A Newton Krylov method for coupling transport with chemistry in porous media

Transcription

1 A Newton Krylov method for coupling transport with chemistry in porous media Laila Amir 1,2 Michel Kern 2 1 Itasca Consultants, Lyon 2 INRIA Rocquencourt Joint work with J. Erhel (INRIA Rennes /IRISA) Funded by GDR MoMaS Groupe de travail «Méthodes numériques» Laboratoire Jacques-Louis Lions 4 décembre 2006 L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 1 / 41

2 Outline 1 Modeling Chemical phenomena Chemical phenomena in aqueous chemistry Modeling equilibrium systems Sorption models 2 Multispecies equilibrium reactive transport Chemical problem Transport equations The coupled system Coupling algorithms 3 Numerical results Pyrite test case Chromatography Simplified MoMaS benchmark L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 2 / 41

3 Outline 1 Modeling Chemical phenomena Chemical phenomena in aqueous chemistry Modeling equilibrium systems Sorption models 2 Multispecies equilibrium reactive transport Chemical problem Transport equations The coupled system Coupling algorithms 3 Numerical results Pyrite test case Chromatography Simplified MoMaS benchmark L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 3 / 41

4 Reactive transport in a porous medium L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 4 / 41

5 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction According to speed of reaction L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

6 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction Heterogeneous Involve different phases Examples: Sorption, precipitation / dissolution,... According to speed of reaction L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

7 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction Heterogeneous Involve different phases Examples: Sorption, precipitation / dissolution,... According to speed of reaction Slow reactions Irreversible, modeled using kinetic law L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

8 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction Heterogeneous Involve different phases Examples: Sorption, precipitation / dissolution,... According to speed of reaction Slow reactions Irreversible, modeled using kinetic law Fast reactions Reversible, modeled using equilibrium L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

9 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction Heterogeneous Involve different phases Examples: Sorption, precipitation / dissolution,... According to speed of reaction Slow reactions Irreversible, modeled using kinetic law Fast reactions Reversible, modeled using equilibrium Depends on relative speed of reactions and transport. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

10 Classification of chemical reactions According to nature of reaction Homogeneous In the same phase (aqueous, gaseous,...) Examples: Acid base, oxydo reduction Heterogeneous Involve different phases Examples: Sorption, precipitation / dissolution,... According to speed of reaction Slow reactions Irreversible, modeled using kinetic law Fast reactions Reversible, modeled using equilibrium Depends on relative speed of reactions and transport. In this talk: Equilibrium reactions, with sorption. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 5 / 41

11 Aqueous reactions: chemical equilibrium Chemical reaction: α 1 C 1 + α 2 C 2 S, in general α i C i 0 i Thermodynamic equilibrium: minimize (change in) Gibb s free energy G = G 0 + RT i α i ln(c i ), R = 8.31 J/K/mol Leads to mass action law ( c α i i = K, K = exp G ) 0 RT i L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 6 / 41

12 Modeling general equilibrium General chemical reactions : N s species, N r reactions N s j=1 ν ij Y j 0, i = 1,..., N r ν ij stoichiometric coefficients. Matrix equation νy = 0 L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 7 / 41

13 Modeling general equilibrium General chemical reactions : N s species, N r reactions N s j=1 ν ij Y j 0, i = 1,..., N r ν ij stoichiometric coefficients. Matrix equation νy = 0 Assumption ν has full rank : Rank ν = N r. Basis for null-space of ν has dimensions N c = N s N r. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 7 / 41

14 Modeling general equilibrium General chemical reactions : N s species, N r reactions N s j=1 ν ij Y j 0, i = 1,..., N r ν ij stoichiometric coefficients. Matrix equation νy = 0 Assumption ν has full rank : Rank ν = N r. Basis for null-space of ν has dimensions N c = N s N r. Partition ν = ( B N ), B R Nr Nr invertible, N R Nc Nr, R = B 1 N ( ) X General solution of νy = 0: Y =, X = RC. C R C Nc, X R Nr. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 7 / 41

15 Example: carbonic acid dissociation Species H 2 O, H 3 O +, OH, H 2 CO 3, HCO 3, CO2 3 Reactions Possible components H 3 O + + OH 2H 2 O, H 2 O + H 2 CO 3 H 3 O + + HCO 3, HCO 3 + H 2O CO H 3 O +. H 2 O, H 3 O +, H 2 CO 3, H 2 O, H 3 O +, HCO 3, H 2 O, H 3 O +, CO 2 3, H 2 O, OH, H 2 CO 3, H 2 O, OH, HCO 3, H 2 O, OH, CO 2 3. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 8 / 41

16 Components: the Morel tableau Species in c: components, in x: secondary species. Rewrite chemical system as (TDB give components, then species) N c j=1 r ij C j X i Components Equ. C 1 C j C Nc constants X 1 R 11 R 1Nc K 1.. X i R i1 R ij R inc K i.. X Ns R 1Ns R NsNc K Ns Total conc. T 1 T i T Nc L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 9 / 41

17 From chemistry to mathematics Each reaction, mass action law ({X i } = activity of X i ) N c {X i } = K i {C j } r ij, i = 1,..., N r j=1 Each component, mass conservation ([X i ] = concentration of X i ) N r T j = [C j ] + r ij [X i ], i=1 j = 1,..., N c L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 10 / 41

18 From chemistry to mathematics Each reaction, mass action law ({X i } = activity of X i ) N c {X i } = K i {C j } r ij, i = 1,..., N r j=1 Each component, mass conservation ([X i ] = concentration of X i ) N r T j = [C j ] + r ij [X i ], j = 1,..., N c i=1 Assume ideal solution [X i ] = {X i } := x (for dissolved species). System of nonlinear algebraic equations log x = R log c + log K T = c + R T x L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 10 / 41

19 Sorption processes Definition Sorption is the accumulation of a fluid on a solid at the fluid solid interface. Main mechanism for exchanges between dissolved species and solid surfaces. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 11 / 41

20 Sorption processes Definition Sorption is the accumulation of a fluid on a solid at the fluid solid interface. Main mechanism for exchanges between dissolved species and solid surfaces. Several possible mechanisms Surface complexation Formation of bond between surface and aqueous species, due to electrostatic interactions. Depends on surface potential. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 11 / 41

21 Sorption processes Definition Sorption is the accumulation of a fluid on a solid at the fluid solid interface. Main mechanism for exchanges between dissolved species and solid surfaces. Several possible mechanisms Surface complexation Formation of bond between surface and aqueous species, due to electrostatic interactions. Depends on surface potential. Ion exchange Ions are exchanged between sorption sites on the surface. Depends on Cationic Exchange Capacity. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 11 / 41

22 Sorption processes Definition Sorption is the accumulation of a fluid on a solid at the fluid solid interface. Main mechanism for exchanges between dissolved species and solid surfaces. Several possible mechanisms Surface complexation Formation of bond between surface and aqueous species, due to electrostatic interactions. Depends on surface potential. Ion exchange Ions are exchanged between sorption sites on the surface. Depends on Cationic Exchange Capacity. Can be modeled as mass action law L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 11 / 41

23 Sorptions isotherms Definition An adsorption isotherm relates F (mol/g) quantity of adsorbed component to its concentration C (mol/l) in the fluid L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 12 / 41

24 Sorptions isotherms Definition An adsorption isotherm relates F (mol/g) quantity of adsorbed component to its concentration C (mol/l) in the fluid Transport of one solute in saturated porous medium, interaction with solid phase : with C t + ρ F t Non-equilibrium df = k(ψ(c) F ) dt Equilibrium F = Ψ(C) + div ( uc D C ) = 0 L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 12 / 41

25 Common equilibrium isotherms Linear Ψ(C) = K d C Langmuir Ψ(C) = κ 1C 1 + κ 2 C Concave near C = 0 Freundlich Ψ(C) = κc p 1 Concave near C = 0, Ψ (0+) = for 0 < p < 1, 0.5 convex near C = 0 for 1 < p,. 0 Analysis for general function Ψ : Knabner, van Duijn Langmuir freundlich Linear L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 13 / 41

26 Examples for linear, Langmuir and Freundlich isotherm Concentration profiles at tmax/3, 2*tmax/3, and tmax Concentration profiles at tmax/3, 2*tmax/3, and tmax Concentration profiles at tmax/3, 2*tmax/3, and tmax Linear isotherm, Pe = Langmuir isotherm 0.9 Freundlich isotherm K L = 6.1/C 0 n F = Pe = Pe = C/C C/C C/C x (cm) x (cm) x (cm) J. A. Cunningham (Texas A&M) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 14 / 41

27 Outline 1 Modeling Chemical phenomena Chemical phenomena in aqueous chemistry Modeling equilibrium systems Sorption models 2 Multispecies equilibrium reactive transport Chemical problem Transport equations The coupled system Coupling algorithms 3 Numerical results Pyrite test case Chromatography Simplified MoMaS benchmark L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 15 / 41

28 Reactions for aqueous/solid system c j aqueous components, s j sorbant components, x i aqueous secondary species, fixed y i secondary species. N c x i S ij c j, j=1 i = 1,..., N x N c N s y j A ij c j + B ij s j, i = 1,..., N y, j=1 j=1 L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 16 / 41

29 Mass action law N c x i = K x i c S ij j, i = 1,..., N x j=1 N c y i = K y i j=1 Use logarithm: linear algebra Mass conservation c A ij j N s j=1 s B ij j, i = 1,..., N y, c + S t x + A T y = T s + B T y = W, T, W : total concentration in components L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 17 / 41

30 The chemical problem System of non-linear equations c + S T x + A T y = T, s + B T y = W, log x = S log c + log K x, log y = A log c + B log s + log K y. Dissolved total: C = c + S T x, Fixed total: F = A T y. Role of chemical model Given totals T (and W, known), split into mobile and immobile total concentrations. C = Φ(T ), F = Ψ(T ) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 18 / 41

31 The chemical problem (2) Take concentration logarithms as main unknowns Nonlinear system H(z) = ( ) T W z = (log c, log s), K = (log K x, log K y ) H(z) = exp(z + S T exp(k + Sz) Jacobian matrix H (z) = diag exp(z) + S T diag(exp(k + Sz)) S L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 19 / 41

32 Numerical solution of nonlinear problem Use globalized Newton s method (line search, trust region). Ion exchange: 6 species, 4 components (vary initial guess) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 20 / 41

33 Transport in a porous medium Diffusion convection equation ω c t D 2 c 2 x + u c x = f for 0 < x < L c = c d at x = 0 D c x = x 0 at x = L c(x, 0) = c 0 (x), 0 < x < L. ω : porosity Assumption D : dispersion coefficient Let L(c) = D 2 c 2 x + u c x. Dispersion tensor independent of species u : Darcy velocity L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 21 / 41

34 Numerical method for transport Space-time finite difference method c n j t x 2 c n+θ = θc n+1 + (1 θ)c n. ω cn+1 j D cn+θ j+1 2cn+θ j + c n+θ j 1 + u cn+θ j c n+θ j 1 x = f n+θ j Implicit scheme Unconditionally stable Upwind scheme (first order in space) θ = 1/2 : Crank Nicolson scheme (2nd order in time) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 22 / 41

35 The coupled system Transport for each species and component x i t y i t + L(x i ) = r x i, = r y i, c j t s j t + L(c j ) = r c j, = r s j, L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 23 / 41

36 The coupled system Transport for each species and component x i t y i t + L(x i ) = r x i, = r y i, c j t s j t + L(c j ) = r c j, = r s j, Eliminate (unknown) reaction rates by using conservation laws: CD equations for totals (T = C + F ) T ic + L(C ic ) = 0, ic = 1,..., N c t T ic ix = Cic ix + F ic ix ic = 1,.., N c and ix = 1,.., N x F ix = Ψ(T ix ) ix = 1,..., N x. Number of transport equations reduced from N x + N y to N c + N s L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 23 / 41

37 Different formulations (1) CC formulation, explicit chemistry (J. Erhel) dc + df + LC = 0 dt dt ( ) C + F H(z) = 0 W F F (z) = 0. + Explicit Jacobian + Chemistry function, no chemical solve Intrusive approach (chemistry not a black box) Precipitation not easy to include L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 24 / 41

38 Different formulations (2) TC formulation, implicit chemistry dt + LC = 0 dt T C F = 0 F Ψ(T ) = 0 + Non-intrusive approach (chemistry as black box) + Precipitation can (probably) be included No explicitjacobian (finite differences) One chemical solve for each function evaluation L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 25 / 41

39 Standard iterative algorithm Block Gauss Seidel method Transport C n+1,k+1 + F n+1,k T n + L(C n+1,k+1 ) = 0, t T n+1,k+1 = C n+1,k+1 + F n+1,k, Chemistry F n+1,k+1 = Ψ(T n+1,k+1 ) Yeh Tripathi (1989), Saaltink et al. (2001), Carrayrou (2001), Dimier, Montarnal et al. (2004). L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 26 / 41

40 DAE approach for CC Coupled system is index 1 DAE (J. Erhel, C. de Dieuleveult) M dy dt + f (y) = 0 L C I 0 I ( C ) y = z, M = 0 0 0, f (y) = C + F H(z) W F F F (z) Use standard DAE software L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 27 / 41

41 Global method for TC C n+1 C n + F n+1 F n + L(C n+1 ) = 0 t t T n+1 = C n+1 + F n+1 F n+1 = Ψ(T n+1 ) C n+1 F n+1 + (I + tl)c n+1 C n F n f T = F n+1 T n+1 C n+1 F n+1 F n+1 Ψ(T n+1 ) = 0. Solve by Newton s method L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 28 / 41

42 Structure of Jacobian matrix (I + tl) 0 I Jacobian : f (C, T, F ) = I I I 0 Ψ (T ) I Ψ (T ) jacobian of chemistry Storage of jacobian matrix is expensive, size of matrix is 3N x N c 3N x N c L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 29 / 41

43 Newton Krylov method Solve the linear system by an iterative method GMRES, TFQMR and BiCGStab require only jacobian matrix by vector products. Can be approximated by finite differences or computed analytically. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 30 / 41

44 Newton Krylov method Solve the linear system by an iterative method GMRES, TFQMR and BiCGStab require only jacobian matrix by vector products. Can be approximated by finite differences or computed analytically. Inexact Newton Approximation of the Newton s direction: f (x k )d + f (x k ) η f (x k ) (0<η<1) Choice of the forcing term η? Keep quadratic convergence (locally) Avoid oversolving the linear system η = γ f (x k ) 2 / f (x k 1 ) 2 (Kelley, Eisenstat and Walker) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 30 / 41

45 Newton Krylov method (2) Computing the jacobian-vector product f f (y + hw) f (y) (y)w h Choice of h? h = 10 7 x w (Kelley). Outstanding issue: preconditioning References Hammond, Valocchi, Lichtner (CMWR, 2002) Knoll, Keyes (JCP, 2004) Mousseau, Knoll (JCP, 2004) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 31 / 41

46 Outline 1 Modeling Chemical phenomena Chemical phenomena in aqueous chemistry Modeling equilibrium systems Sorption models 2 Multispecies equilibrium reactive transport Chemical problem Transport equations The coupled system Coupling algorithms 3 Numerical results Pyrite test case Chromatography Simplified MoMaS benchmark L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 32 / 41

47 Pyrite test case 4 components, 39 aqueuous and 13 fixed species Somewhat artificial without precipitation Use DAE software (Variable time step and order. In Matlab, function ode15s modified to use UMFPACK). C. de Dieuleveult s thesis (INRIA/Andra) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 33 / 41

48 Example: ion exchange Column experiment (Phreeqc ex. 11, Alliances ex. 3) Column contains a solution with 1mmol Na, 0.2mmol K and 1.2mmol NO 3. Inject solution with 1.2mmolCaCl 2. CEC = Chemical system: Ca X CaX 2, log K 1 = 0.8, Na + + X NaX, log K 2 = 0, K + + X KX, log K 3 = 0.7 L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 34 / 41

49 Chromatography: result Concentration at end of column Concentrations (mmol/l) Ca Na Cl K Without diffusion, can be solved semi-analytically (Appelo et al.). Extension to diffusive case? Temps (h) L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 35 / 41

50 Algorithm performance 2000 Comparaison of CPU time 1800 CPU time (second) Global approach Splitting appraoch Iterative Splitting appraoch As mesh is refined Number of Newton iterations remains stable Number of GMRES iterations grows number of nodes For each time step: Block Gauss-Seidel iterations, Newton Krylov 4-7 Newton steps, GMRES steps L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 36 / 41

51 Simplified MoMaS benchmark One dimension, diffusive regime, shorter time period Easy chemistry, Morel tableau X 1 X 2 X 3 X 4 S K C C C C C CS CS Initial: X 2, X 4, inject X 1, X 2, X 3, flush with X 2, X 4. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 37 / 41

52 Simplified benchmark : results Split x = 1/30, t = 2. Split x = 1/50, t = 0.5. Global x = 1/30, t = 2. L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 38 / 41

53 Benchmark: performance Nonlinear residual Nonlinear convergence Residual norm Nonlinear convergence Outer iteration For each time step: 8-9 Newton iterations Function evaluations Number of inner iterations increases for each further Newton iteration function evaluations L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 39 / 41

54 MoMaS benchmark proposal Written by J. Carrayrou (IMFS). International Scientific Committee 1D and 2D geometries 3 levels of chemical difficulty Easy Equilibrium, ion exchange Medium More species, kinetics Hard Precipitation dissolution L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 40 / 41

55 Conclusions Perspectives Formulation of reactive transport within mathematical framework Implementation of Newton Krylov algorithm Preliminary performance tests L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 41 / 41

56 Conclusions Perspectives Formulation of reactive transport within mathematical framework Implementation of Newton Krylov algorithm Preliminary performance tests Areas for further work Numerical analysis of algorithms? Comparison of different formulations, and algorithms Handling of precipitation reactions Handling of per species diffusion coefficient Extension to multi-phase flows L. Amir, M. Kern Coupling transport and chemistry GT Méth. numériques LJLL 41 / 41