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 1V V

V 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 -regioncc c

t

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

ct

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’

injectionKK

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 inclusionsDa

* *2

8 , 2D, cylindrical inclusionsDa

Harmonic average of the eigenvalues

of the closure problem !• Transient and asymptotic solution was also

developped by Rao et al. (1980) for this problemDiscrepancy 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 ( ) :

mixedA

region

cc c c dA

t V

v nD D

*

inclusion ( ) :regionc ct

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