1. Upscaling of Transport Processes in Porous Media with Biofilms in Non-Equilibrium Conditions L. Orgogozo 1, F. Golfier 1, M.A. Buès 1, B. Wood 2, M. Quintard 3 1 Nancy Université - Laboratoire Environnement, Géomécanique et Ouvrages, École Nationale Supérieure de Géologie, Rue du Doyen Marcel Roubault, BP40F-54501 Vandoeuvre-lès-Nancy, France 2 Environmental Engineering, Oregon State University, Corvallis, OR 97331, USA 3 Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France Contact : Laurent.Orgogozo@ensg.inpl-nancy.fr

2. 2 OBJECTIF GENERALE Introduction – Two non equilibrium models – Results – Conclusions and perspectives 2/15 INTRODUCTION Biofilm growth Substrate consumption Substrate availability Biofilm : Biomass bounded to a solid surface (e.g., pore walls in a porous medium) composed of bacterial populations living in extracellular polymeric substances (EPS) Coupling : active transport of the substrate in the porous medium where grows the biofilm

3. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 3/15 SCALES AND PROCESSES Biofilm phase (  ) Diffusion, reaction + Growth Solid phase (  ) Passive phase Coupled transport of substrate A and electron acceptor B (non linear double Monod kinetics reaction) Fluid phase (  ) Convection, diffusion Biofilm growth => modification of hydrodynamic properties (bioclogging)

4. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 4/15 AIM OF THIS WORK -> Simplifying the problem: time uncoupling of growth, transport and flow phenomena -Time scale of biofilm growth is very large compared to time scale of transport -Time scale of relaxation of flow is very small compared to time scale of transport (+ Reynolds number supposed to be small) Upscaling of transport processes from pore scale to Darcy scale Upscaling already done in equilibrium conditions (Wood et al. 2008, Golfier et al. 2009) ->Focus on non equilibrium conditions: two main problematics - Coupling between transport phenomena in each phases - Coupling between transports of solute A and B with non linear kinetics

5. Volume averaging operator and associated theorems (e.g. Whitaker 1999) Microscale equations Introduction – Two non equilibrium models – Results – Conclusions and perspectives 5/15 TRANSPORT MODELLING BY VOLUME AVERAGING Pore and biofilm scale fluid  biofilm ω solid  ll ll ll Representative Elementary Volume Scale L Assumption of separation of scales + macroscopic boundary conditions + closure/microscale problems Macroscale equations R + microscopic boundary conditions

6. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15 TWO NON EQUILIBRIUM MODELS OF TRANSPORT General case : transport in two phases => two-equation model of transport General case 0 0 Fluid Biofilm concentration Interface x Distance to the interface General case : transport in two phases => two-equation model of transport Particular cases : assumption about the relation of the concentration fields of each phase. => One-equation models of transport

7. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15 REACTION RATE LIMITED MODEL General case : transport in two phases => two-equation model of transport Particular cases : assumption about the relation of the concentration fields of each phase. => One-equation models of transport General case Reaction Rate Limited model (RRL model)

8. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15 MASS TRANSFER LIMITED MODEL General case : transport in two phases => two-equation model of transport Particular cases : assumption about the relation of the concentration fields of each phase. => One-equation models of transport General case Mass Transfer Limited model (MTL model)

9. Macroscopic equation of transport Which is defined only in the fluid phase. is the effective dispersion tensor at the macroscale and is the effectiveness factor of the reaction for solute A (stocheometricaly proportionnal for solute B), defined as : Relations between the microscale and the macroscale Introduction – Two non equilibrium models – Results – Conclusions and perspectives 7/15 REACTION RATE LIMITED MODEL Biofilm phase: RRLC assumption Concentration field Quasi-steady state Fluid phase: Gray’s decomposition Closure assumption

10. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 8/15 MASS TRANSFER LIMITED MODEL Macroscopic equation of transport Which is defined only in the fluid phase. is the effective dispersion tensor, is the mass transfer coefficient from fluid phase to biofilm phase and and are non classical convective terms. Relations between the microscale and the macroscale Fluid phase : Gray decomposition Closure assumption

11. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 9/15 CLOSURE PROBLEMS Numerical solving Discretisation scheme : finite volume method Flow equation : Uzawa algorithm Closure equations : convection - first order upwind scheme with antidiffusion dispersion - implicit scheme Non linearities : Picard ’s method Resolution of the linear systems : BiCG_STAB for low Péclet numbers and successive over relaxation method for high Péclet numbers Typical unit cells associated with closure problems

12. Introduction – Two non equilibrium models – Results (RRL) – Conclusions and perspectives 10/15 EFFECTIVENESS FACTOR CALCULATION Comparison between the case of the coupled transport of solutes A and B and the case of uncoupled transports The coupled effectiveness factor is the minimum of the uncoupled effectiveness factors (i.e. the effectiveness factor associated to the limiting reactant) Considered biochemical conditions : Solute A in excess Solute B limiting reactant

13. Introduction – Two non equilibrium models – Results (MTL) – Conclusions and perspectives 11/15 MASS TRANSFER COEFFICIENT CALCULATION Decreasing function of the volume fraction of the fluid phase Increasing function of specific surface of the fluid-biofilm interface Impact of the development of the biofilm

14. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 12/15 DOMAINS OF VALIDITY Calculation of the effective transport properties of the macroscopic medium Biofilm (thickness ) Fluid (thickness ) Solid Comparison between direct simulations of transport at the microscale and upscaled simulations at the macroscale for a stratified porous medium, in the case of a large excess of solute B (uncoupled transport) Direct 2D simulation at the microscale (COMSOL) 1D averaged simulation

15. Péclet numberDamköhler number Introduction – Two non equilibrium models – Results – Conclusions and perspectives 13/15 DOMAINS OF VALIDITY

16. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 14/15 CONCLUSIONS AND PERSPECTIVES Conclusions Simplified non equilibrium models of transport enable to quantify the impact of the biofilm phase on dispersive and reactive properties of the porous medium, in their domains of validity Domains of validity: Mass transfer limited model: Pe > 1 Reaction Rate Limited model: Pe >> Da Da >> 1 (Local Equilibrium Assumption model: Pe < 1 Da < 1) Perspectives Numerical perspectives: Development of a two equation non equilibrium model for the general case of transport Experimental perspectives: Experimental set-up of bidimensionnal reactive transport in a porous medium including a biofilm phase in order to compare numerical and experimental results

17. Thank you for your attention

18. Annexes FULL MICROSCALE PROBLEM Reactive transport of substrate A Reactive transport of electron acceptor B Growth of the biofilm phase Flow of the fluid phase

19. Effective dispersion at the macroscale: closure problem 1 Annexes REACTION RATE LIMITED MODEL: CLOSURES Interfacial flux at the macroscale: closure problem 2 =>

20. Problem 1 Problem 2 With Effective parameters at macroscale : closure problems Annexes MASS TRANSFER LIMITED MODEL: CLOSURES => (+coupling between transport of the two solutes done a posteriori by mass balance)