Upscaling of transport processes in porous media with biofilms in non-equilibrium conditions

Abstract

In this work, we derive two different Darcy-scale models for the transport of biodegradable solutes in porous media containing a microbial biomass that developed under the form of a biofilm. Biofilms are composed of bacterial populations and extra cellular polymeric substances, and grow attached to a solid-fluid interface, e.g. the pore walls of a water-saturated porous medium. We begin with the pore-scale description of mass transport, mass transfer between phases (fluid phase-generally water-and biofilm phase) and biologically-mediated reactions. Then, two limit cases of non-equilibrium transport are identified and characterized. Finally, these processes are upscaled using the method of volume averaging to obtain two different macroscale mass balance models. The models are derived for specific cases of non-equilibrium reactive transport (i.e., spatial concentration gradients may exist in one or both phases), for which mechanisms of mass transfer or reaction kinetics limit the rate of biodegradation. In both cases, a *Manuscript Click here to download Manuscript: Biofilms in Porous Media RRLC MTLC.doc Click here to view linked References 2 one-equation model can be developed even if non-equilibrium conditions exist. The validity domains of the two considered transport models (the Reaction Rate Limited Consumption model-RRLC model-and the Mass Transfer Limited Consumption model-MTLC model) are established in terms of reactive and hydrodynamic conditions of transport (Damköhler number and Péclet number). The RRLC model is found to be consistent with direct numerical simulation (DNS) results at high Péclet and Damköhler numbers, while the applicability of the MTLC model is limited to high Damköhler numbers but low Péclet numbers.

Solute Mass Exchange Between Water Phase and Biofilm for a Single Pore

“…In case 1, Péclet and Damköh-ler numbers are 100 and 10, respectively. This represents a general non-equilibrium condition for solute mass exchange between water phase and biofilm (Orgogozo et al 2010). The comparison of concentration distributions along the tube is shown in Fig.…”

“…Biofilm is treated as a continuum phase. Moreover, it is assumed that water flow through biofilm is negligible (Orgogozo et al 2010). Therefore, with respect to solute transport, both advective and diffusive processes happen in water phase, while in biofilm only diffusion is assumed to occur.…”

“…It is found that ξ * increases dramatically under severe mass-transfer limitations, such as at large Damköhler numbers and small diffusivity ratios. Over the past few years, several researchers (Golfier et al 2009;Hesse et al 2009;Orgogozo et al 2010;Wood et al 2007) have attempted to solve a closure problem of pore-scale concentration fluctuation filed, in order to obtain the effective reaction rate, dispersion tensor, etc. But, each closure problem needs to be numerically solved for pre-specified pore structure and flow condition.…”

“…The employed dimensionless geometric and physical parameters are listed in Table 2. We varied Péclet number, Damköhler number, and biofilm volume fraction to represent three typical transport regimes of reaction-rate limited consumption, mass-transfer limited consumption, and an intermediate regime (Orgogozo et al 2010); we referred to them as case 3, case 2, and case 1, respectively. Because Péclet number, Damköhler number, and biofilm volume fraction are fixed in each case, we can calculate the corresponding specific area and meanwhile obtain the values of dimensionless coefficient ξ * from Eq.…”

“…Under nonequilibrium conditions, where there may be mass-transfer limitation, a so-called biofilm model has been widely used (see, e.g., MacDonald et al 1999;Rittmann 1993). Orgogozo et al (2010) derived two one-equation models for the cases of mass-transfer limited consumption and reaction-rate limited consumption. Alternatively, under non-equilibrium conditions, a general two-equation model (one equation for solute transport in water phase, the other for solute transport in biofilm) may be used.…”