comparison of theory and experiment for solute transport in bimodal heterogeneous porous medium
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Comparison of Theory and Experiment for Solute Transport in Bimodal Heterogeneous Porous Medium. Scaling Up and Modeling for Transport and Flow in Porous Media 2008, Dubrovnik. Fabrice Golfier LAEGO-ENSG, Nancy-Université, France - PowerPoint PPT PresentationTRANSCRIPT
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