University of Nevada

Reno

The Fractional Advection--Dispersion Equation:

Development and Application

A dissertation submitted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy in Hydrogeology

by

David Andrew Benson

Stephen W. Wheatcraft, Dissertation Advisor

May 1998

E 1997, 1998

David Andrew Benson

All Rights Reserved

i

The dissertation of David Andrew Benson is approved:

Dissertation Advisor

Department Chair

Dean, Graduate School

University of Nevada

Reno

May 1998

ii

Dedicated to my father, who taught me how to think about the world,

and to my mother, who taught me how to live in it.

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ACKNOWLEDGEMENTS

No metaphysician ever felt the deficiency of language so much as the grateful.

-- Charles Caleb Colton, Lacon

Oddly, I must first thank my Master’s advisor, David Huntley. When I told him I was considering a Ph.D.,without hesitation he told me to go to UNR and talk to Steve Wheatcraft. I have never received more sageadvice. I went in March 1993, and I had the strange and pleasant feeling that I was not only accepted to theprogram, but I was being actively recruited. I must thank Dr. John Warwick for his part in that feeling. I’malso glad to thank Dr. Warwick for financial and philosophic help over the years. I only wish we had beenin the same building.

From the beginning, Steve Wheatcraft has pushed but never prodded, taught but never instructed, enthusedbut never gladhanded. His humility is endless and his door is never closed. This dissertation was clearlyoutside of my capabilities a few years ago, but I believed in its virtue because he did. Nobody else on earthwho could have planted this seed in my head and had it come to fruition, so I thank him. One of his colleaguessays that every professor should graduate a total of three Ph.D. students -- one to continue his work, one toadvance the science, and one to replace the teacher. I can only say that I am very lucky Steve didn’t followthis piece of advice, since I am number nine.

The yeoman of my committee was Mark Meerschaert. There is no question that this document would not existwithout his help. By shear dumb luck I figured something out about hydrogeology and a member of my com-mittee is an expert in that subject of mathematics. I can’t decide whether to name my first child Levy orMeerschaert. While on the subject of mathematics, I wish to acknowledge the fantastic courses I took (ormerely sat in on) from Jeff McGough. I learned more in those classes than any others I took here at UNR.I hereby officially urge all students at UNR to rely on the valuable resources in the form of Drs. Meerschaertand McGough.

The other members of my committee -- Scott Tyler, Britt Jacobson and Katherine McCall -- did many thingsfor me, not the least of which was to remind me of all of the things that I don’t know or understand. I appreci-ate the time they spent helping me.

I sincerely thank all of the students in the Hydrologic Sciences program. First, the students maintain the highquality of the program and make all of our degrees more valuable. Second, the reputation of the programand hard work of the students bring the best speakers in the world to our campus. I have gained very muchfrom interaction with visiting speakers. Third, my interaction with fellow students has added more refine-ment to the ideas presented in this dissertation than could possibly come frommy ownhead. Being mysound-ing board is an unenviable chore, so I give special thanks to fellow students and colleagues Dr. Anne Carey,Dr. Hongbin Zhan, Dr. Greg Pohll, Maria Dragila, and even Joe Leising.

I thank the Desert Research Institute (DRI) and the generosity of Elizabeth Stout for financial support in theform of the George Burke Maxey fellowship. I also thank the U.S.G.S. and the Mackay School of Minesfor their generosity in the form of scholarships. Thanks also to Dave Prudic and Kathryn Hess at the U.S.G.S.for delivering the Cape Cod data. DRI also paid my salary when I taught the last hurrah of Geol 785 -- Ground-water Modeling. I’m sure I learned more from the students -- Rina Schumer, Dave Decker, David McGraw,and Marija Grabaznjak than I got across to them.

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Many of my old friends kept me in--touch (and in cheap digs!) during this long process, so I thank Tom, Chris,Greggy and Strato, Don (thanks for the deal on the scooter, yeah right!), Laura, and John and Laura.

My mother is the smartest person I know. She should have been the U.S. ambassador to the U.N., but shechose the difficult path of being the mother of her children. Her calm and levelheaded support through theyears has been inspirational. I hope I am able to give one--tenth as much as I received. I must also shatterthe cliche and thank my wife’s parents, Doug and Kathy Guinn, for their unflagging encouragement. Theyare models of thinking, caring citizens.

Finally, I thank my wife Marnee for making so many sacrifices; for leaving her dearest friends and the sunnybeaches of San Clemente and postponing her own dreams of higher education. Your time will come and Iwill remember.

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ABSTRACT

The traditional 2nd--order advection--dispersion equation (ADE) does not adequately describe the movementof solute tracers in aquifers. This study examines and re--derives the governing equation. The analysis startswith a generalized notion of particle movements, since the second--order equation is trying to impart Brow-nian motion on a mathematical plume at any time. If particle motions with long--range spatial correlationare more favored, then the motion is described by Lévy’s family of α--stable densities. The new governing(Fokker--Planck) equation of these motions is similar to the ADE except that the order (α) of the highest deriv-ative is fractional (e.g., the 1.65th derivative). Fundamental solutions resemble the Gaussian except that theyspread proportional to time1/α and have heavier tails. The order of the fractional ADE (FADE) is shown tobe related to the aquifer velocity autocorrelation function.

The FADE derived here is used to model three experiments with improved results over traditional methods.The first experiment is pure diffusion of high ionic strength CuSO4 into distilled water. The second experi-ment is a one--dimensional tracer test in a 1--meter sandbox designed and constructed for minimum hetero-geneity. The FADE, with a fractional derivative of order α = 1.55, nicely models the non--Fickian rate ofspreading and the heavy tails often explained by reactions or multi--compartment complexity. The final ex-periment is the U.S.G.S. bromide tracer test in the Cape Cod aquifer. The order of the FADE is shown tobe 1.6. Unlike theories based on the traditional ADE, the FADE is a “stand--alone” equation since the disper-sion coefficient is a constant over all scales.

A numerical implementation is also developed to better handle the nonideal initial conditions of the CapeCod test. The numerical method promises to reduce the number of elements in a typical numerical modelby orders--of--magnitude while maintaining equivalent scale--dependent spreading that would normally becreated by very fine realizations of the K field.

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TABLE OF CONTENTS

PAGE

ACKNOWLEDGEMENTS .......................................................................................... iii

ABSTRACT ................................................................................................................ v

LIST OF FIGURES ...................................................................................................... viii

LIST OF TABLES ................................................................................................... xi

CHAPTER

1 INTRODUCTION ...................................................................................................... 11.1 Notation and Dimensions ............................................................................. 3

2 CLASSICAL THEORY ........................................................................................... 5

2.1 Advection--Dispersion Equation ................................................................... 52.2 Brownian Motion ....................................................................................... 92.3 The Diffusion Equation and Brownian Motion .......................................... 10

3 STABLE LAWS .................................................................................................... 123.1 Characteristic Functions ............................................................................. 123.2 Stable Distributions (Stable Laws) .............................................................. 233.3 Moments and Quantiles .............................................................................. 28

4 PHYSICAL MODEL ............................................................................................. 20

4.1 Lévy Flights -- Discrete Time ...................................................................... 21Lévy Flights -- Continuous Time ........................................................... 26

4.2 Lévy Walks -- Continuous Time Random Walks ......................................... 27Coupled Space--Time Probability .......................................................... 28

4.3 Velocity Statistical Properties ..................................................................... 36

5 THE FRACTIONAL ADVECTION--DISPERSION EQUATION ......................... 46

5.1 Fractional Fokker--Planck Equation .................................................................. 465.2 Solutions .......................................................................................................... 53

6 EXPERIMENTS .................................................................................................... 60

6.1 High Concentration Diffusion ..................................................................... 606.2 Laboratory--Scale Tracer Test ..................................................................... 646.3 Cape Cod Aquifer ....................................................................................... 67

A Posteriori Estimation of Parameters ................................................ 70Analytic Solutions .............................................................................. 72A Priori Estimation of Parameters ...................................................... 75

7 NUMERICAL APPROXIMATIONS ................................................................... 78

7.1 Motivation ................................................................................................ 787.2 Finite Differences ..................................................................................... 79

vii

8 DISCUSSION OF RESULTS ............................................................................... 84

9 CONCLUSIONS AND RECOMMENDATIONS ................................................. 90

9.1 Conclusions .............................................................................................. 909.2 Recommendations .................................................................................... 91

10 REFERENCES .................................................................................................... 92

APPENDICES ........................................................................................................... 96I FORTRAN LISTINGS ................................................................................... 96

I.1 Program SIMSAS.F .................................................................................. 96I.2 Program ENSEM.F .................................................................................. 98I.3 Program AVEGAM.F ............................................................................... 102I.4 Program WEIER.F ................................................................................... 107I.5 Subroutine CFASTD.F ............................................................................. 108I.6 Subroutine DFASTD.F ............................................................................. 111I.7 Program CVX.F ....................................................................................... 114I.8 Program CVT.F ........................................................................................ 116I.9 Program FRACDISP.F .............................................................................. 118

II STABLE LÉVY MOTION CALCULATIONS ............................................... 121II.1 The Green’s Function Chapman--Kolmogorov Equation for

random walks of random duration ....................................................... 121II.2 Exact Solutions for the transformed α--stable densities ....................... 122II.3 Calculation of power--law transition density Fourier/Laplace

transforms ........................................................................................... 123

III VELOCITY AUTOCOVARIANCE OF LÉVY WALKS ................................ 128III.1 Velocity Autocovariance for Lévy Walks with

Lower Cutoff ...................................................................................... 128III.2 Lévy Walks with Converging Autocovariance ..................................... 131III.3 Autocovariance with Velocity Proportional to

Lévy Walk Size ................................................................................... 132III.4 Full α--stable density ........................................................................... 132

IV FRACTIONAL DERIVATIVES AND THEIR PROPERTIES ....................... 134

V FINITE DIFFERENCE APPROXIMATION OF THEFRACTIONAL DERIVATIVE ....................................................................... 139

viii

LIST OF FIGURES

Figure 1.1 Schematic of the techniques used to obtain solutions to generalizedrandom walks ............................................................................................ 2

Figure 2.1 Illustration of the definition of the divergence of solute flux over many scales.The solid lines denote assumption of local homogeneity and multi--scale,integer--order (classical) divergence. Dashed lines denote continuum--heterogene-ity and the resulting noninteger--order divergence. To reconcile the growth in theinteger divergence (using current theories) from scale a to b, the first orderfluctuations v!C! are approximated by DoC with increasing, spatially local D. ... 6

Figure 3.1 Plots of the distribution function F(x) versus x for several standard symmetricα--stable distributions using a) linear scaling and b) probability scaling. TheGaussian normal (α = 2.0) plots as a straight line using probability scaling for thevertical axis. .................................................................................................. 15

Figure 3.2 Plots of symmetric α--stable densities showing power--law, “heavy” tailedcharacter. a) linear axes, and b) log--log axes. .................................................... 17

Figure 3.3 Expectation of the absolute value of random variable X with a standard,symmetric, α--stable distribution for 0 < α < 2. ................................................. 19

Figure 4.1 Lévy flights in two dimensions. ........................................................................ 23

Figure 4.2 Numerical approximation of one--dimensional, continuous--time, randomLévy walks. .................................................................................................. 24

Figure 4.3 Graphs of the Fourier transform of the particle jump probability (the structurefunction) of a “clustered” walk on a discrete lattice. For this graph, the latticespacing (Δ) was set to unit length. Note the good approximation of the completeWeierstrass function by the exponential function for wave numbers smaller thanthe inverse of the lattice spacing (i.e. k

ix

continuous source. For α < 2, the late--time slope on log--log plots is equal to —α.(c) Half of the scaled instant pulse breakthrough. For α < 2, the late--time slope onlog--log plots is equal to —(1+α). .....................................................................37

Figure 4.9 Graphs of a) the Lévy process, b) the velocity function and c) joint probabilitydistribution of jump length as a function of spatial separation. ......................... 38

Figure 4.10 Log--log and linear plots of the analytical and numerical velocity semivariogramfunctions when the velocity is modeled as proportional to Lévy walk size. Thenumerical result is the ensemble mean of 112 realizations of 1000--jump walksusing a stability index (α) of 1.7......................................................................... 41

Figure 4.11 Log--log and linear plots of the velocity semivariogram for large and small valuesof ν. The value of α used in all plots is 1.7. An exponential model, γ =1--exp(3.8ξ) is plotted for comparison. .............................................................. 42

Figure 4.12 Plot of the scaling prefactor Pα for 1 < α < 2. ....................................................44

Figure 4.13 Maximum expected jump size in discrete standard Gaussian versus near--GaussianLévy process with index of stability (α = 1.99). ................................................ 45

Figure 5.1 Integer and fractional derivatives of two simple power functions. Top row: Integerderivatives of f(x) = x2. Middle row: Integer derivatives of g(x) = x2.33. Bottomrow: Fractional derivatives around the point a=0 of g(x) = x2.33. ....................... 47

Figure 5.2 Comparison of the development of spatially symmetric (dashed lines) andpositively skewed (solid lines) plumes represented by a) continuous source and b)pulse source. Three dimensionless elapsed times (0.1, 1.0, and 10) are shown. Asα gets closer to 2, the skewing diminishes. All curves use α = 1.7 and D = 1.................................................. 54

Figure 6.1 Idealized schematic representation of diffusion via random walk within a highionic strength, high gradient fluid. The random walk occurs within a partially--occupied network. The probability of a walk toward lower concentration (to theright of the figure) is always higher than into higher concentration, where moresites are occupied by other solute ions. At high enough concentrations, the set ofconnected available sites is non--Euclidean, precluding Fickian diffusion. ......... 61

Figure 6.2 Scaled diffusion profiles from Carey’s (1995) experiment: a) scaled by thetraditional (Fickian) square root of time, and b) scaled by time1/α with α = 2.5.The lower curves are also shifted by a mean flux position of x = 1.97 cm. ..........62

Figure 6.3 Closeup of the low--concentration limb of the scaled diffusion profiles fromCarey’s (1995) experiment: a) scaled by the traditional (Fickian) square root oftime, and b) scaled by time1/α with α = 2.5. The lower curves are also shifted by amean flux position of x = 1.97 cm. ....................................................................63

Figure 6.4 Schematic view of the experimental sandbox tracer tests. The flowpathhighlighted by the arrow is analyzed in detail. ...................................................64

x

Figure 6.5 Calculated dispersivities versus distance of probe from source. The flow pathchosen for analysis is shown with the connecting line. The best--fit dashed lineindicates a fractional dispersion index (α) of 1.55. ............................................. 66

Figure 6.6 Plot of normalized concentration versus scaled time for probe 20, test 3 (Burns,1997). A best fit line (implying an underlying Gaussian profile) is typically usedto calculate the apparent dispersivity. Compare this data with the α--stabletheoretical plots in Chapter 3 (Figure 3.1). ........................................................ 67

Figure 6.7 Measured breakthrough “tails” at probes along the flowpath: a) Rescaled by t1/1.55,b) Rescaled by the traditional t1/2. Note the strong skewness that separates theleading and trailing limbs of the plume. Very early and late data show probenoise. ...............................................................................................................68

Figure 6.8 Comparison of traditional and fractional ADEs with the data from probe 3 (x = 55cm) in the sandbox test: a) real time, and b) data tails. Note the large under--prediction of concentration by the traditional ADE at very early and late time. ..69

Figure 6.9 Aerial view of the Cape Cod Br-- plume. The plume deviated from travelling dueSouth by approximately 8_ to the East. Circles are multi--level samplers (MLSs),diamonds are permeameter core samples, and squares are flowmeter tests. ........ 70

Figure 6.10 Calculated plume variance (Garabedian et al. [1991]) along the direction of meantravel. ...............................................................................................................71

Figure 6.11 Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximumconcentrations along plume centerline. Solid lines are solutions to the FADEusing D = 0.14 m1.6/d and classical ADE using asymptotic Fickian D = 0.42 m2/d.a) Early time data. b) Late--time data. Sample times (in days) are shown abovepeaks. ...............................................................................................................73

Figure 6.12 Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximumconcentrations along plume centerline. Solid lines are solutions to FADE andclassical ADE using identical (early--time) dispersion coefficients of 0.13. a) Earlytime data. b) Late--time data. Sample times (in days) are shown above peaks. ... 74

Figure 6.13 Semi--log plots of the plume profile modeled (solid lines) and measured (symbols)at 349 days. (a) Maximum concentration in the y--z plane and (b) average ofvertical samples from the same MLS that from which the maximumconcentrations were measured. The smaller average uses zero for non--detectableconcentration, while the larger average ignores those data. ................................. 76

Figure 6.14 Theoretical dimensionless velocity semivariogram for α = 1.4 and α = 1.8. ...... 77

Figure 7.1 Numerical solution of the FADE with α = 1.6 for a series of times: a) log--logaxes, and b) linear axes. Initial conditions were a “point source” of unit mass atthe node located at x0. The solutions follow the scaling law of the analyticsolution. Note the oscillatory error at the extreme tail ends. ..............................81

Figure 7.2 Comparison of analytic (lines) versus numerical (symbols) solutions of the FADEwith “point source” initial condition. In all solutions, D, t, and Δx set to unity. 82

xi

Figure 7.3 Numerical and analytic solutions of the FADE compared to Cape Cod Br-- plume:a) linear axes, and b) semi--log axes. Numerical model used Δx = 1.0 m and Δt =0.1 days. Both models used α = 1.6 and D = 0.14. Note better fit of thenumerical solution at 13 and 55 days. ................................................................83

Figure 8.1 Comparison of the plume growth predicted by the traditional ADE (ADE), Gelharand Axness (1983) (GA), the fractional ADE (FADE), Mercado (1967) stratifiedflow (M), and Wheatcraft and Tyler (1988) fractal tortuosity model (WT). Theordinate log(Xc2) is roughly equivalent to estimated plume variance. The GAcurve has slope 2:1 at a plume’s origin, transitioning to Fickian 1:1 slope at latetime. ...............................................................................................................85

Figure 8.2 a) Possible values of the velocity parameter (dashed lines) in Carey’s (1995)diffusion experiment. Probable particle behavior as a function of increasingconcentration is shown by the arrow. Variance exponent (VAR ∝ tη) for arrowpath α→ 2 predicted by b) variance equation (4.51) and c) the propagatorequation (4.45). ......................................................................................87

Figure A2.1 Particle travel distance variance when mixed sk2 terms are included in the small--kapproximation of the Laplace--Fourier transformed conditional Lévy walkprobability p(r,t). The dashed line indicates results from simplified form used byBlumen et al. (1989). ....................................................................................125

Figure A3.1 Pareto distribution with lower cutoff. .............................................................. 128

Figure A4.1 Plots of the functions Γ(x) and 1/Γ(x) for 0 < x < 4. Note that n! = Γ(n+1). ....136

Figure A5.1 Flux as a power function of gradient. This is the basis for the numericalapproximation implemented in Chapter 7. ....................................................... 142

LIST OF TABLES

Table 5.1 Error function SERFα(Z) of the symmetric stable distributions for a range of αfrom 1.6 to 2.0. ......................................................................................56

Table 5.2 Error function SERFα(Z) of the symmetric stable distributions for a range of αfrom 0.9 to 1.5. ..................................................................................... 58

Table A5.1 Comparison of analytic and numerical fractional derivatives of power functions.The power function is f(x) = x--α--1 and the derivative is of order α--1. .............. 143

1

CHAPTER 1

INTRODUCTION

When n is a positive integer and if p should be a function of x, the ratio dnp to dxn

can always be expressed algebraically. Now it is asked: what kind of ratio can bemade if n be a fraction? The difficulty in this case can easily be understood. For ifn is a positive integer, dn can be found by continued differentiation. Such a way, how-ever, is not evident if n is a fraction. But the matter may be expedited with the helpof the interpolation of series as explained earlier in this dissertation.

-- Euler (1730)

This study examines the governing equation that is traditionally used to model the movement of dissolvedsolutes in aquifers. The classical governing equation is based on the diffusionequation, whichuses thedefini-tion of divergence. In order for the equation to be defined, a number of assumptions must remain valid. Pri-mary among these is that the dispersion of particles due to differences in velocity should exist as a controlvolume shrinks to zero. Since the velocity fluctuations only arise from disparate aquifer material at a scalethat is large compared to the observation or measurement scale, the classical derivatives in the advection--dis-persion equation are not well--defined. As a result, the classical governing equation does not fully explainthe movement of solutes, and the equation’s parameters are thought to “scale,” or grow larger, with distance.A great deal of effort, in the form of hundreds or thousands of articles, has been expended to explain the scal-ing of parameters. Far less work has been done examining the structure of the governing equation, especiallythe suitability of the differential equation itself. At issue is the structure of the second--order diffusion equa-tion used to model solute spreading as a plume moves. This equation uses mathematical operators whosehidden assumptions are violated when used to model the macroscopic process of solute spreading.

An alternate approach to scaling the parameters is to reformulate the governing equations. Since the equa-tions are models of some underlying process, a good starting point is to generalize the process of solute trans-port to include motions that deviate significantly from the Brownian motion modeled by the diffusion equa-tion. This study examines and generalizes the underlying physical model used to derive the equations ofsolute movement. The generalized motions lead to a new governing equation that uses fractional--order, rath-er than the typical integer--order, derivatives.

Solute transport in subsurface material also can be viewed as a purely probabilistic problem. This viewpointis intimately tied to the classical divergence (Eulerian) point of view through a string of mathematical equiva-lences. Einstein (1908) first explored this method by assuming that a single microscopicparticle wascontinu-ously bombarded by other particles, resulting in a random motion, or a random walk. By taking appropriatelimits (letting nx and nt of the discrete walks go to zero), he found that the resulting Green’s function ofthe probability of finding a particle somewhere in space was a Gaussian (Normal) probability density. TheGreen’s function solution is used to specifies an initial condition of a single particle at the origin. If the mo-tions of a large number of particles are assumed independent, then the particle probability and the concentra-

2

tion of a diffusing tracer are interchangeable. One can also solve the parabolic “diffusion” equation Ct = Cxxand arrive at the same Gaussian Green’s function for a “spike” of tracer placed at the origin.

A series of uniqueness arguments leads to the conclusion that a diffusion equation implies all of the assump-tions of Einstein’s Brownian motion. The most important of these is that Brownian motion implies that aparticle’s motion has little or no spatial correlation, i.e. long walks in the same direction are rare. In orderto use the ADE for spatially correlated velocity fields, a correction is used that forces more dispersion thanthe diffusion equation provides. This is the basis for a scale--dependent, continuously evolving, effectivediffusion coefficient (the dispersion tensor) used to better match the spread of real plumes. Yet the underlyingequation is trying to impart a Gaussian profile (the Green’s function) on a plume at any moment in time. Aquestion naturally arises: What are the equations that describe particle motions with long--range spatial cor-relations? The answer relies on fractional calculus and a class of probability densities first described by Lévy.These stable densities are a superset of the familiar Gaussian and are often called α--stable or Lévy--stable.

This study endeavors to do four important things. First, it is a catalogue of many mathematical techniquesand concepts that are relatively new to the field of hydrology. It is hoped that this text can serve as a stand--alone repository of information related to fractional calculus, Lévy’s Stable Laws, and current techniques inrandom walk studies. For this reason, many derivations that can be found elsewhere are included. Second,this text seeks to unify the derivations from various fields of science and mathematics and provide a standardset of symbology and notation useful to hydrogeologists and others. A number of errors were found duringthe translation and re--derivation process. Many articles refer to erroneous prior results, making an indepen-dent trek through the literature somewhat arduous. Corrections have been made along with the “cataloguing”effort. Third, in the derivation of the governing equation for particle movement in aquifers, this study hasattempted to unify the techniques, concepts, and results of various prior studies (Figure 1.2). By starting withan underlying model of random particle movements that is a superset of traditional Brownian motion, theend result is a generalization of the concept of divergence. In the process, a new and correct derivation ofa fractional ADE is made. Finally, the match between the new theoretical models and experimental data is

Fokker--Planck(advection--dispersion)

Equation

Generalized random walks:Markov Process

Convolution

Fourier/Laplacetransforms

Instantaneousapproximation

Boundary value problemsLévy stable laws

Fourier/Laplacetransforms

Figure 1.2 Schematic of the techniques used to obtain solutions to

generalized random walks.

3

investigated. Some surprises are discovered here, when several systems that are expected to yield classical,Gaussian behavior are better described by the new approach.

Specifically, Chapter 2 is a more extensive review of classical transport theory, including Brownian motion.Chapters 3 and 4 cover Lévy’s α--stable Laws (probability distributions) and Lévy motions, which are super-sets of the corresponding Normal Law and Brownian motion. Using this generalized notion of random mo-tion, the equations of solute transport are derived, starting with the most basic assumption of particle transport-- that a future excursion is unaffected by the previous journey (the Markov property). This gives rise to theChapman--Kolmogorov equation of the space--time evolution of a particle’s position probability.

Two different tacks are used to obtain solutions of the Chapman--Kolmogorov equation (Figure 1.2). Thefirst uses the fact that a convolution is present and transfers to Fourier/Laplace space for solutions. The se-cond stays in real space and solves the instantaneous change in probability resulting in a (Fokker--Planck)differential equation. Similar to equations of conservation of mass, the Fokker--Planck equation is a state-ment of the conservation of probability of a single particle’s whereabouts. Solutions to the partial differentialequation are most easily gained via Fourier and Laplace transforms, so the two methods end up in the sameplace. The two methods generalize the notion of random walks, and rely on fractional calculus (Chapter 5)and the non--Gaussian (Lévy)α--stable laws. The solution space is alsobriefly explored in Chapter5. Chapter6 examines two laboratory and one field experiment to investigate the utility and validity of the fractionalapproach. Chapter 7 contains an ad hoc numerical implementation of the new fractional equation. Becausemany of the theories in this dissertation are relatively new to the field of hydrogeology, a large number ofrecommendations for future work are listed in Chapter 9.

1.1 Notation and Dimensions

α stability exponent in Levy’s stable distributions (also order of fractional space derivative).

β skewness parameter in Levy’s stable distributions: --1 ≤ β≤ 1.

γ(h) semivariogram at a separation of (h).

Γ the Gamma function.

δ(x -- a) Dirac delta function centered at x = a.

Ô(t|r) conditional probability density of a particle transition duration given the excursion length.

λ shorthand notation of 1 + α (see above).

μ shift parameter in Levy’s stable distributions.

η anomalous diffusion exponent.

Ó(h) autocorrelation function at a separation of (h).

σ scale (spread) parameter in Levy’s stable distributions.

ν exponent relating particle walk size and velocity.

ψ(k) characteristic function of a random variable X (i.e., E[eikX]).

Ò Pareto density lower cutoff.

τ mean particle transition duration (T).

ω order of fractional time derivative.

4

ξ separation distance in autocovariance functions.

aL longitudinal dispersivity (L).

A rate of change of a particle’s 1st moment (L T—ω).

B rate of change of a particle’s αth moment (Lα T—ω).

Dqa+ positive--direction fractional derivative of order q with lower limit (a).

Dq+ positive--direction fractional derivative of order q with lower limit (--").

Dqa− negative--direction fractional derivative of order q with upper limit (a).

Dq− negative--direction fractional derivative of order q with upper limit (").

D diffusion or dispersion tensor (Lα T—ω).

E() expectation of a random variable.

ERF(z) the error function.

F(f(x)) Fourier transform of function f(x).

Iqa+ fractional integral of order q integrating from a in the positive direction.

k Fourier variable.

L(f(x)) Laplace transform of function f(x).

N the Gaussian Normal distribution.

Rvv(h) autocovariance of v at a separation of h (L2T--2).

s Laplace variable.

sign(k) sign of the variable (k) times unity (i.e., sign(k) = --1 for k < 0, and 1 otherwise).

SERFα the α--stable error function.

t time.

v velocity (LT--1).

VAR() variance of a random variable.

expectation of a random variable.

d= equal in distribution (as in random variables).

5

CHAPTER 2

REVIEW OF CLASSICAL THEORY

The difference between landscape and landscape is small; but there is a great differ-ence between the beholders.

-- Ralph Waldo Emerson, Nature

2.1 Advection--Dispersion Equation

Nearly all current descriptions of solute transport make use of the Advection--Dispersion Equation (ADE):

∂∂xi− viC + Dij ∂C∂xj = ∂C∂t (2.1)

where C is solute concentration, v and D are the velocity and dispersion tensors (respectively), x is the spatialdomain and t is time. The ADE is based on the classical definition of the divergence of a vector field. Thedivergence is defined as the the ratio of total flux through a closed surface to the volume enclosed by thesurface when the volume shrinks toward zero (c.f., Schey [1992]):

∇ ⋅ J ≡ limV→0

1V

S

J ⋅ ndS (2.2)

where J is a vector field, V is an arbitrary volume enclosed by surface S, and n is a unit normal. Implicitin this equation is that the limit of the integral exists, i.e. the vector function J exists and is smooth as V !0. This is well suited to atomic force vectors such as Maxwell’s equations of electromagnetic fields, sincethe flux is indeed a “point” vector quantity. Conversely, solute dispersion is primarily due to velocityfluctuations that arise only as an observation space grows larger. The ADE is an implementation of Gauss’Divergence Theorem using solute flux as the vector function. In the ADE, J is replaced by the solute flux,so as an arbitrary control volume shrinks, the ratio of total surface flux to volume must converge to a singlevalue. The solute flux (J) is due to the combined effects of mean velocity (advection) and velocityfluctuations or variance (dispersion). The dispersive fluxes for a given volume are averaged in some fashion(volumetric, statistical) and usually approximated by a process using Fick’s first Law, i.e. J = vC -- DoC.Since velocity itself is a variable function of space, as a control volume shrinks, the velocity fluctuations

disappear and the dispersive flux shrinks to zero. Thus, if one uses the definition of divergence in (2.2), theflux cannot contain a dispersive term (except perhaps for molecular diffusion, which is generally negligible).In mathematical terms, the classical divergence of solute flux reduces to:

6

∇ ⋅ (vC − D∇C) ≡ limV→0

1V

S

(vC − D∇C) ⋅ n dS =

limV→0

1V

S

(vC) ⋅ n dS = ∇ ⋅ (vC)

(2.3)

In this setting, the classical divergence theorem is of little use in subsurface hydrology since the boundaryvalue problem for o¡(vC) = --#C/#t is infinitely complex. Because of this complexity, a de facto definitionof divergence has long been used to quantify advection and dispersion. The divergence is associated witha finite volume and is given by the first derivative of total surface flux to volume (Figure 2.1). The dispersioncoefficient tensor does not grow (scale) if the ratio of surface flux to volume is constant over some range ofvolume (solid lines in Figure 2.1). An example is a column of uniform glass beads. At the pore scale, theratio is non--constant and no constant dispersion parameter can be assigned. At some larger scale, the ratio

VOLUME

S

(vC + v′C′) ⋅ ndS

0

slope = div(vC--v′C′)

0

div(vC)

div(vC -- DaoC)

1VS

(vC + v′C′) ⋅ ndS

aa

Figure 2.1 Illustration of the definition of the divergence of solute flux over many scales.The solid lines denote assumption of local homogeneity and multiscale, integer--order(classical) divergence. Dashed lines denote continuum--heterogeneity and the resultingnoninteger--order divergence. To reconcile the growth in the integer divergence (usingcurrent theories) from scale a to b, the first order fluctuations v!C! are approximated byDoC with increasing, spatially local D.

div(vC -- DboC)

bb

SUR

FAC

EFL

UX

VO

LU

ME

TR

ICa) b)

NOTE: SURFACE FLUX =NOTE: VOLUMETRICSURFACE FLUX =

first

VOLUME

derivative

SUR

FAC

EFL

UX

localhomogeneity

7

of total surface flux to volume is constant over a large range of arbitrary volumes and the first derivative (thede facto divergence) is relatively constant. Solute flux within heterogeneous aquifers violates this principlebecause increases in an arbitrary volume result in a growing amount of dispersive flux. The first derivativeof surface flux to arbitrary volume is not constant when a travelling solute plume samples more of the velocityvariations.

When an integer--order divergence is assumed, the ratio of surface flux to volume is forced to take on aconstant value over some volumetric range. This action approximates the monotonically increasing ratio ofsurface flux to volume by a step function (Figure 2.1b). An effective parameter (D) with scaling propertiesis used to account for the fact that the de facto divergence is ill--defined in continually evolving heterogeneity.The parameter D is intimately tied to a specific volume, and the ADE is no longer self--contained with aclosed--form solution for all scales. Estimation techniques include small perturbation solution of a linearizedstochastic ADE and substitution of a local effective parameter D into the ADE for a specific plume size(Gelhar and Axness [1983], Dagan [1984]). More recent suggestions include simple power lawmultiplication of D (Su [1995]), but this leads to an equation that is not dimensionally correct. Thesesolutions suffer primarily from using a special case (integer--order divergence) for a more general problem.The first derivative of the surface flux to volume (Figure 2.1a) is not constant, i.e. the first derivative doesnot account for continuous growth of the surface flux to volume ratio as a plume grows in heterogeneousmedia. A more robust description of the volumetric surface flux growth will be constant over a greater range,perhaps even the entire range expected for a plume’s lifetime. Not only is the dispersion coefficient tied tothe smallest control volume scale, but the scale of measurement (due to concentration averaging) as well.The scale of measurement must be much larger than the scale of heterogeneity in order for the relative sizeof the control volume to approach zero. This is why the Fickian approximation works well in certaininstances. An approach that integrates the measurement scale would also be desirable. For these reasons,the description of solute transport is better suited for fractional derivatives.

Similar arguments also apply to purely statistical treatments of the dispersive fluxes, in which the CentralLimit Theorem (CLT) suggests Gaussian (Fickian) dispersion when the scale of measurement includes a largenumber of independent, finite--variance velocities (c.f., Bhattacharya and Gupta [1990]). The Gaussianvelocity distribution suggests that the probability distribution of travel times can be modelled by a Markovprocess with standard Brownian motion. This yields (through the Kolmogorov forward equation) aFokker--Planck equation of the solute particle probability and therefore concentration. The assumption ofa large scale compared to the velocity fluctuations fulfills the CLT and gives a dispersion tensor that isasymptotically fixed and Fickian dispersion ensues.

As an interesting aside, the average velocity is considered to (roughly) change from an arithmetic to ageometric mean as the amount of heterogeneity encountered increases. This also is a scale effect that ariseswhen hydraulic conductivity appears as a parameter in the Poisson Equation (for an overview ofscale--dependent mean flow, see Gelhar [1993]). The derivatives of velocity are not smooth and are tied anirreducible finite size, or “representative elementary volume” (REV). A fractional derivative approach to thegroundwater flow problem is suggested for further study.

The classic mode of operation with the integer--order derivative formulation (2.1) is to estimate the valuesof the parameters within the ADE at any particular point in a plume’s history. The parameters (D and to alesser extent v) are thought to change as the plume size changes. This is due to the fact that the autocorrelationlengths within the velocity field are large compared to the scale of measurement. In order to have a predictive

8

tool for plume behavior, several theories have arisen to estimate values of “effective” parameters, so that theADE recreates the observed plume moments. These include:

S Volume and statistical averaging (e.g., Gray [1975]; Cushman [1984]);

S Techniques based on a small--perturbation, stochastic differential equation (Gelhar andAxness [1983]; Dagan [1984]; Neuman and Zhang [1990]);

S Power--Law Dispersivity growth via empirical ADE (e.g., Su [1995]) in which Deff = xsD,with s = empirical constant;

S Purely statistical formulation using Kolmogorov forward equation with simple velocityfunction (Bhattacharya and Gupta [1990]).

These methods suffer from various drawbacks due to their inherent assumptions. The first method, volumeaveraging, invokes a hierarchy of distinct scales wherein the averaging length scale is much larger than thescale of perturbation. In other words, the averaged quantity is composed of a relatively homogeneouscollection of smaller (perturbed) quantities. An example is the laboratory scale being much larger than thepore scale. These methods are not valid at the “in--between” scales or in smoothly--varying heterogeneity.

The spectral methods rely on a linearized stochastic ADE that cannot predict spreading when the velocitycontrasts are large. Typically ln(v) is given by a Gaussian normal with a variance less than unity. Thiscondition limits the application of this technique to relatively homogeneous aquifers. The power--lawdispersivity growth (bullet #3 above) does not yield a parameter that is dimensionally correct, thus theparameter does not have a sound physical basis. Moreover, this represents an unjustified, empirical additionof another parameter into the “governing” equation. Berkowitz and Scher (1995) demonstrate that atime--dependent dispersion tensor is also unsound. Purely statistical methods make broad assumptions aboutthe functional basis of a velocity field. Clearly, the work dedicated to evaluating an “effective” parameterhas lost sight of where the actual scaling occurs within (2.1). First, smooth integer derivatives of the fluxdo not exist in natural porous media. Second, dispersion cannot be considered a point flux.

If we looked at how the divergence is defined for solute flux in porous material, we might start with a plotof the total surface flux versus volume for an arbitrary volume at the leading edge of a plume (Figure 2.1).As the volume goes to zero, the surface flux is a real number, and the slope of the line at the origin is o⋅(vC).As the size of the arbitrary volume increases, so does the total surface flux. If the medium is homogeneousover some scale, then the slope of the line (the ratio of surface flux to volume) is a constant (solid lines inFigure 2.1). Within that scale, the dispersion is Fickian and one can assign a divergence of the flux accordingto o⋅(vC -- DoC). Within that length scale range, the divergence is associated with an arbitrary and finitevolume. Since the ratio of surface flux to volume is constant over the range, the value of D appliescontinuously throughout the range. If homogeneity is present in several distinct stages, then the dispersiveflux at all smaller scales are averaged into the effective dispersion coefficient at the largest measured scale.Because the slope is constant within a distinct scale, the first derivative of the surface flux with respect tovolume (not as the volume approaches zero) is used as a de facto definition of divergence.

Typically, plumes at the field scale are in a pre--Fickian stage where an increase in the size of an arbitraryvolume (or measurement size) encloses material with different velocity. This leads to a non--constant ratioof dispersive flux to volume (the curved, dashed line in Figure 2.1a) and an apparent increase in the“divergence” (dashed line in Figure 2.1b). Since many analytic solutions already exist to the classical ADE,it has been advantageous to assume that the non--constant volumetric surface flux can be approximated by

9

a step function wherein each rise is described by a growing D. When an effective parameter D is derivedthrough volume or statistical averaging, it is only valid at that particular volume (or scale). Further increasesin the slope of the dispersive flux (increasing scale) require a new D value. This simply arises because thefirst spatial derivative of the dispersive flux (which defines the de facto divergence) is not constant. Ratherthan assume a step function exists and force D to take on increasing values, one might assume that describingthe evolving dashed curves in Figure 2.1b would more accurately replicate plume histories and give apredictive tool as well. The mathematical tools of fractional calculus are better suited to describing the curvesin Figure 2.1b than the classical (integer--order) divergence. This will be demonstrated in Chapter 5.

The classical ADE is based on the the diffusion equation, which is linked to an underlying physical orprobabilistic model of particle movement. It is instructive to analyze that link before generalizing the notionof particle movements and seeking the governing equation of these generalized movements. The physicalbasis of the diffusion equation is well known to be Brownian motion.

2.2 Brownian Motion

There are several ways to construct a Brownian motion in one or more dimensions. The first and mostintuitive way is to restrict the motions to a regular lattice so that a particle can move in only one directionduring each jump. The probability of moving to an adjacent lattice location is always equally distributed.In one dimension, the position of a particle at time t is a random variable given by X(t). If the distance to thenext lattice point is nx and the time spent in transit is nt, then

X(t) = Δx(X1 + + X[t∕Δt]) (2.4)

where Xi =+ 1 if the ith step is forward− 1 if the ith step is backward

and [t/nt] is the largest integer ≤ t/nt. The probability that Xi = +1 is equal to the probability that Xi = --1,which is 1/2 for symmetric walks.

Denote the expectation of a random variable E[X] and the variance VAR[X]. Since E[Xi] = 0 and VAR[Xi]= E[(Xi)2] = 1, E[X(t)] = 0 and VAR[X(t)] = nx2(t/nt). Now the limit must be carefully defined as nx andnt go to zero. If nx and nt simply go to zero, then VAR[X(t)] converges to zero. If, however,

Δx∕(Δt)1∕2 = c, with (c) a positive constant, then E[X(t)] = 0 and VAR[X(t)] = c2t. By the Central LimitTheorem, as the number of jumps becomes large (i.e. let the increments become very small), X(t) is a Normalrandom variable with zero mean and variance c2t.

Brownian motion is characterized by its independent increments. Since each jump is independent of theprevious jump, for all t1 < t2 < < tn the increments X(tn) − X(tn−1), X(tn−1) − X(tn−2), ... ,X(t2) − X(t1), and X(t1) are also independent and stationary, since the variance of any increment dependsonly on the interval, not on time. The density function for the random variable X(t) is given by

ft(x) = 12Õc2t

e−x2∕2c2t (2.5)

Each increment of finite size X(t + z) -- X(t), where z is a finite constant, is composed of infinitely manysmaller jumps. The increment itself is therefore a Normal random variable with zero mean and variance of

10

c2z. It is easily seen that a Brownian motion is an addition of successive increments that are themselvesindependent, identically distributed (iid) random variables. These variables also have the important featureof finite variance. So the limiting distribution of the sum of a large number of iid finite--variance randomvariables is the Normal distribution. The variance of the sum of independent variables is the sum of theindividual variances.

The term Standard Brownian motion, given the symbol B(t) refers to a Brownian Motion with unit (c). AnyBrownian motion can be related to the standard by B(t) = X(t)/c.

2.3 The Diffusion Equation and Brownian Motion

Several methods are used to relate the diffusion equation and Brownian motion. Solutions to 2--variablepartial differential equations can be facilitated by integral transform in order to remove dependance on oneof the independent variables.

Throughout this text, the Fourier transform F and its inverse F--1 are defined as:

F(f(x)) = f~(k) =

∞

−∞

e−ikxf(x)dx (2.1)

F−1(f~(k)) = f(x) = 1

2Õ∞

−∞

eikxf~(k)dx (2.2)

The pair of functions f(x) and f~(k) are unique. Each function uniquely implies the other. The change of

variable from x → k implies Fourier transform throughout this text.

The Fourier transform of the diffusion equation Ct = DCxx with respect to the space variable is:

∞

−∞

e−ikx ∂C∂t dx = ∞

−∞

e−ikxD ∂2C∂x2

dx (2.3)

If C and its derivatives vanish as |x| ! ∞, than integration by parts twice gives

ddt∞

−∞

e−ikxCdx = − k2DC~ (2.4)

dC~

dt= − k2DC

~ (2.5)

where the tilde indicates the Fourier transformed function.

Given a Dirac delta function initial condition:

C(t = 0, x) = δ(x − 0) (2.6)

C~(t = 0, k) = 1 (2.7)

gives the Gaussian (Normal) density for the Green’s function solution:

11

C~(k, t) = exp(− k2Dt) (2.6)

With inverse transform:

C(x, t) = 12ÕDt

exp(− x2∕2Dt) (2.7)

The width of the concentration profile (the distance between two concentration percentiles) is equal to (Dt)1/2.The Green’s function of the diffusion equation is identical to the solution for Brownian motion where D =c2 = Δx2/Δt. Since Fourier transform pairs are unique, the diffusion equations implies Brownian motion asan underlying probabilistic model.

Another method of relating the diffusion equation and a Brownian motion relies on the fact that thedifferential displacement of particles dX(t) = X(t + dt) − X(t) is Gaussian and satisfies Ito’s stochasticdifferential equation (Bhattacharya and Gupta [1990]) :

dX(t) = f ⋅ dt + g ⋅ dB(t) (2.8)

In one or more dimensions, f represents the drift of the process, or the mean velocity vector. The functiong is a constant tensor of the standard deviation of the Gaussian process X(t). This process satisfies theFokker--Planck equation of the “flow” of probability in time and space:

∂P∂t =

∂∂x (− f ⋅ P) +

∂2∂x2

(g ⋅ P) (2.9)

If many particles are simultaneously released and do not affect each other, the probability and concentrationare interchanged to give the ADE. One can simplify the problem further by describing a mean--removedequation that follows a moving frame of reference that travels at the mean velocity. The diffusion equationis recovered.

If the underlying physical model described above is altered to allow a higher probability of long--rangeparticle transitions, then the 2nd--order diffusion equation is no longer the governing equation of those walks.The link between Brownian motion, the 2nd--order diffusion equation and its Gaussian fundamental solutionis generalized to “Lévy motion,” a fractional--order equation, and fundamental solutions that are superset ofthe Gaussian. These superset probability densities are covered in the next Chapter.

12

CHAPTER 3

STABLE LAWS

All things are difficult before they are easy.

-- Thomas Fuller, Gnomologia

3.1 Characteristic Functions

The properties of many probability distributions are more easily investigated in terms of their characteristicfunction. The characteristic function is a description of the Fourier transform of the probability density func-tion. (Actually it is more akin to the reverse Fourier transform, but this is merely a re--parameterization.)Also useful is the moment generating function, similar to the Laplace--transformed density.

The characteristic function ψ of a random variable X with a density f(x) is given by E[eikX] where E(⋅) isthe expectation:

E(eikX) = ψ(k) = ∞

–∞

eikxf(x)dx (3.1)

The Fourier transform of the density is closely related to the characteristic function by f^(− k) = ψ(k).

The uniqueness of Fourier transform pairs guarantees that the characteristic function defines the density andvice--versa. Unless noted otherwise, the Fourier transforms in this study will place the constant 1/2Õ on theinverse transform to more closely resemble the characteristic function.

For positive domain distribution functions, the one--sided Laplace transform (moment generating function)is useful:

E(esX) = Ô(s) = ∞

0

esxf(x)dx (3.2)

Integration by parts gives the Laplace transform of a cumulative distribution function F(x):

Ô(s)s =

∞

0

esxF(x)dx (3.3)

Inversion of the characteristic functions follows the inverse Fourier transform:

f(x) = 12Õ∞

–∞

e−ikxψ(k)dk (3.4)

13

Or in Laplace space:

f(x) = 12Õiγ+i∞

γ–i∞

e−sxÔ(s)ds (3.5)

where γ is a real number greater than the real component of any singularities in the function Ô(s).

3.2 Stable Distributions (Stable Laws)Chapter 2 contained a demonstration that a Brownian motion can be created by a sum of independent, identi-cally distributed (iid) Normal random variables. It is intuitive that a sum of iid Normal variables would keepthe same distribution after dividing by a normalizing constant. One might wonder if sums of random vari-ables with other distributions maintain the distributions of the individual summands. A large family of thesedistributions were shown to exist by Paul Lévy in 1924. The Normal distribution is merely a member ofLévy’s family of stable distributions. Lévy’s relevant and oft--cited work (1924; 1937) has not been translatedinto English. Lucid summaries and extensions are provided by Feller (1966), Zolotarev (1986), Samorod-nitsky and Taqqu (1994), and Janicki and Weron (1994).

Lévy’s distributions arise when describing a “stable” sum that is distributed identically to the summands.It is easiest to use shifted, or zero--mean random variables, so a scaled sum of (n) zero--mean iid random vari-ables is:

Sn =X1 + X2 + + Xn

cn(3.6)

A number of assumptions about the probability functions are omitted for clarity. See Feller (1966, Ch. XVIIand others) or Körner (1988, Ch. 50) for more complete development.

The characteristic function of a sum of two independent variables X1 and X2 is given by

E(eik(X1+X2)) = E(eikX1eikX2) = ψX1(k)ψX2(k)(3.7)

In a similar manner, the characteristic function of the sum of a sequence of iid Xn is simply (ψX(k))n. We canalso calculate the characteristic function of the expectation of a scaled and shifted variable aY + b:

E(eik(aY+b)) = E(eikaY ⋅ eikb) = eibkψY(ak) (3.8)

Now the scaled sum cnSn in (3.6) can be related to the density of the iid variables Xn:

ψcnSn(k) = ψSn(cnk) = (ψX(k))n (3.9)

Equating the characteristic functions ψX and ψSn and taking logarithms:

logψ(cnk) = n logψ(k) (3.10)

This equality is fulfilled by a power law:

logψ(k) = Akα (3.11)

where A is a constant and the value of the exponent α is limited to 0≤α≤2 (Feller [1966]).

The equality in (3.10) will only be true when cnα = n, or cn = n1/α. With these constraints, the scaled sumand summands are identically distributed. The result is a generalization of the Central Limit Theorem forsum of n random variables (X) that are iid:

14

Sn d=X1 + X2 + + Xn

n1∕α(3.12)

An entire family of distributions that includes the Gaussian is described when the value of the exponent αranges from 0 < α≤ 2 (Feller [1966, Ch. XVII]). The constant A can be complex (indicating skewness),and the variable can have non--zero mean, so the characteristic functions of these α--stable distributions takethe general form (Samorodnitsky and Taqqu [1994]):

ψ(k) = exp(–|k|ασα 1–iβsign(k) tan(Õα∕2) + iμk) α ≠ 1 (3.13)

where the parameters σ, β andμ describes the spread, the skewness and the location of the density, respective-ly. The sign(k) function is --1 for k < 0 and 1 otherwise. The characteristic function for α = 1 (the Cauchydistribution) is slightly different and will not be listed for clarity.

When the density is symmetric, the skewness parameter (β) is zero, and the symmetric characteristic functionis:

ψ(k) = exp(− σα|k|α+ iμk) (3.14)

A standard α--stable density function has unit “spread” and is centered on the origin, so σ = 1 and μ = 0. Itis a simple matter to show that forα > 1, E(X) = μ. The mean is undefined forα≤ 1. A standard, symmetricα--stable distribution (SSαS) is characterized by the compact formula:

ψ(k) = exp(− |k|α) (3.15)

In this form it is easy to see that the Gaussian (Normal) density is α--stable with α = 2. Note, however, thatwhen the scale factor of the stable law σ = 1, the standard deviation of the Normal (α = 2) distribution (N)

is 2 :

N(k) = exp− 2σ2k2 + iμk (3.16)

The most important feature of the α--stable distributions (3.13) is the characteristic exponent (also called theindex of stability) α. The value of α determines how “non--Gaussian” a particular density becomes. As thevalue of α decreases from a maximum of 2, more of the probability density shifts toward the tails. Figure3.1 shows the standard α--stable distribution functions for α= 1.6, 1.8, 1.9, and 2. Note that the distributionsappear very Gaussian in untransformed coordinates, and that the difference lies in the relative weight presentin the tails. For probabilities between 1 and 99 percent, the different distributions appear near--normal.

Non--standard (σ≠1 andμ≠ 0) stable distribution functions (F) and densities (f) are related to their standardcounterparts by the relations:

Fαβ(x, σ, μ) = Fαβ(x − μ)σ , 1, 0 (3.17)

fαβ(x, σ, μ) =1σ fαβ(x − μ)σ , 1, 0 (3.18)

Cauchy and Lévy sought closed--form formulas for the stable densities (in real, not Fourier, space) for allvalues of α. They found that direct inversion of the characteristic function ψ(k) is only possible when α =½, 1, or 2. A number of accurate approximations are available for other values. Sinceψ(k) is known exactly,

15

99.9

90

99

70

50

30

10

1

0.1--10 --5 5 100

α = 2.0 1.9 1.8 1.6

PRO

BA

BIL

ITY

(PE

RC

EN

T)

Figure 3.1 Plots of the distribution function F(x) versus x for several standard sym-metric α--stable distributions using a) linear scaling and b) probability scaling. TheGaussian normal (α= 2.0) plots as a straight line using probability scaling for the ver-tical axis.

x

--10 --5 5 100

PRO

BA

BIL

ITY

(PE

RC

EN

T)

x

0

20

40

60

80

100

(a)

(b)

16

a fast numerical Fourier inversion can yield accurate densities. The Fourier inversion formula also has manyreal--valued integral representations that yield quick numerical solutions (c.f., McCulloch [1994, 1996]; Zo-lotarev [1986]). In particular, McCulloch (1996) gives the integral representation of the standard forms forσ =1 and μ = 0 of the cumulative probability function (Fαβ):

Fαβ(x) = C(α, Ò) +sign(1 − α)

2

1

−Ò

exp− x* αα−1 Uα(Ô, Ò)dÔ (3.19)

where

x* = c*x

c* = 1 + β tan(Õα∕2)2−1

2α

Ò = 2Õα tan−1β tan(Õα∕2)

C(α, Ò) = 1, α > 1(1 − Ò)∕2, α< 1Uα(Ô, Ò) = sin Õ2 α(Ô+ Ò)cos Õ2Ô

α1−α

The densities are obtained by differentiating the cumulative probabilities with respect to x. Note McCul-loch’s (1996) mistaken standard density (fαβ) that should read:

fαβ(x) =x*

1α−1αc*

2|1 − α| 1

−Ò

Uα(Ô, Ò)exp− |x*| αα−1 U(Ô, Ò)dÔ (3.20)

Equations (3.19) and (3.20) were coded using a simple trapezoidal rule to return values of the distribution(Figure 3.1) and the density (Figure 3.2) for various values of α. Listings of the FORTRAN subroutines(DFASTD.F and CFASTD.F) are given in Appendix I.

Several series expansions of the standard α--stable densities are listed in readily--available recent documents(c.f., Feller [1966]; Nikias and Shao [1995]; Janicki and Weron [1994]). Bergstrom (1952) and Feller arecredited with independently deriving similar expansions. Feller (1966) also gives series expansions for aslightly different parameterization, using γ to quantify the skewness, rather than β:

fαγ(x) = 1Õx∞

k=0

Γ(kα+ 1)k! (

− x)−kα sin Õk2 (γ− α) 0 ≤ α < 1(3.21)

fαγ(x) = 1Õx∞

k=0

Γ(kα−1 + 1)k! (

− x)k sin Õk2α (γ− α) 1 < α ≤ 2(3.22)

Feller’s skewness parameter γ is obtained by equating the canonical form (3.13) to his equivalent representa-tion of the standard characteristic function:

17

10--1

10--2

10--3

10--4

10--5

10--6

10--7

100

10 100110--1

Figure 3.2 Plots of symmetric α--stable densities showing power--law “heavy” tailedcharacter. a) linear axes, and b) log--log axes.

1.8

1.4

1.2

1.6

2.0 (Gaussian)

--5.0 --3.0 --1.0 1.0 3.0 5.00.0

0.2

0.4

α = 2.0(Gaussian)

α = 1.4

(a)

x − μσ

σ ⋅ fαx − μσ α =

σ ⋅ fαx − μσ

(b)

18

ψ(k) = exp− |k|αeiÕ(sign k)γ∕2 (3.23)

Resulting in (Samorodnitsky and Taqqu [1994]):

γ =

− 2Õ arctan(β tan(Õα∕2)) 0 < α< 1

2Õ arctan(β tan(Õ(α− 2)∕2)) 1 < α < 2

(3.24)

For symmetric densities, setting γ=0 in (3.22) yields a formula that converges with reasonably few terms evenwith large arguments. This expansion will be used throughout this study:

fα(x) = 1Õ∞

k=0

(− 1)k(2k + 1)! Γ

2k + 1α + 1x2k 1 < α ≤ 2 (3.25)Portions of this study require a generator of random variables that share anα--stable distribution. Janicki andWeron (1994) gives an algorithm for generating a standard, symmetric stable random variate X based on auniform random variable V on (--Õ/2,Õ/2) and an exponential variable W with unit mean:

X = sin(αV)(cos(V))1∕α

⋅ cos(V − αV)W

(1−α)∕α (3.26)3.3 Moments and Quantiles

It is interesting to note the behavior of the α--stable densities in the large x limit. The simplest characteristicfunction of an α--stable random variable can be approximated for small k (large x) by ψ(k) = exp(--|k|α) ≈1 -- |k|α, with an inverse transform f(x)≈ Cx--1--α. Figure 3.2b is a log--log plot of the positive half of severalof the α--stable densities, clearly showing the power--law tail behavior. The Gaussian density lacks the pow-er--law tail, although theα--stable family represents a continuum. Asα approaches 2, the power--lawbehavioronly becomes evident at very large values of |x|. It has been shown that the power--law tail behavior is presentwith any values of α, σ and β (Samorodnitsky and Taqqu [1994]).

The moments of a distribution with density f(x) can be defined by the integral

μr ≡ ∞

−∞

xrf(x)dx (3.27)

The first several integer moments are historically those that are studied in physical sciences. In order for amoment to exist, the integral (3.27) must converge. Since at least one of the tails of any α--stable densityfollows a power law for large |x| (Feller [1966]), we can integrate (3.27) to check for convergence. At the

tails we have lim|x|→∞

C|x|r−α which is finite only if r < α. So the moments higher than the real number α do

not exist for these distributions. In particular, the variance and standard deviation are undefined for all α--stable distributions except the Gaussian, when α = 2.

The infinite variance of α--stable laws can aid in their detection. The calculated variance of a series of α--stable random variables will not tend to converge using standard variance estimators. The failure of this esti-

19

mator to converge will require a large population as α grows closer to 2, since the probability of extreme val-ues is only slightly larger than predicted by the Normal distribution.

The fact that the variance of an α--stable random variable is infinite does not preclude measurement of the“spread” of the density of the variable. Nikias and Shao (1995) advocate the use of fractional moments (anyrth moment with r < α). Janicki and Weron (1994) use quantiles of the distribution when investigating thespread of an α--stable process. The quantiles qp are defined here as F--1(p) with its pair F--1(1--p), where pis a desired probability. Thus for a random variable X with distribution F(x), the quantiles qp are the pointsx where F(x) = p and F(x) = 1--p. The probabilities 0.159 and 0.841 are typically used for the Normal distribu-tion since these numbers correspond to the mean one standard deviation. These quantiles will be usedin this study for convenience.

Another useful formula gives the value of the moments of order less than α. Nikias and Shao (1995) showthat the fractional lower--order moments of a symmetric SαS variable X are calculated by the formula:

E(|X|r) =2r+1Γr+12 Γ(− r∕α)

α Õ Γ(− r∕2)σr (3.28)

In particular, the expectation of the absolute value of X (r = 1) reduces to

E(|X|) =2Γ(1 − 1∕α)

Õ σ(3.29)

Figure 3.3 is a plot of E(|X|) for 1

20

CHAPTER 4

PHYSICAL MODEL

The process of irregular motion which we have to conceive of as the heat--content ofa substance will operate in such a manner that the single molecules of a liquid willalter their positions in the most irregular manner thinkable.

-- Einstein (1908)

The stochastic process of Brownian motion was reviewed in Chapter 2. Brownian motion is a continuoustime random walk (CTRW) with Gaussian increments that is also the limit process ofuncorrelated, unit jumpson a lattice. The movement of a particle in aquifer material clearly does not follow Brownian motion sincegeologic material is deposited in continuous, correlated units. A particle travelling faster than the mean atsome instant is much more likely to still be travelling faster than the mean some later time due to the spatialautocorrelation of aquifer hydraulic conductivity. The same is true for particles travelling slower then themean velocity. This suggests that particle excursions that deviate significantly from the mean are much morelikely than traditional Brownian motion can model. This Chapter examines another (superset) model of par-ticle random walks that accommodates these large deviations from the mean particle trajectory.

Many other random physical processes are characterized by extreme and/or persistent behavior (the Josephand Noah effects coined by Mandelbrot and Wallis [1968]) for which Brownian motion is an inadequate mod-el. Notable among these is the dispersion of a passive scalar in near--turbulent (chaotic) and turbulent flow(see Klafter et al. [1996] for a survey). In these flows, a particle tends to spend long periods of time trappedin vortices that are essentially stagnant with respect to mean flow. Mixing within a vortex may in fact followBrownian motion, but a particle can occasionally escape and travel with high velocity “jets” between vortices(Weeks, et al. [1995]). These relatively rare, high velocity events represent a heavier--tailed probability dis-tribution for the particle excursions. These particle motions are described by Lévy flights and Lévy walks,which are similar to Brownian motion but differ in the probability distribution of the jumps. Rather thanhaving Gaussian increments, they have Lévy’s α--stable, or power--law (Pareto) distribution increments. Itis instructive to analyze how these Lévy motions can arise as a limit process of jumps on a lattice, just as wasdone with Brownian motion in Chapter 2. This analysis leads to a more general model of random walks ona lattice and provides a link between the memory of a fractional derivative and the memoryless property ofrandom walks and Markov processes.

The particle “propagator” describes the probability of finding a particle somewhere in space at some time.Solving the equations for the propagator start with the mathematical representation of a single particle re-leased at the Cartesian origin. This propagator is a surrogate for concentration if it represents a large quantityof independent solute “particles.” Since a contaminant mass placed in an aquifer is composed of a huge num-ber of these particles, the propagator density is “filled in” by the solute particles. So the task of deriving agoverning equation for the movement of an instantaneously released slug of solute tracer is reduced to solvingthe equations for the propagator. This equivalence often will be used in this study.

21

It is instructive to follow the development of this propagator, starting again with random jumps on a lattice.With prior knowledge of the α--stable distributions, one might expect that a particle will be given a higherpropensity to make longer excursions than a particle experiencing Brownian motion. If thewalks areuncorre-lated with respect to time, they must be spatially correlated in order to embark on these longer walks. Brow-nian motion’s unit walks on a lattice are uncorrelated in space (although it will be shown that a Brownianmotion defined by Gaussian increments has some very short--range spatial correlation). Thus the differencebetween Brownian motion, and its superset Lévy motion, is the range of spatial correlation. This is shownin Section 4.3. The first two Sections (4.1 and 4.2) are primarily a review of current theories that are applica-ble to solute transport, with minor corrections to the originals where indicated. The final Section (4.3) alsoincludes a new derivation of the statistical properties of a particle undergoing Lévy walks to enable estimationof certain parameters from aquifer characteristics.

4.1 Lévy Flights -- Discrete Time

Hughes et al. (1981) describe a random walk on a one--dimensional infinite lattice. The probability of findingthe particle at lattice position j after n jumps is denoted Pn(j). Let p(m) be the probability of jumping m latticesites during a single step. The Markov property of jump independence dictates that the probability of findingthe walker at site j at the next step is the sum of the transition probabilities from all other lattice sites (j′) multi-plied by the probability of being at those sites. This is stated mathematically in the Chapman--Kolmogorovequation:

Pn+1(j) = ∞

j′=−∞p(j − j′)Pn(j′) (4.1)

This is a convolution, so the Fourier--transformed probabilities are used where:

P~

n(k) = ∞

j=−∞eikjPn(j) (4.2)

p~(k) = ∞

m=−∞eikmp(m) (4.3)

The Fourier transformed walk probability p~(k) is know as the “structure function.” Montroll and Weiss(1965) solve the convolution with a Green’s function equation, i.e. using an initial condition that a walk start-ing at the origin has a delta function initial probability: P0(i) = δ(i--0). The resulting probability Pn(l) is calledthe “propagator” since it describes the n--step spatial evolution of a single event at the origin. By inductionand the definition of convolution, (4.1) and the initial condition gives:

P~

n(k) = (p~(k))n (4.4)

The inverse transform gives the spatial probability density of a walker after n steps:

Pn(j) = 12Õ2Õ

0

e−ikj(p(k))ndk (4.5)

22

A Brownian motion must also be described by (4.4). One way to recover Brownian motion is to restrict par-ticle movement to 1 lattice position in either direction of the current (mth) position using the Dirac delta func-tion distribution (Hughes et al. [1981]):

p(m) = 12

(δ(m + 1) + δ(m − 1)) (4.6)

The transformed probability is:

p~(k) = eik + e−ik

2= cos k (4.7)

When the number of transitions becomes large (n→∞),

P~(k∕ n ) = (p~(k∕ n ))n = cos(k∕ n )n = 1 − k22n + O(1∕n)

n

≈ exp− k22 (4.8)Fourier inversion gives the Gaussian profile:

Pn(j∕ n ) ≈ 12Õ2Õ

0

exp(− ikj) exp− k22dk = 1

2Õexp(− j2∕2) (4.9)

A change to the real variable x = j/(n1/2) results, for large n, in the approximation

Pn(x) = 12Õn

exp(− x2∕2n) (4.10)

which is normal with zero mean and variance (n). Another more general way to generate the Brownian motion

is to define each jump by a random variable with finite variance m2 where ⋅ denotes expectation:

m2 = ∞

l=−∞m2p(m) (4.11)

The resulting limiting value of the transformed step probability is Gaussian:

p~(k) ≈ 1 − m2 k22≈ exp− m2 k2

2 (4.12)

And the overall trajectory density of an individual walker with any finite--variance transition probability is

Pn(j) = 12Õm2n

exp − j22m2n (4.13)

The distinguishing characteristic of all Brownian motions is held in (4.6) and (4.11). If the distance that arandom walker instantaneously travels has finite variance, then the random walk is asymptotically Gaussian.A number of researchers have investigated the limit process that results when each jump is 1) random withinfinite variance, and/or 2) not instantaneous. Both of these modifications lead to models that simply andconcisely describe a wealth of macroscopic, non--Gaussian processes (Bouchard [1995]). These real--worldphenomena include faster--than Fickian dispersion (often called superdiffusion) in chaotic to turbulent flow(Shlesinger, et al. [1987]; Weeks, at al. [1995]), structure of DNA (Stanley, et al. [1995]), movement of

23

charges in semiconductors (c.f., Geisel [1995] and references within), and quantum Hamiltonian systems(Zaslavsky [1994a]).

Consider a random walk in which each particle jump has a travel distance probability that isα--stable. Thesewalks would favor larger deviations from the mean because of the tail--heavy density of the distributions.A symmetric jump probability has a purely real Fourier transform:

p~(k) = exp(− σα|k|α) (4.14)

And the probability propagator is also an α--stable distribution, by virtue of its characteristic function:

P~

n(k) = exp(− nσα|k|α) ⇔ P~

n(k∕n1∕α) = exp(− σα|k|α) (4.15)

These random motions are named Lévy flights since the particles are instantaneously moved from point topoint. These random diffusion paths have infinite variance. Figure 4.1 shows a series of points in two dimen-sions that follow a Lévy flight. The distance of each flight is an α--stable random variable and the directionis a uniform random [0,2Õ] variable. (In this case, the direction changes were limited to multiples of Õ/2 toshow simulated movement on an orthogonal lattice). The left plot (Figure 4.1) shows a Lévy flight with acharacteristic exponent of 1.9, which is not too dissimilar to a Brownian motion (α=2.0). The rightmost plotuses a more tail--heavy distribution with α=1.7. Note that these flights are characterized by clusters that areseparated by relatively infrequent, long--distance jumps. The set of turning points is a random fractal, andmagnification of the clusters shows the same clustered behavior on increasingly smaller scales. Figure 4.2shows the cumulative displacement of a one--dimensional walker undergoing Lévy flights with exponentsof α=1.7 and 1.9. The variance of both processes are infinite, but the “spread” of these processes can be ana-lyzed using quantiles. Note that within the definition of the Levy flights, a “particle” does not visit the pointsin space between turning points, i.e. the connecting lines in Figures 4.1 and 4.2. Rather, a particle is instantlymoved from turning point to turning point.

--50 0 50

α = 1.9

--50

--50--100

0 50 100 150

α = 1.7

Figure 4.1 Lévy flights in two dimensions.

50

0

100

150

200

24

--50

50

150

100

0

20001000500 1500

Figure 4.2 Numerical approximation of one--dimensional, continuous--time, random Lévy walks.

α=1.7α=1.9

X(t)

t

The symmetricα--stable random variables used to create random walks were generated using equation (3.26)and identical seeds for the random number generator, hence the similarity of the traces using value of α of1.7 and 1.9 (Figures 4.1 and 4.2). A listing of the FORTRAN code (SIMSAS.F) used to generate the Levyflights is given in Appendix I.

It was shown (4.6) -- (4.13) that a single--lattice jump becomes a Brownian motion (i.e. a Bernoulli processbecomes a Gaussian process) as the number of jumps or trials becomes large. Hughes et al. (1981) describea simple clustering symmetric walk that approximates a Lévy process in the same way. They use jumps ona lattice in which the probability of travelling a step of length x is

p(x) = a − 12a∞n=0

a−n(δ(x − Δbn) + δ(x + Δbn)) (4.16)

where b2 > a > 1 and $ > 0 is the lattice spacing. If b is an integer, then jumps of size 1, b, b2 ... bn latticepositions are allowed. The larger jumps are less likely by a factor of a--n. On average, a cluster of (a) jumpsof length 1 are linked by a jump of length b. About (a) of these clusters are linked by a jump of length b2,and so on in a classically fractal manner. The structure function of this walk probability is (Hughes, et al.[1981]):

p~(k) = a − 1a ∞

n=0a−ncos(Δbnk) (4.17)

which is the everywhere--continuous, nowhere (integer) differentiable, self--similar Weierstrass function.This function has been shown to be fractionally differentiable up to order ln(a)/ln(b) (Kolwankar and Gangal[1996]). A quick check also shows that this density has an infinite variance.

Equation 4.16 follows the scaling relationship

25

p~(k) = 1a p~(bk) + a − 1a cos(Δk)

(4.18)

Hughes et al. (1981) show that the asymptotic (k → 0) behavior is satisfied by an exponential function:

p~(k) ≈ exp(− C|k|α) where α= ln(a)ln(b) and C = −(Δα∕τ) Õ2

Γ(α) sin(Õα∕2)(4.19)

In fact, for wave numbers smaller than that of the lattice spacing, the exponential function that describes anα--stable transition density matches quite well (Figure 4.3). By the Tauberian Theorem (Feller [1966]), thetails of the densities (4.17) and (4.19) are identical. Note that Δα/τ is a constant, analogous to the constant(Δx)2/t defined for Brownian motion.

The probability propagator is also asymptotically (in the domain of attraction of) an α--stable density, sinceits Fourier transform follows:

P~

n(k∕n1∕α) = p~(k∕n1∕α)n≈ exp(− C|k|α) (4.20)

which indicates that the propagator is asymptotically invariant after scaling by n1/α.

0.0 5.0 10.0--0.5

0.0

0.5

1.0

wave number (k)

p(k)

Weierstrass Function (4.17)

Exponential (4.19)

Figure 4.3. Graphs of the Fourier transform of the particle jump probability (thestructure function) of a “clustered” walk on a discrete lattice. For this graph, the latticespacing (Δ) was set to unit length. Note the good approximation of the completeWeierstrass function by the exponential function for wave numbers smaller than theinverse of the lattice spacing (i.e. k

26

Levy Flights -- Continuous TimeTo generalize this random walk to non--integer jump sizes and real--valued time, first allow jumps of non--in-teger size, i.e. b∈9 > 1. An integral replaces the summation for the n--step propagator:

Pn+1(x) = ∞

−∞

p(x − x′)Pn(x′)dx′ (4.21)

If each jump takes an equal amount of time (τ) to complete, then the change in probability over a single jumpis

Pn+1(x) − Pn(x)τ =

∞

−∞

1τ (p(x − x′) − δ(x − x′))Pn(x′)dx′

(4.22)

Noting that the limit as τ→ 0 is the time derivative of the propagator, then the motion can be generalizedto a continuous time function. If we denote the time of the nth step as t, we have

∂P(x, t)∂t =

∞

−∞

limτ→0

1τ (p(x − x′) − δ(x − x′))P(x′, t)dx′

(4.23)

The integral now requires that the step sizes dx′ are infinitesimally small, which requires that the lattice spac-ingΔ→ 0. The rate at which the spacing shrinks must depend on how τ→ 0. In Fourier space, the last resultbecomes

∂P~ (k, t)∂t = limτ,Δ→0

1τ (p

~(k) − 1))P~ (k, t) (4.24)

Two options arise for the transition density, finite or infinite variance. The finite variance case should recoverBrownian motion (i.e. solve the diffusion equation). This requires taking the limitsΔ and τ so that the expres-sion is not trivially 0 or infinity, i.e. Δ2/τ is a constant. The finite--variance structure function where eachjump is $j in size is now

p~(k) ≈ 1 − Δ2j2 k22

(4.25)

leading to the evolution of the propagator:

∂P~ (k, t)∂t = limτ,Δ→0

− Δ2j22τP~ (k, t) ≡ − DP~(k, t) (4.26)

which is the Fourier transform of the integer--order diffusion equation with a diffusion coefficient definedby

D = limτ,Δ→0

Δ2j22τ

(4.27)

To generalize the walk, Hughes et al. (1981) use the structure function for infinite variance walks. The limitsof Δ and τ are once again taken so that zero or infinity do not result, i.e. $α/τ = constant. Now the structurefunction is approximated by 1 -- C|$k|α, so

27

limτ,Δ→0

− Cτ |Δk|

αP(k, t) = limτ,Δ→0

− CΔα

τ |k|αP(k, t) (4.28)

And the probability propagator follows the equation

∂P~ (k, t)∂t = − Dα|k|

αP~(k, t) (4.29)

and is therefore α--stable:

P~(k, t) = exp(− Dαt|k|α) (4.30)

Hughes at al. (1981) use the full structure function equation (not listed here for simplicity) to derive the lastequation with a complete description of the diffusion coefficient for Lévy flights:

Dα = limΔ,τ→0

Δατ ⋅

Õ2Γ(α) sin(αÕ∕2)

(4.31)

The velocity of a particle undergoing the continuous time random walks (including Brownian motion) have

infinite velocity (v) whenα>1, since v = limΔ,τ→0

Δτ = limΔ,τ→0

Δα ⋅ Δ1−ατ = limΔ→0