1. Laboratoire Environnement, Géomécanique & Ouvrages Comparison of Theory and Experiment for Solute Transport in Bimodal Heterogeneous Porous Medium Fabrice Golfier LAEGO-ENSG, Nancy-Université, France Brian Wood Environmental Engineering, Oregon State University, Corvallis, USA Michel Quintard IMFT, Toulouse, France Scaling Up and Modeling for Transport and Flow in Porous Media 2008, Dubrovnik

2. 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 method Does such a representation always yield an upscaled model that works?

3. Large scale modeling 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

4. Darcy-scale equations

5. Upscaling Closure relations Macroscopic equations: Closure variables Effective coefficients are given by a series of steady-state closure problems

6. Example of closure problem Closure problem for related to the source : Calculation performed on a simple periodic unit cell in a first approximation geometry of the interface needed steady-state assumption !

7. Experimental Setup Zinn et al. (2004) Experiments Parameters High contrast,  =1800 0.505 0.004/0.0040.0004/0.00041.320.66 Low contrast,  =300 0.505 0.002/0.0020.0002/0.00021.260.63 Parameters calibrated from direct simulations Two dimensional inclusive heterogeneity pattern 2 different systems 2 different flowrates ‘Flushing mode’ injection

8. Concentration fields and elution curves

9. Comparison with large-scale model 1 rt -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 ?

10. Impact of pore-scale geometry No significant improvement!!

11. 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 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

12. 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.

13. Mixed model: Formulation Limitations: –convection negligible in  -region –deviation term neglected at Interfacial flux Valuable assumptions if  high

14. 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) Concentration fields for both regions at t=500 mn (  =300 – Q=0.66mL/mn) Simulation performed with COMSOL M.

15. Mixed model: Results Improved agreement even for the case  = 300 where convection is an important process But a larger computational effort is required !!

16. 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