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
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
2/15
OBJECTIF GENERALE
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
consumption
Substrate
2
availability
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 ()
Solid phase ()
Diffusion, reaction
Fluid phase ()
Passive phase
+ Growth
Convection, diffusion
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
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
5/15
TRANSPORT MODELLING BY VOLUME AVERAGING
Pore and biofilm
scale
Assumption of separation of scales
Representative
Elementary Volume Scale
fluid 
biofilm ω
solid 
l
l
R
l
Volume averaging operator and associated
theorems (e.g. Whitaker 1999)
Microscale equations
+ microscopic boundary conditions
L
Macroscale equations
+ macroscopic boundary conditions
+ closure/microscale problems
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
TWO NON EQUILIBRIUM MODELS 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
concentration
Interface
Biofilm
Fluid
0
x
0 Distance to the interface
General case
6/15
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
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)
6/15
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
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)
6/15
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
REACTION RATE LIMITED 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
Fluid phase:
Biofilm phase:
Gray’s decomposition
RRLC assumption
Closure assumption
Concentration field
Quasi-steady state
7/15
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
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
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
9/15
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)
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
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
)
Introduction – Two non equilibrium models – Results – Conclusions and perspectives
DOMAINS OF VALIDITY
Damköhler number
Péclet number
13/15
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
Flow of the fluid phase
Reactive transport of substrate A
Reactive transport of electron acceptor B
Growth of the biofilm phase
Annexes
REACTION RATE LIMITED MODEL: CLOSURES
Effective dispersion at the macroscale: closure problem 1
=>
Interfacial flux at the macroscale: closure problem 2
=>
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)