1. US Army Corps of Engineers ® Engineer Research and Development Center Reactive Transport (3): Solve Biogeochemistry with The Primitive Approach Pearce Cheng (x3699) Last Update: October 8, 2008

2. US Army Corps of Engineers ® Engineer Research and Development Center Outline A Generic Biogeochemical System Primitive Approach Advantages and Disadvantages with The Primitive Approach Solve Biogeochemistry with The Primitive Approach

3. US Army Corps of Engineers ® Engineer Research and Development Center A Generic Biogeochemical System No. of species = M No. of reactions = N r – Fast reversible (equilibrium) reactions – Slow reversible/irreversible (kinetic) reactions – Fast irreversible (one way instantaneous) reactions where = the ( M x M ) identity matrix; = the vector representing the M species concentrations [M/L 3 ]; = the ( M x N r ) reaction stoichiometry matrix with & as components; = the vector representing the N r reaction rates [M/(t  L 3 )].

4. US Army Corps of Engineers ® Engineer Research and Development Center Example for Demonstration (R1)  a fast reversible (equilibrium) reaction (R2) (R3)  a slow reversible (kinetic) reaction (R4)  a slow irreversible (kinetic) reaction (R5) NTA: sodium nitrilotriacetate; C 6 H 9 NO 6 ; a chelating agent used in detergents to improve their cleaning ability  a fast reversible (equilibrium) reaction (R6)  a fast irreversible (one way instantaneous) reaction

5. US Army Corps of Engineers ® Engineer Research and Development Center Primitive Approach

6. US Army Corps of Engineers ® Engineer Research and Development Center Advantages with The Primitive Approach Straightforward in terms of formulation and solution strategy Easy to be incorporated into reactive transport equations

7. US Army Corps of Engineers ® Engineer Research and Development Center Disadvantages with The Primitive Approach (1/2) The time step size must be sufficiently (or infinitely) small to resolve fast/equilibrium reactions  DAE approach The integration of the primitive rate equations, due to numerical errors, does not guarantee the conservation of the total mass of component chemical/species, which is necessary.  DAE approach There is no way to define the subtraction or addition of infinity if more than one equilibrium reactions are involved  DAE approach

8. US Army Corps of Engineers ® Engineer Research and Development Center Disadvantages with The Primitive Approach (2/2) Even if all reactions are slow/kinetic reactions, their rates are coupled via the concentration-versus-time curves of all species. They cannot be formulated and parameterized one reaction by one reaction independently of each other (most difficult issue)  Identify the kinetic variables associated with the given experimental concentration-versus-time curves [Co] [P]

9. US Army Corps of Engineers ® Engineer Research and Development Center The Primitive Approach Can Be Used When … Fast reversible (equilibrium) reactions do not exist. – Linear equilibrium assumption may be used for limited cases (e.g., RT3D strategy) Reaction networks are adequately constructed. Time steps used are sufficiently small to resolve all slow reactions. Special treatments to account for fast irreversible reactions are incorporated when necessary.

10. US Army Corps of Engineers ® Engineer Research and Development Center Solve Biogeochemistry with The Primitive Approach Solve Reaction Network with the Newton Method – Compute numerical Jacobian (generic approach) – Compute analytical Jacobian (when all reactions can be represented in analytical form) Compute adaptive time steps to avoid negative concentrations Employ special treatments to handle fast irreversible reactions

11. US Army Corps of Engineers ® Engineer Research and Development Center Compute Adaptive Time Steps For each chemical species, check the contribution from all slow reactions: – Scenario 1:  – Scenario 2:  Examine and exclude unnecessary reactions: – If ( is a small positive number, e.g., 10 -6 ), then (a) Remove the reaction that yields the greatest value of (b) Re-calculate by excluding the reaction just removed (c) If still exists, repeat (a) and (b) until Set Note: A simplified approach is actually adopted.

12. US Army Corps of Engineers ® Engineer Research and Development Center Special Treatments to Handle Fast Irreversible Reactions Fast irreversible reactions are achieved immediately when compared to the slow reactions  determine available reactants for fast irreversible reactions to occur The resultant chemical distribution after accounting for fast irreversible reactions is used to compute biogeochemistry where only slow reactions are taken into account within each nonlinear iteration.

13. US Army Corps of Engineers ® Engineer Research and Development Center Computation Flow Chart Account for Fast Irreversible Reactions Compute Adaptive Time Steps

14. US Army Corps of Engineers ® Engineer Research and Development Center Demonstration Example: ( 10 species & 4 reactions) (R1)  a slow reversible reaction (R2)  a slow irreversible reaction (R3)  a fast irreversible reaction (R4)  a slow reversible reaction (R5)  a fast irreversible reaction

15. US Army Corps of Engineers ® Engineer Research and Development Center Set Working Concentration before Adjustment …

16. US Army Corps of Engineers ® Engineer Research and Development Center Adjust Species Concentrations Associated with Fast Irreversible Reactions

17. US Army Corps of Engineers ® Engineer Research and Development Center Adjust Species Concentrations Associated with Fast Irreversible Reactions   (R3)  a fast irreversible reaction (R5)  a fast irreversible reaction

18. US Army Corps of Engineers ® Engineer Research and Development Center Set New Initial Concentration before Nonlinear Iterations

19. US Army Corps of Engineers ® Engineer Research and Development Center Set Working Concentrations for Nonlinear Iterations …

20. US Army Corps of Engineers ® Engineer Research and Development Center Compute Reaction Rates for Slow Reactions

21. US Army Corps of Engineers ® Engineer Research and Development Center Compute Reaction Rates for Slow Reactions (R1)  a slow reversible reaction (R2)  a slow irreversible reaction (R4)  a slow reversible reaction Note: The reaction rates are computed based on the working concentration: Note: The reaction rates can also be computed with the given formulae when necessary.

22. US Army Corps of Engineers ® Engineer Research and Development Center Time Step Adaption

23. US Army Corps of Engineers ® Engineer Research and Development Center Check the overall reaction rate for each species, e.g., for the i - th species Adjustment is necessary only when the overall reaction rate is found negative: Time Step Adaption (A Simplified Approach) Otherwise, temporarily de-activate the reactions that would consume the i-th species for the time being.

24. US Army Corps of Engineers ® Engineer Research and Development Center Special Treatment for Fast Irreversible Reactions

25. US Army Corps of Engineers ® Engineer Research and Development Center Governing Equations (R1) (R2) (R3) (R4) (R5)  slow reversible  slow irreversible  fast irreversible  slow reversible  fast irreversible

26. US Army Corps of Engineers ® Engineer Research and Development Center Special Treatment to Account for Fast Irreversible Reactions (R3)  a fast irreversible reaction (R5)  a fast irreversible reaction (When R 1 > 0)  (When R 1  0) (When R 4 < 0) (When R 4  0) 

27. US Army Corps of Engineers ® Engineer Research and Development Center Compute Residuals

28. US Army Corps of Engineers ® Engineer Research and Development Center Compute Residuals (R1) (R2) (R3) (R4) (R5)  slow reversible  slow irreversible  fast irreversible  slow reversible  fast irreversible

29. US Army Corps of Engineers ® Engineer Research and Development Center Compute Jacobians

30. US Army Corps of Engineers ® Engineer Research and Development Center Compute Jacobians Compute Numerical Jacobians (When C j > C m ) (When C j  C m )

31. US Army Corps of Engineers ® Engineer Research and Development Center Full-Pivoting Direct Solver Reference: Numerical Recipes, 2 nd Edition, byW.H. Press, S.A. Teukolsky, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery

32. US Army Corps of Engineers ® Engineer Research and Development Center Check Convergence Convergence reached if (When C j old > C m ) where (When C j old  C m )

33. US Army Corps of Engineers ® Engineer Research and Development Center Test Example

34. US Army Corps of Engineers ® Engineer Research and Development Center Summary Major Difficulties in Solving Biogeochemistry A Preprocessor with Reaction-Based Approach to Overcome Major Difficulties