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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 - PowerPoint PPT Presentation
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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, USA3Institut de Mécanique des Fluides de Toulouse, Allée du Professeur Camille Soula, 31400 Toulouse, France
Contact : [email protected]
2
OBJECTIF GENERALEIntroduction – 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
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)
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
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 AVERAGINGPore and biofilm
scale fluid
biofilm ω
solid
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
Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15
TWO NON EQUILIBRIUM MODELS OF TRANSPORTGeneral case : transport in two phases
=> two-equation model of transport
General case
0
0
Fluid Biofilm
con
cen
trat
ion
Interface
x
Distance to the interface
General case : transport in two phases
=> two-equation model of transportParticular cases : assumption about the relation of the concentration fields of each phase.
=> One-equation models of transport
Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15
REACTION RATE LIMITED MODELGeneral case : transport in two phases
=> two-equation model of transportParticular 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)
Introduction – Two non equilibrium models – Results – Conclusions and perspectives 6/15
MASS TRANSFER LIMITED MODELGeneral case : transport in two phases
=> two-equation model of transportParticular 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)
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
Introduction – Two non equilibrium models – Results – Conclusions and perspectives 8/15
MASS TRANSFER LIMITED MODELMacroscopic 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
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
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
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 phaseIncreasing function of specific surface of the fluid-biofilm interface
Impact of the development of the biofilm
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
Péclet numberDamköhler number
Introduction – Two non equilibrium models – Results – Conclusions and perspectives 13/15
DOMAINS OF VALIDITY
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
Thank you for your attention
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
Effective dispersion at the macroscale: closure problem 1
Annexes
REACTION RATE LIMITED MODEL: CLOSURES
Interfacial flux at the macroscale: closure problem 2
=>
=>
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)