Upscaling of Transport Processes in Porous Media with Biofilms in Non-Equilibrium Conditions

1. Upscaling of Transport Processes in Porous Media with Biofilms in Non-Equilibrium Conditions L. Orgogozo1, F. Golfier1, M.A. Buès1, B. Wood2, M. Quintard3 1Nancy Université - Laboratoire Environnement, Géomécanique et Ouvrages, École Nationale Supérieure de Géologie, Rue du Doyen Marcel Roubault, BP40F-54501 Vandoeuvre-lès-Nancy, France 2Environmental Engineering, Oregon State University, Corvallis, OR 97331, USA 3Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France Contact : Laurent.Orgogozo@ensg.inpl-nancy.fr

2. OBJECTIF GENERALE Introduction – Two non equilibrium models – Results – Conclusions and perspectives 2/15 INTRODUCTION 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) Biofilm growth Coupling : active transport of the substrate in the porous medium where grows the biofilm Substrate availability Substrate consumption

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

4. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 4/15 AIM OF THIS WORK Upscaling of transport processes from pore scale to Darcy scale -> 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 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. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 5/15 TRANSPORT MODELLING BY VOLUME AVERAGING Assumption of separation of scales Pore and biofilm scale Representative Elementary Volume Scale fluid  l biofilm ω l Volume averaging operator and associated theorems (e.g. Whitaker 1999) L l solid  R Microscale equations Macroscale equations + macroscopic boundary conditions + closure/microscale problems + 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 : 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 Interface concentration Fluid Biofilm Distance to the interface 0 x 0 General case

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. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 7/15 REACTION RATE LIMITED MODEL Fluid phase: Gray’s decomposition Closure assumption 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 Biofilm phase: RRLC assumption Concentration field Quasi-steady state

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 Typical unit cells associated with 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

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 • Considered biochemical conditions : • Solute A in excess • Solute B limiting reactant The coupled effectiveness factor is the minimum of the uncoupled effectiveness factors (i.e. the effectiveness factor associated to the limiting reactant)

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

14. Introduction – Two non equilibrium models – Results – Conclusions and perspectives 12/15 DOMAINS OF VALIDITY 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) Biofilm (thickness ) Solid Fluid (thickness ) Calculation of the effective transport properties of the macroscopic medium 1D averaged simulation

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

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 < Da Da >> 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 Flow of the fluid phase Reactive transport of substrate A Reactive transport of electron acceptor B Growth of the biofilm phase

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

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