Telephone: (916) 978-5066. Fax: (916) 978-5094. E-mail: knelson@ mp.usbr.gov. (1) Gustafsson, O.; Gschwend, P. M. Limnol. Oceanogr. 1997, 42, 519-528. (2) Harvey, R. W.; Garabedian, S. P. Environ. Sci. Technol. 1991, 25, 178-185. 2173 Langmuir 2005, 21, 2173-2184 10.1021/la048404i CCC: $30.25 © 2005 American Chemical Society Published on Web 02/10/2005