1. A modified Lagrangian-volumes method to simulate nonlinearly and kinetically adsorbing solute transport in heterogeneous media J.-R. de Dreuzy, Ph. Davy, K. Besnard Géosciences Rennes Evaluation of Lagrangian methods for simulating reactive transport

2. Outline 1.Objectives 2.Numerical method development General advantages and drawbacks of particle methods A Lagrangian Volume (LV) method for simulating reactive transport The LV method for Freundlich adsorption in 1D Analysis of Local Equilibrium Assumption (LEA) 3.Sorption simulation under kinetic and non linear conditions Double peak formation and analysis Heterogeneous 2D porous media 4.Extension Simulation of mixing and dispersion

3. Objectives  Influence of heterogeneity on reactive transport processes.  Influence of heterogeneity range and correlations  How to define equivalent reaction coefficients?  Simulation of a large range of reaction systems  easy switch between reaction systems (e.g. denitrification) fractured mediabounded correlationslarge correlations

4. Outline 1.Objectives 2.Numerical method development General advantages and drawbacks of particle methods A Lagrangian Volume (LV) method for simulating reactive transport The LV method for Freundlich adsorption in 1D Analysis of Local Equilibrium Assumption (LEA) 3.Sorption simulation under kinetic and non linear conditions Double peak formation and analysis Heterogeneous 2D porous media 4.Extension Simulation of mixing and dispersion

5. General advantages and drawbacks of particle methods  Advantages  No numerical diffusion for simulating the advective term  Existing methods for treating particle diffusion and dispersion  Time effective for obtaining coarse solutions  Relative ease of implementation  Drawbacks  Improvement of precision  Coupling between transport and reaction  Definition of concentration from particles

6. Lagrangian-Volumes model in 1D  Segmentation of flow in elementary volumes (water slices) LV are tracked in the domain like in classical particles tracking schemes LV contain solutes as concentrations LV interact with the solid and between themselves  Existence of two scales: LV scale and mesh scale Increase of precision without additional intra-mesh discretization of S u, flow velocity Elementary Lagrangian volumes C = C 0 C = 0 Injected concentrations (pulse) C = 0 C time C0C0 Meshes

7. Lagrangian Volumes with Adsorption Computation of the chemical flux  at the LV scale along the pore surface Modification of the solute concentration C according to   : amount of solute sorbing per unit surface area per unit time (  <0: desorption)  : surface to volume ratio

8. Kinetically-controlled Freundlich sorption C concentration in solution (mass/m 3 ) S concentration sorbed (mass/m 2 ) a kinetic constant of reaction (s ‑ 1 ) K a reaction capacity n Freundlich (n=1 or n < 1)

9. Transport and Sorption coupling  Advective-dispersive term simulated like in classical particle tracking Water mass conservation remains valid globally but not necessarily locally Solute mass is always conserved both globally and locally C aqueous concentration (in mass/m 3 ) S sorbed concentration adsorbée (in mass/m 2 ) u flow velocity (m/s),  surface/volume ratio (m ‑ 1 ), a reaction rate (s ‑ 1 ), K a distribution coefficient n Freundlich coefficient (n < 1)

10. Within the mesh Mesh scale homogenization of S when the LV leaves the mesh Decoupling of transport and reactivity at LV scale Batch reaction at LV scale Contact time between LV and corresponding solid  t=  x/u  x: characteristic length of the LV xx t0t0 titi t i+1 t n-1 numerical integration on [t i,t i+1 [ differential equation on [t i,t i+1 [ n successive integrations between t 0 and t n-1 Mesh scale homogenization of S at t n-1 {Jump+React} i=0,…,n-1 + homogenization of S

11. Validation cases 3 parameters: Freundlich exponent (n), kinetic constant (  ), heterogeneity (  )

12. Kinetically-controlled linear sorption n = 1,  Ka=4,  =1 days -1, analytical solutions for spatial moments [e.g. Dagan et Cvetkovic, 1993]

13. Asymptotic concentration profiles C/C 0 scales asymptotically as t 1/(1-n) Jaekel et al. [1996] n=0.9,  =1 day -1, u=1 m/day,  K a C 0 n-1 =4, LV/Mesh volume=10 -2

14. Analysis of Local Equilibrium Assumption   reaction <  advection  First steps of LV algorithm  n=1, Ka=1  Analytical solution  Injection form constant  Velocity reduced by 1+Ka  S≠ 0 where C≠ 0  Right retardation factor  LV induce a spreading D/U= 0,1 m  Other cause of numerical diffusion  homogenization of S at the mesh scale  Induces a slight reduction of diffusion

15. Analysis of Local Equilibrium Assumption  Solutions to lower the numerical diffusion  Mesh refinement  reaction >  advection  Use of an analytical solution of the reactive transport equation at the mesh scale  What does the LEA assumption imply?  S≠ 0 where C≠ 0 although solutes are delayed  instantaneous homogenization of solute concentration within the fluid phase

16. Outline 1.Objectives 2.Numerical method development General advantages and drawbacks of particle methods A Lagrangian Volume (LV) method for simulating reactive transport The LV method for Freundlich adsorption in 1D Analysis of Local Equilibrium Assumption (LEA) 3.Sorption simulation under kinetic and non linear conditions Double peak formation and analysis Heterogeneous 2D porous media 4.Extension Simulation of mixing and dispersion

17. Extension to non-linear kinetically controlled sorption reactions 3 parameters: Freundlich exponent (n), kinetic constant (  ), heterogeneity (s)

18. τ = 0,5  = 1.5τ = 2,2 double peak τ = 3 Existence of a transient double peak due to kinetic conditions What are its effects, how long does it occur? Is it necessary to take it into account?  = 5 n = 0,8 K* = 1,6 C (in solution) S (adsorbed)

19. At short term, the sole effect is kinetic Kinetic >> non-linearity Double peak effects amplified by the non-linearity non linearity ~ kinetic K*=4 Effect of non-linearity (n) on pulse injection Asymptotic behaviour of the non linear equilibrium case non linearity >> kinetic

20. Quantification of double-peak occurrence

21. 2D Heterogeneous media particlesconcentration  =1, n=0.9, D=0,  Ka=1,  2 =1.5

22. 2D Heterogeneous media permeability field with bounded correlation (  2 =1.5)

23. Outline 1.Objectives 2.Numerical method development General advantages and drawbacks of particle methods A Lagrangian Volume (LV) method for simulating reactive transport The LV method for Freundlich adsorption in 1D Analysis of Local Equilibrium Assumption (LEA) 3.Sorption simulation under kinetic and non linear conditions Double peak formation and analysis Heterogeneous 2D porous media 4.Extension Simulation of mixing and dispersion

24. Mixing  Mixing is required for simulating reactions in solution  Implementation  Mixing is performed between Lagrangian Volumes leaving a mesh and their predecessor on the duration needed for crossing the mesh

25. Validation on the 1D advection-diffusion case diffusion time in the LV < advection in the mesh D/(ud mesh )=10 -3 D/(ud mesh )=10 -2

26. CONCLUSION: Advantages and drawbacks of the Lagrangian-Volume method  Drawbacks  Numerical dispersion in equilibrium conditions  Advantages  Decoupling at the LV scale allowing the direct implementation of various chemical sets and reactions  Simulation of large domains  Differentiation of dispersion and mixing at the local scale fractured mediabounded correlationslarge correlations