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 : Laurent.Orgogozo@ensg.inpl-nancy.fr

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)