Laboratoire Environnement, Géomécanique & Ouvrages

Comparison of Theory and Experiment for Solute Transport in Bimodal

Heterogeneous Porous Medium

Fabrice Golfier LAEGO-ENSG, Nancy-Université, FranceBrian Wood Environmental Engineering, Oregon State

University, Corvallis, USAMichel Quintard IMFT, Toulouse, France

Scaling Up and Modeling for Transport and Flow in Porous Media 2008, Dubrovnik

Introduction

• Highly heterogeneous porous medium: medium with high variance of the log-conductivity

• Multi-scale aspect due to the heterogeneity of the medium.

• Transport characterized by an anomalous dispersion phenomenon: Tailing effect observed experimentally

• Different large-scale modeling approaches :non-local theory (Cushman & Ginn, 1993), stochastic approach (Tompson & Gelhar, 1990), homogenization (Hornung, 1997), volume averaging method (Ahmadi et al., 1998; Cherblanc et al., 2001).

• First-order mass transfer model (with a constant mass transfer coefficient) is the most usual methodDoes such a representation always yield an upscaled model that works?

Large scale modeling

1 1

V VV V

c c dV c c dV

c c c c

First-order mass transfer model obtained from volume averaging method (Ahmadi et al., 1998; Cherblanc et al., 2003, 2007 )

Objective: Comparison of Theory and Experiment for two-region systems where significant mass transfer effects are

present

Case under consideration:Bimodal porous medium

Volume fractions of the two regions

-region

-region

Darcy-scale equations

* in the -regionc

c c ct

v D

* in the -regionc

c ct

v D

B.C.1 at c c A

* *B.C.2 at c c A n D n D

Upscaling

• Closure relations

• Macroscopic equations:

* 1 1 *

ConvectionDispersion Inter-phase mass transferAccumulation

Accumulat

Matrix ( )

Inclusion ( )

region

cc c c c

t

region

c

t

vD

* 1 1 *

ConvectionDispersion Inter-phase mass transferion

c c c c vD

c c c c c r c c

b b

c c c c c r c c

b b

Closure variables

Effective coefficients are given by a series of steady-state closure problems

Example of closure problem

Closure problem for related to the source :

Calculation performed on a simple periodic unit cell in a first approximation

* * 1 v b v D b D c

B.C.1 at A b b

* 1 v b D b c

* * *B.C.2 at A n D b n D n D b

Periodicity i i b r l b r b r l b r

0 0

b b

geometry of the interface needed

b c

* *

b v bD D I

steady-state assumption !

Experimental SetupZinn et al. (2004) Experiments

Parameters

High contrast, =1800

0.505 0.505 0.004/0.004 0.0004/0.0004 1.32 0.66

Low contrast, =300 0.505 0.505 0.002/0.002 0.0002/0.0002 1.26 0.63

, ,/L L , ,/T T highQ lowQ3cm /min3cm /minmm

Parameters calibrated from direct simulations

Two dimensional inclusive heterogeneity pattern

• 2 different systems• 2 different flowrates• ‘Flushing mode’

injectionK

K

33.5%

66.5%

Concentration fields and elution curves

Comparison with large-scale model

• 1rt-order mass transfer theory under-predicts the concentration at short times and over-predicts at late times

• Origin of this discrepancy?– Impact of the unit cell

geometry ?– Steady-state closure

assumption ?

300

1800

Impact of pore-scale geometry

No significant improvement!!

Steady state closure assumption

• Special case of the two-equation model (Golfier et al., 2007) :– convective transport neglected within the inclusions– negligible spatial concentration gradients within the

matrix– inclusions are uniform spheres (or cylinders) and are

non-interacting* *

2

15, 3D, spherical inclusionsD

a

* *2

8, 2D, cylindrical inclusionsD

a

Harmonic average of the eigenvalues

of the closure problem !

• Transient and asymptotic solution was also developped by Rao et al. (1980) for this problem

Discrepancy due to the steady-state closure assumption

Analytical solution of the associated closure problem

Discussion and improvement

• First-order mass transfer models:– Harmonic average for * forces the zeroth, first and

second temporal moments of the breakthrough curve to be maintained (Harvey & Gorelick, 1995)

– Volume averaging leads to the best fit in this context !!

• Not accurate enough?– Transient closure problems– Multi-rate models (i.e., using more than one

relaxation times for the inclusions)– Mixed model : macroscale description for mass

transport in the matrix but mass transfer for the inclusions modeled at the microscale.

Mixed model: Formulation

• Limitations:– convection negligible in -region– deviation term neglected at

B.C.1 at c c A

* *B.C.2 at c c A n D n D

1 1

* *,

matrix ( ) :

mixed

A

region

cc c c dA

t V

v nD D

*

inclusion ( ) :region

cc

t

D

A c

Interfacial flux

Valuable assumptions if high

Mixed model: Simulation

• Dispersion tensor : solution of a closure problem (equivalent to the case with impermeable inclusions)

• Representative geometry (no influence of inclusions between themselves is considered)

* *,

1mixed

V

dVV

b v bD D I

Concentration fields for both regions at t=500 mn ( =300 – Q=0.66mL/mn)

Simulation performed with COMSOL M.

Mixed model: Results

• Improved agreement even for the case = 300 where convection is an important process

• But a larger computational effort is required !!

Conclusions

• First-order mass transfer model developed via volume averaging:

– Simple unit cells can be used to predict accurate values for *, even for complex media.

– It leads to the optimal value for a mass transfer coefficient considered constant

– Reduction in complexity may be worth the trade-off of reduced accuracy (when compared to DNS)

• Otherwise, improved formulations may be used such as mixed models