Anomalous Dispersion Particle tracking velocimetry and image processing for classical and non-classical dispersion theories J. Ramirez, T. Weinstein, S. Harrington, K. Bardsley, Y. Wu, A. Cagnioncle and L. Donado John Cushman, Monica Moroni and Natalie Kleinfenter Anomalous Dispersion OUTLINE Introduction 1. Mathematical Theory 2. 3. 4. Group 6 Experiments Objectives Image Processing

Classical Dispersion Generalized Dispersion Data Analysis Summary & Conclusions Summer School in Geophysical Porous Media 2 Anomalous Dispersion Experimental Setup Group 6 Summer School in Geophysical Porous Media 3 Anomalous Dispersion Objectives Group 6 Determine if and when dispersion becomes Fickian

(classical) using Particle Tracking Velocimetry (PTV) Calculate the generalized dispersion coefficient Other Questions At what scale is the medium homogeneous? At what scale is the medium heterogeneous? Summer School in Geophysical Porous Media 4 Anomalous Dispersion Original Photos Group 6 Summer School in Geophysical Porous Media 5

Anomalous Dispersion Filtered Noise Group 6 Summer School in Geophysical Porous Media 6 Anomalous Dispersion Final Image for Analyses Group 6 Summer School in Geophysical Porous Media 7 Anomalous Dispersion Centroid Tracking t3 t2 t1 toll t0 Dmax

Group 6 Summer School in Geophysical Porous Media 8 Anomalous Dispersion Trajectories Group 6 Summer School in Geophysical Porous Media 9 Anomalous Dispersion OUTLINE Introduction 1. Mathematical Theory 2.

3. 4. Group 6 Experiments Objectives Image Processing Classical Dispersion Generalized Dispersion Data Analysis Summary & Conclusions Summer School in Geophysical Porous Media 10 Anomalous Dispersion Lagrangian Description of Dispersion Position and velocity at time t of a particle initially located at x0 X (t) = X (t;x0) 2 R3; dX (t) V (t) = : dt

Displacements: Y (t) X (t) X (0) Group 6 Summer School in Geophysical Porous Media 11 Anomalous Dispersion Two assumptions: T hemotion of theparticleisdictated by t-independent transition probabilities f : Z P(X (t+) 2 A) = f (X (t);x; ) dx; for all t > 0 A Steady-state assumption hV (t)i = hV i Stationarity (Homogeneity) Group 6 Summer School in Geophysical Porous Media 12 Anomalous Dispersion Lagrangian Description of Dispersion

Probability distribution of displacements: Zt Y (t) = X (t; x0) x0 = V (s; x0) ds 0 G(y; t) = P(Y (t) = y) = h(y (X (t) X (0)))i Z f (x0; X (t); t) P(X (0) = x0) dx0 = Group 6 Summer School in Geophysical Porous Media 13 Anomalous Dispersion Dispersion: Classical Approach Y (t) = hY i + Y 0(t); V (t) = hV i + V 0(t) Zt Y 0(t) = V 0(s; x0) ds hY (t)i = hV i t; dh(Yi0(t))2i dt

0 d 0 2 (Yi (t)) = 2hYi0(t)Vi0(t)i = dt Z t Vi0(s)Vi0(t) ds = 2 0 Zt Zt CV 0(s) ds hVi0(s)Vi0(t)i ds = 2 = 2 i 0 Group 6 Summer School in Geophysical Porous Media 0 14 Anomalous Dispersion

dh(Yi 0(t))2i dt Z =2 0 t CV 0(s) ds i as t ! 0: h(Yi0(t))2i htVi 0(0) tVi0(0)i = hV 02i t2 as t ! 1 : dh(Yi 0(t))2i dt Group 6 Z 2 0 Fickian 1 CV 0(s) ds 2D i i i Summer School in Geophysical Porous Media

15 Anomalous Dispersion Non-Classical Approach: Generalized Dispersion Coefficient Dispersion \ at scale" k: dA(t) = iL A(t); dt A k (t) = ei kX (t) ; iL = V r x WHAT DA?: D ^ t) = G(k; ei k X (t) e Group 6 i k X (0) E D

E i k X (t) ei k X (0) e = CA k (t) Summer School in Geophysical Porous Media 16 Anomalous Dispersion From projection operator theory: there exists a memory kernel K A k such that, dCA k = ihL A(0); A(0)i CA k (t) dt Z 0 t K^A k () CA k (t ) d Zt ^ dG(k; t) ^ t) ^ K^A k () G(k;t ) d = ik hV i G(k;

dt 0 Group 6 Summer School in Geophysical Porous Media 17 Anomalous Dispersion Generalized dispersion coecient: ^ ) k = K^A () k D(k; k Lets check it out Zt ^ dG(k; t) ^ t) ^ )k] G(k; ^ t ) d [kD(k; = ikhV i G(k; dt 0 Z tZ dG(x; t) D(y; )r G(x y; t ) dy d = hV i r G(x; t)+r dt 0

Group 6 Summer School in Geophysical Porous Media 18 Anomalous Dispersion ? Estimating ^ ) k K A k = k D(k; Volterra equation for K A k Zt 2 ^ ^ @ G @ G ^A (t) K^A k (t) = (k; t) K (k; t ) d k

2 @t @t 0 D ^ t) = ei kX (t) e G(k; Group 6 i kX (0) E Summer School in Geophysical Porous Media From data 19 Anomalous Dispersion OUTLINE Introduction 1. Mathematical Theory

2. 3. 4. Group 6 Experiments Objectives Image Processing Classical Dispersion Generalized Dispersion Data Analysis Summary & Conclusions Summer School in Geophysical Porous Media 20 Anomalous Dispersion Velocity Correlation coefficients in the transverse and longitudinal directions for two different mean velocities 1.2 1 Transverse Longitudinal

0.8 Transverse 0.6 Longitudinal 0.4 0.2 0 -0.2 1 3 5 7 9 11 13 15 17 19

21 -0.4 -0.6 Time lag (s) Group 6 Summer School in Geophysical Porous Media 21 Anomalous Dispersion Mean Velocity Transverse direction hV (t)i 0 -0.01 0 5 10 15 20 25 t

-0.02 -0.03 -0.04 -0.05 v4 v2 -0.06 -0.07 Group 6 Summer School in Geophysical Porous Media 22 Anomalous Dispersion Correlation of velocity Transverse direction hV (t)V (t + )i Group 6 Summer School in Geophysical Porous Media 23

Anomalous Dispersion Displacement variance $\la (Y_z'(t))^2 100$ ra} Longitudinal direction \ pe o l S 10 1 1 pe o Sl = 2. .0 1 =

t 0 10 V2 V4 100 0.1 Variance of displacement Group 6 Summer School in Geophysical Porous Media 24 Anomalous Dispersion Intermediate scattering function Transverse direction k= ^ )) Re(G(k; increases

1.2 1 0.8 0.6 0.4 0.2 0 1 Group 6 2 2 d 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 d=0.05 d=0.1 d=0.4 d=0.8 d=1.4

d=2.0 d=6.0 d=9.0 d=13.0 d=19.0 Summer School in Geophysical Porous Media 25 Anomalous Dispersion Intermediate scattering function ^ )) I m(G(k; Transverse direction 0.08 0.06 0.04 0.02

0 -0.02 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -0.04 -0.06 d=0.05 d=6.0 Group 6 d=0.1 d=6.0 d=0.4 d=9.0 d=0.8 d=13.0 d=1.4 d=19.0 d=2.0 Summer School in Geophysical Porous Media d=2.0 d=2.0 26

Anomalous Dispersion Intermediate scattering function ^ )) Re(G(k; Longitudinal direction 1.5 1 0.5 0 -0.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

-1 d=0.05 d=2.0 Group 6 d=0.1 d=6.0 d=0.4 d=9.0 d=0.8 d=13.0 Summer School in Geophysical Porous Media d=1.4 d=19.0 27 Anomalous Dispersion Intermediate scattering function Longitudinal direction ^ )) I m(G(k; 1 0.8 0.6 0.4

0.2 0 -0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 -0.4 -0.6 d=0.05 Group 6 d=0.1 d=0.4

d=0.8 d=1.4 d=2.0 d=6.0 Summer School in Geophysical Porous Media d=9.0 d=13.0 k= d=19.0 2 d 28 Anomalous Dispersion Generalized dispersion coefficient Transversal direction ^x (k; ) D 0.12 0.1

0.08 0.06 0.04 0.02 0 -0.02 0 4 8 12 16 20 -0.04 -0.06 -0.08 -0.1 d=0.05 d=0.4 d=1.4 d=6.0

d=13.0 k= 2 d Generalized dispersion tensor should equal to the velocity covariance in the Fickian limit. Group 6 Summer School in Geophysical Porous Media 29 Anomalous Dispersion Generalized dispersion tensor longitudinal direction ^z (k; ) D k= 2 d increases Group 6

Summer School in Geophysical Porous Media 30 Anomalous Dispersion OUTLINE Introduction 1. Mathematical Theory 2. 3. 4. Group 6 Experiments Objectives Image Processing Classical Dispersion Generalized Dispersion Data Analysis

Summary & Conclusions Summer School in Geophysical Porous Media 31 Anomalous Dispersion Summary & Conclusions Examined a mathematical theory aimed at describing non-Fickian (anomalous) dispersion. Analyzed experimental data to examine the mean, variances, classical and non-classical measures of dispersion. From that analysis, we concluded that on the observed spatial scales the transport is anomalous even though the medium is homogeneous. Other measures of dispersion are needed to describe anomalous dispersion. Group 6 Summer School in Geophysical Porous Media

32 Anomalous Dispersion 5. References Moroni M, Cushman JH Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. II. Experiments PHYSICS OF FLUIDS 13 (1): 81-91 JAN 2001 Cushman JH, Moroni M Statistical mechanics with three-dimensional particle tracking velocimetry experiments in the study of anomalous dispersion. I. Theory PHYSICS OF FLUIDS 13 (1): 75-80 JAN 2001 Moroni M, Cushman JH, Cenedese A A 3D-PTV two-projection study of pre-asymptotic dispersion in porous media which are h eterogeneous on the bench scale INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE 41 (3-5): 337-370 FEB-MAR 2003 Monica Moroni, Natalie Kleinfelter and John H. Cushman Analysis of dispersion in porous media via matched-index particle tracking velocimetry experiments ARTICLE ADVANCES IN WATER RESOURCES. In Press, Corrected Proof, Available online 31 March 2006

IGUCHI K STARTING METHOD FOR SOLVING NONLINEAR VOLTERRA INTEGRAL-EQUATIONS OF OF SECOND KIND COMMUNICATIONS OF THE ACM 15 (6): 460& 1972 Group 6 Summer School in Geophysical Porous Media 33 Anomalous Dispersion Questions?? Group 6 Summer School in Geophysical Porous Media 34