Reactive Transport in Porous Media

Transcription

1 Reactive Transport in Porous Media Daniel M. Tartakovsky Department of Mechanical & Aerospace Engineering University of California, San Diego Alberto Guadagnini, Politecnico di Milano, Italy Peter C. Lichtner, Los Alamos National Laboratory

2 Outline 1. Mathematics of Chemical Reactions 2. Uncertainty Quantification & Stochastic PDEs 3. PDF Methods 4. Upscaled effective reaction rate 5. Conclusions

3 Mathematics of Chemical Reactions Any general reversible chemical reaction with n reacting species A 1, A 2,...A n can be represented as α 1 A 1 + α 2 A α m A m α m+1 A m α n A n The concentration of species A i is denoted by C i [A i ], i = 1,..., n; C = mass of species volume of solution = [M] [L 3 ] Each reaction process satisfies a (nonlinear) rate equation dc i dt = R(C 1, C 2,...C n ), i = 1,..., n

4 Reactive Transport Transport equations: Mass conservation Advection Molecular diffusion Kinematic dispersion Chemical reaction ω C i t = (D C i) (uc i ) + R(C 1,..., C n ), i = 1,..., n

5 Example 1: Adsortption The mass concentration of the adsorbed substance: C 2 = mass of adsorbed solute unit mass of solid The mass of solids in a unit volume of a porous medium: (1 ω)ρ s = unit mass of solids unit volume of porous media The mass of a substance bound to solids in a unit volume: (1 ω)ρ s C 2

6 Example 1: Adsorption (cndt.) The change in mass per unit volume per unit time: If adsorption is instantaneous, C 2 R = (1 ω)ρ s, (R 0) t C 2 K d C 1, K d the distribution coefficient Transport equation (retardation coefficient), ωr C 1 t = (D C 1) (uc 1 ), R c = ω ω ρ sk d

7 Example 2: Dissolution Dissolution αa A (s), C [A] Reaction rate equation (effective reaction rate) Transport equation R dc dt = αk(cα C α eq) ω C t = (D C) (uc) αk(cα C α eq)

8 Reaction coefficients Relation to the structure of a porous medium K d, R c, k = F 1 (surface areas) = F 2 (pore structure) Effective reaction rate k k 0 a, a (1 ω)λ l k 0 laboratory measured rate constant a specific surface area λ roughness factor l characteristic grain size

9 Outline 1. Mathematics of Chemical Reactions 2. Uncertainty Quantification & Stochastic PDEs 3. PDF Methods 4. Upscaled effective reaction rate 5. Conclusions

10 Uncertainty in Environmental Modeling Wisdom begins with the acknowledgment of uncertainty of the limits of what we know. David Hume (1748), An Inquiry Concerning Human Understanding Most physical systems are fundamentally stochastic (Wiener, 1938; Frish, 1968; Papanicolaou, 1973; Van Kampen, 1976): Model & Parameter uncertainty Heterogeneity Lack of sufficient data Measurement noise Experimental errors Interpretive errors Randomness as a measure of ignorance

11 Alternative Frameworks for UQ Ignore parameter uncertainty Fuzzy logic Interval mathematics Probabilistic/stochastic approaches Geostatistics Monte Carlo simulations Moment equations PDF methods Etc. Upscaling (homogenization)

12 Stochastic Framework Reaction rate k(x) Random field k(x, ω) Ergodicity Transport equations become stochastic Solutions are given in terms of PDFs

13 Probabilistic Description of k Relationship between the distributions of the grain size l and effective rate constant k: p k (k)dk p l (l)dl Cubic grains p k (k) = 6λφ sk 0 k 2 p l Spherical grains p l (l) lσ 2 l ( ) 6λφs k 0 k Normalized distribution (pdf) of k 1 e (ln l µ l) 2 /(2σl 2 ) π Normalized k

14 Stochastic Upscaling Meso-scale transport equation: c t = (vc) + f k(c), f k = αk(c α C α eq) Uncertain (random) parameters: velocity v & reaction rates k Effective transport equation: Reynolds decomposition c t = (v effc) αk eff (c α C α eq) k = k + k, k kp k (k)dk, k = 0 Stochastic upscaling vs. deterministic upscaling

15 Traditional Approaches Effective transport equation: Note that c t = (v effc) αk eff (c α C α eq) k eff k k eff = k eff (k, σ 2 k, l k,...) Standard approaches: k eff k Linearization f κ (c) = f κ (c) +...

16 Outline 1. Mathematics of Chemical Reactions 2. Uncertainty Quantification & Stochastic PDEs 3. PDF Methods 4. Upscaled effective reaction rate 5. Conclusions

17 PDF Approach Motivation To avoid the linearization To obtain complete statistics Raw distribution: Π(c, C; x, t) δ(c(x, t) C) Probability density function (PDF): p c (C; x, t) = Π(c, C; x, t) Concentration moments: c n (x, t) = C n p c (C; x, t)dc

18 PDF Equation & Closures Stochastic PDE for the raw distribution in R 4 : x = (x 1, x 2, x 3, C) T Π t = (ṽπ) ṽ = (v 1, v 2, v 3, f κ ) T Deterministic PDE for PDF Closure approximations p t = ( ṽ p) ṽ Π Direct interaction approximation (Kraichnan, 1987) Large-eddy simulations (Koch and Brady, 1987) Weak approximation (Neuman, 1993) Closure by perturbation (Cushman, 1997)

19 Outline 1. Mathematics of Chemical Reactions 2. Uncertainty Quantification & Stochastic PDEs 3. PDF Methods 4. Upscaled effective reaction rate 5. Conclusions

20 Example: One-Dimensional Batch System Mesoscale transport equation Exact solution for PDF dc dt = ακ(cα C α eq) c(0) = C 0 Find c[κ(y )], where Y is multivariate Gaussian Find p c (C) = (dy/dc)p(y) Probability Density Function pc(c,t) t=0.01 t=0.1 t=0.25 t=0.5 t=1 t=1.25 t= C

21 Effective (Upscaled) Rate Constant Find mean concentration c(t) Define k eff as the coefficient in dc dt = αk eff ( c α C α eq) α = 1 α = 2 Upscaled reaction rate, k eff k eff (0) = k Time, t

22 Perturbation Expansion for PDF Gaussian mapping, e.g., κ = f(y ) where Y is Gaussian Small parameter σ 2 Y p = p (0) + p (1) +... p t = ( ṽ p) ṽ Π p(0) t = [ ṽ (0) p (0)] Randomness in initial and boundary conditions is easily incorporated Solution for the concentration PDF (à la Kubo expansion) p(c; t) = dy dc n=0 σ 2n Y (2n)!! δ(2n) (Y Y )

23 Mean Concentration for Linear Kinetics (α = 1) Mild heterogeneity (σ 2 Y = 0.5) 1 Normalized mean concentration n = 0 n = 1 n = 2 n = 3 n = Dimensionless time

24 Mean Concentration for Linear Kinetics (α = 1) Mild-to-moderate heterogeneity (σ 2 Y = 1.0) 1 Normalized mean concentration n = 0 n = 1 n = 2 n = 3 n = Dimensionless time

25 Mean Concentration for Non-linear Kinetics α = 2 Mild heterogeneity (σ 2 Y = 0.5) 2 Normalized mean concentration n = 0 n = 1 n = 2 n = 3 n = Dimensionless time

26 Mean Concentration for Non-linear Kinetics α = 2 Mild-to-moderate heterogeneity (σ 2 Y = 1.0) 5 Normalized mean concentration n = 0 n = 1 n = Dimensionless time

27 LED Approximation of PDF Equations Transport equation c t = (vc) + f κ(c) in R 3 Unclosed PDF equation p t = ( ṽ p) ṽ Π in R 4 Large Eddy Diffusivity (LED) closure p t = ũ i p + [ ] p D ij x i x j x i Dispersion coefficient t D ij ( x, t) ṽ i( x)ṽ j(ỹ) G( x, ỹ, t τ)dỹdτ 0 Ω

28 Lagrangian-Eulerian Description Governing equations: c t = (vc) + f k(c) Π t = (ṽπ) Random Green s function G( x, ỹ, t τ) Explicit expression G τ + ṽ ỹg = δ( x ỹ)δ(t τ) G( x, ỹ, t τ) = [1 H(τ t)]δ(ỹ x L ), x L = x + ṽ(t τ) Mean Green s function: G( x, ỹ, t τ) = G( x, ỹ, t τ)

29 Effective parameters Normalized dispersion coefficient Exponential Gaussian Spherical Normalized effective velocity Exponential Gaussian Spherical Normalized time Normalized time 7 2 Exponential Gaussian Spherical 6 Exponential Gaussian Spherical Normalized dispersion coefficient Normalized effective velocity Normalized time Normalized time

30 Conclusions Most physical systems are under-determined due to Heterogeneity Lack of sufficient data Measurement noise Wisdom begins with the acknowledgment of uncertainty of the limits of what we know. David Hume (1748), An Inquiry Concerning Human Understanding