1. 1Recent developments in Reactive Transport, Alicante, Spain. October, 2005 RECENT DEVELOPMENTS IN REACTIVE TRANSPORT Jesus Carrera Technical University of Catalonia (UPC) Barcelona, Spain In collaboration with Carlos Ayora, Maarten Saaltink, Xavier SánchezVila, Michela Desimoni …

2. 2Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Contents –Is Reactive transport needed? An example –Can be understood? –Can be solved efficiently? … and the answer is YES

3. 3Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Calcite dissolution in coastal aqf. Mixture of calcite equilibrated with salt and fresh waters can be under or oversaturated with respect to calcite (often under) To simulate this effect, consider 1D diffusion experiment freshwater calcite saltwater

4. 4Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Simple mixing (no transport) Saturation Index (SI) Dissolution rate (controlled by diffusion) Reaction Rate SI & r Mixing leads to maximum undersaturation for around 20% fresh water Dissolution rate proportional to Diff coeff. and maximum at the fresh water end

5. 5Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Speciation Dissolution causes diffusion of CO 2 (acidity) at the freshwater end, which drives further dissolution

6. 6Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Sensitivity to CO 2 Reducing concentration of CO 2 at the freshwater end, causes an increase in subsaturation. Therefore, one would expect an increase in dissolution rate However, dissolution rate is dramatically reduced

7. 7Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Summary of 1D diffusion exp The interplay between transport and reactions is non-trivial. Saturation index calculations are needed but they fail to indicate 1)how much calcite is dissolved, which is controlled by mixing rate, 2)nor where (or under which conditions) dissolution rate is maximum. Simulating reactive transport is needed to understand the fate of reacting solutes!

8. 8Recent developments in Reactive Transport, Alicante, Spain. October, 2005 The ingredients of reactive transport Flow –Momentum conservation –Water mass conservation K, T, S, recharge B.C’s, geometry Dispersivity, porosity Equilibrium and rate constants Transport –Solute mass conservation –Advection –Diffusion/Dispersion Reactive transport –Chemical reactions Equilibrium Kinetic

9. 9Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Chemical reactions: Stoichiometric matrix Assume a chemical system Stoichiometric Matrix Reaction rate: Mass balance Let r i be the number of moles of reactants that evolve into products for the i-th reaction Primary Secondary Constant Ac. The columns of S can be viewed as the contribution of reactions to each species How to evaluate r?

10. 10Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Equilibrium reactions: Mass Action Law Traditional notation Matrix notation Notice that neither water nor calcite enter in the mass action law, as their activity is constant However, for equilibrium reactions, it is not possible to write the reaction rate explicitly

11. 11Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Formulation of Reactive transport problems n s transport equations n r algebraic equations Looks awful! ( n r + n s unknowns at every point) Seek tricks and/or simplifications

12. 12Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Components: Linear combinations of species that remain unaltered by equilibrium reactions The basic trick: components Choose component matrix U, such that Then, n s -n r transport equations. (A good choice of U allows these equations to be decoupled!)

13. 13Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Example Chemical system Stoichiometric Matrix Components matrix Components

14. 14Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Procedure 1.Define chemical system and components 2.Solve transport equations for components (and/or primary species) 3.Speciation: Compute species concentrations from components (and/or primary species) 4.Substitute species back into transport equations to obtain reaction rates

15. 15Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Assume 2 species (e.g. SO 4 2- and Ca 2+ ) in eq. with gypsum Analytical solution for 2 species Transport equations where (1) (2) (1)-(2) yields: Step 2: Solve transport of u Reaction Stoichiometric matrix Components: is conservative! Step 1: Chemical system

16. 16Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Analytical solution for 2 species Step 3: Speciation Solve Together with Plugging C 2 into We obtain Step 4: Compute r Transport Chemistry

17. 17Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Distance from the peak of u reaction rate, r/  r/  u u Distance from peak of u Spatial distribution of reaction rate Spatial distribution of reaction rate is more controlled by mixing, than chemistry

18. 18Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Mixing of several waters Step 3: Speciation Can be very complex, but Where  is the mixing ratio (possibly a vector) Plugging C 2 into We obtain Step 4: Compute r That is, 1) Transport   Use any code (e.g., PHREEQE, RETRASO,…) to compute speciation

19. 19Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Solution of binary system for pulse input water 2 water 1 mixture c 1 (e.g., SO 4 2- ) c 2 (e.g., Ca 2+ )   Dimensionless [SO 4 2- ] reaction rate distance, (a) (b) Dimensionless [Ca 2+ ]

20. 20Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Dimensionless reaction rate x-distance from the plume centre Dimensionless y-distance Spatial distribution of reaction rate

21. 21Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Integrated reaction rate for varying initial pulse 30 40 Dimensionless reaction rate Dimensionless distance from the plume centre reaction rate distance from plume centre -30 -20

22. 22Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Spatial distribution of total precipitate Vy/D T Vx/D L 00.50 0 0.5 1.5 2.0 1.0 0 20 40 80 - 115 60 Vy/D T 0 0.5 1.5 2.0 1.0 Vx/D L 0 0.50 0 0.02 0.04 0.08 0.06

23. 23Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Chemical systems paradigms Types of reaction Paradigm Tank* Canal** River** * HomogeneousHeterogeneous SystemsFast (equilibrium) Slow (kinetic) Fast (equilibrium) Slow (kinetic) Aquifer****

24. 24Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Classification os chemical species (1) Primary species(2) Secondary species (1) first N s -N e columns of S last N e columns of S Mobile (a) aqueous (primary) species(2) aqueous (secondary) first N s -N e -N k columns of S (not specifically labeled) (f) kinetic aqueous species. solutes in the 2nd block of S k Immobil e (k) kinetic minerals (& gases)(q) minerals (& gases) in equilib. minerals (& gases) in the 2nd block of S k

25. 25Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Conclusions Is Reactive transport needed? –Reaction (rate, where, when, under which conditions) are controlled by transport. Can be understood? –All it takes is to understand components –The difficult part is to choose the relevant species and reactions. Can be solved efficiently? –Similar effort as conservative transport

26. 26Recent developments in Reactive Transport, Alicante, Spain. October, 2005 But Reactions are driven by disequilibrium Disequilibrium is driven by actual mixing We need to know how to evaluate actual mixing! Current stochastic transport theories fail to do so!

27. 27Recent developments in Reactive Transport, Alicante, Spain. October, 2005 Reverse time weigthing Flow Equation Water mass balance = _ Adv. form Tpt. Eqn Flow Eqn. x C k+1 Div. form Tpt. Eqn Solute mass balance = = = = + _ _ _ Mass Final Initial Mass Mass balance mass mass In out __ =

28. 28Recent developments in Reactive Transport, Alicante, Spain. October, 2005 infiltration case (INF).

29. 29Recent developments in Reactive Transport, Alicante, Spain. October, 2005 evaporation case (EVA).

30. 30Recent developments in Reactive Transport, Alicante, Spain. October, 2005 alternating evaporation-infiltration case (ALT)