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.
iii
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 sage
advice. I went in March 1993, and I had the strange and pleasant feeling that I was not only accepted to the
program, but I was being actively recruited. I must thank Dr. John Warwick for his part in that feeling. I’m
also glad to thank Dr. Warwick for financial and philosophic help over the years. I only wish we had been
in the same building.
From the beginning, Steve Wheatcraft has pushed but never prodded, taught but never instructed, enthused
but never gladhanded. His humility is endless and his door is never closed. This dissertation was clearly
outside of my capabilities a few years ago, but I believed in its virtue because he did. Nobody else on earth
who could have planted this seed in my head and had it come to fruition, so I thank him. One of his colleagues
says that every professor should graduate a total of three Ph.D. students -- one to continue his work, one to
advance the science, and one to replace the teacher. I can only say that I am very lucky Steve didn’t follow
this 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 exist
without his help. By shear dumb luck I figured something out about hydrogeology and a member of my committee is an expert in that subject of mathematics. I can’t decide whether to name my first child Levy or
Meerschaert. While on the subject of mathematics, I wish to acknowledge the fantastic courses I took (or
merely 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. Meerschaert
and McGough.
The other members of my committee -- Scott Tyler, Britt Jacobson and Katherine McCall -- did many things
for me, not the least of which was to remind me of all of the things that I don’t know or understand. I appreciate the time they spent helping me.
I sincerely thank all of the students in the Hydrologic Sciences program. First, the students maintain the high
quality of the program and make all of our degrees more valuable. Second, the reputation of the program
and hard work of the students bring the best speakers in the world to our campus. I have gained very much
from interaction with visiting speakers. Third, my interaction with fellow students has added more refinement to the ideas presented in this dissertation than could possibly come from my own head. Being my sounding 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 the
form of the George Burke Maxey fellowship. I also thank the U.S.G.S. and the Mackay School of Mines
for 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 -- Groundwater 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.
iv
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 she
chose the difficult path of being the mother of her children. Her calm and levelheaded support through the
years has been inspirational. I hope I am able to give one--tenth as much as I received. I must also shatter
the cliche and thank my wife’s parents, Doug and Kathy Guinn, for their unflagging encouragement. They
are models of thinking, caring citizens.
Finally, I thank my wife Marnee for making so many sacrifices; for leaving her dearest friends and the sunny
beaches of San Clemente and postponing her own dreams of higher education. Your time will come and I
will remember.
v
ABSTRACT
The traditional 2nd--order advection--dispersion equation (ADE) does not adequately describe the movement
of solute tracers in aquifers. This study examines and re--derives the governing equation. The analysis starts
with a generalized notion of particle movements, since the second--order equation is trying to impart Brownian motion on a mathematical plume at any time. If particle motions with long--range spatial correlation
are 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 derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions resemble the Gaussian except that they
spread proportional to time1/α and have heavier tails. The order of the fractional ADE (FADE) is shown to
be 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 experiment is a one--dimensional tracer test in a 1--meter sandbox designed and constructed for minimum heterogeneity. The FADE, with a fractional derivative of order α = 1.55, nicely models the non--Fickian rate of
spreading and the heavy tails often explained by reactions or multi--compartment complexity. The final experiment is the U.S.G.S. bromide tracer test in the Cape Cod aquifer. The order of the FADE is shown to
be 1.6. Unlike theories based on the traditional ADE, the FADE is a “stand--alone” equation since the dispersion coefficient is a constant over all scales.
A numerical implementation is also developed to better handle the nonideal initial conditions of the Cape
Cod test. The numerical method promises to reduce the number of elements in a typical numerical model
by orders--of--magnitude while maintaining equivalent scale--dependent spreading that would normally be
created by very fine realizations of the K field.
vi
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS ..........................................................................................
iii
ABSTRACT ................................................................................................................
v
LIST OF FIGURES ......................................................................................................
viii
LIST OF TABLES ...................................................................................................
xi
CHAPTER
1
INTRODUCTION ......................................................................................................
1.1 Notation and Dimensions .............................................................................
1
3
2
CLASSICAL THEORY ...........................................................................................
5
2.1 Advection--Dispersion Equation ...................................................................
2.2 Brownian Motion .......................................................................................
2.3 The Diffusion Equation and Brownian Motion ..........................................
5
9
10
3
STABLE LAWS ....................................................................................................
3.1 Characteristic Functions .............................................................................
3.2 Stable Distributions (Stable Laws) ..............................................................
3.3 Moments and Quantiles ..............................................................................
12
12
23
28
4
PHYSICAL MODEL .............................................................................................
20
4.1 Lévy Flights -- Discrete Time ......................................................................
Lévy Flights -- Continuous Time ...........................................................
4.2 Lévy Walks -- Continuous Time Random Walks .........................................
Coupled Space--Time Probability ..........................................................
4.3 Velocity Statistical Properties .....................................................................
5
6
THE FRACTIONAL ADVECTION--DISPERSION EQUATION .........................
46
5.1 Fractional Fokker--Planck Equation ..................................................................
5.2 Solutions ..........................................................................................................
46
53
EXPERIMENTS ....................................................................................................
60
6.1 High Concentration Diffusion .....................................................................
6.2 Laboratory--Scale Tracer Test .....................................................................
6.3 Cape Cod Aquifer .......................................................................................
A Posteriori Estimation of Parameters ................................................
Analytic Solutions ..............................................................................
A Priori Estimation of Parameters ......................................................
7
21
26
27
28
36
NUMERICAL APPROXIMATIONS ...................................................................
7.1 Motivation ................................................................................................
7.2 Finite Differences .....................................................................................
60
64
67
70
72
75
78
78
79
vii
8
DISCUSSION OF RESULTS ...............................................................................
84
9
CONCLUSIONS AND RECOMMENDATIONS .................................................
90
9.1 Conclusions ..............................................................................................
9.2 Recommendations ....................................................................................
90
91
10 REFERENCES ....................................................................................................
92
APPENDICES ...........................................................................................................
I FORTRAN LISTINGS ...................................................................................
I.1 Program SIMSAS.F ..................................................................................
I.2 Program ENSEM.F ..................................................................................
I.3 Program AVEGAM.F ...............................................................................
I.4 Program WEIER.F ...................................................................................
I.5 Subroutine CFASTD.F .............................................................................
I.6 Subroutine DFASTD.F .............................................................................
I.7 Program CVX.F .......................................................................................
I.8 Program CVT.F ........................................................................................
I.9 Program FRACDISP.F ..............................................................................
96
96
96
98
102
107
108
111
114
116
118
II STABLE LÉVY MOTION CALCULATIONS ...............................................
II.1
The Green’s Function Chapman--Kolmogorov Equation for
random walks of random duration .......................................................
II.2
Exact Solutions for the transformed α--stable densities .......................
II.3
Calculation of power--law transition density Fourier/Laplace
transforms ...........................................................................................
121
121
122
123
III VELOCITY AUTOCOVARIANCE OF LÉVY WALKS ................................
III.1
Velocity Autocovariance for Lévy Walks with
Lower Cutoff ......................................................................................
III.2
Lévy Walks with Converging Autocovariance .....................................
III.3
Autocovariance with Velocity Proportional to
Lévy Walk Size ...................................................................................
III.4
Full α--stable density ...........................................................................
128
IV FRACTIONAL DERIVATIVES AND THEIR PROPERTIES .......................
134
V FINITE DIFFERENCE APPROXIMATION OF THE
FRACTIONAL DERIVATIVE .......................................................................
139
128
131
132
132
viii
LIST OF FIGURES
Figure 1.1
Schematic of the techniques used to obtain solutions to generalized
random 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--heterogeneity and the resulting noninteger--order divergence. To reconcile the growth in the
integer divergence (using current theories) from scale a to b, the first order
fluctuations 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 symmetric
α--stable distributions using a) linear scaling and b) probability scaling. The
Gaussian normal (α = 2.0) plots as a straight line using probability scaling for the
vertical axis. .................................................................................................. 15
Figure 3.2
Plots of symmetric α--stable densities showing power--law, “heavy” tailed
character. 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, random
Lévy walks.
.................................................................................................. 24
Figure 4.3
Graphs of the Fourier transform of the particle jump probability (the structure
function) of a “clustered” walk on a discrete lattice. For this graph, the lattice
spacing (Δ) was set to unit length. Note the good approximation of the complete
Weierstrass function by the exponential function for wave numbers smaller than
the inverse of the lattice spacing (i.e. k<1). ....................................................... 25
Figure 4.4
Particle velocity as a function of Lévy walk magnitude and conditional walk time
parameter (ν). .................................................................................................. 29
Figure 4.5
Regions of applicability of particle travel distance variance
calculations. ....................................................................................................... 32
Figure 4.6
Regions of applicability of propagator expressions. ...........................................33
Figure 4.7
Spatial distribution of plumes predicted by equation (4.54) for α = 1.7 (solid lines)
versus 2.0 (dashed lines) at four different dimensionless times (t = 0.1, 1.0, 10,
and 50). All curves use D = 1. The continuous source curve (a) is 1 -- the CDF,
while the “pulse” contamination source (b) is the density. Note that these curves
can be scaled and represented by a single curve for all time. The distance between
two concentrations (the apparent dispersivity for x16--vt) grows∝t1/α. .............. 35
Figure 4.8
Breakthrough of a contaminant plume at a fixed point in space with α = 1.5, 1.7,
and 2.0. (a) Real time for x = 10, v = 1. (b) Half of the scaled tails from a
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 on
log--log plots is equal to —(1+α). .....................................................................37
Figure 4.9
Graphs of a) the Lévy process, b) the velocity function and c) joint probability
distribution of jump length as a function of spatial separation. ......................... 38
Figure 4.10
Log--log and linear plots of the analytical and numerical velocity semivariogram
functions when the velocity is modeled as proportional to Lévy walk size. The
numerical result is the ensemble mean of 112 realizations of 1000--jump walks
using a stability index (α) of 1.7......................................................................... 41
Figure 4.11
Log--log and linear plots of the velocity semivariogram for large and small values
of 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--Gaussian
Lévy process with index of stability (α = 1.99). ................................................ 45
Figure 5.1
Integer and fractional derivatives of two simple power functions. Top row: Integer
derivatives of f(x) = x2. Middle row: Integer derivatives of g(x) = x2.33. Bottom
row: 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) and
positively 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. As
α 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 high
ionic strength, high gradient fluid. The random walk occurs within a partially-occupied network. The probability of a walk toward lower concentration (to the
right of the figure) is always higher than into higher concentration, where more
sites are occupied by other solute ions. At high enough concentrations, the set of
connected available sites is non--Euclidean, precluding Fickian diffusion. ......... 61
Figure 6.2
Scaled diffusion profiles from Carey’s (1995) experiment: a) scaled by the
traditional (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 from
Carey’s (1995) experiment: a) scaled by the traditional (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. ....................................................................63
Figure 6.4
Schematic view of the experimental sandbox tracer tests. The flowpath
highlighted by the arrow is analyzed in detail. ...................................................64
x
Figure 6.5
Calculated dispersivities versus distance of probe from source. The flow path
chosen for analysis is shown with the connecting line. The best--fit dashed line
indicates 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 used
to calculate the apparent dispersivity. Compare this data with the α--stable
theoretical 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 the
leading and trailing limbs of the plume. Very early and late data show probe
noise. ...............................................................................................................68
Figure 6.8
Comparison of traditional and fractional ADEs with the data from probe 3 (x = 55
cm) 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 due
South 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 mean
travel. ...............................................................................................................71
Figure 6.11
Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximum
concentrations along plume centerline. Solid lines are solutions to the FADE
using 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 above
peaks. ...............................................................................................................73
Figure 6.12
Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximum
concentrations along plume centerline. Solid lines are solutions to FADE and
classical ADE using identical (early--time) dispersion coefficients of 0.13. a) Early
time 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 of
vertical samples from the same MLS that from which the maximum
concentrations were measured. The smaller average uses zero for non--detectable
concentration, 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--log
axes, and b) linear axes. Initial conditions were a “point source” of unit mass at
the node located at x0. The solutions follow the scaling law of the analytic
solution. Note the oscillatory error at the extreme tail ends. ..............................81
Figure 7.2
Comparison of analytic (lines) versus numerical (symbols) solutions of the FADE
with “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 the
numerical solution at 13 and 55 days. ................................................................83
Figure 8.1
Comparison of the plume growth predicted by the traditional ADE (ADE), Gelhar
and Axness (1983) (GA), the fractional ADE (FADE), Mercado (1967) stratified
flow (M), and Wheatcraft and Tyler (1988) fractal tortuosity model (WT). The
ordinate log(Xc2) is roughly equivalent to estimated plume variance. The GA
curve has slope 2:1 at a plume’s origin, transitioning to Fickian 1:1 slope at late
time. ...............................................................................................................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 increasing
concentration is shown by the arrow. Variance exponent (VAR ∝ tη) for arrow
path α → 2 predicted by b) variance equation (4.51) and c) the propagator
equation (4.45).
......................................................................................87
Figure A2.1
Particle travel distance variance when mixed sk2 terms are included in the small--k
approximation of the Laplace--Fourier transformed conditional Lévy walk
probability p(r,t). The dashed line indicates results from simplified form used by
Blumen 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 numerical
approximation implemented in Chapter 7. ....................................................... 142
LIST OF TABLES
Table 5.1
Error function SERFα(Z) of the symmetric stable distributions for a range of α
from 1.6 to 2.0.
......................................................................................56
Table 5.2
Error function SERFα(Z) of the symmetric stable distributions for a range of α
from 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 be
made if n be a fraction? The difficulty in this case can easily be understood. For if
n is a positive integer, dn can be found by continued differentiation. Such a way, however, is not evident if n is a fraction. But the matter may be expedited with the help
of 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 dissolved
solutes in aquifers. The classical governing equation is based on the diffusion equation, which uses the definition of divergence. In order for the equation to be defined, a number of assumptions must remain valid. Primary among these is that the dispersion of particles due to differences in velocity should exist as a control
volume shrinks to zero. Since the velocity fluctuations only arise from disparate aquifer material at a scale
that is large compared to the observation or measurement scale, the classical derivatives in the advection--dispersion equation are not well--defined. As a result, the classical governing equation does not fully explain
the 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 scaling of parameters. Far less work has been done examining the structure of the governing equation, especially
the suitability of the differential equation itself. At issue is the structure of the second--order diffusion equation used to model solute spreading as a plume moves. This equation uses mathematical operators whose
hidden 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 equations are models of some underlying process, a good starting point is to generalize the process of solute transport to include motions that deviate significantly from the Brownian motion modeled by the diffusion equation. This study examines and generalizes the underlying physical model used to derive the equations of
solute movement. The generalized motions lead to a new governing equation that uses fractional--order, rather than the typical integer--order, derivatives.
Solute transport in subsurface material also can be viewed as a purely probabilistic problem. This viewpoint
is intimately tied to the classical divergence (Eulerian) point of view through a string of mathematical equivalences. Einstein (1908) first explored this method by assuming that a single microscopic particle was continuously bombarded by other particles, resulting in a random motion, or a random walk. By taking appropriate
limits (letting nx and nt of the discrete walks go to zero), he found that the resulting Green’s function of
the probability of finding a particle somewhere in space was a Gaussian (Normal) probability density. The
Green’s function solution is used to specifies an initial condition of a single particle at the origin. If the motions 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 = Cxx
and 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 assumptions of Einstein’s Brownian motion. The most important of these is that Brownian motion implies that a
particle’s motion has little or no spatial correlation, i.e. long walks in the same direction are rare. In order
to use the ADE for spatially correlated velocity fields, a correction is used that forces more dispersion than
the diffusion equation provides. This is the basis for a scale--dependent, continuously evolving, effective
diffusion coefficient (the dispersion tensor) used to better match the spread of real plumes. Yet the underlying
equation is trying to impart a Gaussian profile (the Green’s function) on a plume at any moment in time. A
question naturally arises: What are the equations that describe particle motions with long--range spatial correlations? 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 techniques
and 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 in
random 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 standard
set of symbology and notation useful to hydrogeologists and others. A number of errors were found during
the translation and re--derivation process. Many articles refer to erroneous prior results, making an independent 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 has
attempted to unify the techniques, concepts, and results of various prior studies (Figure 1.2). By starting with
an underlying model of random particle movements that is a superset of traditional Brownian motion, the
end result is a generalization of the concept of divergence. In the process, a new and correct derivation of
a fractional ADE is made. Finally, the match between the new theoretical models and experimental data is
Generalized random walks:
Markov Process
Convolution
Instantaneous
approximation
Fokker--Planck
(advection--dispersion)
Equation
Fourier/Laplace
transforms
Lévy stable laws
Fourier/Laplace
transforms
Boundary value problems
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 supersets of the corresponding Normal Law and Brownian motion. Using this generalized notion of random motion, 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 the
Chapman--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). The
first uses the fact that a convolution is present and transfers to Fourier/Laplace space for solutions. The second 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 statement of the conservation of probability of a single particle’s whereabouts. Solutions to the partial differential
equation are most easily gained via Fourier and Laplace transforms, so the two methods end up in the same
place. 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 also briefly explored in Chapter 5. Chapter
6 examines two laboratory and one field experiment to investigate the utility and validity of the fractional
approach. Chapter 7 contains an ad hoc numerical implementation of the new fractional equation. Because
many of the theories in this dissertation are relatively new to the field of hydrogeology, a large number of
recommendations 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).
D qa+
positive--direction fractional derivative of order q with lower limit (a).
D q+
positive--direction fractional derivative of order q with lower limit (--").
D qa−
negative--direction fractional derivative of order q with upper limit (a).
D q−
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).
I qa+
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 difference 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):


∂ − v C + D ∂C = ∂C
i
ij ∂x
∂x i
∂t
j
(2.1)
where C is solute concentration, v and D are the velocity and dispersion tensors (respectively), x is the spatial
domain and t is time. The ADE is based on the classical definition of the divergence of a vector field. The
divergence is defined as the the ratio of total flux through a closed surface to the volume enclosed by the
surface when the volume shrinks toward zero (c.f., Schey [1992]):
∇ ⋅ J ≡ lim 1
V→0 V
 J ⋅ ndS
(2.2)
S
where J is a vector field, V is an arbitrary volume enclosed by surface S, and n is a unit normal. Implicit
in 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, since
the flux is indeed a “point” vector quantity. Conversely, solute dispersion is primarily due to velocity
fluctuations 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 single
value. The solute flux (J) is due to the combined effects of mean velocity (advection) and velocity
fluctuations 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), the
flux 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) ≡ lim 1
V→0 V
(vC − D∇C) ⋅ n dS =
S
lim 1
V
V→0
(2.3)
(vC) ⋅ n dS = ∇ ⋅ (vC)
S
In this setting, the classical divergence theorem is of little use in subsurface hydrology since the boundary
value problem for o¡(vC) = --#C/#t is infinitely complex. Because of this complexity, a de facto definition
of divergence has long been used to quantify advection and dispersion. The divergence is associated with
a finite volume and is given by the first derivative of total surface flux to volume (Figure 2.1). The dispersion
coefficient tensor does not grow (scale) if the ratio of surface flux to volume is constant over some range of
volume (solid lines in Figure 2.1). An example is a column of uniform glass beads. At the pore scale, the
ratio is non--constant and no constant dispersion parameter can be assigned. At some larger scale, the ratio
a)
b)
first
derivative
local
homogeneity
div(vC - D boC)
VOLUMETRIC
SURFACE FLUX
SURFACE FLUX
slope = div(vC-- v′C′)
div(vC - D aoC)
div(vC)
0 a
b
0 a
b
VOLUME
VOLUME
NOTE: SURFACE FLUX =
(vC + v′C′) ⋅ ndS
S
NOTE: VOLUMETRIC
SURFACE FLUX =
1
V
(vC + v′C′) ⋅ ndS
S
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 resulting
noninteger--order divergence. To reconcile the growth in the integer divergence (using
current theories) from scale a to b, the first order fluctuations v!C! are approximated by
DoC with increasing, spatially local D.
7
of total surface flux to volume is constant over a large range of arbitrary volumes and the first derivative (the
de facto divergence) is relatively constant. Solute flux within heterogeneous aquifers violates this principle
because increases in an arbitrary volume result in a growing amount of dispersive flux. The first derivative
of surface flux to arbitrary volume is not constant when a travelling solute plume samples more of the velocity
variations.
When an integer--order divergence is assumed, the ratio of surface flux to volume is forced to take on a
constant value over some volumetric range. This action approximates the monotonically increasing ratio of
surface flux to volume by a step function (Figure 2.1b). An effective parameter (D) with scaling properties
is 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 a
closed--form solution for all scales. Estimation techniques include small perturbation solution of a linearized
stochastic 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 law
multiplication of D (Su [1995]), but this leads to an equation that is not dimensionally correct. These
solutions 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 does
not account for continuous growth of the surface flux to volume ratio as a plume grows in heterogeneous
media. 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 to
the 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 size
of the control volume to approach zero. This is why the Fickian approximation works well in certain
instances. 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 Central
Limit Theorem (CLT) suggests Gaussian (Fickian) dispersion when the scale of measurement includes a large
number of independent, finite--variance velocities (c.f., Bhattacharya and Gupta [1990]). The Gaussian
velocity distribution suggests that the probability distribution of travel times can be modelled by a Markov
process with standard Brownian motion. This yields (through the Kolmogorov forward equation) a
Fokker--Planck equation of the solute particle probability and therefore concentration. The assumption of
a large scale compared to the velocity fluctuations fulfills the CLT and gives a dispersion tensor that is
asymptotically fixed and Fickian dispersion ensues.
As an interesting aside, the average velocity is considered to (roughly) change from an arithmetic to a
geometric mean as the amount of heterogeneity encountered increases. This also is a scale effect that arises
when hydraulic conductivity appears as a parameter in the Poisson Equation (for an overview of
scale--dependent mean flow, see Gelhar [1993]). The derivatives of velocity are not smooth and are tied an
irreducible finite size, or “representative elementary volume” (REV). A fractional derivative approach to the
groundwater flow problem is suggested for further study.
The classic mode of operation with the integer--order derivative formulation (2.1) is to estimate the values
of the parameters within the ADE at any particular point in a plume’s history. The parameters (D and to a
lesser extent v) are thought to change as the plume size changes. This is due to the fact that the autocorrelation
lengths 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 the
ADE 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 and
Axness [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 velocity
function (Bhattacharya and Gupta [1990]).
These methods suffer from various drawbacks due to their inherent assumptions. The first method, volume
averaging, invokes a hierarchy of distinct scales wherein the averaging length scale is much larger than the
scale of perturbation. In other words, the averaged quantity is composed of a relatively homogeneous
collection of smaller (perturbed) quantities. An example is the laboratory scale being much larger than the
pore 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 velocity
contrasts are large. Typically ln(v) is given by a Gaussian normal with a variance less than unity. This
condition limits the application of this technique to relatively homogeneous aquifers. The power--law
dispersivity growth (bullet #3 above) does not yield a parameter that is dimensionally correct, thus the
parameter does not have a sound physical basis. Moreover, this represents an unjustified, empirical addition
of another parameter into the “governing” equation. Berkowitz and Scher (1995) demonstrate that a
time--dependent dispersion tensor is also unsound. Purely statistical methods make broad assumptions about
the functional basis of a velocity field. Clearly, the work dedicated to evaluating an “effective” parameter
has lost sight of where the actual scaling occurs within (2.1). First, smooth integer derivatives of the flux
do 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 plot
of 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 homogeneous
over some scale, then the slope of the line (the ratio of surface flux to volume) is a constant (solid lines in
Figure 2.1). Within that scale, the dispersion is Fickian and one can assign a divergence of the flux according
to o⋅(vC -- DoC). Within that length scale range, the divergence is associated with an arbitrary and finite
volume. Since the ratio of surface flux to volume is constant over the range, the value of D applies
continuously throughout the range. If homogeneity is present in several distinct stages, then the dispersive
flux 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 to
volume (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 arbitrary
volume (or measurement size) encloses material with different velocity. This leads to a non--constant ratio
of 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 derived
through volume or statistical averaging, it is only valid at that particular volume (or scale). Further increases
in the slope of the dispersive flux (increasing scale) require a new D value. This simply arises because the
first spatial derivative of the dispersive flux (which defines the de facto divergence) is not constant. Rather
than assume a step function exists and force D to take on increasing values, one might assume that describing
the evolving dashed curves in Figure 2.1b would more accurately replicate plume histories and give a
predictive tool as well. The mathematical tools of fractional calculus are better suited to describing the curves
in 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 or
probabilistic model of particle movement. It is instructive to analyze that link before generalizing the notion
of particle movements and seeking the governing equation of these generalized movements. The physical
basis 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 most
intuitive way is to restrict the motions to a regular lattice so that a particle can move in only one direction
during 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 the
next lattice point is nx and the time spent in transit is nt, then
X(t) = Δx(X 1 +  + X [t∕Δt])
+ 1
where X i = 
− 1
(2.4)
if the i th step is forward
if the i th 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 and
nt 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 Limit
Theorem, as the number of jumps becomes large (i.e. let the increments become very small), X(t) is a Normal
random variable with zero mean and variance c2t.
Brownian motion is characterized by its independent increments. Since each jump is independent of the
previous jump, for all t 1 < t 2 <  < t n the increments X(t n) − X(t n−1) , X(t n−1) − X(t n−2) , ... ,
X(t 2) − X(t 1) , and X(t 1) are also independent and stationary, since the variance of any increment depends
only on the interval, not on time. The density function for the random variable X(t) is given by
f t(x) =
1 e −x 2∕2c 2t
2Õc 2t
(2.5)
Each increment of finite size X(t + z) -- X(t), where z is a finite constant, is composed of infinitely many
smaller 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 themselves
independent, identically distributed (iid) random variables. These variables also have the important feature
of finite variance. So the limiting distribution of the sum of a large number of iid finite--variance random
variables is the Normal distribution. The variance of the sum of independent variables is the sum of the
individual variances.
The term Standard Brownian motion, given the symbol B(t) refers to a Brownian Motion with unit (c). Any
Brownian 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--variable
partial differential equations can be facilitated by integral transform in order to remove dependance on one
of the independent variables.
Throughout this text, the Fourier transform F and its inverse F- 1 are defined as:
∞
~
F(f(x)) = f(k) =
e
−ikx
f(x)dx
(2.1)
ikx~f(k)dx
(2.2)
−∞
∞
F −1(f(k)) = f(x) = 1
2Õ
~
e
−∞
~
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
−ikx
−∞
D ∂ C2 dx
∂x
2
(2.3)
If C and its derivatives vanish as |x| ! ∞, than integration by parts twice gives
∞
d
dt
e
−ikx
~
Cdx = − k 2DC
(2.4)
−∞
~
~
dC = − k 2DC
dt
(2.5)
where the tilde indicates the Fourier transformed function.
Given a Dirac delta function initial condition:
C(t = 0, x) = δ(x − 0)
~
C(t = 0, k) = 1
gives the Gaussian (Normal) density for the Green’s function solution:
(2.6)
(2.7)
11
~
C(k, t) = exp(− k 2Dt)
(2.6)
With inverse transform:
C(x, t) =
1 exp(− x 2∕2Dt)
2ÕDt
(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 as
an underlying probabilistic model.
Another method of relating the diffusion equation and a Brownian motion relies on the fact that the
differential displacement of particles dX(t) = X(t + dt) − X(t) is Gaussian and satisfies Ito’s stochastic
differential 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 function
g is a constant tensor of the standard deviation of the Gaussian process X(t). This process satisfies the
Fokker--Planck equation of the “flow” of probability in time and space:
∂P = ∂ (− f ⋅ P) + ∂ 2 (g ⋅ P)
∂t
∂x
∂x 2
(2.9)
If many particles are simultaneously released and do not affect each other, the probability and concentration
are interchanged to give the ADE. One can simplify the problem further by describing a mean--removed
equation that follows a moving frame of reference that travels at the mean velocity. The diffusion equation
is recovered.
If the underlying physical model described above is altered to allow a higher probability of long--range
particle 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 solution
is generalized to “Lévy motion,” a fractional--order equation, and fundamental solutions that are superset of
the 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 characteristic
function. The characteristic function is a description of the Fourier transform of the probability density function. (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(⋅) is
the expectation:
∞
e
E(e ikX) = ψ(k) =
(3.1)
ikxf(x)dx
–∞
^
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 and
vice--versa. Unless noted otherwise, the Fourier transforms in this study will place the constant 1/2Õ on the
inverse transform to more closely resemble the characteristic function.
For positive domain distribution functions, the one--sided Laplace transform (moment generating function)
is useful:
∞
E(e sX) = Ô(s) =
 e f(x)dx
sx
(3.2)
0
Integration by parts gives the Laplace transform of a cumulative distribution function F(x):
Ô(s)
s =
∞
 e F(x)dx
sx
(3.3)
0
Inversion of the characteristic functions follows the inverse Fourier transform:
∞
f(x) = 1
2Õ
e
–∞
−ikxψ(k)dk
(3.4)
13
Or in Laplace space:
γ+i∞
f(x) = 1
2Õi

e −sxÔ(s)ds
(3.5)
γ–i∞
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, identically distributed (iid) Normal random variables. It is intuitive that a sum of iid Normal variables would keep
the same distribution after dividing by a normalizing constant. One might wonder if sums of random variables with other distributions maintain the distributions of the individual summands. A large family of these
distributions were shown to exist by Paul Lévy in 1924. The Normal distribution is merely a member of
Lévy’s family of stable distributions. Lévy’s relevant and oft--cited work (1924; 1937) has not been translated
into English. Lucid summaries and extensions are provided by Feller (1966), Zolotarev (1986), Samorodnitsky 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 variables is:
X 1 + X 2 +  + X n
cn
Sn =
(3.6)
A number of assumptions about the probability functions are omitted for clarity. See Feller (1966, Ch. XVII
and 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(e ik(X1+X 2)) = E(e ikX 1e ikX2) = ψ X1(k)ψ X 2(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 can
also calculate the characteristic function of the expectation of a scaled and shifted variable aY + b:
E(e ik(aY+b)) = E(e ikaY ⋅ e ikb) = e ibkψ Y(ak)
(3.8)
Now the scaled sum cnSn in (3.6) can be related to the density of the iid variables Xn:
ψ cnS n(k) = ψ S n(c nk) = (ψ X(k)) n
(3.9)
Equating the characteristic functions ψX and ψSn and taking logarithms:
log ψ(c nk) = n log ψ(k)
(3.10)
This equality is fulfilled by a power law:
log ψ(k) = Ak α
(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 sum
and summands are identically distributed. The result is a generalization of the Central Limit Theorem for
sum of n random variables (X) that are iid:
14
d X 1 + X 2 +  + X n
Sn =
n 1∕α
(3.12)
An entire family of distributions that includes the Gaussian is described when the value of the exponent α
ranges 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 take
the general form (Samorodnitsky and Taqqu [1994]):
ψ(k) = exp(–|k| ασ α 1–iβsign(k) tan(Õα∕2) + iμk) α ≠ 1
(3.13)
where the parameters σ, β and μ describes the spread, the skewness and the location of the density, respectively. The sign(k) function is --1 for k < 0 and 1 otherwise. The characteristic function for α = 1 (the Cauchy
distribution) 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 function
is:
ψ(k) = exp(− σ α|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. It
is a simple matter to show that for α > 1, E(X) = μ. The mean is undefined for α ≤ 1. A standard, symmetric
α--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, that
when the scale factor of the stable law σ = 1, the standard deviation of the Normal (α = 2) distribution (N)
is 2:
N(k) = exp− 2σ 2k 2 + iμk
(3.16)
The most important feature of the α--stable distributions (3.13) is the characteristic exponent (also called the
index of stability) α. The value of α determines how “non--Gaussian” a particular density becomes. As the
value of α decreases from a maximum of 2, more of the probability density shifts toward the tails. Figure
3.1 shows the standard α--stable distribution functions for α = 1.6, 1.8, 1.9, and 2. Note that the distributions
appear very Gaussian in untransformed coordinates, and that the difference lies in the relative weight present
in the tails. For probabilities between 1 and 99 percent, the different distributions appear near--normal.
Non--standard (σ ≠1 and μ ≠ 0) stable distribution functions (F) and densities (f) are related to their standard
counterparts by the relations:
(x −σ μ) , 1, 0
(3.17)

(3.18)
F αβ(x, σ, μ) = F αβ
1 f (x − μ) , 1, 0
f αβ(x, σ, μ) = σ
αβ
σ

Cauchy and Lévy sought closed--form formulas for the stable densities (in real, not Fourier, space) for all
values 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. Since ψ(k) is known exactly,
15
100
PROBABILITY (PERCENT)
(a)
80
60
40
20
0
--10
--5
0
x
99.9
α = 2.0 1.9
1.8
99
PROBABILITY (PERCENT)
10
5
1.6
(b)
90
70
50
30
10
1
0.1
--10
--5
0
x
5
10
Figure 3.1 Plots of the distribution function F(x) versus x for several standard symmetric α--stable distributions using a) linear scaling and b) probability scaling. The
Gaussian normal (α = 2.0) plots as a straight line using probability scaling for the vertical axis.
16
a fast numerical Fourier inversion can yield accurate densities. The Fourier inversion formula also has many
real--valued integral representations that yield quick numerical solutions (c.f., McCulloch [1994, 1996]; Zolotarev [1986]). In particular, McCulloch (1996) gives the integral representation of the standard forms for
σ =1 and μ = 0 of the cumulative probability function (Fαβ):
1
sign(1 − α)
F αβ(x) = C(α, Ò) +
2
 exp− x
α
* α−1

U α(Ô, Ò) dÔ
(3.19)
−Ò
where
x * = c *x

c * = 1 + β tan(Õα∕2)
2

1
−2α
2 tan −1β tan(Õα∕2)
Ò = Õα


1, α > 1
C(α, Ò) = (1 − Ò)∕2, α < 1
sin Õ2 α(Ô + Ò)
U α(Ô, Ò) =
cos Õ2 Ô

α
1−α
The densities are obtained by differentiating the cumulative probabilities with respect to x. Note McCulloch’s (1996) mistaken standard density (fαβ) that should read:
1
x *α−1αc *
f αβ(x) = |
2 1 − α|
1
 U (Ô, Ò)exp− |x |
α
*
α
α−1
U(Ô, Ò)dÔ
(3.20)
−Ò
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 are
credited with independently deriving similar expansions. Feller (1966) also gives series expansions for a
slightly different parameterization, using γ to quantify the skewness, rather than β:
1
f αγ(x) = Õx
∞
 Γ(kαk!+ 1) (− x)−kα sin Õk2 (γ − α)
0≤α<1
(3.21)
1<α≤2
(3.22)
k=0
1
f αγ(x) = Õx
∞
Õk (γ − α)
 Γ(kα−1k! + 1) (− x)k sin 2α
k=0
Feller’s skewness parameter γ is obtained by equating the canonical form (3.13) to his equivalent representation of the standard characteristic function:
17
0.4
(a)

x−μ
σ ⋅ fα σ

α = 2.0
(Gaussian)
0.2
α = 1.4
0.0
- 5.0
- 3.0
- 1.0
1.0
3.0
5.0
x−μ
σ
100
10- 1
10- 2

x−μ
σ ⋅ fα σ

10- 3
(b)
α=
10- 4
1.2
1.4
1.6
10- 5
10- 6
10- 7
10- 1
2.0 (Gaussian)
1
10
1.8
100
Figure 3.2 Plots of symmetric α--stable densities showing power--law “heavy” tailed
character. a) linear axes, and b) log--log axes.
18
ψ(k) = exp− |k| αe iÕ(sign k)γ∕2
(3.23)
Resulting in (Samorodnitsky and Taqqu [1994]):
 −Õ 2 arctan(β tan(Õα∕2))
γ = 2
Õ arctan(β tan(Õ(α − 2)∕2))
0<α<1
(3.24)
1<α<2
For symmetric densities, setting γ=0 in (3.22) yields a formula that converges with reasonably few terms even
with large arguments. This expansion will be used throughout this study:
1
f α(x) = Õ
∞
k
+ 1 + 1x 2k
 (2k(−+1)1)!
Γ2k α
1<α≤2
(3.25)
k=0
Portions of this study require a generator of random variables that share an α--stable distribution. Janicki and
Weron (1994) gives an algorithm for generating a standard, symmetric stable random variate X based on a
uniform random variable V on (--Õ/2,Õ/2) and an exponential variable W with unit mean:
X=

sin(αV)
cos(V − αV)
⋅
1∕α
W
(cos(V))

(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 characteristic
function 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 several
of the α--stable densities, clearly showing the power--law tail behavior. The Gaussian density lacks the power--law tail, although the α--stable family represents a continuum. As α approaches 2, the power--law behavior
only becomes evident at very large values of |x|. It has been shown that the power--law tail behavior is present
with 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 ≡
 x f(x)dx
r
(3.27)
−∞
The first several integer moments are historically those that are studied in physical sciences. In order for a
moment to exist, the integral (3.27) must converge. Since at least one of the tails of any α--stable density
follows a power law for large |x| (Feller [1966]), we can integrate (3.27) to check for convergence. At the
tails we have lim C|x| r−α which is finite only if r < α. So the moments higher than the real number α do
|x|→∞
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 values 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 (any
rth moment with r < α). Janicki and Weron (1994) use quantiles of the distribution when investigating the
spread of an α--stable process. The quantiles qp are defined here as F - 1(p) with its pair F - 1(1--p), where p
is a desired probability. Thus for a random variable X with distribution F(x), the quantiles qp are the points
x where F(x) = p and F(x) = 1--p. The probabilities 0.159 and 0.841 are typically used for the Normal distribution since these numbers correspond to the mean  one standard deviation. These quantiles will be used
in this study for convenience.
Another useful formula gives the value of the moments of order less than α. Nikias and Shao (1995) show
that the fractional lower--order moments of a symmetric SαS variable X are calculated by the formula:

2 r+1Γr+1
2 Γ(− r∕α) r
σ
E(|X| ) =
α Õ Γ(− r∕2)
(3.28)
r
In particular, the expectation of the absolute value of X (r = 1) reduces to
E(|X|) =
2Γ(1 − 1∕α)
σ
Õ
(3.29)
Figure 3.3 is a plot of E(|X|) for 1 < α < 2 illustrating the fact that for the standard symmetric α--stable distributions with 1.4<α<2.0, the expected value of |X| lies between 1.1 and 2.0.
100
E(|X|)
10
1
1.0
1.2
1.4
1.6
1.8
2.0
α
Figure 3.3 Expectation of the absolute value of random variable X with a standard, symmetric, α-stable distribution for 0<α<2.
20
CHAPTER 4
PHYSICAL MODEL
The process of irregular motion which we have to conceive of as the heat--content of
a substance will operate in such a manner that the single molecules of a liquid will
alter 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 continuous
time random walk (CTRW) with Gaussian increments that is also the limit process of uncorrelated, unit jumps
on a lattice. The movement of a particle in aquifer material clearly does not follow Brownian motion since
geologic material is deposited in continuous, correlated units. A particle travelling faster than the mean at
some instant is much more likely to still be travelling faster than the mean some later time due to the spatial
autocorrelation of aquifer hydraulic conductivity. The same is true for particles travelling slower then the
mean velocity. This suggests that particle excursions that deviate significantly from the mean are much more
likely than traditional Brownian motion can model. This Chapter examines another (superset) model of particle 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 Joseph
and Noah effects coined by Mandelbrot and Wallis [1968]) for which Brownian motion is an inadequate model. 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 trapped
in vortices that are essentially stagnant with respect to mean flow. Mixing within a vortex may in fact follow
Brownian 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 distribution 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 than
having Gaussian increments, they have Lévy’s α--stable, or power--law (Pareto) distribution increments. It
is instructive to analyze how these Lévy motions can arise as a limit process of jumps on a lattice, just as was
done with Brownian motion in Chapter 2. This analysis leads to a more general model of random walks on
a lattice and provides a link between the memory of a fractional derivative and the memoryless property of
random 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 released at the Cartesian origin. This propagator is a surrogate for concentration if it represents a large quantity
of independent solute “particles.” Since a contaminant mass placed in an aquifer is composed of a huge number of these particles, the propagator density is “filled in” by the solute particles. So the task of deriving a
governing equation for the movement of an instantaneously released slug of solute tracer is reduced to solving
the 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 higher
propensity to make longer excursions than a particle experiencing Brownian motion. If the walks are uncorrelated with respect to time, they must be spatially correlated in order to embark on these longer walks. Brownian motion’s unit walks on a lattice are uncorrelated in space (although it will be shown that a Brownian
motion defined by Gaussian increments has some very short--range spatial correlation). Thus the difference
between Brownian motion, and its superset Lévy motion, is the range of spatial correlation. This is shown
in Section 4.3. The first two Sections (4.1 and 4.2) are primarily a review of current theories that are applicable to solute transport, with minor corrections to the originals where indicated. The final Section (4.3) also
includes a new derivation of the statistical properties of a particle undergoing Lévy walks to enable estimation
of 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 finding
the particle at lattice position j after n jumps is denoted Pn(j). Let p(m) be the probability of jumping m lattice
sites during a single step. The Markov property of jump independence dictates that the probability of finding
the walker at site j at the next step is the sum of the transition probabilities from all other lattice sites (j′) multiplied by the probability of being at those sites. This is stated mathematically in the Chapman--Kolmogorov
equation:
∞

P n+1(j) =
p(j − j′)P n(j′)
(4.1)
j′=−∞
This is a convolution, so the Fourier--transformed probabilities are used where:
∞

~
P n(k) =
e ikjP n(j)
(4.2)
e ikmp(m)
(4.3)
j=−∞
∞
p~ (k) =

m=−∞
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 starting at the origin has a delta function initial probability: P0(i) = δ(i--0). The resulting probability Pn(l) is called
the “propagator” since it describes the n--step spatial evolution of a single event at the origin. By induction
and 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:
2Õ
P n(j) = 1
2Õ
e
0
−ikj(
n
p(k)) dk
(4.5)
22
A Brownian motion must also be described by (4.4). One way to recover Brownian motion is to restrict particle movement to 1 lattice position in either direction of the current (mth) position using the Dirac delta function distribution (Hughes et al. [1981]):
p(m) = 1 (δ(m + 1) + δ(m − 1))
2
(4.6)
The transformed probability is:
−ik
ik
= cos k
p~ (k) = e + e
2
(4.7)
When the number of transitions becomes large (n→∞),

2
P(k∕ n) = (p~ (k∕ n)) n = cos(k∕ n) n = 1 − k + O(1∕n)
2n
~

n

2
≈ exp − k
2

(4.8)
Fourier inversion gives the Gaussian profile:
2Õ
P n(j∕ n) ≈ 1
2Õ
 exp(− ikj) exp−2k dk = 2Õ1 exp(− j ∕2)
2
2
(4.9)
0
A change to the real variable x = j/(n1/2) results, for large n, in the approximation
P n(x) =
1 exp(− x 2∕2n)
2Õn
(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 m 2 where  ⋅
m  =
2
∞

m 2p(m)
 denotes expectation:
(4.11)
l=−∞
The resulting limiting value of the transformed step probability is Gaussian:

p~ (k) ≈ 1 − m 2 k ≈ exp − m 2 k
2
2
2
2

(4.12)
And the overall trajectory density of an individual walker with any finite--variance transition probability is
P n(j) =
 
− j2
1
exp
2m 2n
2Õm2n
(4.13)
The distinguishing characteristic of all Brownian motions is held in (4.6) and (4.11). If the distance that a
random 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 with
infinite variance, and/or 2) not instantaneous. Both of these modifications lead to models that simply and
concisely describe a wealth of macroscopic, non--Gaussian processes (Bouchard [1995]). These real--world
phenomena 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 is α--stable. These
walks 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(− σ α|k| α)
(4.14)
And the probability propagator is also an α--stable distribution, by virtue of its characteristic function:
~
~
P n(k) = exp(− nσ α|k| α) ⇔ P n(k∕n 1∕α) = exp(− σ α|k| α)
(4.15)
These random motions are named Lévy flights since the particles are instantaneously moved from point to
point. These random diffusion paths have infinite variance. Figure 4.1 shows a series of points in two dimensions that follow a Lévy flight. The distance of each flight is an α--stable random variable and the direction
is a uniform random [0,2Õ] variable. (In this case, the direction changes were limited to multiples of Õ/2 to
show simulated movement on an orthogonal lattice). The left plot (Figure 4.1) shows a Lévy flight with a
characteristic exponent of 1.9, which is not too dissimilar to a Brownian motion (α=2.0). The rightmost plot
uses a more tail--heavy distribution with α=1.7. Note that these flights are characterized by clusters that are
separated by relatively infrequent, long--distance jumps. The set of turning points is a random fractal, and
magnification of the clusters shows the same clustered behavior on increasingly smaller scales. Figure 4.2
shows the cumulative displacement of a one--dimensional walker undergoing Lévy flights with exponents
of α=1.7 and 1.9. The variance of both processes are infinite, but the “spread” of these processes can be analyzed using quantiles. Note that within the definition of the Levy flights, a “particle” does not visit the points
in space between turning points, i.e. the connecting lines in Figures 4.1 and 4.2. Rather, a particle is instantly
moved from turning point to turning point.
200
α = 1.9
150
α = 1.7
100
50
0
- 50
- 50
0
50
--100
- 50
0
50
Figure 4.1 Lévy flights in two dimensions.
100
150
24
150
α=1.9
α=1.7
100
X(t)
50
0
- 50
500
1000
t
1500
2000
Figure 4.2 Numerical approximation of one--dimensional, continuous--time, random Lévy walks.
The symmetric α--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 of
1.7 and 1.9 (Figures 4.1 and 4.2). A listing of the FORTRAN code (SIMSAS.F) used to generate the Levy
flights 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 process
becomes a Gaussian process) as the number of jumps or trials becomes large. Hughes et al. (1981) describe
a simple clustering symmetric walk that approximates a Lévy process in the same way. They use jumps on
a lattice in which the probability of travelling a step of length x is
p(x) = a − 1
2a
∞
 a−n(δ(x − Δbn) + δ(x + Δbn))
(4.16)
n=0
where b2 > a > 1 and $ > 0 is the lattice spacing. If b is an integer, then jumps of size 1, b, b2 ... bn lattice
positions are allowed. The larger jumps are less likely by a factor of a- n. On average, a cluster of (a) jumps
of 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]):
1
p(k) = a −
a
~
∞
 a−ncos(Δbnk)
(4.17)
n=0
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
1
p~ (k) = 1a p~ (bk) + a −
a 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 =
(Δ α∕τ) Õ2
ln(a)
and C = −
ln(b)
Γ(α) sin(Õα∕2)
(4.19)
In fact, for wave numbers smaller than that of the lattice spacing, the exponential function that describes an
α--stable transition density matches quite well (Figure 4.3). By the Tauberian Theorem (Feller [1966]), the
tails 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, since
its Fourier transform follows:
n
P n(k∕n 1∕α) = p~ (k∕n1∕α) ≈ exp(− C|k| α)
~
(4.20)
which indicates that the propagator is asymptotically invariant after scaling by n1/α.
1.0
b = 1.12
a = 1.2
α = 1.61
0.5
Exponential (4.19)
p(k)
0.0
Weierstrass Function (4.17)
--0.5
0.0
5.0
wave number (k)
10.0
Figure 4.3. Graphs of the Fourier transform of the particle jump probability (the
structure function) of a “clustered” walk on a discrete lattice. For this graph, the lattice
spacing (Δ) was set to unit length. Note the good approximation of the complete
Weierstrass function by the exponential function for wave numbers smaller than the
inverse of the lattice spacing (i.e. k<1).
26
Levy Flights -- Continuous Time
To generalize this random walk to non--integer jump sizes and real--valued time, first allow jumps of non--integer size, i.e. b∈9 > 1. An integral replaces the summation for the n--step propagator:
∞
P n+1(x) =
 p(x − x′)P (x′)dx′
(4.21)
n
−∞
If each jump takes an equal amount of time (τ) to complete, then the change in probability over a single jump
is
∞
P n+1(x) − P n(x)
=
τ
 1τ (p(x − x′) − δ(x − x′))P (x′)dx′
n
(4.22)
−∞
Noting that the limit as τ → 0 is the time derivative of the propagator, then the motion can be generalized
to a continuous time function. If we denote the time of the nth step as t, we have
∞
 lim 1τ (p(x − x′) − δ(x − x′))P(x′, t)dx′
∂P(x, t)
=
∂t
(4.23)
τ→0
−∞
The integral now requires that the step sizes dx′ are infinitesimally small, which requires that the lattice spacing Δ → 0. The rate at which the spacing shrinks must depend on how τ → 0. In Fourier space, the last result
becomes
~
~
∂P(k, t)
= lim 1τ (p~ (k) − 1))P(k, t)
∂t
τ,Δ→0
(4.24)
Two options arise for the transition density, finite or infinite variance. The finite variance case should recover
Brownian motion (i.e. solve the diffusion equation). This requires taking the limits Δ and τ so that the expression is not trivially 0 or infinity, i.e. Δ2/τ is a constant. The finite--variance structure function where each
jump is $j in size is now
p~ (k) ≈ 1 − Δ 2j 2 k
2
2
(4.25)
leading to the evolution of the propagator:


− Δ 2j 2 ~
~
∂P(k, t)
= lim
P(k, t) ≡ − DP(k, t)
∂t
2τ
τ,Δ→0
~
(4.26)
which is the Fourier transform of the integer--order diffusion equation with a diffusion coefficient defined
by
Δ 2j 2
τ,Δ→0 2τ
D = lim
(4.27)
To generalize the walk, Hughes et al. (1981) use the structure function for infinite variance walks. The limits
of Δ and τ are once again taken so that zero or infinity do not result, i.e. $α/τ = constant. Now the structure
function is approximated by 1 -- C|$k|α, so
27
α
lim −τ C |Δk| αP(k, t) = lim − C Δτ |k| αP(k, t)
τ,Δ→0
τ,Δ→0
(4.28)
And the probability propagator follows the equation
~
~
∂P(k, t)
= − D α|k| αP(k, t)
∂t
(4.29)
and is therefore α--stable:
~
P(k, t) = exp(− D αt|k| α)
(4.30)
Hughes at al. (1981) use the full structure function equation (not listed here for simplicity) to derive the last
equation with a complete description of the diffusion coefficient for Lévy flights:
α
Õ
D α = lim Δτ ⋅
2Γ(α) sin(αÕ∕2)
Δ,τ→0
(4.31)
The velocity of a particle undergoing the continuous time random walks (including Brownian motion) have
Δ α ⋅ Δ 1−α = lim CΔ 1−α = ∞. Lévy motions
=
infinite velocity (v) when α>1, since v = lim Δ
lim
τ
Δ→0
Δ,τ→0 τ Δ,τ→0
with α<1 are called “ballistic,” and correspond to wave--equation type behavior. This study will concentrate
on Lévy motions with α>1. In subsequent sections, other formalisms are introduced that use continuous time
as a basis of particle motion rather than the limit of a discrete time process above.
Since the probability propagator is a density, calculation of its second moment is done easily in Fourier space
since the variance of a random variable X with a density of f(x) follows
∞
~
 d2 ∞

d 2f(k)
−ikx
2
2
=−
E(X ) =
x f(x) =− 2
e
f(x)
dk 2 k=0
dk


−∞
−∞
k=0



(4.32)
An α--stable Lévy flight (4.30) has an infinite variance for all values of time greater than zero since the second
derivative contains terms with |k|α- 2.
4.2 Lévy Walks -- Continuous Time Random Walks
Some researchers believe that while the variance of many diffusion processes grows nonlinearly with time,
it maintains a finite value. It is unclear whether measurements of, say, solute concentrations are reliable indicators of a finite variance, since only the tail of the propagator (akin to concentration at very large values of
|x|) causes a diverging variance. In the case of dissolved solutes, a diverging variance may require accurate
measurement of concentrations many orders of magnitude smaller than an injected tracer. The sample variance of an α--stable process will have finite variance that grows according to the number of samples raised
to the power 1/α (we show equivalent growth as a function of time later in this Section). Nevertheless, controlled two--dimensional chaotic flow studies (Weeks et al. [1995]) and numerical simulation of iterated functions that mimic Lévy motion (Zumofen and Klafter [1993]) have prompted a search for a process that maintains a finite variance, albeit a variance that grows as a power of time other than one:
r 2  ∝ t η
(4.33)
28
where r is the magnitude of a particle’s position vector. This is one of the characteristics of fractional Brownian motion (fBm); however, fBm requires Gaussian increments, which are not observed in these motions.
Rather, the increments of the processes under study are distributed like α--stable Laws.
Another formalism that has been introduced (Montroll and Weiss [1965]; Geisel [1995]) uses continuous
time as the basic motion, rather than an ending limit process of a discrete time model as described in the previous Section. The continuous time model will be described here because of the attractiveness of the particle
motion description to reactive solute transport problems. The premise of the continuous time models is that
a particle moves at a constant velocity through a series of independent paths that require a random amount
of time to complete. Thus the path lengths are random variables with distributions that are either α--stable
or Pareto (see Appendix III for a definition of the Pareto law).
It has long been known that a particle in a simple random walk on a disordered lattice (for example, a Sierpinski carpet) results in anomalous diffusion with η<1 (c.f., Giona and Roman [1992a, 1992b] and references
within). In order to define a continuous time random walk (CTRW) with η>1, the Lévy flights must be used,
but a particle no longer makes an instantaneous transition from point to point. Instead, each transition requires some time to complete. A variety of methods are used to limit the infinite excursion velocity of the
Lévy flights, including random waiting periods at turning points (Zumofen and Klafter [1993]) or description
of the velocity as deterministic (coupled) or random (decoupled) functions of each flight’s length (Shlesinger,
et al. [1982]; Klafter, et al. [1987]). These models are particularly attractive for solute transport in an aquifer,
since a particle undergoing macroscopic dispersion has a measurable velocity along its trajectory. It is also
an appropriate model for reacting or sorbing solutes, since the waiting time invokes random trapping within
a random velocity field. For simplicity, this study will principally examine non--reactive solutes and rely on
velocity functions. The velocity function model of Lévy walks is physically realistic and related to the field-measurable velocity within an aquifer.
With simple deterministic velocity functions, a particle “scheduled” for a rare long trip by an α--stable p(r)
must take some finite time to travel. Suitable velocity functions preclude the possible visitation of an infinity
of points over a specified time interval. The superdiffusion exponent η>1 is a function of the Lévy index α
and the velocity function. Since the particle must move along the entire trail between the turning points (Figure 4.1), these CTRWs are named “Lévy walks” to distinguish them from the “flights” of the previous section.
The most useful property of the Lévy walks is that the velocity of a particle is defined along the entire trajectory. The theoretical spatial autocorrelation of a particle undergoing Lévy walks can be calculated and matched
with the measurable spatial autocorrelation of velocity within an aquifer. These procedures are developed
in Section 4.3.
Coupled space--time probability
We are ultimately interested in the function P(x,t) which is the probability density associated with a particle
at location x and time t. There is a possibility of moving to point x from a former location (x1), and we denote
this transition density p(x,t|x1,t1). This density is a conditional probability and specifies the probability that
the particle will move to position x at time t given the fact that the particle is at location x1 at the previous
time t1. We will denote this transition density p(x--x1;t--t1) denoting movement from x1 to x in the interval
t--t1. When possible, we use the shorthand notation p(r,t) for this density, indicating the probability density
function of a jump of length r requiring a time span (t) to complete. A popular methodology in the study of
chaotic flow (Shlesinger, et al. [1982]; Klafter, et al. [1987]) assumes that the transition probability is governed by a coupled travel time and distance probability distribution:
29
p(r, t) = Ô(t|r)p(r)
(4.34)
where Ô(t|r) is the conditional probability that a particle will take time t to complete a walk of length r, given
that such a walk occurs. Klafter et al. (1987) and Blumen et al. (1989) also use a functional velocity form,
but choose a walk time that is a power function of the distance:
Ô(t|r) = δ(|r| − t ν)
(4.35)
So the time required for a walk is equal to |r|1/ν resulting in a velocity v(r) = |r|1-- (1/ν) (Figure 4.4). The joint
(and coupled) space--time jump probability are equivalently represented (Zumofen and Klafter [1993]):
p(r, t) = δ(|r| − t ν)p(r) = 1 δ(t − r 1∕ν)p(t)
2
(4.36)
where p(r) or p(t) are Pareto or α--stable.
The probability propagator can be directly solved via this space--time jump probability through several similar derivations (Klafter et al. [1987]; Shlesinger et al. [1982]) that are primarily based on the CTRW model
of Montroll and Weiss (1965). First recognize that the jumps now have a random duration, so that each walk
occurs at a random time. This transition time (walk duration) probability is related to the space--time walk
probability by summing over all possible walk lengths:
20.0
ν=100
ν=∞
Velocity
ν=10
10.0
ν=2
C
0.00.0
ν=1
ν=1/2
10.0
Walk Size
20.0
Figure 4.4 Particle velocity as a function of Lévy walk magnitude and conditional walk
time parameter (ν).
30
∞
p(t) =

p(r, t)
(4.37)
r=−∞
A straightforward replacement of the summations used here with integrals would describe transitions anywhere in space, not just on a lattice. The survival probability of a particle at any position x is
t
Φ(t) = 1 −
 p(τ)dτ
(4.38)
0
with a Laplace transform of
Φ(s) =
1 − p(s)
s
(4.39)
Within this text, the Laplace transformed functions are denoted by a change of variable from t to s. Following
Shlesinger et al. (1982) and Klafter et al. (1987), an intermediate function q(x,t) is introduced that describes
the probability density of a particle reaching the turning point x of a Lévy walk at time t + dt. This density
is the result of the space--time Chapman--Kolmogorov Equation that states that the particle can move from
all other lattice points in the correct amount of (prior) time:
q(x, t) =
t
∞

x′=−∞
 q(x′, τ)p(x − x′; t − τ)dτ + δ(x − 0)δ(t − 0)
(4.40)
0
where the last term represents the initial conditions of the particle (x=0 at t=0).
Now the total probability that a particle is located at x at time t is obtained by integrating the probability of
being at x in the past q(x,t--τ) times the probability of staying at the site Φ(t):
t
P(x, t) =
 q(x, t − τ)Φ(τ)dτ
(4.41)
0
Taking Laplace transforms of the last two convolutions allows combination of the two equations (Appendix
II.1):
t
P(x, t) =
  P(x′, τ)p(x − x′; t − τ)dτ + Φ(t)δ(x − 0)
x′
(4.42)
0
Taking Fourier and Laplace transforms, the space and time convolutions give:
~
~
P(k, s) = P(k, s)p~ (k, s) + Φ(s)
(4.43)
Simplifying,
~
P(k, s) =
Φ(s)
1 − p~ (k, s)
Using (4.39), both p~ (k, s) and Φ(s) are defined by the joint (coupled) space--time jump probability:
(4.44)
31
~
P(k, s) =
1 − p(s)
s − sp(k, s)
(4.45)
The behavior of the second moment of the propagator is of principle interest in the study of dispersion, since

~
x2
2
 = − ∂ P(k,2 t)
∂k

k=0
= L−1

− p~ kk(k, s) 1 − p(s)
s
(1 − p~ (k, s)) 2

(4.46)
k=0
This reduces to studying the transforms of the transition probability distribution p(r,t) and in particular, the
chosen velocity function v(r), since p(r) is a known α—stable density. However, direct analytic solutions may
not be tractable when p(r) is α--stable (Appendix II). Following Klafter et al. (1987) and Blumen et al. (1989),
one can assume that for sufficiently large time and distance, the probabilities are described as a power--law
Zeta (Zipf) distribution for jumps on a lattice or by the Pareto distribution for real--valued excursions, i.e.
p(r, t) = δ(|r| − t ν)|r| −1−α ≡ δ(|r| − t ν)|r| −λ
(4.47)
In addition to limiting the speed at which particles may travel, the delta function imposes a maximum jump
size (upper cutoff). A finite time means that, through the delta function, infinite jumps occur with zero probability. For the power functions to be probability densities, they must also have a lower cutoff since zero raised
to a negative constant is infinity. The cutoff is also physically justified since the act of measurement imposes
a filter that averages the smallest movements (Cushman [1984]). Also, a positive norming constant C (which
is a function of the lower cutoff) is used to bring the area under the density to unity. Calculation of the propagator and variance requires the calculation of three transforms. The propagator requires p(k,s) that is valid
over a wide range of k, while the variance requires an expression that is valid at very small values of k, so
that the second derivative can be calculated and evaluated at k = 0.
Several of the transforms developed in this study differ from those given by Klafter et al. (1987) and Blumen
et al. (1989), so a full derivation is given (Appendix II) for completeness. To first order, we have for the transform p(s):
p(s) ≈

1 − C Òs λν−1
1 − τs
1 < λν < 2
2 < λν
(4.48)
in agreement with Blumen et al. (1989). In the second case, the constant τ is simply the mean waiting time
per step, which is finite only for 2 < λν.
Two different approximations are needed for p(k,s): one for very small k (to calculate the particle’s position
variance) and one valid over a large range of k (to calculate the propagator). A more simply manipulated
transform of the former combines the waiting time distribution and the transition density:
ν(λ − 2) > 1
 C2
p~ (k, s) − p(s) ≈ − k 2 ⋅  ν(λ−2)−1
ν(λ − 2) < 1
C3s
(4.49)
In the last case, when ν(λ -- 2) < 1, the transform does not converge at s = 0. This suggests that in the region
ν(λ -- 2) < 1, the variance of the particle transition distance diverges to infinity as time becomes asymptotically
32
large. Since we have two realms of mean transition time (finite and infinite) by equation (4.48), and two
realms of transition distance variance (finite and infinite) from the last equation, we have four distinct cases
for the transformed density as k → 0. One region is practically eliminated because for λ < 4, the conditions
ν(λ -- 2) < 1 and νλ < 2 are mutually exclusive (Figure 4.5). The remaining three regions have approximate
transition density transforms of:
Region V1
 1 − τs − C2k2

ν(λ−2)−1
2
~
Region V2
p(k, s) ≈  1 − τs − C 3k s
1 − CÒsνλ−1 − C3k2sν(λ−2)−1 Region V3

(4.50)
where Region V1 has 2 < νλ and 2 < ν(λ -- 2); region V2 has 2 < νλ and ν(λ -- 2) < 2; and, region V3 has 1
< νλ < 2 and ν(λ -- 2) < 2. Using (4.46) one has for the asymptotic (large time) variance of the propagator
for the same regions:
t
t2−ν(λ−2)
r 2  ∝ 

 t2ν
Region V1
(4.51)
Region V2
Region V3
One can now see the effect of the delta function conditional probability on the calculated variance. A Lévy
flight with exponent α < 2 (λ < 3) will have infinite variance. The delta function truncates the density and
appears to do so in a manner that allows many Lévy walks to mimic Fickian growth (Region V1). Since the
10.0
8.0
Region V1
Region V2
r 2 ∝ t
r 2 ∝ t 2−ν λ−2
6.0
ν
ν = 1∕(λ − 2)
4.0
Region V3
2.0
0.0
1.0
ν = 2∕λ
ν = 1∕λ
r 2 ∝ t 2ν
2.0
λ
3.0
4.0
Figure 4 5 Regions of applicability of particle travel distance variance calculations
33
density diverges at s = 0 in this region, so will the variance. This is due to the fact that the delta function
conditional probability at s = 0 would allow any jump size at infinite time, and the α--stable jump density
is no longer truncated. If (s) is always greater than zero, then time is always finite and so too are the possible
excursion distances, i.e. the jump probability is truncated and finite. One might expect that the walk would
approach a Gaussian variable, since each truncated walk has finite variance; however, as time grows larger,
more and more of the heavy “tails” are added (by the delta function relating walk distance and time), so the
variance of each walk approaches infinity. Thus the walk would be expected to remain “stable” but non-Gaussian, even though each transition has finite variance.
In order to calculate the propagator, the transform p(k,s) needs to be accurate over a large range of k. Our
calculations differ from Klafter et al. (1987) and Blumen et al. (1989). The complete development is listed
in Appendix II. We get to first order:
 1 − τs − C2|k|λ−1

νλ−1 −
C 2|k| λ−1
p~ (k, s) ≈ 1 − C Òs
 1 − C sνλ−1 − C1k2
Ò

Region P1
(4.52)
Region P2
Region P3
where the regions of applicability are now defined (Figure 4.6) by: Region P1 has λ < 3 and 2 < νλ; Region
P2 has λ < 3 and 1 < νλ < 2; and, Region P3 has 3 < λ and 1 < νλ < 2.
These transition densities for Regions P1 through P3 in Figure 4.6 can be simply characterized as 1) quick,
long walks, 2) slow, long walks, and 3) slow, short walks, respectively. With respect to the transition densities, we should expect that the first density would result in faster--than Fickian growth (superdiffusion), the
5.0
4.0
Brownian
Motion
Region P1
3.0
ν
2.0
1.0
0.0
1.0
Region P2
Region P3
2.0
3.0
4.0
λ
Figure 4 6 Regions of applicability of propagator expressions
34
third density would result is slower growth (subdiffusion) and the middle could show either result. Using
(4.45), the three transition densities give for the propagator:
 τs + Cτ |k|λ−1
2

CÒs νλ−2

P(k, s) = C s νλ−1 + C |k| λ−1
2
Ò
νλ−2

C Òs
 νλ−1
 C Òs + C 1 k 2
Region P1
(4.53)
Region P2
Region P3
where the first two are for infinite--variance excursion distances (λ < 3). Laplace inversion of the first propagator (Lévy walks with finite mean transition time, 2 < νλ) is straightforward. Defining the constant dispersion coefficient D = C2/τ = cos(Õα/2)Γ(1--α)/τ and recalling that λ = α + 1 gives an α--stable density for the
Region P1 propagator:
P(k, t) = exp(− Dt|k| α)
(4.54)
This implies (from the Lévy density scale parameter σ) that the density is invariant after scaling by (Dt)1/α.
The corollary of this statement is that the distance between any two arbitrary quantiles of the density grows
linearly with t1/α. This includes the subset Brownian motion (α = 2), wherein the distance between two quantiles is a constant times the standard deviation; therefore, the probability density spreads in space as a function
of t1/2 (and the variance obviously grows linearly with time).
The propagator is also the solution for the concentration of a Dirac delta function “spike” of dissolved solute
composed of non--interacting molecules (for a compact discussion see the notes by Fürth [1956]). Similarly,
a continuous--source solute plume is characterized by a step--function initial condition, which is an integrated
the delta function initial condition. For reasonable transport distances (Ogata and Banks [1961]), the integrated delta function (propagator) solution reasonably models a the continuous--source solution (Figure 4.7).
The equivalence between the propagator and concentration is used here and elsewhere in this dissertation.
The second propagator in (4.53) describes walks with infinite mean excursion duration and infinite variance
walk distance. Inverting the second propagator is aided, not fortuitously, by formulas based on fractional
derivatives (Miller and Ross [1993, Chapter V, eq. 5.19]). To first order, the small k propagator for Lévy
walks with 1 < νλ < 2 can be approximated by
P(k, t) =

1 exp − C 2 t νλ−1|k| λ−1
C1
Γ(νλ)

(4.55)
This propagator is also α--stable in space with a density that is scale--invariant with t(νλ- 1)/α, since the Lévy
scale parameter σ = t (νλ−1)C 1∕C2
1∕α
. The density will appear to scale at a rate somewhere between t0
(no growth when νλ = 1) to t1/α, with an apparent variance that is the square of the scaling rate (i.e., from t0
to t2/α). Depending on the relative magnitude of ν and α, the rate of spatial growth can appear subdiffusive,
35
1.0
(a)
t = 10.
0.8
C/C0
t = 0.1
t = 50.
t = 1.0
(x16 --vt)10
0.6
(x16 --vt)50
0.4
0.2
0.16
0.0
--20.0
--10.0
0.0
x--vt
10.0
20.0
t = 0.1
0.3
(b)
0.2
C
t = 1.0
0.1
t = 10.
t = 50.
0.0
--20.0
--10.0
0.0
x--vt
10.0
20.0
Figure 4.7. Spatial distribution of concentration predicted by equation (4.54) for α = 1.7
(solid lines) versus 2.0 (dashed lines) at four different dimensionless times (t = 0.1, 1.0, 10,
and 50). All curves use D = 1. The continuous source curves (a) are 1 -- the CDF, while the
“pulse” contamination source (b) is the density. Note that these curves can be scaled and
represented by a single curve for all time. The distance between two concentrations (the
apparent dispersivity for x16--vt) grows∝t1/α.
36
superdiffusive, or “Fickian,” although if α<2, the density will approximate an α--stable, not Gaussian. This
propagator will most likely find an application in solutes that sorb to aquifer solids.
The third density in (4.53) represents subdiffusion (infinite mean walk duration and finite walk distance variance); however, the Laplace inversion does not give a “stable--in--time” density. This case will not be explored in this dissertation. It is listed here to suggest further investigation of this regime:
1
P(x, s) = Õ
P(x, t) =


C Òs νλ+1
exp(− |x|)
C1
C Ò exp(− |x|)
t (νλ−1)∕2
C 1 ÕΓ((νλ + 1)∕2))
(4.56)
(4.57)
The heavy--tailed concentration profiles at a specified time are also predicted in the “breakthrough” curves
of concentration versus time at a fixed point in space (Figure 4.8). If we consider the propagator (4.54), which
is a symmetric α--stable density, the solution is invariant after making the coordinate transform (vt -x)/(Dt)1/α. In these coordinates, the density and cumulative distribution function are symmetric, and the “ascending” limb is identical to the descending limb. The positive half of the density tails (i.e. the rise or fall
of the concentration from a point source) have a slope of --1--α in log--log plots (Figure 4.8c). The breakthrough curve for a continuous source is unity minus the integral of the density (Figure 4.8a). When shifted
by the mean travel time and properly scaled (i.e. (vt -- x)/(Dt)1/α), and plotted on log--log axes, the concentration tails from a continuous source have a slope --α for sufficiently long time. Note that the Gaussian solution
(for any dispersion coefficient) predicts very rapid decreases in concentration as a plume passes. The α--stable
plumes will have concentrations in the tails that are orders--of--magnitude greater than Fick’s Law (the 2nd--order diffusion equation) allows.
The solutions (4.51) and (4.53), which use different asymptotic transition densities, indicate different observable growth of the spread of a particle (or concentration) undergoing Lévy walks. Equation (4.51) integrates
over the entire spatial domain (k → 0), so all possible particle walks are “accounted for.” However, the solution requires asymptotic long times. Equation (4.53) assumes α--stable tails for all space (i.e., no truncation),
therefore infinite variance. The sample variance of (4.53) will appear to spread at a rate of the square of the
particle’s excursion distance standard deviation (or the distance between arbitrary concentrations for a tracer).
For the first propagator, this implies that the measured variance would spread at a rate proportional to t2/α,
rather than t or tν- ν(α- 1) predicted by the variance estimates (4.51). Equation (4.51) requires asymptotically
long time and complete measurement of all possible walks, suggesting that the spreading rate predicted by
(4.53) will be more readily observed. Note that the first propagator for the Lévy walks (4.53) is identical to
the propagator for Lévy flights.
One can summarize the behavior of all random walks based on this Section’s definitions. Loosely speaking,
an individual walk can be distant or not so distant, and the time to complete the walk can be quick or slow.
Short, slow walks lead to slower--than--linear growth of the variance. Quick, long walks lead to infinite variance. Combinations of quick, short walks or slow, long walks lead to finite variance that is a power function
of time. This latter category includes the subset of Brownian motion.
4.3 Velocity Statistical Properties
Previous studies (c.f., Geisel [1995]) have focused on the time--correlation of velocity when the duration of
each walk is a random variable, i.e. the particle travels at a constant velocity in random directions. Measure-
37
1.0
α = 2.0
1.7
Figure 4.8. Breakthrough of a
contaminant plume at a fixed point
in space with α = 1.5, 1.7, and 2.0. (a)
Real time for x = 10, v = 1. (b) Half
of the scaled tails from a 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 on log--log plots is
equal to —(1+α).
1.5
(a)
0.8
C/C0
0.6
C0
(Dt) 1∕α
v=1
D=1
x = 10
0.4
α=
1.5
1.7
2.0
0.2
0.0
0.0
10.0
20.0
30.0
time
40.0
50.0
1
(b)
10- 1
1--C/C0
10- 2
α=
2.0
10- 3
10- 4
10- 1
100
vt − x
(Dt) 1∕α
1.7
1.5
102
10
1
(c)
C0
(Dt) 1∕α
10- 1
10- 2
10- 3
α=
10- 4
10- 1
2.0 1.7 1.5
100 vt − x
(Dt) 1∕α
10
102
38
ments of aquifer velocity are typically fixed in space and the time--correlation is not observable. Instead, we
are interested in the spatial autocorrelation of velocity, since this is directly related to the hydraulic conductivity, a field measurable quantity (Dagan [1989], Rubin and Dagan [1992]).
Since the distance traveled during a walk is independent of any previous walks, the “velocity” of the walks
are uncorrelated from step to step. If each step requires a finite constant time to complete, say τ, then the
velocity is different from step to step but constant within a single walk, and the longer flights will create longer correlations of velocities with respect to distance. Since measurements of velocity within an aquifer are
made at fixed locations on space, we are interested in the spatial autocovariance of the velocity random field,
represented by the random Lévy walk.
The spatial velocity autocorrelation function, hence the semivariogram, is readily calculated, since the velocity is constant within a single jump. This makes calculation of the joint probability f v x,v x+ξ(a, b) straightforward by a conditioning argument (Figure 4.9). Define Ri as the random displacement length of a moving
particle during the ith segment of a Lévy walk. The longer segments occupy more of the length of the trajecto-
(b)
(a)
Xt
vx
x+ξ
x
Rx = Ri
x
0
x x+ξ
1 2 3 4 ...
Rx
i
(c)
fvx(y)
fvx,vx+ξ(y,b)
0
0
ξ
Rx
Figure 4.9 Graphs of a) the Lévy process, b) the velocity function and c) joint
probability distribution of jump length as a function of spatial separation.
39
ry, so the probability of the random walk length with respect to space (Rx) is weighted by the length of the
walk (Figure 4.3b). Thus the random variable Rx has a density C¡r¡fRi(r) where C is a normalizing constant
equal to 1/E(Ri). For any given lag (ξ), the probability that both of the points x and (x + ξ) fall within the
same jump Rx is linearly related to the jump size (Figure 4.9c). If x, chosen at random, falls within a certain
walk, it has a uniform probability of landing at any place within the walk. Therefore, given a set value of
ξ, the joint probability drops linearly with ξ. For example, if ξ is equal to 1/2 the length of a walk, then x
can be randomly chosen within the first half of the walk and both x and x+ξ are within that walk, but if x is
chosen in the second half of the walk, x+ξ is in the next (independent) walk. If ξ is greater than Rx, the probability that both x and (x + ξ) are within a walk of size Rx is obviously zero, and the joint probability is the
product of the individual probabilities. As ξ goes to zero, the probability that both x and x + ξ are in a jump
of size Rx is merely fRx(r)δ(Rx+ξ -- Rx). The relationship of the joint probability to the conditioned probability
(p(x,y)=p(x|y)p(y)) allows computation of the joint probability:
f v x,v x+ξ(y, b) =
fv (y)(1 − ξ∕r)δ(y − b) + fv (y)fv

f v (y)f v (b)

x
x
x
(b)ξ∕r
x+ξ
r≥ξ
(4.58)
r<ξ
x+ξ
The general form of the autocovariance can be expressed in terms of the joint density or of the velocity functional dependence on the Lévy walk size (see Appendix III for details). If the walks are independent and symmetric, the autocovariance becomes equivalently
∞
bf
R vv(ξ) =
2
ν
v x(b)(1 − ξ∕b ν−1)db
(4.59)
ξ1−1∕ν
∞
R vv(ξ) =
 g (r) f
2
R x(r)(1 − ξ∕r)dr
(4.60)
ξ
where g(r) is the required functional relationship between velocity and walk size.
The simplest velocity function is that the jump length is proportional to velocity (vx = Rx/τ). A more general
form is developed in Appendix III. The probability density of the α--stable random walk size Ri is not known
except by its Fourier transform. Solutions can be approached two ways. First, the density can be approximated by the tail density, or Ri µ C r- 1-- α for walks larger than some small cutoff Ò. The small cutoff allows
the power--law density to converge as r → 0. A full explanation and derivation is included in Appendix 2.
The integral in the last equation does not converge, so an upper bound (MN) is also assigned to the jump
length. This is physically justified by bounded velocity and walk distance. We use the subscript N to imply
that the maximum jump distance may depend on the number of jumps. The marginal velocity density has
two forms, one valid below the cutoff (ξ<Ò) and the power--law tail density for larger distance and velocity
(ξ>Ò). For ξ<Ò, the autocovariance is (Appendix III):
R VV(ξ) = C

3−α
2−α
ξM N
(− 1 − α)Ò3−α (1 + α)ξÒ 2−α ξÒ −1−α M N
+
+
+
−
4(3 − α)
(3)(2 − α)
12
3−α
2−α
With a velocity variance of

(4.61)
40
3−α


(− 1 − α)Ò 3−α M N
+
VAR(v) = C
3 − α
 4(3 − α)

(4.62)
When the largest jump size is much greater than the lower cutoff, the semivariogram reduces to:
 
(4.63)
(1 − ξ∕r)dr
(4.64)
ξ
γ v(ξ) ≈ 3 − α
2 − α MN
For the case ξ>Ò one has
MN
R vv(ξ) = C
r
2−α
ξ
M 3−α − ξMN 2−α +

ξ 3−α
R vv(ξ) = C N

 3 − α 2 − α (3 − α)(2 − α)
 
 
ξ
ξ
− 1
γ v(ξ) ≈ 3 − α
2 − α MN
2 − α MN
(4.65)
3−α
(4.66)
A second approach uses an exact series expansion of fRx(r) for 1≤α≤2:
1
f Rx(α)(x) = Õ
∞
k
+ 1 + 1x 2k
 (2k(−+1)1)!
Γ2k α
(4.67)
k=0
We derive (Appendix III) the autocorrelation function and semivariogram using this density for the case when
the velocity is proportional to jump size (2/ν = 0) to find:
∞
ξ
 f(k)− ξ M3+2k + (3+2k)(4+2k)

3+2k
4+2k
N
γ v(ξ) =
k=0
∞
(4.68)
 f(k) M4+2k
4+2k
N
k=0
Convergence of the last expression generally requires fewer than five terms. The two semivariogram functions (4.66) and (4.68) using different walk densities agree best at larger lags (Figure 4.10). Numerical results
based on α--stable random walks closely agree with the full series density. Simulations were obtained by
using an ensemble mean of 112 realizations of a 1,000--jump Lévy walk. The lag was scaled by the expected
value of MN, not the sample mean. For speed of convergence, a walk was rejected if its maximum jump size
was 50 times greater than the expected value, resulting in the loss of approximately 1 percent of the generated
walks.
A remarkable feature of the Lévy walk velocity semivariogram for large values of ν is the resemblance to
the commonly used exponential model (Figure 4.11). The parameter of 3.8 used in the exponential model
was obtained from the coefficient in equation (4.66). The semivariogram functions for particles with any
value of the velocity parameter (ν) have also been derived (Appendix III). With this parameter, an inverse
relationship between velocity and walk size can be specified. Such a relationship might be expected for reac-
41
1
γ(ξ)
0.1
Eq. (4.66)
Eq. (4.68)
Numerical
0.1
1
ξ
MN
1.0
0.8
0.6
γ(ξ)
0.4
Eq. (4.66)
Eq. (4.68)
Numerical
0.2
0.0
0.0
0.5
1.0
ξ
MN
Figure 4.10 Log--log and linear plots of the analytical and numerical velocity
semivariogram functions when the velocity is modeled as proportional to Lévy walk
size. The numerical result is the ensemble mean of 112 realizations of 1000--jump
walks using a stability index (α) of 1.7.
42
1
0.1
-2
γ(ξ) 10
ν=2
exponential
10- 3
10- 4 - 5
10
ν=∞
10- 4
10- 3
10- 2
0.1
1
LAG (ξ) (dimensionless)
1
0.8
0.6
γ(ξ)
ν=2
0.4
exponential
ν=∞
0.2
00
0.2
0.4
0.6
0.8
1
LAG (ξ) (dimensionless)
Figure 4.11. Log--log and linear plots of the velocity semivariogram for large and
small values of ν. The value of α used in all plots is 1.7. An exponential model, γ =
1--exp(3.8ξ) is plotted for comparison.
43
tive solutes, although it is likely that a decoupled velocity probability will be needed in this case. For
ν<2/(3--α), the velocity autocorrelation function converges with an infinite upper bound.
One should note that the linear velocity function results in a diverging particle displacement variance unless
a largest walk size is imposed. Once this cutoff is imposed, the variance is finite and an infinite number of
walks will eventually yield a Gaussian propagator (Mantegna and Stanley [1995]). The time required to
achieve a Gaussian propagator is a function of the largest walk cutoff and α, and may be arbitrarily long.
Since the cutoff may be an important parameter in the study and modeling of solute transport, its estimation
from field data must be addressed.
Given a finite large number of Lévy walks, the largest expected walk is obviously larger for smaller values
of α. Since the density tails are power functions, one might expect that the largest walk would be very large
for smaller values of α (Figure 4.1). This simple velocity model would then predict that the spatial velocity
autocorrelation function goes to zero at a point that is directly related to the characteristic exponent α. It is
important, therefore, to estimate the relationship between the largest jump size (ΜΝ) and other statistical
properties of the velocity. The maximum expected jump size in N jumps is strongly dependent on the characteristic exponent of the distribution. They can be calculated using order statistics (Samorodnitsky and Taqqu
[1994]) or by theorems of Regular Variation (Feller [1966], Ch. VIII; Leadbetter et al. [1983]). A concise
description of extreme values in heavy--tailed series is given by Anderson and Meerschaert (1997) and partially listed below. If a vector of iid variables Xt have the same α--stable distribution function F(x), then the maximum (MN) of a series of these variables converges in probability to a random variable Z with a known distribution function G(x):
MN
max(X 1, X 2, , X N)
⇒ Z ~ G(x)
aN ≡
aN
(4.69)
If the tail 1 -- F(x) is regularly varying with index α (as with the α--stable laws), then the limit distribution
of MN is called a type II max--stable distribution (Leadbetter et al. [1983]) of the form G(x) = exp(--Cx- α)
for C, x, and α > 0. Similar to Lévy’s stable distributions of sums of series, the norming constants for the
max--stable domain of attraction satisfy aN= N1/αLN, where LN is a slowly varying function of N, so that aN
≈ N1/α for large N. For a large number of samples (or Lévy walks in the present case), the maximum value
MN in N jumps converges to a random variable with the probability distribution:
P(M N < x) = P(Z < x∕N 1∕α) = exp(− C(x∕N 1∕α) −α)
(4.70)
The value of C is sometimes called the dispersion of the random variables X, and is related to the scale factor
(σ) of the α--stable random variable by the relationship
C=
(1 − α)σ α
Γ(2 − α)cos(Õα∕2)
(4.71)
Equation (4.70) means that any percentile or value of the distribution can be chosen (the mean, for example)
and this value is unchanged, after a rescaling by N1/α, for any number of variables. For convenience, the
expected maximum jump size will be used: E(Z) = C1/αΓ(1--1/α). Then the expected maximum jump MN
is given by N1/αE(Z):

N(1–α)
E(M N) = Γ(1 − 1∕α)σ
Γ(2–α) cos(Õα∕2)

1∕α
(4.72)
44
Combining constants into Pα, the expected maximum jump size is simply E(MN) = PαN1/α. We now have
an expression for the maximum expected jump size in terms of the expected jump size (M1), the index of the
process (α), and the number of jumps (N), which is analogous to the elapsed time of the diffusion process.
For standard Lévy motions, σ=1 and Pα is easily calculated (Figure 4.12). One should note that the prefactor
does not vary appreciably in the range 1.2<α<2.0 compared to the other component of the scaling factor: N1/α.
For simplicity, one might choose a linear fit (Figure 4.5) of Pα over the range 1.4<α<2.0 and specify that:
(4.73)
E(M N) ≈ (5 − 2.3α)N 1∕α
For reference, the maximum of a sequence of iid Normal random variables with variance VAR(X) is described
by a Type--I maximum distribution: G(x) = exp(--e- x). The scaling constant aN = (VAR(X) log n)1/2. Therefore, the expected maximum of a sequence of Normal random variables scales according to (VAR(X) log
N)1/2γ, where γ ≈ 0.58 (Euler’s constant) is the expectation of the Type--I maximum distribution. As N becomes large, the maximum expected jump in a Brownian motion is much smaller than in a Lévy walk with
α approaching 2.0, since the Brownian motion scales like ≈(log N)1/2, while the Lévy motion scales like
≈N1/2 (Figure 4.13). As a result, the spatial autocovariance of velocity within a Lévy walk is expected to
have much greater persistence than in a Brownian Motion, even when the indices of stability are nearly equal.
If we assume, for simplicity, that (4.66) fairly represents the semivariogram function, then (4.73) is substituted for the upper bound:




ξ
ξ
(3 − α)
1
−
γ V(ξ) =
(2 − α) (5 − 2α)N 1∕α
2 − α (5 − 2α)N 1∕α
3−α
(4.74)
Now the semivariogram of velocity is a function of two parameters: α and N. The field velocity (or hydraulic
conductivity as a first--order approximation, see Gelhar and Axness [1993]; Rubin and Dagan [1992]) semi-
4.0
Pα
2.0
0.0
1.0
1.2
1.4
α
1.6
1.8
2.0
Figure 4.12 Plot of the scaling prefactor Pα for 1.0<α<2.0.
45
102
α = 1.99
E(MN)
10
Gaussian Process
1 2
10
103
104
105
Number of Jumps
Figure 4.13 Maximum expected jump size in discrete standard Gaussian versus near--Gaussian Lévy
process with index of stability (α = 1.99).
variogram yields the range of autocorrelation (MN or simply N), and α can (theoretically) be determined a
priori.
46
CHAPTER 5
THE FRACTIONAL ADVECTION--DISPERSION EQUATION
Suit the action to the word, the word to the action; with this special observance, that
you o’erstep not the modesty of nature.
- William Shakespeare, Hamlet
A number of excellent texts describe the long history and analytic properties of fractional derivatives and
fractional differential equations (Oldham and Spanier [1974]; Miller and Ross [1993]; Samko et al. [1993]).
Analysis of fractional derivatives is also finding exposure in recent mainstream texts (c.f., Debnath [1995]).
A brief review of the derivation of fractional derivatives and a summary of the useful properties is given in
Appendix IV.
For an illustration of how fractional derivatives relate to the definition of divergence in the context of solute
transport, consider two simple functions ƒ(x) = x2 and g(x) = x2.33 (Figure 5.1). The 1st derivative ƒ′(x) =
2x and ƒ′′(x) = 2. In this case, all of the information about the function is held in a constant. The derivatives
deduce how much curvature is in a function of another variable by stripping off successive levels of curvature.
The integer derivatives describe the curvature of well--behaved (integer--power) functions, but do not fare so
well with a rational--powered function. After taking two derivatives, we have not reduced the amount of
information needed, since the second derivative still depends on (x). If a fractional differential operator
(FDO) is chosen that scales similarly to the function, then the curvature is reduced to a constant, and all of
the information is “stored” in the order of the derivative and that constant. In this case, the 1.33rd and 2.33rd
derivatives of g(x) return well behaved (linear or constant) functions (Figure 5.1). If a plume is travelling
through material with evolving heterogeneity, then a fractional divergence will account for the increased dispersive flux over a larger range of measurement scale (compare Figures 2.1 and 5.1). In addition, a fractional
divergence could be used to integrate the smallest scale of measurement (Appendix IV).
5.1 Fractional Fokker--Planck Equation
A Fokker--Planck equation (FPE) describes the change of probability of a random function in space and time,
so it is naturally used to describe solute transport. The FPE is a statement about the conservation of probability that a particle will occupy a specific location. At any particular time, the sum of the probabilities at all
locations must equal unity. So if the probability changes in one location from one moment to the next, the
probability must also change in the vicinity to conserve probability. An ensemble of particles (or a large number) can fulfill the probabilities and the FPE becomes an equation of the conservation of mass.
Derivation of an FPE starts with a simple mathematical statement of how a random measure changes state
from one moment to the next, after some event has occurred. In this case, we are interested in the probability
that a particle has moved from location x1 to x3 in the time t1 to t3, or p(x3--x1;t3--t1). The particle must move
through an intermediate location x2, so this probability can be found by summing over all possible intermediate points x2. The Markov property dictates that a particle’s movements are independent of past movements,
so the probability of making both transitions (x1 to x2 to x3) is the product of the single transition probabilities,
giving the Chapman--Kolmogorov equation:
47
integer-- order derivatives
6.0
6.0
6.0
f(x) = x 2
f !!(x) = 2
f !(x) = 2x
4.0
4.0
2.0
2.0
2.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
2.0
4.0
1.0
2.0
1.0
2.0
integer-- order derivatives
6.0
6.0
6.0
g!(x) = 2.33x 1.33
g(x) = x 2.33
g!!(x) = 3.1x 0.33
4.0
4.0
4.0
2.0
2.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
2.0
fractional derivatives
6.0
6.0
g(x) =
x 2.33
D 1.33
0 g(x)
6.0
Γ(3.33)
=
x
Γ(0)
≈ 2.8x
4.0
4.0
4.0
2.0
2.0
2.0
0.0
0.0
1.0
2.0
0.0
0.0
1.0
Γ(3.33)
Γ(1)
≈ 2.8
D 2.33
0 g(x) =
2.0
0.0
0.0
1.0
2.0
Figure 5.1 Integer and fractional derivatives of two simple power functions. Top
row: Integer derivatives of f(x) = x2. Middle row: Integer derivatives of g(x) = x2.33.
Bottom row: Fractional derivatives around the point a=0 of g(x) = x2.33.
48
p(x 3 − x 1; t 3 − t 1) =
 p(x − x ; t − t ) p(x − x ; t − t )dx
3
2
3
2
2
1
2
1
2
(5.1)
The relationship between this transition density and the particle position density (i.e. the propagator for a
single particle) is that the particle position density has moved from (and must incorporate) the initial conditions. The particle’s initial position is x0 at time t0. Placing this density into the Chapman--Kolmogorov
equation yields
p(x − x 0; t) =
 p(x − x ; t − t ) p(x − x ; t − t )dx
2
2
2
0
2
0
2
(5.2)
This equation is a special case of equation (4.40) in which the transition times are constant, so the time of
the prior transition (t2) is not a random variable and can be given a fixed value. Realizing that the propagator
is the transition from the initial condition to the present time gives a shorthand notation of the density p(x
-- x0; t) = P(x,t). Using this shorthand notation, setting t--t2 to Δt and replacing x2 with y gives a compact
form:
P(x, t) =
 p(x − y; Δt)P(y, t)dy
(5.3)
This equation suggests that a direct solution to P(x,t) is not tractable since the value of Δt is completely arbitrary. By taking infinitesimal values of Δt, however, we will know the change in P(x,t) over a very recent
and short time period, resulting in a differential equation. This is the key in constructing a differential FPE
of the probability flow. The limits of the particle transition probability must be cleverly and correctly identified. One should expect that a particle that travels along fractal paths and requires power--law times to complete individual walks (the Lévy walks of Chapter 4) will have different limiting behavior than a typical Gaussian process. To illustrate, the classical FPE will be derived first.
The infinitesimal time transition density can be expanded to first order:
p(x − x 0; t + Δt) = p(x − x 0; t) + Δt
∂p(x − x0; t)
∂t
(5.4)
Rearranging and taking the limit:
∂p(x − x 0; t)
= lim 1 p(x − x 0; t + Δt) − p(x − x 0; t)
∂t
Δt→0 Δt
(5.5)
The density p(x--x0; t+Δt) can be replaced by the Markov relation:
p(x − x 0; t + Δt) =
 p(x − y; Δt) p(y − x ; t − t )dy
0
0
(5.6)
Placing (5.6) into (5.5) and recalling that p(x -- x0; t) ≡ P(x,t) gives the differential probability change:
∂P(x, t)
= lim 1
∂t
Δt→0 Δt

p(x − y; Δt)P(y, t)dy − P(x, t)
The instantaneous transition density has the following limit:

(5.7)
49
lim p(x − y; Δt) = δ(x − y)
(5.8)
Δt→0
which means that as the transition time disappears, the probability goes to unity that the particle stays at position y (y=x). This probability can be expanded using standard Taylor series for small values of Δt:
p(x − y; Δt) = δ(x − y) + A(y; Δt)δ′(x − y) + 1 B(y; Δt)δ′′(x − y) + 
2!
(5.9)
where A(y;Δt) and B(y,Δt) are functions that describe the behavior of the instantaneous transition probability
to second order. Based on the results of Chapter 4, we should expect that these are related to the mean and
variance for a Gaussian process. They can be formally expressed as such in the forms where we denote the
transition distance x--y as Δy,
(x − y)p(x − y; Δt)dx = Δy 
(5.10)
(x − y) p(x − y; Δt)dx = Δy 
(5.11)
A(Δy; Δt) =
B(Δy; Δt) =
2
2
Now limits are taken that may or may not exist in non--Gaussian particle movement:
A(y) ≡ lim 1 Δy 
Δt→0 Δt
(5.12)
B(y) ≡ lim 1 Δy 2
Δt→0 2Δt
(5.13)
Placing the small time expansion of p(x--y;Δt) (5.9) into (5.7) yields
∂P(x, t)
= lim 1
∂t
Δt→0 Δt

(δ(x − y) + A(Δy; Δt)δ′(x − y) +

1 B(Δy; Δt)δ′′(x − y) P(y, t)dy − P(x, t)
2!

(5.14)
Integrating the three terms (directly, and by parts once and twice, respectively) gives

2
∂P(x, t)
= lim 1 − ∂ (A(Δx; Δt)P(x, t) + 1 ∂ 2 (B(Δx; Δt)P(x, t))
∂t
∂x
2 ∂x
Δt→0 Δt

(5.15)
If the limits of the functions A and B exist, then this reduces to the classical FPE for random particle movement:
∂P = − ∂ (AP) + ∂ 2 (BP)
∂t
∂x
∂x 2
(5.16)
in which A is the mean particle instantaneous velocity and B is the instantaneous particle variance over time.
The functions A(Δy;Δt) and B(Δy;Δt) must be chosen so that they grow smaller at the same rate as Δt. Then
when the limits are taken, A and B are not trivially zero or infinity. Instead, the coefficient A is a measure
50
of the expectation of the transition density. Its derivative with respect to time is merely the drift of the process,
or the tendency for non--zero mean movement. In the context of solute movement, this is the mean groundwater velocity. The coefficient B is a measure of the instantaneous spread of the transition probability. In solute
transport parlance, this is the dispersion of the particle transport.
It was shown earlier that arbitrary power functions do not have constant first or second derivatives. Moreover,
random particle motions may have infinite second moments. It seems likely that fractional derivatives of
the instantaneous transition probability may yield well--behaved limits. Zaslavsky (1994a) introduces an
approximation to the transition density that relies on the fractional--order moments that do exist. Although
his derivation may be useful in certain instances, it neglects several important features. It is listed presently
for completeness. Zaslavsky (1994a) introduced a first--order approximation of the fractional derivative of
the instantaneous transition probability as an analog to (5.5):
∂ ωP(x, t)
= lim 1 ω p(x − x 0; t + Δt) − p(x − x 0; t)
∂t ω
Δt→0 (Δt)
(5.17)
where 0 < ω ≤ 1 is the scaling exponent for time (i.e. if particle movements are fractal in time, or if the mean
particle transition time is infinite). Since the second moment of the transition density is infinite for Lévy
flights, Zaslavsky (1994a) describes the instantaneous density based on the highest finite moment (α):
B(x; Δt)
1
ω = lim (Δt) ω
Δt→0 (Δt)
Δt→0
B(x) ≡ lim

 |x − y| p(x − y; Δt)dy = lim |Δx|
(Δt)
α
α
Δt→0
ω
(5.18)
Zaslavsky also takes a rational--order (α/2) moment for the function A(x,Δt) and gives the following fractional expansion of the transition density:
δ(x − y) + B(Δy; Δt) D α+δ(x − y)
p(x − y; Δt) = δ(x − y) + A(Δy; Δt)D α∕2
+
(5.19)
And a fractional FPE of
∂ ωP = ∂ α∕2 AP + ∂ α BP
∂t ω
∂(− x) α
∂(− x) α∕2
(5.20)
Several corrections to this derivation are needed if solute transport in aquifer material is the process to be
modeled. First, we use the first--order approximation of the fractional derivative given by Kolwankar and
Gangal (1996) for the time derivative. This adds a minor constant to the approximation of the time derivative:
∂ ωP(x, t)
Γ(1 + ω) 
= lim
p(x − x 0; t + Δt) − p(x − x 0; t)
ω
ω
∂t
Δt→0 (Δt)
(5.21)
Replacing the density with the Markov relation (5.6) into the fractional time derivative limit gives a fractional--in--time analog to (5.7):
∂ ωP(x, t)
Γ(1 + ω)
= lim
ω
∂t
Δt ω
Δt→0

p(x − y; Δt)P(y, t)dy − P(x, t)

(5.22)
Second, when the particle mean velocity is finite, the re--definition of the first moment is superfluous (and
may actually be inaccurate). In this case, the definition of A(Δx; Δt) remains the first moment of the transition
hΔxi. Bringing the constant from the time derivative into this expression slightly modifies the traditional
definition of A (for finite--mean transition lengths) to:
51
Γ(1 + ω)  
Δx
Δt
Δt→0
(5.23)
A(x) ≡ lim
Finally, a fractional derivative of the Dirac delta function (Appendix IV) results in one--sided functions depending on the direction of the derivative. As a result, Zaslavsky’s expansion is a totally skewed transition
density. In order to describe transitions that are possible in both the positive and negative directions, we must
include another term. By using a constant (β) below, the skew of the density is completely described. The
constant from the fractional time derivative can be placed in the definition of the constants that are now based
on fractional moments:
Γ(1 + ω)B(x; Δt)
Γ(1 + ω)
= lim
ω
ω
(Δt)
Δt→0
Δt→0 (Δt)
B(x) ≡ lim
 |x − y| p(x − y; Δt)dy =
α
Γ(1 + ω)
|Δx|α 
ω
Δt→0 (Δt)
lim
(5.24)
We see here the dimension of the function B: Lα ¡ T- ω. Note also that the gamma function constants go to
unity when omega is unity (i.e. the first time derivative). We also find that additional constants (gamma functions) have been ignored by Zaslavsky (1994a, 1994b, 1995). These constants are needed so that the instantaneous density integrates to unity and has the correct moments. Combining these with the drift of the process
gives a complete description of the instantaneous transition density for Levy walks:
p(x − y; Δt) = δ(x − y) − A(y; Δt)δ′(x − y) +
1 (1 − β) B(y; Δt) D α δ(x − y) + 1 (1 + β) B(y; Δt) D α δ(x − y)
2
2
Γ(α + 1) +
Γ(α + 1) −
(5.25)
where 1 < α ≤ 2 is the scaling exponent in 1--dimensional space. The last two terms can be directly evaluated
as proportional to (|x--y|)- 1-- α (Appendix IV), which shows the Pareto--like distribution of the instantaneous
transition approximation. For symmetric jumps, β = 0. In this study, the scaling exponent is a single constant,
although an interesting, and open, question concerns the behavior of particles with different scaling exponents (α1, α2, α3) in the three principal movement directions (see for example, Meerschaert and Scheffler
[1998]). Placing the expansion into (5.22) and using the formula for inner products in Appendix IV on the
fractional derivatives gives the fractional FPE:
∂ ωP = ∂ AP + 1 (1 − β) ∂ α BP + 1 (1 + β) ∂ α BP
∂t ω
∂x
∂xα
∂(− x) α
2
2
(5.26)
Within this definition, the functions A and B are truly meant to be constants. If the variance of the particle
transition probability is infinite, then the nonlinear growth of the particle propagator should be incorporated
within the fractional derivative. The derivative is defined so that it correctly captures the scaling of the transition density. To accurately model particle transport, one need only estimate the order of the fractional derivatives. The derivation supposes that the instantaneous density can be approximated by the 1st and αth moments, just as the traditional FPE uses the 1st and 2nd moments.
This is a reasonable derivation for transition densities that have finite first and infinite second moments
(1<α≤2). Note that when a particle undergoes finite variance walks with finite mean waiting time, α=2,
ω=1, and the fractional FPE reduces to the traditional integer--order FPE (5.16).
One should also be aware that Zaslavsky’s fractional spatial derivative on AP (5.19) would mean that the
function A represents the (α/2)th moment of the transition density, rather than the first moment (i.e. the mean
52
velocity). If, in fact, the mean velocity is a property that scales continuously in space, then a fractional derivative may be more appropriate. Many studies are concerned with the large--scale evolution of mean velocity
(see Gelhar [1993] for a discussion). The current study will not address this open and important property
of mean groundwater flow that would consider that validity of the integer--order continuity (Poisson) equation for fractal media. For faster--than--Fickian dispersion in groundwater, we believe that the velocity can
be considered a constant and an integer--order time derivative (representing finite mean transition time) is
most appropriate. As previously discussed, for independent solute “particles,” the probability propagator is
replaced by concentration (Bhattacharya and Gupta [1990]), and the governing equation for solute movement
(the fractional ADE) simplifies to:
∂C = − v ∂C + 1 (1 + β)D α (DC) + 1 (1 − β)D α (DC)
−
+
∂t
∂x
2
2
(5.27)
where D α− and D α+ signify the negative and positive direction fractional derivatives, respectively (Appendix
IV). The dimensions of the dispersion coefficients in (5.27) are Lα ¡ T- 1. To maintain consistent notation
with the results in previous Chapters, we combine the constant used to describe the time rate of change of
a particle’s αth moment with the constant cos(Õα/2) to define the dispersion coefficient: D = Bcos(Õα/2).
The reason for this notational change will become clear in the next Section. For symmetric transitions, β =
0. Defining the symmetric operator
2∇ α ≡ D α+ + D α−
(5.28)
and using a mean--removed equation (i.e. shifting coordinates by the mean travel distance = vt) gives the mass
balance equation for symmetric dispersion (or diffusion) using a fractional divergence:
∂C = D∇ αC
∂t
(5.29)
Thus the physical meaning of fractional divergence postulated in Chapter 2 has been rigorously derived based
on random particle displacements. For α = 2, this collapses to the classical parabolic diffusion equation.
In summary, the traditional derivation of the FPE relies on expansion of the instantaneous particle transition
density that uses the first and second moments of the transitions. For transitions that follow a power--law
(Pareto or α--stable) density, the second moment is infinite and the traditional FPE is ill--defined. The fractional FPE is based on a transition density expansion based on the highest finite moment of the transitions,
which also happens to be the order of the Lévy stability index (α). An important feature of the fractional
derivation is that it contains the classical result as a subset.
We add that particles that follow a truncated power--law distribution will have finite variance, but will approximate the power--law density for quite some time before evolving to a Gaussian (Mantegna and Stanley
[1995]). In other words, a few truncated Lévy walks still look like a Lévy walk. Only a large number of
truncated Lévy walks looks like a Gaussian. This means that the fractional FPE will be a better model of
a finite--variance, truncated Lévy walk until particles have moved many times the length scale of the largest
transitions. After reaching that scale, a 2nd--order diffusion equation would be more representative. The
number of transitions required to move from one regime to the next is a function of the stability index and
the truncation cutoff. An open question is the scale at which a plume undergoing truncated power--law transitions will converge to a Gaussian, second--order equation. The previous Chapter showed that the size of the
truncation, or cutoff, grows larger within continuously evolving heterogeneous media, and a transition to the
53
Gaussian is never reached. Numerical simulation of transport within aquifers with a specified largest scale
of hydraulic conductivity correlation would shed light on this question.
5.2 Solutions
Solutions to common solute transport boundary value problems (BVP) are gained through Laplace or Fourier
transforms in a manner similar to Ogata and Banks (1961). We will solve the BVP for instantaneous injection
of a “spike” of solute, i.e. the Green’s function. The fractional--in--time exponent (ω) is set to unity, since
a solution to a fractional time and space equation is ungainly, and spatial correlations give rise to a fractional-in--space governing equation. The fractional--in--space equation (5.27) is solved via Fourier transform (Appendix IV):
~
~
~
~
dC
(k, t) = ikvC + 1 (1 + β)(− ik) αBC(k, t) + 1 (1 − β)(ik) αBC(k, t)
2
dt
2
(5.30)
An ODE with solution:

~
C(k, t) = exp 1 (1 + β)(− ik) αBt + 1 (1 − β)(ik) αBt + ikvt
2
2


= exp 1 (1 + β)Bte −i(sign(k))Õα∕2|k| α + 1 (1 − β)Bte i(sign(k))Õα∕2|k| α + ikvt
2
2

= exp 1 Bt|k| α(1 + β)e −i(sign(k))Õα∕2 + (1 − β)e i(sign(k))Õα∕2 + ikvt
2

= exp 1 Bt|k| α(2cos(Õα∕2) − 2βi sin(Õ(sign(k))α∕2)) + ikvt
2



~
C(k, t) = expcos(Õα∕2)Bt|k| α(1 − iβ(sign(k))tan(Õα∕2)) + ikvt
(5.31)
(5.32)
where the identities i = eiÕ/2 and eiÒ = cosÒ + isinÒ have been used. Recalling the definition of the dispersion
coefficient based on B we have:
~
C(k, t) = exp− Dt|k| α(1 − iβ(sign(k))tan(Õα∕2)) + ikvt
(5.33)
This Fourier transform does not have a closed--form inverse. However, putting it in the form of the characteristic function (substituting --k for k), the density can be manipulated into the canonical form of the characteristic function for α--stable densities (3.13):
C(− k, t) = exp− Dt|k| α[1 − iβsign(k)tan(Õα∕2)] − ikvt
(5.34)
where the positive constant σ = (Dt)1/α, indicating a stable density that is shifted by the mean (vt) and invariant upon scaling by t1/α (Figure 5.2). This result is especially interesting (and novel, to the author’s knowledge) because the entire family of stable densities is generated from the governing equation. The constant-source solution (Figure 5.2a) is unity minus the CDF. The skewness that results from higher probability of
particles moving either ahead or behind the mean diminishes as α gets closer to 2. When α = 2, the solution
to the classical ADE is recovered.
A solution of the simplified symmetric fractional divergence equation (5.29) results in
C(k, t) = exp(− Dt|k| α)
(5.35)
54
1.0
t = 0.1
t = 1.0
t = 10.
0.8
C/C0
α = 1.7
D = 1.0
0.6
0.4
0.2
0.0
--20.0
--10.0
0.0
10.0
20.0
x - vt
t = 0.1
0.3
α = 1.7
D = 1.0
0.2
C
t = 1.0
0.1
t = 10.
0.0
--20.0
--10.0
0.0
x - vt
10.0
20.0
Figure 5.2. Comparison of the development of spatially symmetric (dashed lines)
and positively 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. As
α gets closer to 2, the skewing diminishes. All curves use α = 1.7 and D = 1.
55
This parallels the results gained directly via Fourier--Laplace transforms in the previous Section. The fractional FPE provides an important extension to asymmetric transitions that have not been gained via the integral transform methods in Chapter 4. A survey of most field sites would show highly skewed hydraulic conductivity histograms and a propensity for skewed plumes. Moreover, we believe that sorbing solutes will
show a propensity toward maximum skewness, since transitions will be favored in the direction behind the
mean velocity. The corollary statement is that transitions in the forward direction (into clean aquifer solids)
will be truncated.
The solution to the classical ADE with the continuous source initial condition is generally written in “closed
form” using the error function. The error function itself is twice the integral of the positive half of a Gaussian
density with variance = 1/2, or standard deviation of 2 ∕2:
z
ERF(z)
=
2
 1Õ exp(− x )dx
2
(5.36)
0
This integral has no algebraic formula, so it is numerically estimated and tabulated. The step function BVP
using the classical ADE is reasonably approximated by (Ogata and Banks [1961]):
C=

C 0
1 − ERF x − vt
2
2 Dt


(5.37)
For continuity with this widely--used formula, a similar solution for the FADE is given by:
C=



C0
vt
1 − SERF α x −1∕α
2
(Dt)
(5.38)
where we define the α--stable error function (SERFα) function similarly to the error function, i.e., twice the
integral of a symmetric α--stable density from 0 to the argument (z):
z
SERF α(z) = 2
 f (x)dx
α
(5.39)
0
where fα(x) is the standard, symmetric, α--stable density. The factor of 2 in the denominator of the SERF
argument has been dropped from equation (5.38) for simplicity. The values of the SERFα(z) function have
been tabulated over a range of arguments from zero to ten and for values of α from 0.9 to 2.0 incremented
by 0.1 (Tables 5.1 and 5.2). Note that the definition of SERFα (z) uses a standard distribution which, for α
= 2.0, is a Gaussian with standard deviation of 2. Since the error function is a Gaussian with standard deviation of 2 ∕2, ERF(z) and SERF2.0(z) are related by:
ERF(z) = SERF 2.0(2z)
(5.40)
If one does not have access to a computer in order to compile the numerical integrators listed in Appendix
I, it is a simple matter to estimate the concentration within a symmetric plume (including a Gaussian) at any
point in space and time using (5.38) and the data in Table 5.1.
56
Table 5.1 Error function SERFα(Z) of the symmetric stable distributions.
Z
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.20
3.40
3.60
3.80
α = 2.0
0.02820
0.05637
0.08447
0.11246
0.14032
0.16800
0.19547
0.22270
0.24966
0.27633
0.30265
0.32863
0.35421
0.37938
0.40411
0.42839
0.45218
0.47548
0.49826
0.52049
0.56331
0.60384
0.64201
0.67778
0.71113
0.74208
0.77064
0.79688
0.82086
0.84267
0.86240
0.88017
0.89609
0.91028
0.92286
0.93397
0.94372
0.95224
0.95965
0.96606
0.97631
0.98375
0.98905
0.99275
α = 1.9
0.02824
0.05644
0.08457
0.11259
0.14045
0.16814
0.19561
0.22282
0.24975
0.27636
0.30262
0.32851
0.35398
0.37903
0.40362
0.42773
0.45134
0.47443
0.49697
0.51897
0.56124
0.60116
0.63865
0.67370
0.70630
0.73647
0.76426
0.78973
0.81297
0.83407
0.85315
0.87032
0.88571
0.89945
0.91166
0.92247
0.93200
0.94039
0.94773
0.95415
0.96460
0.97246
0.97834
0.98272
α = 1.8
0.02830
0.05656
0.08474
0.11280
0.14071
0.16842
0.19589
0.22310
0.25000
0.27657
0.30276
0.32856
0.35392
0.37883
0.40326
0.42718
0.45058
0.47343
0.49572
0.51743
0.55907
0.59828
0.63500
0.66923
0.70097
0.73027
0.75720
0.78182
0.80425
0.82459
0.84297
0.85952
0.87436
0.88763
0.89946
0.90998
0.91931
0.92757
0.93487
0.94131
0.95201
0.96031
0.96677
0.97182
α = 1.7
0.02839
0.05674
0.08501
0.11314
0.14111
0.16887
0.19637
0.22358
0.25047
0.27699
0.30312
0.32882
0.35406
0.37882
0.40306
0.42677
0.44993
0.47250
0.49449
0.51588
0.55679
0.59518
0.63102
0.66431
0.69509
0.72342
0.74938
0.77307
0.79462
0.81415
0.83179
0.84769
0.86196
0.87477
0.88623
0.89647
0.90561
0.91377
0.92104
0.92753
0.93849
0.94725
0.95429
0.95999
α = 1.6
0.02853
0.05702
0.08540
0.11365
0.14171
0.16954
0.19710
0.22434
0.25122
0.27771
0.30377
0.32937
0.35447
0.37905
0.40308
0.42654
0.44941
0.47167
0.49331
0.51431
0.55437
0.59180
0.62662
0.65884
0.68853
0.71577
0.74067
0.76336
0.78397
0.80264
0.81951
0.83473
0.84845
0.86079
0.87189
0.88187
0.89084
0.89891
0.90618
0.91272
0.92396
0.93317
0.94077
0.94709
57
Table 5.1 continued. Error function SERFα(Z) of the symmetric stable distributions for a
range of α from 1.6 to 2.0.
Z
α = 2.0
4.00
0.99528
4.20
0.99698
4.40
0.99810
4.60
0.99881
4.80
0.99927
5.00
0.99955
5.20
0.99972
5.40
0.99982
5.60
0.99988
5.80
0.99996
6.00
0.99998
6.20
1.0
6.40
1.0
6.60
1.0
6.80
1.0
7.00
1.0
7.20
1.0
7.40
1.0
7.60
1.0
7.80
1.0
8.00
1.0
8.20
1.0
8.40
1.0
8.60
1.0
8.80
1.0
9.00
1.0
9.20
1.0
9.40
1.0
9.60
1.0
9.80
1.0
10.00 1.0
α = 1.9
0.98598
0.98842
0.99027
0.99167
0.99276
0.99363
0.99432
0.99488
0.99535
0.99575
0.99609
0.99639
0.99665
0.99688
0.99708
0.99727
0.99743
0.99758
0.99772
0.99784
0.99795
0.99806
0.99815
0.99824
0.99833
0.99840
0.99847
0.99854
0.99860
0.99866
0.99871
α = 1.8
0.97580
0.97896
0.98150
0.98357
0.98528
0.98670
0.98791
0.98894
0.98982
0.99060
0.99128
0.99188
0.99242
0.99290
0.99333
0.99372
0.99407
0.99439
0.99469
0.99496
0.99521
0.99544
0.99565
0.99585
0.99603
0.99620
0.99636
0.99651
0.99665
0.99678
0.99690
α = 1.7
0.96465
0.96850
0.97171
0.97441
0.97671
0.97868
0.98039
0.98188
0.98319
0.98435
0.98538
0.98631
0.98714
0.98789
0.98857
0.98919
0.98976
0.99028
0.99076
0.99120
0.99161
0.99198
0.99234
0.99266
0.99297
0.99325
0.99352
0.99377
0.99400
0.99422
0.99443
α = 1.6
0.95239
0.95688
0.96072
0.96402
0.96689
0.96939
0.97160
0.97355
0.97529
0.97684
0.97824
0.97950
0.98065
0.98169
0.98264
0.98352
0.98432
0.98506
0.98574
0.98637
0.98696
0.98751
0.98802
0.98850
0.98894
0.98936
0.98975
0.99012
0.99047
0.99080
0.99111
58
Table 5.2 Error function SERFα(Z) of the symmetric stable distributions for a range of α
from 0.9 to 1.5.
Z
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
3.00
3.20
3.40
3.60
3.80
α = 1.5
0.02873
0.05740
0.08597
0.114380
0.14258
0.17053
0.19816
0.22545
0.25235
0.27881
0.30480
0.33028
0.35521
0.37959
0.40336
0.42653
0.44905
0.47093
0.49214
0.51268
0.55173
0.58806
0.62169
0.65268
0.68114
0.70718
0.73093
0.75253
0.77214
0.78992
0.80601
0.82056
0.83372
0.84562
0.85638
0.86612
0.87494
0.88293
0.89020
0.89680
0.90832
0.91794
0.92606
0.93295
α = 1.4
0.02900
0.05794
0.08676
0.11540
0.14380
0.17191
0.19968
0.22704
0.25396
0.28040
0.30630
0.33164
0.35638
0.38050
0.40396
0.42675
0.44886
0.47027
0.49097
0.51096
0.54880
0.58381
0.61607
0.64568
0.6277
0.69748
0.71999
0.74045
0.75903
0.77590
0.79120
0.80509
0.81771
0.82918
0.83961
0.84912
0.85780
0.86573
0.87299
0.87965
0.89141
0.90141
0.90998
0.91739
α = 1.3
0.02939
0.05870
0.08787
0.11683
0.14552
0.17386
0.20179
0.22926
0.25623
0.28263
0.30842
0.33358
0.35806
0.38185
0.40491
0.42724
0.44882
0.46965
0.48972
0.50903
0.54542
0.57890
0.60958
0.63763
0.66321
0.68651
0.70771
0.72698
0.74452
0.76047
0.77500
0.78824
0.80033
0.81138
0.82150
0.83079
0.83932
0.84717
0.85442
0.86111
0.87306
0.88338
0.89235
0.90020
α = 1.2
0.02993
0.05977
0.08943
0.11884
0.14790
0.17656
0.20472
0.23233
0.25934
0.28568
0.31133
0.33623
0.36036
0.38370
0.40624
0.42796
0.44887
0.46896
0.48825
0.50674
0.54139
0.57308
0.60199
0.62832
0.65228
0.67409
0.69394
0.71201
0.72849
0.74355
0.75731
0.76992
0.78149
0.79213
0.80193
0.81098
0.81935
0.82711
0.83431
0.84100
0.85307
0.86362
0.87291
0.88112
α = 1.1
0.03070
0.06128
0.09163
0.12166
0.15125
0.18031
0.20877
0.23654
0.26356
0.28979
0.31518
0.33970
0.36333
0.38605
0.40788
0.42881
0.44885
0.46802
0.48634
0.50383
0.53645
0.56611
0.59306
0.61755
0.63981
0.66008
0.67856
0.69544
0.71088
0.72504
0.73804
0.75002
0.76107
0.77128
0.78075
0.78954
0.79771
0.80534
0.81245
0.81911
0.83120
0.84189
0.85140
0.85989
α = 0.9
0.03344
0.06665
0.09938
0.13144
0.16263
0.19283
0.22192
0.24982
0.27650
0.30194
0.32615
0.34915
0.37097
0.39166
0.41127
0.42985
0.44745
0.46412
0.47993
0.49497
0.52269
0.54773
0.57040
0.59100
0.60977
0.62693
0.64267
0.65714
0.67049
0.68284
0.69429
0.70493
0.71485
0.72411
0.73278
0.74091
0.74855
0.75574
0.76252
0.76892
0.78072
0.79133
0.80094
0.80966
59
Table 5.2 continued. Error function SERFα(Z) of the symmetric stable distributions for a
range of α from 0.9 to 1.5.
Z
α = 1.5
4.00
0.93885
4.20
0.94394
4.40
0.94836
4.60
0.95223
4.80
0.95564
5.00
0.95866
5.20
0.96135
5.40
0.96376
5.60
0.96592
5.80
0.96788
6.00
0.96965
6.20
0.97126
6.40
0.97273
6.60
0.97408
6.80
0.97533
7.00
0.97647
7.20
0.97753
7.40
0.97851
7.60
0.97942
7.80
0.98026
8.00
0.98105
8.20
0.98179
8.40
0.98248
8.60
0.98313
8.80
0.98374
9.00
0.98431
9.20
0.98485
9.40
0.98535
9.60
0.98584
9.80
0.98629
10.00 0.98672
α = 1.4
0.92383
0.92947
0.93443
0.93883
0.94275
0.94626
0.94942
0.95227
0.95486
0.95721
0.95936
0.96133
0.96315
0.96482
0.96636
0.96779
0.96912
0.97036
0.97151
0.97259
0.97360
0.97455
0.97544
0.97628
0.97707
0.97782
0.97852
0.97919
0.97982
0.98042
0.98099
α = 1.3
0.90711
0.91324
0.91870
0.92358
0.92798
0.93195
0.93555
0.93883
0.94183
0.94458
0.94711
0.94944
0.95159
0.95359
0.95545
0.95718
0.95879
0.96030
0.96172
0.96304
0.96429
0.96547
0.96658
0.96762
0.96861
0.96955
0.97044
0.97128
0.97209
0.97285
0.97358
α = 1.2
0.88844
0.89499
0.90088
0.90620
0.91103
0.91543
0.91945
0.92313
0.92652
0.92965
0.93255
0.93523
0.93773
0.94005
0.94222
0.94426
0.94616
0.94795
0.94964
0.95123
0.95272
0.95414
0.95548
0.95675
0.95796
0.95910
0.96019
0.96123
0.96222
0.96316
0.96406
α = 1.1
0.86753
0.87443
0.88068
0.88638
0.89158
0.89636
0.90075
0.90481
0.90856
0.91204
0.91529
0.91831
0.92113
0.92378
0.92626
0.92859
0.93079
0.93286
0.93482
0.93667
0.93843
0.94009
0.94167
0.94318
0.94461
0.94597
0.94727
0.94852
0.94971
0.95084
0.95193
α = 0.9
0.81763
0.82494
0.83166
0.83786
0.84360
0.84894
0.85390
0.85854
0.86288
0.86695
0.87077
0.87437
0.87777
0.88098
0.88402
0.88689
0.88963
0.89223
0.89470
0.89705
0.89930
0.90145
0.90350
0.90547
0.90735
0.90916
0.91089
0.91255
0.91415
0.91569
0.91718
60
CHAPTER 6
EXPERIMENTS
The sciences do not try to explain, they hardly even try to interpret, they mainly make
models. By a model is meant a mathematical construct which, with the addition of
certain verbal interpretations, describes observed phenomena. The justification of
such a mathematical construct is solely and precisely that it is expected to work.
-- John Von Neumann
The motivation for the development of the theories within this work was to provide a simplified model of
solute transport. The fractional ADE (FADE) predicts concentration versus time and distance in closed form,
once the scaled α--stable density (fundamental solution) is known. In this spirit, several interesting experiments will be analyzed in the simplest way possible. A typical questions that a contaminant hydrogeologist
wishes to answer is “How far and how fast will a tracer move?” As a first approximation, this reduces most
problems to one spatial dimension (1--D). The following experiments will be treated as such. The questions
associated with multiple dimensions and averaging will be left open. The 1--D FADE is given by:
∂C = − v ∂C + 1 (1 + β) ∂ α BC + 1 (1 − β) ∂ α BC
∂t
∂x
∂x α
∂(− x) α
2
2
(6.1)
Three experiments are analyzed in this Chapter. Two of these are intuitively expected to follow Fick’s Law.
The first concerns diffusion within pure liquid with a step--function initial condition and a large difference
in ionic strength. The second is a 1--D tracer test in a laboratory--scale (1 m) sandbox. The sandbox was
constructed with very uniform sand in such a way that heterogeneity was minimized. Observing any non-Fickian nature in these tracer tests would suggest that the α--stable transport is ubiquitous in real--world problems. The final test uses data collected by the U.S.G.S. during a 511--day long tracer test within a sand and
gravel aquifer on Cape Cod.
6.1 High Concentration Diffusion
Carey (1995) and Carey et al. (1995) investigated the validity of Fick’s Law in modeling ionic transport within high--ionic strength liquids with ions of different valence. To isolate the separate mechanisms of strength
and valence, several experiments were conducted in which a single salt (copper sulfate) at high concentration
was allowed to diffuse into distilled water. A solute’s rate of diffusion in aqueous solution typically decreases
as the solute concentration increases, since the solute has fewer “free” sites to visit on a random walk. A
typical approach to modeling diffusion across high concentrations gradients is to use a concentration--dependent diffusion coefficient (Crank [1975]). A number of functional forms of the diffusion coefficient versus
concentration have been explored (c.f., references within Crank [1979]). The solution to boundary value
problems using these functional relationships are generally non--Gaussian in shape, yet they are invariant after scaling by (time)1/2. This arises when the general solution can be decomposed into the superposition of
an infinite number of fundamental solutions (c.f., Strang [1986]) that are of the form exp(x2/Dt).
Carey et al. (1995) found that a numerical solution using Miller et al.’s (1980) values of CuSO4 diffusion
coefficients at various concentrations was not able to accurately model the diffusion experiments; nor was
61
a Fickian model with an arbitrarily lowered coefficient. Carey et al. (1995) point out that the previously published values of CuSO4 diffusion coefficients were measured using small initial concentration differences
(0.03 mol/L) across the initial step function, and postulate that the concentration difference in their experiment (0.4 mol/L) may lead to unmodeled processes. Indeed, the underlying second--order Fokker--Planck
diffusion equation is based on a symmetric particle transition density. In essence, the traditional diffusion
equation using a concentration--dependent coefficient accounted for the fact that a molecule had less available
transition sites as the concentration increased (or conversely a smaller mean free path), but did not allow for
a basically different (skewed, non--Gaussian) transition probability density. If the aqueous solution is idealized as a percolation network, then as the number potential sites available for transitions are filled by other
Cu and SO4 ions, the remaining connected set is fractal. Moreover, the dimension of the set is reasonably
constant over a large occupation density, yet the transition probability is skewed in favor of the less--occupied
(fresh water) direction (Figure 6.1) The fractional diffusion equation is an attractive alternative model for
Carey’s experiment for two reasons: 1) the walks are fractals with dimension related to the Lévy index (and
therefore the order of the fractional FPE), and 2) skewed transitions are allowed in the construction of the
governing equation. It is an open question whether the applicability of a fractional FPE for pure diffusion
can be made rigorous through, say, statistical mechanical arguments.
The high ionic strength should cause a randomly walking ion to disperse more slowly than it would in distilled
water (called subdiffusion in previous Chapters), so one should expect that the index of stability, or the order
of fractional diffusion, should be greater than two. Thus the diffusion profile (a plot of concentration versus
distance) at any time should be a single curve after scaling the distance axis by time1/α. For reference, diffusion following the classical second--order equation should be represented by a single curve when the distance
is divided scaled by the square root of time (Crank [1974]). An empirical match of the concentration profiles
over time yields a scaling index of approximately 2.5 (Figure 6.2). Since the fractional model also allows
a skewed transition density, the “mean” and “median” concentrations are not identical. Physically, this suggests that the point of maximum solute flux (the median) is shifted several centimeters behind the initial step
function (Figure 6.3). The improvement of the fractional equation’s predictions are especially realized in the
Figure 6.1. Idealized schematic representation of diffusion via
random walk within a a high ionic strength, high gradient
fluid. The random walk occurs within a partially--occupied
network.
The probability of a walk toward lower
concentration (to the right of the figure) is always higher than
into higher concentration, where more sites are occupied by
other solute ions. At high enough concentrations, the set of
connected available sites is non--Euclidean, precluding
Fickian diffusion.
-- site occupied by solute ion
- unoccupied site
62
CONCENTRATION (mol/L)
0.40
0.30
α = 2 (Fickian)
μ = 2 cm
0.20
0.10
0.00
--1.0
0.0
x−μ
t 1∕α
0.0
CONCENTRATION (mol/L)
0.40--1.0
1.0
1.0
0.30
α = 2.5
μ = 1.97 cm
0.20
0.10
0.00
--1.0
0.0
1.0
Elapsed Measurement Time (hrs)
0.8
4.2
7.6
11.0
14.5
18.0
Figure 6.2 Scaled diffusion profiles from Carey’s (1995) experiment: a) scaled by the
traditional (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.
Elapsed
Measurement
Time (hrs)
CONCENTRATION (mol/L)
63
0.20
α = 2 (Fickian)
μ = 2 cm
0.10
0.00
0.0
0.8
0.1
0.2
4.2
0.3
0.4
x−μ
t 1∕2
0.5
0.6
11.0
14.5
18.0
CONCENTRATION (mol/L)
7.6
0.20
α = 2.5
μ = 1.97 cm
0.10
0.00
0.0
0.1
0.2
0.3
0.4
x−μ
t 1∕2.5
0.5
0.6
Figure 6.3 Closeup low--concentration limb of the scaled diffusion profiles from Carey’s
(1995) experiment: a) scaled by the traditional (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.
64
leading edge of the solute profile (Figure 6.3). A discussion of the slower--than--Fickian scaling is contained
in Chapter 8.
6.2 Laboratory--Scale Tracer Test
The nonlinear problem of density--coupled flow and transport, embodied in the problem of saltwater intrusion
into freshwater aquifers, has generated renewed interest. Because of difficulties in obtaining analytic solutions (Ségol [1994]) and accurate numerical approximations (Croucher and O’Sullivan [1995]; Benson et al.
[1998]) there are questions concerning the validity of the governing equations (Carey [1995]) and the previous “verification” of numerical codes. In short, the numerical solutions to a standard seawater intrusion
problem do not yield identical results, and none of them have been shown to match either an analytic solution
or a physical experiment. In order to provide a benchmark experiment against which the numerical codes
could be tested, a laboratory--scale sandbox (Figure 6.4) was constructed at the Desert Research Institute for
the purpose of recreating Henry’s (1964) standard seawater intrusion problem. Henry’s problem requires a
constant dispersion coefficient (i.e. Fickian dispersion), so the sandbox was designed and built using as homogeneous a porous medium as possible (Burns [1997]). Before the density--coupled experiments were run,
a number of simple tracer tests were conducted to estimate the basic characteristics of the sand. These tracer
tests unexpectedly showed non--Gaussian breakthrough curves (Burns, 1997) predicted in previous Chapters.
The present study explores the possibility of α--stable transport within the sandbox, stressing the fact that the
experiments were conducted without any knowledge of non--Gaussian transport.
In a typical 1--D laboratory tracer test, the velocity is held constant and large enough to neglect molecular
dispersion. The classical ADE is used to model the breakthrough curve:
∂C = − v ∂C + a v ∂ 2C
L
∂x
∂t
∂x 2
(6.2)
where aL is the longitudinal dispersivity, which is a measure of the medium’s intrinsic propensity to disperse
a passive scalar in transport. Fickian transport refers to transport within a medium in which aL remains a
constant throughout a plume’s history, yielding a constant coefficient on the second--order dispersion term.
“Homogeneous”
Fine sand
26
22
23
24
25
27
CONDUCTIVITY
18 PROBES
19 14 10 6
20
21
15
16
17
11
12
13
7
8
9
3
4
5
1
2
Figure 6 4 Schematic view of the experimental sandbox tracer tests
highlighted by the arrow is analyzed in detail (after Burns 1996)
The flowpath
65
This would be expected for transport at larger--than--pore scales in a column (or sandbox) of perfectly mixed,
homogeneous sand (Taylor [1953], De Josselin De Jong [1958]).
The second--order equation predicts a Gaussian density for an instantaneous, Dirac delta function solute injection (Carslaw and Jaeger [1959]):
C=

(x − vt) 2
1
exp
4a Lvt
4Õa Lvt

(6.3)
A continuous tracer has a breakthrough curve that is a shifted Gaussian distribution function. The curves are
translated by a distance vt, which is the mean travel distance within the column. The quantity (2aLvt)1/2 is
analogous to the standard deviation of a the graph of the concentration versus distance, so the distance between any two concentration levels (Xc) in a plume grows proportional to (aLt)1/2. If the rate of growth is
faster, then typically aL is made to absorb the increase since the second--order diffusion equation can only
afford growth proportional to t1/2.
It was shown previously that Xc in an α--stable plume should grow proportional to t1/α. If the dispersivity
is thought to grow as a power function of the mean travel distance or time during a particular test (i.e. aL ∝
tm), then the value of the Levy index (α) can be directly calculated:
(a Lt) 1∕2 ∝ (tmt) 1∕2 ∝ X c ∝ t 1∕α
(6.4)
Some algebra gives the expression for α in terms of the slope (m) of the increase of apparent dispersivity
versus time on a log--log graph:
α=
2
m+1
(6.5)
The Fickian result is recovered if the dispersivity does not increase with scale. Then m = 0 and α = 2.
A series of passive tracer tests were conducted in the sandbox in order to estimate the single value of aL for
the sandbox. Details of the experiment are given by Burns (1997). The value of dispersivity at each of 23
conductivity probes was markedly different, with a general increase of the values with mean travel distance
(Figure 6.5). Moreover, the mean travel velocity was found to vary with depth within the sandbox, indicating
the presence of fining--upward sequences that were created during sand emplacement. These sequences are
not visible because of the high degree of sand uniformity (Burns, 1997). The initial indication of α--stable
transport within the sandbox lies within the heavy--tailed breakthrough curves (Figure 6.6). When the concentration is normalized and plotted on a probability axis versus either scaled time (or distance) on a normal
axis, a Gaussian plume appears as a straight line (Pickans and Grisak, 1981). The slope of the line is proportional to aL, so this method is commonly used to estimate the dispersivity of the transport medium. An α-stable plume plotted in the same manner will appear nearly Gaussian throughout the middle of the breakthrough curve, but will also show higher tail probabilities, presenting a sigmoid shape.
This heavy--tailed breakthrough is typically explained and subsequently modeled by kinetic reactions (refs)
or multiple “compartments” into which the solute can partition (Coats and Smith [1964]; van Genuchten and
Wierenga [1976]; Brusseau et al., [1989]) with different rates of Gaussian transport in each compartment.
These models typically can be tuned to provide excellent data fits, and in some cases are based on physical
principles. These models are inherently more complex. In order to model skewed, heavy--tailed plumes, a
number of new (generally empirical) parameters are needed. Yet solute transport is concerned with a single
APPARENT DISPERSIVITY (cm)
66
slope conversion
guide
2
α = 1.0
α = 1.2
8 4
α = 1.5
0.1
17
α=2
Fickian
13
5
1
9
16
24
23
25
α = 1.55
1
19
21
14
10
15 11
7 3
20
10
TEST SCALE (cm)
100
Figure 6.5 Calculated dispersivities versus distance of probe from source. The flow
path chosen for analysis is shown with the connecting line. The best fit dashed line
indicates a fractional diffusion index (α) of 1.55.
property: the distribution of particle velocities, however the distribution arises. A non--Gaussian distribution
should be modeled by a non--Gaussian model, and the fractional ADE is equipped to describe both heavy tails
and skewed transitions. An interesting and open question is whether the simple fractional ADE can reproduce
a range of results from other, more complex, multi--parameter Gaussian models.
The apparent dispersivities along the chosen flowpath (Figure 6.5) indicate an α value of 1.55. As a result,
the breakthrough curves should be scale--invariant after shifting by the mean travel time and dividing by
time1/1.55. By plotting the tails of the distribution, i.e. C/C0 for the leading edge and 1.0 -- C/C0 for the trailing
edge versus the absolute value of (t -- tmean), the skewness of a plume is immediately apparent (Figure 6.7).
The algorithm for the the ordinate is the minimum of C/C0 and 1--C/C0. Proper time rescaling shows reasonably good agreement throughout the entire plume history, particularly the trailing edge. One can empirically
estimate the value of a by plotting the breakthrough curves with the abscissa scaled by different values of t1/α.
If α is too large, the downstream curves plot to the right (Figure 6.7b). Too small a value of α results in downstream curves shifted to the left. This technique indicates that the value of α for this test should be slightly
larger than 1.55. To maintain continuity with the estimate from the rate of increase of the apparent dispersivity (Figure 6.5), a value of 1.55 will be used in the analytic model.
67
99.99
99.9
99
C/C0 (%)
90
70
50
30
10
1
0.1
0.01
--1.0
--0.5
0.0
t − t mean
(t meant) 1∕2
0.5
1.0
Figure 6.6 Plot of normalized concentration versus scaled time for probe 20, test 3 (after
Burns, 1997). A best fit line (implying an underlying Gaussian profile) is typically used
to calculate the apparent dispersivity. Compare this data with the α--stable theoretical
plots in Chapter 3 (Figure 3.1b).
Once the index of differentiation is known, a predictive model of transport is handily gained. One need only
generate a single density (or CDF for a continuous tracer test) and scale this density for any time or distance.
A simple numerical integrator has been written to generate concentration versus time (at a point in space)
or distance (at a specific time) in the FORTRAN codes CVT.F and CVX.F; both are included in Appendix
I. Several standard densities using α = 1.55 and various skewness parameters (β) were generated to achieve
the observed separation of the leading and trailing tails. A value of β = --0.5 provided a reasonable separation
(Figure 6.8). Compared to the classical ADE solution, the fractional ADE model more accurately represents
the heavy tails observed at all of the probe locations. More important, the fractional ADE is “self--contained”
with a constant dispersion coefficient. One need not recalculate or look up a different value of the dispersion
coefficient at a specific distance to generate the concentration profile. The fractional derivative is responsible
for the faster--than--Fickian plume growth.
6.2 Cape Cod Aquifer
In July 1985 approximately 7.6 m3 of tracer was introduced into a sand and gravel aquifer in Cape Cod, Massachusetts. The injected tracer contained 640 mg/L of the relatively non--reactive bromide (Br- ) ion, as well
as reactive (sorbing) Li+ and MoO42-- ions. LeBlanc et al (1991) and Garabedian et al. (1991) document the
tracer test and the characteristics of the plume, such as estimated first and second moments. Over 650 multi--
68
MIN(C/C0,1--C/C0)
1
0.1
(a)
Probe
Distance (cm)
0.01
15.5 (probe 25)
26.6 (probe 15)
0.001
0.01
0.1
|t − t mean|
t 1∕1.55
1
35.5 (probe 11)
45.6 (probe 7)
55.6 (probe 3)
MIN(C/C0,1--C/C0)
1
0.1
(b)
0.01
0.001
0.001
0.01
|t − t mean|
t 1∕2
0.1
1
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 the leading and trailing limbs of the plume. Very early and late data
show probe noise.
69
1.0
FICKIAN
(2nd - order PDE)
0.8
α--stable
(α=1.55)
0.6
C/C0
0.4
0.2
0.0
100
DATA
110
120
130
TIME (min)
1
Trailing edge
DATA
0.1
C/C0
Leading edge
0.01
0.001
FICKIAN
(2nd - order PDE)
α--stable
(α=1.55)
0.01
0.1
|vt − x|
(Dt) 1∕α
1
10
Figure 6.8 Comparison of traditional and fractional ADEs with the data from probe
3 (x = 55 cm) in the sandbox test: a) real time, and b) data tails. Note the large
underprediction of concentration by the traditional ADE at very early and late time.
70
level samplers (MLS) were installed to monitor the plume (Figure 6.9) for over 511 days . The Br- plume
extended well beyond the MLS array after 511 days, creating an effective time cutoff for analysis of the nonreactive plume. Each MLS (the numerous small circles in Figure 6.9) consisted of 15 sampling ports at different depths, resulting in the collection of a huge number of data points in x--y--z--time coordinates.
For simplicity, the positive x--direction will refer to the mean plume movement direction of roughly 8_ East
of South (Figure 6.9). The deviations that the plume made from this line are small enough that the difference
between the actual travel distance and the distance projected onto this x--axis are negligible. This problem
has been simplified further by reducing the 15 vertical samples at each MLS into 3 pieces of data: the maximum concentration, the average of all samples, and the average of all samples above the detection limit. Each
of these pieces of information carries biases that will be discussed in more detail later in this study. In short,
the maximum concentration will be considered the true peak concentration at a given point in horizontal
space, while the averages represent concentrations that might be expected in a well that is screened across
most of the aquifer. Thus the 3--D problem is reduced to 2--D.
Finally, since the MLS array generally consists of a series of MLS arraigned perpendicular to the flow in order
to laterally “bracket” the plume (Figure 6.9), the maximum concentration observed within a specific travel
distance range is taken to represent the peak concentration for that distance. This procedure gives a 1--D picture of the plume at any sampling time. The peak concentrations in 3--D are projected to the x--axis and a
series of 1--D snapshots of the plume’s “core” result.
A Posteriori Estimation of Parameters
Garabedian et al. (1991) calculated the variance of the plume roughly along the x--direction and concluded
that the growth was linear after 83 days. This has important implications, since a linear growth of the variance
implies a Fickian governing equation. A plot of the variance along the travel direction on log--log axes indicates that the growth appears nonlinear for most, if not all, of the plume’s history (Figure 6.10). One should
50
50
N
13
55
111
237
349
x
461
0
0
Flowmeter
Permeameter
--50
0
50
100
150
200
250
DISTANCE ALONG MEAN FLOW DIRECTION (m)
--50
300
Figure 6.9 Aerial view of the Cape Cod Br- plume. The plume deviated from travelling due
South by approximately 8_ to the East. Circles are multi--level samplers (MLSs), diamonds
are permeameter core samples, and squares are flowmeter tests.
71
CALCULATED VARIANCE (m2)
1000
α = 1.5
α = 1.6
α = 1.75
α = 2.0
100
α
2
10
1
1
10
100
1000
CALCULATED MEAN TRAVEL DISTANCE (m)
Figure 6.10 Calculated plume variance (Garabedian et al. [1991]) along the direction of mean travel.
also realize that the calculation of plume variance is strongly influenced by detection limits. As a plume
grows, a larger fraction goes below detection limits or is simply never sampled. Reasons for not sampling
the plume tails are financial (why collect samples with a high probability of containing solute below detection
limits?) and practical when the plume simply travels past the detection array (see day 461, Figure 6.9). Since
the calculated variance would strongly weight these neglected values, one must assume that the calculated
variance is underestimating the “real” variance more strongly at later times. Of course, the α--stable plumes
developed theoretically in previous Chapters have infinite variance, yet the mass of the plume that leads to
the non--converging variance is far--flung and at very low concentrations. One would need solutes that were
detectable at very low concentrations and a large network of wells to demonstrate the non--converging variance (if it existed).
The Levy index (α), or the order of fractional dispersion, is estimated to be 1.6 from the log--log plot of apparent plume variance versus mean travel distance (Figure 6.10). This slope is evident after just a few plume
measurements. The dispersion coefficient for the FADE is discerned by generating a standard (σ = 1) stable
density and plotting this density alongside the field data measured at some point in time. The lateral shift
required on log--log plot is equal to σ = (Dt)1/α.
A simpler method recognizes that the an α--stable plume is nearly Gaussian close to the center of mass, so
the early--time estimates of the diffusion coefficient based on standard methods will give a reasonable estimate. The distance (call it X13) between the leading and trailing edge concentrations of approximately 13
percent of the maximum concentration of a pulse source covers 4 sigma of the Gaussian density (denoted here
as σG to differentiate from the similar scale factor in the α--stable density). One fourth of X13 is equal to
(2Dt)1/2, i.e. X13/4 = σG = (2Dt)1/2. Equating the Gaussian and α--stable solutions over this distance gives
72
(Dt) 1∕α ≈ (Dt) 1∕2 =
X 13
4 2
(6.6)
where the square root of 2 in the last term arises from the standard α--stable density being identical to a Gaussian with standard deviation of 2. From this we can estimate the dispersion coefficient (which will remain
constant) for the FADE:
  1t
X 13
D≈
5.7
α
(6.7)
The first four rounds of sampling (13, 33, 55 and 83 days) give estimates of 0.19, 0.18, 0.17 and 0.16 m1.6/d.
The numbers from these early sampling rounds are probably somewhat inflated by the porosity--corrected
injection volume (LeBlanc, et al. [1991]) of 4×4×1.2m (20 m3), which would tend to enlarge the initial
“instantaneous” pulse. Therefore, a slightly lower value of 0.14 m1.6/d will be used for the FADE model.
Garabedian et al. (1991) use the plume variance data (Figure 6.10) to infer an asymptotic, or Fickian, aL value
of 0.96 m. With the nearly steady velocity of 0.43 m/d, the asymptotic, Fickian dispersion coefficient is 0.42
m2/d. One should observe in the previous discussion that the units of the dispersion coefficient imply the
order of the FADE: The classical units of L2/T imply second--order differentiation.
Analytic Solutions
Several sets of analytic solutions have been generated using the aquifer parameters estimated a posteriori
in the previous section. The simplest 1--D equations to solve are:
∂C + v ∂C − D ∂ 2C = C x δ(t, x)
0 0
∂x
∂t
∂x 2
(6.8)
∂C + v ∂C − D∇ 1.6C = C x δ(t, x)
0 0
∂t
∂x
(6.9)
where C0x0δ(t,x) denotes the initial solute concentration (C0) spread over some injection distance x0 which
is mathematically concentrated into a delta function “spike.” This number is the area under all of the concentration versus distance curves and should coincide with the injected concentration times the initial size
of the injected mass. A value of 0.2 gm/cm2 is used in all of the solutions. The value is slightly less than
LeBlanc et al.’s (1991) estimated concentration × length in the x--direction of 640 mg/L×400 cm = 0.256
gm/cm2 and accounts somewhat for the lateral mass loss as the plume moves. In both equations, v = 0.43
m/d, and in the Fickian equation, the asymptotic dispersion coefficient of 0.42 m2/d is used. The FADE uses
a dispersion coefficient of 0.14 m1.6/d. A plot of the analytic solutions (equation 5.34 with α = 2.0 and 1.6)
of concentration versus distance at 8 sampling rounds shows the basic differences between the models (Figure
6.11). Visual inspection shows that, as expected, the asymptotic Fickian model overestimates dispersion at
early time (Figure 6.11a), while the FADE solution nicely models data from all periods. In order to get better
early--time data fits from the Fickian model one can use a dispersion coefficient similar to that of the FADE
(i.e. 0.14 m2/d), but the modeled plume is under--dispersed at the end of the test (Figure 6.12b). Virtually
identical data fits from the Fickian and FADE models could be generated at any time, but a unique dispersion
coefficient would be needed at each time for the Fickian equation.
If one accepts the FADE’s α--stable solutions as a fair representation of the plume, then one must also accept
the implication of heavy tails. The Cape Cod data was not collected with this type of analysis in mind, so
73
600.0
CONCENTRATION (mg/L)
13
α = 1.6 (D = 0.14)
Fickian (D = 0.42)
400.0
55
200.0
111
0.0
0.0
50.0
DISTANCE FROM SOURCE (m)
100.0
140.0
α = 1.6 (D = 0.14)
Fickian (D = 0.42)
CONCENTRATION (mg/L)
120.0
100.0
80.0
60.0
203
349
511
40.0
20.0
0.0
50.0
100.0
150.0
200.0
250.0
DISTANCE FROM SOURCE (m)
300.0
Figure 6.11 Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximum
concentrations along plume centerline. Solid lines are solutions to FADE using 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 above peaks.
74
600.0
CONCENTRATION (mg/L)
13
Fickian (D = 0.14)
α = 1.6 (D = 0.14)
400.0
55
200.0
0.0
0.0
CONCENTRATION (mg/L)
140.0
20.0
40.0
60.0
DISTANCE FROM SOURCE (m)
80.0
203
Fickian (D = 0.14)
α = 1.6 (D = 0.14)
120.0
100.0
(a)
111
349
511
80.0
(b)
60.0
40.0
20.0
0.0
50.0
100.0
150.0
200.0
250.0
DISTANCE FROM SOURCE (m)
300.0
Figure 6.12 Simple analytic models of the Cape Cod plume in 1--D. Symbols are maximum
concentrations along plume centerline. Solid lines are solutions to FADE and classical ADE
using identical (early--time) dispersion coefficients of 0.15. a) Early time data. b) Late--time
data. Sample times (in days) are shown above peaks.
75
most MLSs were not sampled if the concentration was thought to be near the detection limit of 0.01 mg/L.
One sampling period (349 days) is an exception to this general rule, as many MLSs behind the main plume
body were sampled and analyzed (Figure 6.13). To emphasize the tail characteristics, the concentrations measured at this time are plotted on a log--axis. The Fickian model with D = 0.42 m2/d and the FADE model
with D = 0.14 m1.6/d are shown on both plots. The maximum concentration measured in vertical planes
roughly perpendicular to flow are shown in the uppermost plot. An MLS that contained Br- below the detection limit in all 15 vertical samples was excluded from the plot of maximums (Figure 6.13a), resulting in the
loss of 4 points. Each of the discarded points was “surrounded” in the forward and backward directions by
at least two detectable concentrations.
Vertical averages of data from the MLS that contained the maximum concentration at a specific distance are
shown in the lower plot (Figure 6.13b). The higher valued average uses only the Br- concentrations above
the detection limit, and the lower average uses all measured values (including zeros for Br- concentrations
below 0.01 mg/L). The second average is presented to roughly represent the concentrations that would be
measured in a conventional monitoring well that intersects a continuous section of aquifer material. Of course
a sample drawn from such a well preferentially samples the higher conductivity layers, while the MLSs supply an equal amount of groundwater from all ports, so these averages are merely a guide. The higher average
that discards all non--detectable concentrations is meant to counteract the effect of variable vertical spacing
of the MLS ports along the length of the plume at 349 days. One could imagine the effect of using all of the
vertical data from an errant MLS that placed probes every ten, rather than every 1, meter from the water table
to a depth of 150 m -- most if not all would be below detection limits, significantly lowering the average value.
The two analytic solutions plotted with the averages also use a source that is diluted by a factor of five.
Inspection of the tail data (Figure 6.13) leads to the conclusion that the Fickian mode vastly under--predicts
the concentrations of Br- on the trailing (well--sampled) edge of the plume. Another interesting feature is
the area between the tails and the peak, where the Fickian model fits the data more closely than the FADE.
No effort was made to iteratively fit the models to the measured data. A lower value of dispersion coefficient
would tighten the FADE solution to this portion of the concentration profile without significantly changing
the predictions for other portions of the plume. The lack of data at the leading edge of the plume precludes
a confident judgement about which solution is more accurate.
A Priori Estimation of Parameters
To minimize the disturbance of the natural aquifer while installing nearly 10,000 sampling points, little aquifer material was removed from the plume path. Yet current analytic solutions that are based on the traditional
ADE (Gelhar and Axness [1983]; Dagan [1984]) require a knowledge of the statistical properties of the random field of the hydraulic conductivity (thus the velocity, through various arguments). The reason that the
velocity autocorrelation estimates are needed is because no other data will give an estimate of the long--term
plume behavior modeled by the ADE. The correlation data give an estimate of the asymptotic (long--term)
dispersion coefficients. Unlike the FADE, in which an early--time estimate of D and α can be used for the
duration of plume travel, the early--time D offers no information that the traditional (second--order) ADE can
use at late time. The hydraulic conductivity (K) data (Figure 6.9) were collected from a number of wells and
soil cores located roughly 20 to 40 m west of the plume centerline and 70 to 120 m south of the plume source
(Hess et al. [1991]).
An analysis of the data (not reproduced here) show that the variance at the smallest separation distance (lag)
in the horizontal direction is on the order of 50 percent of the maximum variance (Hess et al. [1991]). Stated
76
100
CONCENTRATION (mg/L)
10
1
(a)
α = 1.6
10- 1
?
10- 2
10- 3
α = 2.0
(Fickian)
10- 4
10- 5
10- 6
0.0
100.0
200.0
300.0
DISTANCE FROM INJECTION POINT (m)
100
CONCENTRATION (mg/L)
10
1
(b)
α = 1.6
10- 1
10- 2
10- 3
10- 4
10- 5
10- 6
0.0
α = 2.0
(Fickian)
100.0
200.0
300.0
DISTANCE FROM INJECTION POINT (m)
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
of vertical samples from the same MLS that from which the maximum concentrations
were measured. The smaller average uses zero for non--detectable concentration,
while the larger average ignores those data.
77
differently, the smallest lag measured is roughly half the reported “correlation scale.” Further, since the boreholes were placed at regular intervals, only several lags smaller than the correlation scale incorporate enough
data pairs to be represented. Recalling the theoretical results for estimation of α from the velocity variogram
data (Figure 6.14), it seems impossible that the measurements will be able to distinguish a value of the order
of differentiation in the FADE. It also seems very unlikely that the K data from any site will be able to distinguish between values of α, given the methodology developed in this dissertation. This is a topic that warrants
further study, but based on the discussion in the previous paragraph, it is a minor point when the applicability
of the FADE to field sites is considered. A few early--time measurements of the plume give ample information (D and α) for FADE solutions at all times.
1.0
α = 1.4
α = 1.8
γv(h)
0.5
0.00.0
0.5
1.0
DIMENSIONLESS SEPARATION (h)
Figure 6.14 Theoretical dimensionless velocity semivariogram for α = 1.4 and α = 1.8.
78
CHAPTER 7
NUMERICAL APPROXIMATIONS
It’s about time, it’s about space ...
- John Doe, Exene Cervenka, I must not think bad thoughts
7.1 Motivation
The analytic solutions presented in previous Chapters suffer from the limitations of most analytic solutions.
The solutions are only gained when parameters are constant, and relatively simple initial and boundary
conditions (ICs and BCs) are imposed on the PDEs. In the real world, parameters such as velocity are
spatially or temporally variable, and the contamination is introduced into an aquifer according to a history
very unlike a Dirac delta or step function. In order to accurately predict natural movement or cleanup under
these complex or variable conditions, a numerical model is required. A numerical model discretizes the
transport domain (space and time) into user--specified subsets. Each subset must have all of the parameters,
ICs and BCs needed by the governing equation, yet each subset may have different or unique values. Present
models of solute transport are based on the classical second--order ADE, so the spatial subsets (elements or
blocks) are given the parameters of velocity and diffusion coefficient tensor and the initial and boundary
concentration information.
The user then has a choice of how information is placed into parameters and discretized. A very fine grid
can have detailed velocity information and the dispersion tensor contains only “local” information about
solute spreading. The detailed velocity field is itself responsible for the “non--local” or large--scale spreading.
Conversely, a coarse grid has little velocity information, and the user must choose a dispersion tensor that
contains this information. However, this dispersion tensor is not valid across a range of scales. When
accuracy is desired, the natural tendency is to use a fine grid with detailed velocity information. But where
does this information come from? Typically, the statistical properties of an aquifer’s hydraulic conductivity
(K) distribution are measured at several points. Then a random field is generated that honors both the
statistical properties and the value of K at known points. The hydraulic head field is numerically solved,
giving the velocity parameter field for the ADE numerical model. Along with the solute ICs and BCs, a
concentration solution is generated. Since this is a single realization of an infinite number of random K and
velocity fields that fit the data, a number of realizations are generated and the process is repeated, giving an
ensemble of concentration realizations.
This rather lengthy and computationally taxing exercise might be avoided if the coarse grid could contain
an equivalent amount of information. The trouble with the coarse grid is that the dispersion tensor cannot
fully represent the spreading at various scales. The fractional ADE (FADE) developed in Chapter 5 is based
on the notion that the order of the derivative describes the spread of particles following Gaussian and
non--Gaussian motions. The dispersion tensor (coefficient in 1--D) is a constant at all scales. So too is the
mean velocity. The fractional derivative in the governing equation is “responsible” for the non--local
spreading, and fine discretization of the velocity parameter is unnecessary. A coarsely discretized numerical
79
model of the FADE should be expected to give equivalent results (in terms of information) as a finely
discretized numerical solution of the classical ADE. Further, since the FADE includes the non--Gaussian
underlying probability distributions of particle transitions, the distinct heavy--tailed plumes attributed to
long--range velocity autocorrelation could be directly simulated without a large computational effort.
7.2 Finite Differences
The classical ADE is a mixed hyperbolic (advection) and parabolic (dispersion) PDE. Numerical
approximations are typically gained by splitting the operators. The hyperbolic portion is relatively difficult
to solve accurately because of the lower order of spatial differentiation. The simplest algorithms truncate
derivatives on the same order as the dispersion term, so accurate solution of the advection term either requires
very fine discretization (by fine Eulerian grids or many Lagrangian particles within an Eulerian grid) or
higher--order corrections. Further, the stability of the advection solution is typically limited to much smaller
timesteps then the dispersion calculations, so very simple and fast methods are stable and accurate for the
dispersion term. Finite difference methods are certainly the simplest to implement. The dispersion portion
of the ADE can be approximated in 1--D by:
∂C = D ∂ 2C ⇔
∂t
∂x 2
− C ti
C t+Δt
i
Δt
=D
C ?i+1−C ?i
C ?i−C ?i−1
−
Δx i+1∕2
Δx i−1∕2
(7.1)
Δx i
where the spatial discretization (node number) is denoted by the subscript and, for simplicity of notation, D
has been assumed spatially constant. The question mark in some superscripts implies a choice that can be
made in the approximation: If concentrations are taken from time (t), the calculation is called explicit. If the
question marks are replaced by (t + Δt), the calculation is implicit, and requires a simultaneous solution at
all of the nodes. Since the finite difference approximation is linear, the implicit approximation is solved by
reduction of the resulting matrix equation.
An attractive candidate for the finite difference approximation of a fractional derivative for 1 ≤ α ≤2 would
be (Appendix V):
∂C = D∇ αC ⇔
∂t
− C ti
C t+Δt
i
Δt
=D
(C ti+1−C ti) α−1
(Δx i+1∕2)α−1
−
(C ti−C ti−1)α−1
(Δx i−1∕2)α−1
(7.2)
Δx i
Note that an explicit formulation is specified, since the nonlinear implicit equation cannot be solved without
iteration. Simplifying further by using a constant grid spacing of Δx:
− C ti = DΔtα (C ti+1 − C ti) α−1 − (C ti − C ti−1) α−1
C t+Δt
i
Δx
(7.3)
Note that a simple generalization would allow waiting times with infinite mean (Chapter 4), or a fractional
time derivative (Chapter 5) approximated by:
∂ γC
∂t γ
≈
Ct+Δt
− C ti
i
γ
(7.4)
Δt γ
Resulting in:
Ct+Δt
− C ti
i
γ
γ
= DΔtα (C ti+1 − C ti) α−1 − (C ti − C ti−1) α−1
Δx
(7.5)
80
Let the constant DΔtγ/Δxα be denoted r. Solving for the unknown:
= C ti + r(C ti+1 − C ti) α−1 − r(C ti − C ti−1) α−1
C t+Δt
i
1∕γ
(7.6)
If γ=1 and α=2, this reduces to the explicit finite difference approximation of the diffusion equation. The
last equation can be coded quite easily in 2-- or 3--D and placed into an existing flow and transport model (see
program FRACDISP.F in Appendix I), with one caveat: the sign of the concentration difference is removed
from the power operations.
What should follow is a detailed investigation of the stability, convergence, and truncation error of the
numerical approximation. This will be left to a later study. Here it will suffice to observe the behavior of
the solutions and use them if they perform as expected. Therefore, the numerical solutions are considered
heuristic and await rigorous derivation.
The analytic solutions to the symmetric FADE have two main characteristics. First, the solutions are identical
after scaling by t1/α. Second, the solutions have “heavy” power tails in which the concentration declines as
a function of time or distance raised to the power (--α) for a continuous source and (--1--α) for a point source.
A point source was approximated by setting the concentration in all nodes except one (at a position labeled
x0) to zero. The solution should tend toward an α--stable density. Like the analytic solution, the numerical
results tend to converge to a single solution when the distance and concentration are scaled by t1/α (Figure
7.1). Similar scaling (not shown) is seen for all 1<α≤2. Note that the tails follow a power--law relationship
(Figure 7.1b) but the slope is not in perfect agreement with the α--stable densities, i.e. the slope is not --α.
Not shown on the plots are the concentration curves for different discretization values of Δx and Δt. The
curves for all values are coincident. It is well known that the stability criterion for the 1--D diffusion equation
is r = DΔt/Δx2 < 1/2. All of the numerical solutions of the FADE show oscillations at the extreme tail for
all values of r down to 0.01. Decreasing the timestep size moves the oscillations father down the tail, but
they still remain at all values of r investigated.
The power--law tail is also seen with other values of α (Figure 7.2). In general, for 1.5<α<2, the numerical
tails have somewhat lower concentration than the analytic solutions. Of course the numerical solutions all
have heavier tails than the classical second--order solution. For very low values of α, the tail concentrations
appear higher than those predicted by the analytic solution. A proper explanation of this discrepancy will
probably require a careful analysis of the finite--difference truncation error. However, the numerical results
do contain the desired scaling properties and the non--Gaussian, heavy tails. Fortuitously, the agreement
between the analytic and numerical solutions appears best in the range exhibited by the Br- tracer plume at
the Cape Cod site (Chapter 6).
The analytic solutions for the Cape Cod Br- plume tended to under--predict the plume concentrations at early
time. This can be ascribed to the initial injection volume, which is very different from a Dirac delta function
used in the analytic solution (Chapter 5). The delta function has no initial width, so the concentration
decreases immediately. A slug with some finite width will “decay” more slowly at its center, in essence
because there is little or no concentration gradient there at early time. So a numerical solution with the initial
2 m wide slug of Br- at 640 mg/L should maintain relatively high concentrations for early time, and converge
to the analytic solution as time progresses (Figure 7.3). While the numerical solution does not predict quite
81
10
10- 1
10- 2
10- 3
1.0
10- 4
(Dt) 1∕αC
10- 5
C0
10- 6
10- 7
0.316 minutes
1.0 minutes
1.6
3.16 minutes
10.0 minutes
10- 8
10- 9
10- 10
0.1
1
10
x − x0
100
1000
(Dt) 1∕α
0.50
0.40
0.316 minutes
1.0 minutes
0.30
3.16 minutes
10.0 minutes
(Dt) 1∕αC
C0
0.20
0.10
0.00
0.0
1.0
2.0
3.0
x − x0
4.0
5.0
6.0
(Dt) 1∕α
Figure 7.1 Numerical solution of the FADE with α = 1.6 for a series of times: a) log--log axes, and
b) linear axes. Initial conditions were a “point source” of unit mass at the node located at x0. The
solutions follow the scaling law of the analytic solution. Note the oscillatory error at the extreme
tail ends.
82
as tail--heavy a plume as the analytic solution, the match is still reasonable and better than a Fickian (classical
ADE) solution at 349 days.
The implications of the numerical solution are very important: It predicts concentrations in the leading edge
of a plume that are many orders--of--magnitude greater than the classical ADE will predict. Many chemicals
that threaten drinking water supplies are toxic at concentrations well below their solubility. Benzene, a
carcinogen with a solubility of approximately 1800 mg/L at room temperature, requires cleanup if detected
above about 0.0003 mg/L in groundwater. The FADE, in analytic and numerical forms, will predict toxic
concentrations in downstream wells far earlier than previous methods. It will also predict that much more
time is required to clean an aquifer to a specified level of residual concentrations.
This Chapter has merely introduced an idea for a numerical implementation of fractional derivatives. The
“quick and dirty,” explicit, finite--difference solution served the dual purpose of improving Cape Cod plume
predictions and suggesting a vast simplification (without loss of information) of current, fine--grid,
Monte--Carlo methods. For example, a 10--fold reduction of resolution in each spatial dimension translates
to a speedup of greater than 1,000 times in 3--D (since linear solvers are not linear with the number of nodes).
However, the notion of the finite difference operator used herein runs contrary to previous definitions and
expansions (Appendix V). Since the fractional derivative implies “global” dependence, a more accurate
representation may require an unreasonable number of nodes for quick solution. Since the fully--defined
fractional derivative is a linear operator, finite element methods might be readily applied with user--controlled
truncation error. This numerical emphasis deserves a more detailed investigation.
1
10- 1
10- 2
10- 3
10- 4
10- 5
10- 6
(Dt) 1∕αC 10- 7
C0
10- 8
10- 9
10- 10
10- 11
10- 12
10- 13
10- 14
10- 15
0.1
α = 1.4
α = 1.8
α = 1.6
α = 2.0
1
10
x − x0
100
1000
(Dt) 1∕α
Figure 7.2 Comparison of analytic (lines) versus numerical (symbols) solutions of the FADE
with “point source” initial condition. In all solutions, D, t, and Δx set to unity.
83
CONCENTRATION (mg/L)
600
13 days
400
(a)
55
200
203
349
511
0
0
100
200
300
DISTANCE FROM INJECTION WELL (m)
103
CONCENTRATION (mg/L)
(b)
102
101
100
10- 1
10- 2
10- 3
10- 4
0
100
200
300
DISTANCE FROM INJECTION WELL (m)
Figure 7.3 Numerical (thin lines) and analytic solutions (thick lines) of the FADE compared to Cape
Cod Br-- plume (symbols): 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 improved fit of the numerical solution
at 13 and 55 days.
84
CHAPTER 8
DISCUSSION OF RESULTS
The classification of the constituents of a chaos, nothing less here is essayed.
- Hermann Melville, Moby Dick
The starting point of the derivation of the fractional governing equations was the Chapman--Kolmogorov
equation for particle walks that are random in space and either random or uniform in time. This equation
states that a particle’s next movement is independent of past movements, or “memoryless.” For finite--variance particle excursion lengths, this leads to a Gaussian propagator and a second--order (diffusion) governing
equation. In order to force longer--range spatial correlation into this equation, three things can be done: 1)
scale the dispersion parameter, 2) use a non--Markovian formulation, or 3) specify independent jumps with
long--range spatial correlation. The first method is embodied in the methods of Gelhar and Axness (1983)
and Dagan (1984). The second method leads to difficult mathematical manipulation (Cushman, et al. [1994])
and limited applicability. The third method (the subject of this dissertation) leads to a simple and intuitive
governing equation and straightforward application.
To get a solution we first took convolutions to get an equation for the probability propagator (akin to concentration for a pulse solute injection). The propagator is only a function of the particle’s joint space--time
transition density. To solve the equation in Fourier--Laplace space, simplifications were made, such as symmetry of walks, and a functional relationship between velocity and transition length. The propagator thus
obtained is a symmetric α--stable density. The use of a functional velocity dependence imposes a cutoff of
the jump size that grows with time. The variance of the propagator is made finite by this cutoff, but since
it is an increasing number, the variance continues to grow. This can be explained physically by a hydraulic
conductivity (K) semivariogram that increases for all lags. As a plume grows, it samples more disparate velocities, so the variability of higher or lower K material that a particle might encounter continues to increase.
We next substituted an instantaneous approximation of the transition density in the Chapman--Kolmogorov
equation to arrive at a FPE. The only assumption about the transition density was that the αth moment existed
and that the fractional Taylor series was a reasonable expansion for small time. Using the boundary value
problem of a pulse injection, we once again found an α--stable density. The general nature of the derivation
accounts for unequal probabilities of a particle moving either faster or slower than the mean. This fractional
FPE leads naturally to the definition of a fractional divergence which is spatially or temporally non--local.
A necessary and important component of these derivations is that Brownian motion and Fick’s 2nd Law are
subsets. Finite variance walks ultimately result in Gaussian plumes. An open question is how a fixed upper
cutoff on the α--stable jump size probability affects these results. This is likely to be tractable, since a gate
function on the probability distributions is readily handled by Fourier transforms. This procedure would naturally lead to results that include a lower cutoff that represents a finite observation or measurement scale.
The variance and propagator should follow the present results for a certain time period before transitioning
85
to a Fickian regime. Montegna and Stanley (1995) show this behavior for the probability of a particle’s return
to the origin for Lévy flights with fixed maximum jump length (i.e. using a truncated α--stable transition density).
An open question concerns the simplifications made in order to analytically derive the Fourier--Laplace transformed transition density in Chapter 4. The first is that the Pareto density is a reasonable approximation of
an α--stable. Second, symmetric transitions ahead and behind the mean were used. Both greatly simplify
the transforms; but the difference for the velocity semivariogram is significant. There is a need to investigate
the form of the variance and propagator using numerical transforms of skewed and/or exact series representations of the α--stable densities and investigate the regions of validity of the simplified expressions.
The first “field--scale” solute transport model (Mercado [1967]) assumed perfect stratification of the aquifer’s
random permeability and no transverse mixing (similar to Taylor’s [1953] “fast” tube flow). The result is
a linear increase of the width of the vertically averaged concentration profile. The separation (call it Xc) between the distances traveled by a plume’s mean concentration and an arbitrary concentration grows linearly
with the mean travel distance. It is more common to see plots of plume variance (Xc2) so the Mercado model
predicts a slope of 2 on a log--log plot (Figure 8.1). This equivalent to “ballistic” motion described in Chapter
4, since every layer is experiencing piston, or wave equation, flow. Therefore, the FADE is a model of the
average Mercado aquifer as the order of differentiation α → 1. If α is exactly equal to unity, the FADE is
M
WT
1
1
log(X 2c)
D
FADE
α
GA
ADE
2
1
1
1
log(mean travel distance)
Figure 8.1 Comparison of the plume growth predicted by the traditional ADE (ADE), Gelhar and
Axness (1983) (GA), the fractional ADE (FADE), Mercado (1967) stratified flow (M), and Wheatcraft
and Tyler (1988) fractal tortuosity model (WT). The ordinate log(Xc2) is roughly equivalent to
estimated plume variance. The GA curve has slope 2:1 at a plume’s origin, transitioning to Fickian
1:1 slope at late time.
86
singular and has no solution. Note that the Fickian solution (α = 2 with a constant dispersion coefficient)
predicts linear growth of the variance with travel distance.
Gelhar and Axness (1983) use a scaled dispersion coefficient that predicts Mercado--type spreading at the
birth of a plume, and Fickian--type spreading as the plume travel distance → ∞ (Figure 8.1). When and
whether the transition to Fickian (α = 2) behavior takes place in real aquifers are debatable questions (refs).
The Wheatcraft and Tyler (1988) model, based on fractal tortuosity, predicts faster--than--Mercado spreading.
Finally, the fractional ADE predicts spreading at any speed between Fickian to Mercado. The plot of calculated plume variance from the Cape Cod site (Figure 6.10) shows close adherence to the FADE predictions
over about 1½ decades of plume travel distance.
The fundamental solutions given in this paper have three primary features: heavy tails, nonlinear growth of
variance or apparent variance, and single--equation solutions valid across many scales. This leads to two a
posteriori methods of estimating α. The concentration in the leading or trailing edge of a plume is predicted
to be a power function of time or distance. Estimation of the exponent parameter α from a single observation
point would be possible using a tracer that is detectable over several orders of magnitude so that plotting concentration versus time or distance on log--log axes gives a line with a simple linear function of α as the slope.
The second method would use multiple points at several times and estimate α from the the spread of the particle density, which is proportional to t1/α (Figure 8.1). This method was successfully used on the Cape Cod
data.
A very important point must be be made concerning the value of this a posteriori data. Since the dispersion
coefficient is constant, is is known immediately. The value of alpha is discerned after only two or three early-time measurements. These two pieces of information are all that is needed to predict the plume configuration
at all future times. This is in stark contrast to current field scale theories that use the second--order equation.
Early--time measurements of the dispersion coefficient do not give any information about the asymptotic
(plume travel distance → ∞) dispersion coefficient. In order to have a predictive tool, these theories require
information about the hydraulic conductivity autocovariance. This information only comes from a large
number of flowmeter tests or permeameter tests on core samples. This type of information is largely superfluous to the FADE at a typical field site. Even when these data are available, the estimates of asymptotic
dispersion coefficient for the second--order ADE are not particularly accurate. For the Cape Cod site, the
asymptotic longitudinal dispersion coefficient based on permeameter and flowmeter tests were approximately 0.15 and 0.3 m2/d, compared to the observed field value of 0.4 m2/d (Hess, et al. [1992]). The values of
the fractional dispersion coefficient from the first 4 sampling periods at the site (chapter 6) were 0.19, 0.18,
0.17 and 0.16 m1.6/d, all reasonably close to the “real” value of 0.14 m1.6/d. The large volume of the injected
fluid was largely responsible for the inflated early--time values.
Using the FADE on two laboratory experiments yielded surprising results. First, the FADE was able to more
accurately model a pure diffusion process. Carey et al.’s (1995) experiment of the diffusion of high ionic
strength CuSO4 into distilled water did not follow the classical scaling law dictated by the second--order diffusion equation. The width of the transition zone grew proportional to t1/α where α ≈ 2.5, rather than the classical α ≈ 2. The scaling index is greater than 2, which requires one of the following explanations.
First, if the distance travelled by a diffusing particle follows a power law (Chapter 4), then the longer walks
must suffer a velocity penalty. Referring to equation (4.53), the first propagator (Region P1 in Figure 8.2a)
is ruled out because it predicts faster--than--Fickian spreading. The second propagator implies that ν has a
value between 0.6 and 1.0 (Region P2 in Figure 8.2a) since by equation (4.55), ν = (0.4α + 1)/(α + 1). Another
87
2
3.0
Eq.
(4.51)
b)
a)
ν
1
Brownian
Motion
Region P1
2.0
00
ν
1
η
2
2
1.0
Region P2
Eq.
(4.53)
c)
ν
Region P3
1
0.0
0.0
1.0
α
2.0
3.0
00
1
η
2
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 increasing concentration is shown by the
arrow Variance exponent (VAR ∝ tη) for arrow path α → 2 predicted by b) variance equation (4 51)
and c) the propagator equation (4 53)
possibility is that the power law for particle excursion distance has a value of α > 2 This particular propagator
was not examined in detail The variance equation (4 51) indicates that the diffusion experiment is in the
region P3 and that the parameter ν = 0 4 (Figure 8 2a) The resulting variance of the concentration profile
would grow proportional to t2 2 5 = t0 8
The probable evolution of the governing behavior of the particles as the total concentration increases is shown
by the arrow (Figure 8 2a) An infinitely dilute solution follows Gaussian increments (α = 2) with ν ≤ 1
At this point the variance and propagator equations predict Brownian motion As the concentration increases the effective velocity parameter decreases taking the asymptotic solutions into realms of subdiffusion while maintaining walk distance increments that are Gaussian or nearly so (i e α approaches 2) The
variance equations developed by Klafter et al (1987) and Blumen et al (1989) curiously indicate that the
variance should grow faster--than--Fickian for 0 5 < ν < 1 (Figure 8 2b) This leads to the possibly erroneous
conclusion that Carey’s (1995) experiment is not in Region P2 of Figure 8 2a The apparent variance shown
by the propagator equation (4 53) developed in this dissertation shows the expected systematic decrease in
the growth rate that is expected with a decrease in ν (Figure 8 2c) As a result particle excursions that are
Gaussian or nearly so but suffer a velocity penalty show the expected subdiffusion All of the values of
ν less than unity imply that longer excursions take place at a much slower rate than shorter ones (Figure 4 4)
a notion that is counter--intuitive compared to the long excursions that happen within aquifer material because
of higher velocities
88
The fundamental (Green’s function) solution to a diffusion equation with a fractional spatial derivative of
order α > 2 is still a density that scales with t1/α. However, it is not the density of a stable variable, since
it has finite variance. A sum of these variables converges to a Gaussian. An important and open question
is how fast the convergence occurs for a finite number of particle transitions. In other words, does Ct =
Do2.5C appear quasi--stable with index 2.5 for a long period before converging to a Gaussian? It is unknown
whether the experiment converged to Gaussian (Fickian) behavior very late in the test.
It is also possible that the particle motions have infinite mean duration, and that the solution might be predicted by the FADE using a fractional time derivative (Giona and Roman [1992a, 1992b]) and a second--order
spatial derivative. Fractional--in--time processes have not been explored in this dissertation. Another alternative is that the Lévy walks of the copper and sulfate ions inhabit region P1 (i.e., α < 2) in three dimensions,
but the projection into one dimension increases the apparent scaling index. Imagine a particle on a long excursion perpendicular to the gradient. The particle would appear to be motionless in 1--D. The particle movements might appear to have an index α > 2, implying eventual Fickian behavior, yet the underlying process
remains Lévy--stable. The deceptively simple experiment conducted by Carey et al. (1995) invites more detailed study.
Only slightly less surprising was the α--stable character of the laboratory sandbox experiments. Building a
sandbox with predictable, known characteristics is generally regarded as a difficult task. The sandbox, designed to be homogeneous throughout, showed extensive tailing and a value of α on the order of Cape Cod
(1.55 and 1.6, respectively). The heavy--tailed data typically would be modelled by a multi--compartment
Fickian model (i.e. mobile and immobile water phases) with exchange coefficients between compartments.
The FADE is similar in its conception, but much simpler in its inception. The FADE is based on the heavy-tailed, skewed velocities that the Gaussian density lacks. The multi--compartment models force a bimodal
distribution that may not apply in many instances. A bimodal distribution is not indicated in this experiment,
where a conservative tracer moved through clean, fine quartz sand.
The ad hoc numerical implementation of the fractional advection--dispersion equation (Chapter 7) proved
useful for addressing the non--ideal initial conditions at the Cape Cod site. The numerical approximation was
shown to have several of the desirable features of the FADE’s analytic solution. These include the nonlinear-with time spreading and power--law tails. Unfortunately the power--law tails were not the right power law.
A “quick and dirty” analysis of the convergence properties of the approximation (Appendix V) suggests that
two or three point approximations will always have this behavior, and computationally expensive seven or
nine point approximations would properly incorporate the spatial dependence of the fractional derivatives.
A Galerkin finite element method may prove useful in providing a fast numerical implementation of the
FADE, since integration of the linear fractional differo--integral operator should be straightforward.
The potential gains that a proper numerical model of the FADE would yield are enormous. Currently, a modeler who wishes to accurately model a plume at all stages must input a very fine--scale description of the random K field. The modeler gives a local value of the dispersion tensor to each element. As the plume grows
and encounters more of the correlated K field, it spreads in a faster--than Fickian manner. In order to model
a plume that traverses many of the K correlation scales, the modeler is forced to use many elements in every
spatial dimension. Further, the model’s timestep size is roughly inversely proportional to element size. So
a traditional numerical model accurate across all scales might require thousands to millions of nodes and
executions on the order of teraflops. Conversely, within a numerical model of the FADE, the derivatives are
responsible for plume spreading, and very fine--scale descriptions of the K field are superfluous. The modeler
89
would input only the coarsest observable K distribution, and the tracer would disperse properly at smaller
scales due to the underlying probability distribution that the fractional derivatives solve. The number of elements required in each Cartesian dimension might be reduced by more than an order of magnitude, reducing
model size by factors of thousands or more and reducing execution times to trivial numbers. This is especially
appealing for the recent increased interest in basin--scale (100’s of kilometers) modeling of geologic processes.
It is unknown at present whether anisotropic spreading will be adequately handled by a 2nd--rank tensor of
constants. Detailed multi--directional tracer tests may indicate that an an extension to 3--D requires a vector
α which is different in any dimension. Current investigations into heavy--tailed random vectors (exemplified
by a 3--D plume) suggest that the dominant index of stability tends to overwhelm the detection of other “subordinates” (Meerschaert and Scheffler [1998]) unless measurements are made precisely in the principle directions of the α vector. This might obviate the detailed study of a vector form of α, since field--scale validation
would be difficult for a meandering plume.
90
CHAPTER 9
CONCLUSIONS AND RECOMMENDATIONS
9.1 CONCLUSIONS
Classical descriptions of field--scale solute dispersion based on the 2nd--order diffusion equation (Gelhar and
Axness [1983]; Dagan [1984]) are burdened by two assumptions: 1) the velocity contrasts are small, and 2)
the travel distance is large compared to a typical velocity correlation length. These assumptions arise because
the diffusion equation is essentially a restatement of the Central Limit Theorem -- a large number of finite-variance particle trajectories (random variables) must be added before a Gaussian appears. In reality, tracer
“particles” released into real aquifers experience large velocity contrasts along their trajectories. Recent theories devised to explain non--Fickian dispersion in turbulent and chaotic systems (c.f., Shlesinger et al. [1982];
Klafter et al. [1987]; Zaslavsky [1994]) begin with the assumption that particle excursion distances and velocities are likely to have large, even infinite, variance. Since the trajectories are stable variables, these transitions converge to their limit distribution immediately. The result is that a tracer test needn’t sample large
volumes before it can be described by a single, self--contained equation. The governing equation of the stable
transitions -- the fractional advection--dispersion equation -- is valid over all scales.
The theoretical results presented here are attractive for three primary reasons:
S
The underlying model of particle motion is based on large, even infinite, variance of the
velocity and excursion distance;
S
The fundamental solution (a density) spreads proportional to t1/α with 1 ≤ α ≤ 2, a result
that is ubiquitous in field tracer studies, and;
S
The fundamental solution predicts higher concentrations in the plume tails than do classical
theories. This result is also often reported in field studies.
The work in turbulence (e.g. Shlesinger et al. [1982], Klafter et al. [1987]) has been extended to transport
in subsurface materials with an analysis of the spatial autocovariance of the velocity field, which theoretically
allows an a priori estimate of the Lévy stability (fractional divergence) parameter based on the velocity semivariogram. Analysis of the Cape Cod aquifer indicates that the a priori estimates, which require independent
analysis of aquifer hydraulic conductivity spatial autocorrelation, not reliable; yet they are also largely superfluous. Observation of a plume at several early times gives all the information needed by the FADE. Conversely, early observations do not give information regarding the asymptotic value of the dispersion tensor
required to make long--term predictions using the theories based on the second--order ADE. This is regarded
as one of the most important implications of the FADE -- it is easily applied to any site without extensive
measurement of permeability.
Using the FADE on two other laboratory experiments yielded surprising results. First, the FADE was able
to more accurately model a pure diffusion process. Carey’s (1995) experiment of the diffusion of high ionic
strength CuSO4 into distilled water did not follow the classical scaling law dictated by the second--order diffusion equation. The width of the transition zone grew proportional to t1/α where α ≈ 2.5, rather than the classical α ≈ 2. This simple experiment indicates the potentially widespread occurrence of “fractional” dispersive
processes.
91
For simplicity, we have limited this discussion to transport in one dimension. Extensions to higher dimensions are straightforward if the Lévy stability index α is the same value (with different scaling) in the principle
directions. However, an extension to a vector α may prove challenging.
9.1 RECOMMENDATIONS
Based on the information within this dissertation, the following recommendations for future studies are suggested:
S
Extend the results to reactive (sorbing) solutes. This will probably require the use of a
random velocity that is somewhat decoupled from the Lévy walk size. It will also result in
asymmetric walks, since sorption and desorption occur at different rates. The results can be
compared to the Lithium plume that was released simultaneously with the bromide plume
at Cape Cod.
S
Extend the FADE to three dimensions, including the possibility of a different stability index
(α) in each direction. This derivation can include the lower and upper cutoffs on the Levy
walk size to model the limitation of measurement size imposed by observation wells and the
maximum possible walk length dictated by aquifer parameters.
S
Compare the FADE solutions to kinetic or multi--compartment models.
S
Generate numerical aquifer simulations to further validate the FADE solutions. Specifically,
do the aquifer statistical characteristics generate accurate numbers for α? What are the limits
of the α--stable behavior? Are the autocorrelation functions derived in this dissertation
accurate? What are the effects of detection limits and fixed wellfields on the calculated
plume variance?
S
Apply the FADE to the other well--studied tracer tests that are readily available, including
the Borden site in Canada and the Columbus Air Force Base in Mississippi.
S
Rigorously derive the numerical approximation using finite differences and finite element
methods.
S
Investigate the possibility of a fractional Poisson equation as a governing equation of
groundwater flow and an explanation of scale--dependent hydraulic conductivity.
92
CHAPTER 10
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APPENDIX I
FORTRAN SOURCE CODES
I.1 Program SIMSAS.F - A program to generate symmetric Levy flights and diffusions
parameter(nsize=10000,im=139968,ia=3877,ic=29573)
dimension sasvec(nsize),vec(nsize)
character*40 fileout
write(6,*)’enter output filename: ’
read(*,10)fileout
10
format(a)
open(unit=12,file=fileout)
write(*,*)’enter alpha, number of jumps (<10000): ’
read(*,*)alpha,njumps
jran=512
xpos=0.0
ypos=0.0
xdiff=0.0
c----- prepare an x,y,X(t) file of a 2-d Levy flight and Levy diffusion.
c----- First, pack a vector with symmetric a-stable iid variables:
call sas(alpha,sasvec,vec,njumps)
c----- now generate quick and dirty random direction using a uniform
c----- random number on (0,1):
do 100 i=1,njumps
jran=mod(jran*ia+ic,im)
ran=float(jran)/float(im)
100
if(ran.gt.0.5)then
xpos=xpos+sasvec(i)
else
ypos=ypos+sasvec(i)
endif
xdiff=xdiff+sasvec(i)
write(12,*)xpos,ypos,xdiff
continue
stop
end
subroutine sas(a,rvec1,rvec2,nsize)
dimension rvec1(nsize),rvec2(nsize)
open(16,file=’uniform’)
pi=3.141592654
eps=1.e-30
idum=5145
c----- first fill two vectors with uniform (0,1) i.i.d.s
do 10 i=1,nsize
call uniform(ran0,idum)
rvec1(i)=ran0
call uniform(ran0,idum)
rvec2(i)=ran0
c
write(16,*)1.0,rvec1(i),rvec2(i)
10
continue
c----- convert to uniform (-pi/2,pi/2) and exponential with mean 1:
do 30 i=1,nsize
rvec1(i)=pi*(rvec1(i)-0.5)
97
rvec2(i)=-1.0*log(rvec2(i))
30
continue
c----- convert to standard symmetric stable of order a (alpha)
c----- see Janicki and Weron, Stat. Sci, (9), pp.109-126
j=0
do 40 i=1,nsize
uni=rvec1(i)
expo=rvec2(i)
c1=sin(a*uni)
c2=cos(uni)**(1.0/a)
c3=cos(uni-a*uni)
if((abs(c2).gt.eps).and.(expo.gt.eps))then
j=j+1
rvec1(j)=(c1/c2)*(c3/expo)**((1-a)/a)
endif
write(16,*)rvec1(j)
40
continue
nsize=j
return
end
subroutine uniform(ran0,idum)
c---- modified from Numerical Recipes (Press et al. [1992])
parameter(ia=16807,im=2147483647,am=1./im,iq=127773,
1
ir=2836,mask=123459876)
k=idum/iq
idum=ia*(idum-k*iq)-ir*k
if(idum.lt.0)idum=idum+im
ran0=am*idum
return
end
98
I.2 Program ENSEM.F -- calculates the ensemble velocity autocorrelation function in Levy
walks for v(x) = f(Z), v=velocity, Z = symmetric a--stable random variate. Includes analytic solution for v(x) = Z
parameter(nsize=100000,im=139968,ia=3877,ic=29573)
dimension sasvec(nsize),vec(nsize),velo(10*nsize),
1
enscorr(500)
character*40 fileout1,fileout2
write(*,*)’enter seed int., alpha, # of Xi s, time incr., # of r
1ealizations: ’
read(*,*)idum,alpha,njumps,tau,nens
open(unit=22,file=’ensout’)
do 50,ne=1,nens
jfillmax=0
npoints=50
c----- First, clear then pack a vector with symmetric a-stable
c----- iid variables:
do 10 i=1,nsize
10
sasvec(i)=0.0
call sas(alpha,sasvec,vec,njumps,idum)
c----- now generate the velocity profile to see what the
c----- spatial autocorrelation looks like.
ncount=0
nactual=0
do 100 i=1,njumps
nactual=nactual+1
jfill=int(0.5+100.0*abs(sasvec(i)))
if(ncount+jfill.gt.10*nsize) goto 555
if(jfill.gt.jfillmax)jfillmax=jfill
do 110 j=1,jfill
ncount=ncount+1
velo(ncount)=sasvec(i)/tau
110
continue
100 continue
c----- figure out the running average jfillmax:
c----- and use the first realization to gauge the scale:
if(ne.eq.1)nplag=max(1,ncount/800)
c----- new try - adjust scale first, then get lags close
nblah=int(0.5+float(jfillmax/float(npoints)))
nplag=max(1,nblah)
avxmax=(avxmax*float(ne-1)+float(jfillmax))/float(ne)
555 variance=0.0
c----- figure the mean velocity
sum=0.0
do 250 n=1,ncount
sum=sum+velo(n)
250
continue
xpect=sum/float(ncount)
do 260 n=1,ncount
variance=variance+(velo(n)-xpect)**2.0
260 continue
c
variance=variance/float(ncount-1)
c
write(6,*)variance,jfillmax
99
nlags=0
do 200 nlag=nplag,2*jfillmax,nplag
nlags=nlags+1
corr=0.0
do 210 i=1,ncount-nr
corr=corr+velo(i)*velo(i+nlag)
210 continue
c----- keep track of the running ensemble correlation
enscorr(nlags)=(enscorr(nlags)*float(ne-1)+corr/variance)/ne
200 continue
write(*,*)nactual,xpect,avxmax,enscorr(3)
50
continue
c----- figure out what the scaled lags are:
do 300 k=1,nlags
x=float(k)/float(npoints)
c
x=float((k-1)*nscale+1)/avxmax
anal1=1.0
anal2=1.0
if(x.le.1.0)anal1=x*(3.-alpha)/(2.-alpha)1
(x**(3.-alpha))/(2.-alpha)
if(x.le.1.0) then
sum1=0.0
sum2=0.0
sum3=0.0
do 310 m=0,20
cnt=float(2*m)
c1=gammln(1.0+(cnt+1.0)/alpha)
c2=gammln(1.0+cnt)
if(c1-c2.gt.85.)goto 311
fm=(-1.0)**m*exp(c1-c2)
sum1=sum1+fm/(cnt+3.)
sum2=sum2+fm*x**(cnt+4.0)/((cnt+4.)*(cnt+3.))
sum3=sum3+fm/(cnt+4.)
310
continue
311
anal2=(x*sum1-sum2)/sum3
endif
300 write(22,*)x,1.0-enscorr(k),anal1,anal2
stop
end
subroutine sas(a,rvec1,rvec2,nsize,idum)
dimension rvec1(nsize),rvec2(nsize)
open(16,file=’uniform’)
pi=3.141592654
eps=1e-40
c----- first fill two vectors with uniform (0,1) i.i.d.s
do 10 i=1,nsize
call uniform(ran0,idum)
rvec1(i)=ran0
call uniform(ran0,idum)
rvec2(i)=ran0
c
write(16,*)1.0,rvec1(i),rvec2(i)
10
continue
c----- convert to uniform (-pi/2,pi/2) and exponential with mean 1:
do 30 i=1,nsize
rvec1(i)=pi*(rvec1(i)-0.5)
100
rvec2(i)=-1.0*log(rvec2(i))
30
continue
c----- convert to standard symmetric stable of order a (alpha)
c----- see Janicki and Weron, Stat. Sci, (9), pp.109-126
j=0
do 40 i=1,nsize
uni=rvec1(i)
expo=rvec2(i)
c1=sin(a*uni)
c2=cos(uni)**(1.0/a)
c3=cos(uni-a*uni)
if((abs(c2).gt.eps).and.(expo.gt.eps))then
j=j+1
rvec1(j)=(c1/c2)*(c3/expo)**((1-a)/a)
endif
write(16,*)rvec1(j)
40
continue
nsize=j
return
end
subroutine uniform(ran0,idum)
parameter(ia=16807,im=2147483647,am=1./im,iq=127773,
1
ir=2836,mask=123459876)
k=idum/iq
idum=ia*(idum-k*iq)-ir*k
if(idum.lt.0)idum=idum+im
ran0=am*idum
return
end
c
c
c
c
c
c
c
function gammln(xx)
real gammln,xx
From numerical recipes (Press et al. [1992])
- returns the log of gamma(xx).
If this number is bigger than about 85, the exp of it will kill
most computers, so catch it in the main program.
real*8 cof(6),ser,stp,tmp,x,y,half,one,fpf
data cof,stp/76.18009173d0,-86.50532033d0,24.01409822d0,
1
-1.231739516d0,.120858003d-2,-.536382d-5,2.50662827465d0/
data half,one,fpf /0.5d0,1.0d0,5.5d0/
cof(1)=76.18009172947146d0
cof(2)=-86.50532032941677d0
cof(3)=24.01409824083091d0
cof(4)=-1.231739572450155d0
cof(5)=.1208650973866179d-2
cof(6)=-.5395239384953d-5
stp=2.5066282746310005d0
half=0.5d0
one=1.0d0
fpf=5.5d0
x=xx
y=x
tmp=x+fpf
tmp=(x+half)*log(tmp)-tmp
101
10
ser=1.000000000190015d0
do 10 j=1,6
y=y+one
ser=ser+cof(j)/y
continue
gammln=tmp+log(stp*ser/x)
return
end
102
I.3 Program AVEGAM.F -- calculates the ensemble velocity autocorrelation function in
Levy walks for v(x) = f(Z), v=velocity, Z = symmetric a--stable random variate. Includes
analytic solution for v(x) = Z. This program calculates gamma at the same lags (x) for
each realization, then interpolates the value of x for a given value of gamma. It then averages the interpolated x value at each known gamma for the various realizations. This is in
contrast to the program ensem, which just averages the value of gamma at the known lag
values.
5
parameter(nsize=80000,im=139968,ia=3877,ic=29573)
dimension sasvec(nsize),vec(nsize),velo(200*nsize),
1
xcorrav(500),corr(500)
character*40 fileout1,fileout2
write(*,*)’enter output filename: ’
read(*,5)fileout1
format(a)
write(*,*)’enter seed int., alpha, # of Xi s, # of realizations,
1 tolerance for each realization: ’
read(*,*)idum,alpha,jumpwant,nens,tol
pi=3.141592654
a=alpha
avemax=0.0
nreject=0
filler=max(200.,1.e5/float(jumpwant))
c---- figure out the expected jump size
c=(1.-a)/(exp(gammln(2.-a))*cos(pi*a/2.))
palpha=exp(gammln(1.-1./a))
expjump=filler*palpha*(c*float(jumpwant))**(1./a)
ne=0
777 jfillmax=0
ncorrpt=40
nxpoints=25
njumps=jumpwant
c----- First, clear then pack a vector with symmetric a-stable
c----- iid variables:
do 10 i=1,nsize
10
sasvec(i)=0.0
call sas(alpha,sasvec,vec,njumps,idum)
ncount=0
nactual=0
do 100 i=1,njumps
nactual=nactual+1
jfill=int(0.49999+filler*abs(sasvec(i)))
if(ncount+jfill.gt.100*nsize) goto 555
if(jfill.gt.jfillmax)jfillmax=jfill
do 110 j=1,jfill
ncount=ncount+1
velo(ncount)=sasvec(i)
110
continue
100 continue
c----- Reject walks with jumps outside a tolerance Also,
c----- reject non-full series:
555 if(
1
(float(jfillmax).lt.(1.-.5*tol)*expjump).or.
1
(float(jfillmax).gt.(1.+.5*tol)*expjump).or.
103
2
(nactual.lt.(njumps-5)))then
nreject=nreject+1
goto 777
endif
ne=ne+1
c----- figure out the running average jfillmax (the first try was
c----- empirical, now I use the expected value of the largest walk):
write(*,*)ne,nreject,expjump,jfillmax,nactual
c----- figure the mean velocity
variance=0.0
sum=0.0
do 250 n=1,ncount
sum=sum+velo(n)
250
continue
xpect=sum/float(ncount)
c----- assume zero mean
xpect=0.0
do 260 n=1,ncount
variance=variance+(velo(n)-xpect)**2.0
260 continue
variance=variance/float(ncount)
write(6,*)variance,xpect
c---- Figure the correlation in the series. If the lag goes past the
c---- end, wrap to the beginning to avoid edge alias.
nplag=int(jfillmax/float(nxpoints))
nlags=0
do 200 nlag=nplag,2*jfillmax,nplag
nlags=nlags+1
corr(nlags)=0.0
lagcnt=0
do 210 i=1,ncount
lagcnt=lagcnt+1
next=i+nlag
if(next.gt.ncount)next=i+nlag-ncount
corr(nlags)=corr(nlags)+(velo(i)-xpect)*(velo(next)-xpect)
c
corr(nlags)=corr(nlags)+(velo(i))*(velo(next))
210 continue
c----- Could use two varieties of autocorrelation (Gelhar’s book),
c----- but with wrapping, it doesn’t matter:
c
200
corr(nlags)=corr(nlags)/(variance*float(lagcnt))
corr(nlags)=corr(nlags)/(variance*float(ncount))
continue
avemax=(avemax*float(ne-1)+float(jfillmax))/float(ne)
c---- figure the average value of x that corresponds to a level of the
c---- semivariogram. Use linear interpolation of the log(x) and gamma.
c---- Don’t go past gamma of 1.0 for now.
dx=float(jfillmax)/(expjump*float(nxpoints))
back1=0.0
back2=0.0
do 400 m=1,ncorrpt
corrlev=float(ncorrpt-m)/float(ncorrpt)
104
do 410 n=1,nlags
if(corr(n).lt.corrlev)then
xup=float(n)*dx
corrback=1.
if(n.gt.1)corrback=corr(n-1)
xcorr=xup-dx*(corr(n)-corrlev)/(corr(n)-corrback)
goto 405
endif
c----- catch the ones that don’t go all the way to zero by either
c----- zooming straight to zero or extrapolating the last two
c----- interpolated points:
xcorr=back1
c
xcorr=back1+back1-back2
410 continue
405 back2=back1
back1=xcorr
xcorrav(m)=(xcorrav(m)*float(ne-1)+xcorr)/float(ne)
write(*,901)n,corrback,corrlev,corr(n),xcorr,xup
400 continue
901 format(i4,5f9.4)
c----- figure out what the scaled lags are and overwrite the file
c----- created after the last realization:
open(unit=22,file=fileout1)
do 300 k=1,ncorrpt
x=xcorrav(k)
c
x=float(k*nplag)/avemax
c
x=float(k*nplag)/expjump
anal1=1.0
anal2=1.0
if(x.le.1.0)anal1=x*(3.-alpha)/(2.-alpha)1
(x**(3.-alpha))/(2.-alpha)
if(x.le.1.0) then
sum1=0.0
sum2=0.0
sum3=0.0
do 310 m=0,20
cnt=float(2*m)
c1=gammln(1.0+(cnt+1.0)/alpha)
c2=gammln(1.0+cnt)
if(c1-c2.gt.85.)goto 311
fm=(-1.0)**m*exp(c1-c2)
sum1=sum1+fm/(cnt+3.)
sum2=sum2+fm*x**(cnt+4.0)/((cnt+4.)*(cnt+3.))
sum3=sum3+fm/(cnt+4.)
310
continue
311
anal2=(x*sum1-sum2)/sum3
endif
write(22,*)x,1.0-float(ncorrpt-k)/float(ncorrpt),anal1,anal2
300
continue
close(22)
c----- If you have enough realizations, then figure the analytic and
c----- print. otherwise, start generating more realizations
if(ne.lt.nens)goto 777
stop
end
105
c----- This subroutine fills a vector (rvec1) with
c----- iid SaS random variates.
subroutine sas(a,rvec1,rvec2,nsize,idum)
dimension rvec1(nsize),rvec2(nsize)
pi=3.141592654
eps=1e-30
c----- first fill two vectors with uniform (0,1) i.i.d.s
do 10 i=1,nsize
call uniform(ran0,idum)
rvec1(i)=ran0
call uniform(ran0,idum)
rvec2(i)=ran0
c
write(16,*)1.0,rvec1(i),rvec2(i)
10
continue
c----- convert to uniform (-pi/2,pi/2) and exponential with mean 1:
do 30 i=1,nsize
rvec1(i)=pi*(rvec1(i)-0.5)
rvec2(i)=-1.0*log(rvec2(i))
30
continue
c----- convert to standard symmetric stable of order a (alpha)
c----- see Janicki and Weron, Stat. Sci, (9), pp.109-126
j=0
do 40 i=1,nsize
uni=rvec1(i)
expo=rvec2(i)
c1=sin(a*uni)
c2=cos(uni)**(1.0/a)
c3=cos(uni-a*uni)
if((abs(c2).gt.eps).and.(expo.gt.eps))then
j=j+1
rvec1(j)=(c1/c2)*(c3/expo)**((1-a)/a)
endif
40
continue
nsize=j
return
end
subroutine uniform(ran0,idum)
c--- From numerical recipes (Press et al. [1992])
c--- good for a few million
parameter(ia=16807,im=2147483647,am=1./im,iq=127773,
1
ir=2836,mask=123459876)
k=idum/iq
idum=ia*(idum-k*iq)-ir*k
if(idum.lt.0)idum=idum+im
ran0=am*idum
return
end
c-c-c-c--
function gammln(xx)
real gammln,xx
From numerical recipes (Press et al. [1992] p. 207)
returns the log of gamma(xx).
If this number is bigger than about 85, the exp of it will kill
most computers, so catch it in the main program.
106
10
real*8 cof(6),ser,stp,tmp,x,y,half,one,fpf
cof(1)=76.18009172947146d0
cof(2)=-86.50532032941677d0
cof(3)=24.01409824083091d0
cof(4)=-1.231739572450155d0
cof(5)=.1208650973866179d-2
cof(6)=-.5395239384953d-5
stp=2.5066282746310005d0
half=0.5d0
one=1.0d0
fpf=5.5d0
x=xx
y=x
tmp=x+fpf
tmp=(x+half)*log(tmp)-tmp
ser=1.000000000190015d0
do 10 j=1,6
y=y+one
ser=ser+cof(j)/y
continue
gammln=tmp+log(stp*ser/x)
return
end
107
I.4 Program WEIER.F - calculates the Weierstrass structure function of clustered walks
on a discrete lattice. The user must specify the constants b, λ (l), lattice spacing (del), and
number of points to calculate (np) within a wavenumber range (range). See Chapter 4.
100
200
real k,l
write(*,*)’Enter b>l>1, DELTA, # of points and the k range: ’
read(*,*)b,l,del,np,range
pi=3.1415926
a=log(l)/log(b)
c=0.7107*pi*del**a/(2.0*exp(gammln(a))*sin(a*pi/2.0))
write(*,*)’alpha and c = ’,a,c
do 200 kk=0,np
k=range*kk/np
sum=cos(del*k)
term=1.0
denom=1.0
do 100 n=1,300
denom=denom*l
term=term*b
sum=sum+cos(k*del*term)/denom
continue
sum=(1.0-1.0/l)*sum
write(*,*)k,sum,exp(-1.0*c*k**a)
continue
stop
end
function gammln(xx)
real gammln,xx
c modified from numerical recipes (Press et al [1992]) p. 157
real*8 cof(6),ser,stp,tmp,x,y,half,one,fpf
cof(1)=76.18009172947146d0
cof(2)=-86.50532032941677d0
cof(3)=24.01409824083091d0
cof(4)=-1.231739572450155d0
cof(5)=.1208650973866179d-2
cof(6)=-.5395239384953d-5
stp=2.5066282746310005d0
half=0.5d0
one=1.0d0
fpf=5.5d0
x=xx
y=x
tmp=x+fpf
tmp=(x+half)*log(tmp)-tmp
ser=1.000000000190015d0
do 10 j=1,6
y=y+one
ser=ser+cof(j)/y
10
continue
gammln=tmp+log(stp*ser/x)
return
end
108
I.5 Subroutine CFASTD.F -- generates the value of the standard α--stable distribution
function F(t) at a point. The user must specify alpha (a), beta (b), the point at which the
function is evatualted (t), and the number of integration points.
c---- Generates the standard a-stable distribution function f(t) (called cf(t)
here). User specifies 0<alpha<=2, skewness (-1<=beta<=1) and number of points
for numerical integration by trapezoidal rule. Integrals given by McCulloch
and Zolotarev 2.2.11 and 2.2.18 (pp. 71 etc.) Note - this should be changed
to Gauss quadrature for speed and accuracy.
c----------------------------------------------------------------------subroutine cfastd(cf,a,beta,t,np)
implicit real*8 (a-h,o-z)
implicit integer (i-n)
real*8 neg,gam
logical max
data pi/3.141592653589793d0/
one=1.0d0
two=2.0d0
neg=-1.0d0
half=0.5d0
zero=0.0d0
small=1.0d-10
big=1.0d200
halfpi=pi/two
enat=2.718281828459045d0
c---- If alpha is very near 1, go to a whole different thing
if(abs(a-one).lt.small) goto 888
if((abs(beta)-one).gt.small)then
write(6,*)’ Beta has been specified outside [-1,1]! Try again.’
stop
endif
max=.false.
if((abs(beta)-one).lt.small)max=.true.
if((a.lt.zero).and.(abs(beta-neg).lt.small))then
cf=zero
return
endif
t2=t
b=beta
if(t.lt.zero) then
b=neg*beta
t2=neg*t
endif
cstar=(one+(b*tan(halfpi*a))**two)**(neg*half/a)
x=cstar*t2
theta=(two/(pi*a))*datan(b*tan(halfpi*a))
eps=one
c=(one-theta)/two
if(a.gt.one)then
eps=neg
c=one
endif
pow1=a/(one-a)
pow2=a/(a-one)
109
c---- Trapezoidal integration. When j=1 figure at lower limit,
c---- when np, upper limit.
cf=zero
temp=zero
nip=np
if(abs(x).lt.one)nip=5*np
if(abs(x).lt.1.0d-1)nip=10*np
if(abs(x).lt.1.0d-2)nip=20*np
if(abs(x).lt.1.0d-3)nip=30*np
dgam1=(one+theta)/dble(nip-1)
do 15 j=1,nip
gam=neg*theta+dgam1*dble(j-1)
c---- calculate U given the problems at the integration limits (see
c---- McCulloch, maximally skewed paper, p. 16)
u=big
if(j.eq.nip)then
if(a.gt.one)then
u=zero
c---- This is from McCulloch
if(abs(b+one).lt.small)u=(a-one)*a**(a/(one-a))
c---- This is from Maple
c
if(abs(b+one).lt.small)u=(one-a)*(neg*a)**(a/(one-a))
endif
c---- This is from McCulloch
if(a.lt.one)u=(one-a)*a**(a/(one-a))
else
u1=sin(halfpi*a*(gam+theta))
u2=cos(halfpi*gam)
if(abs(u2).lt.1.0d-50)u2=1.0d-50
u3=u1/u2
if(u3.gt.big)u3=big
u4=cos(halfpi*(gam*(a-one)+a*theta))
u5=u4/u2
if((a.gt.one).and.(u3.gt.big)) then
u3=big
elseif((a.lt.one).and.(abs(u3).lt.1.0d-99)) then
u3=big
else
u3=u3**pow1
endif
u4=cos(halfpi*(gam*(a-one)+a*theta))
u=u3*u4/u2
endif
if(u.gt.big)u=big
intarg=u*neg*x**pow2
if(intarg.gt.900.d0)intarg=500.d0
f=dexp(u*neg*x**pow2)
15
cf=cf+f+temp
temp=f
continue
cf=c+half*half*dgam1*eps*cf
if(t.lt.zero)cf=one-cf
return
110
c---- calculate the Cauchy distribution
888 t2=t
b=beta
if(beta.lt.zero) then
t2=neg*t2
b=neg*beta
endif
x=halfpi*t2+(b*log(halfpi))
c---- Trapezoidal integration from -1 to 1.
cf=zero
temp=zero
nip=np
if(abs(x).lt.one)nip=5*np
dgam1=two/dble(nip-1)
do 25 j=1,nip
gam=neg+dgam1*dble(j-1)
u=big
if(j.eq.1)then
if(b.gt.zero)u=zero
if(abs(b-one).lt.small)u=one/enat
elseif(j.eq.nip)then
if(b.lt.zero)u=zero
if(abs(b+one).lt.small)u=one/enat
else
u1=tan(halfpi*gam)*halfpi*(gam+one/b)
u2=halfpi*(one+b*gam)/cos(halfpi*gam)
if(u2.gt.big)u2=big
if(u1.gt.500.d0)then
u=big
else
u=u2*exp(u1)
endif
endif
25
f=exp(u*neg*exp(neg*x/b))
cf=cf+f+temp
temp=f
continue
cf=half*half*dgam1*cf
if(beta.lt.zero)cf=one-cf
return
end
111
I.6 Subroutine DFASTD.F -- generates the value of the standard α--stable density function
f(t) at a point. The user must specify alpha (a), beta (b), the point at which the function is
evatualted (t), and the number of integration points.
c---- Generates the standard a-stable density f(t) (called df(t) here). User
specifies 0<alpha<=2, skewness (-1<=beta<=1) and number of points for numerical integration by trapezoidal rule. Integrals given by McCulloch and Zolotarev 2.2.11 and 2.2.18 (pp. 71 etc.) Note - this should be changed to Gauss
quadrature for speed and accuracy.
c---------------------------------------------------------------------subroutine dfastd(df,a,b,t,np)
implicit real*8 (a-h,o-z)
implicit integer (i-n)
real*8 neg,gam
logical max
data pi/3.141592653589793d0/
one=1.0d0
two=2.0d0
neg=-1.0d0
half=0.5d0
zero=0.0d0
small=1.0d-9
halfpi=pi/two
enat=2.718281828459045d0
big=1.0d200
c---- Right now, I don’t have any use for lam and shft
c---- They’ll be used for non-standard functions
c---- Use x for calculations, not t.
c---- If alpha is very near 1, go to a whole different thing (Cauchy)
if(abs(a-one).lt.small) goto 888
if(abs(a-two).lt.small)then
df=half*exp(neg*t*t/4.0d0)/pi**half
return
endif
if((abs(b)-one).gt.small)then
write(6,*)’ Beta has been specified outside [-1,1]! Try again.’
stop
endif
max=.false.
if((abs(b)-one).lt.small)max=.true.
c
t2=t
cstar=(one+(b*tan(halfpi*a))**two)**(neg*half/a)
x=cstar*t2
theta=(two/(pi*a))*datan(b*tan(halfpi*a))
if(t.lt.zero)theta=neg*theta
pow1=a/(one-a)
pow2=a/(a-one)
write(*,*)b,b2
c---- Trapezoidal integration.
c---- when np, upper limit.
df=zero
temp=zero
if(np.lt.50)np=50
When j=1 figure at lower limit,
112
nip=np
if(abs(x).lt.one)nip=5*np
if(abs(x).lt.1.0d-1)nip=10*np
if(abs(x).lt.1.0d-2)nip=50*np
if(abs(x).lt.1.0d-3)nip=100*np
dgam1=(one+theta)/dble(nip-1)
do 15 j=1,nip
gam=neg*theta+dgam1*dble(j-1)
c---- calculate U given the problems at the integration limits (see
c---- McCulloch, maximally skewed paper, p. 16)
u=big
if(j.eq.nip)then
if(a.gt.one)then
u=zero
c---- This is from McCulloch
if(abs(b+one).lt.small)u=(a-one)*a**(a/(one-a))
c---- This is from Maple
c
if(abs(b+one).lt.small)u=(one-a)*(neg*a)**(a/(one-a))
endif
c---- This is from McCulloch
if(a.lt.one)u=(one-a)*a**(a/(one-a))
else
u1=sin(halfpi*a*(gam+theta))
u2=cos(halfpi*gam)
if(abs(u2).lt.1.0d-50)u2=1.0d-50
u3=u1/u2
if(u3.gt.big)u3=big
u4=cos(halfpi*(gam*(a-one)+a*theta))
if((a.gt.one).and.(u3.gt.big)) then
u3=big
elseif((a.lt.one).and.(abs(u3).lt.1.0d-99)) then
u3=big
else
u3=u3**pow1
endif
u4=cos(halfpi*(gam*(a-one)+a*theta))
u=u3*u4/u2
endif
arg=u*neg*abs(x)**pow2
if((arg.gt.500.d0).or.(abs(u).lt.1.0d-190))then
f=zero
else
f=u*exp(u*neg*abs(x)**pow2)
endif
df=df+f+temp
temp=f
15
continue
c
df=dgam1*df*0.25d0*a*(abs(x)**(one/(a-one)))/(cstar*abs(one-a))
df=dgam1*df*0.25d0*a*cstar*(abs(x)**(one/(a-one)))/abs(one-a)
return
c---- calculate the Cauchy density
888
if(abs(b).lt.small)then
df=half/(pi*pi/4.0d0+t*t)
return
113
endif
x=halfpi*t+(b*log(halfpi))
c---- Trapezoidal integration from -1 to 1.
df=zero
temp=zero
nip=np
if(abs(x).lt.one)nip=5*np
if(abs(x).lt.1.0d-1)nip=10*np
if(abs(x).lt.1.0d-2)nip=20*np
if(abs(x).lt.1.0d-3)nip=30*np
dgam1=two/dble(nip-1)
25
c
do 25 j=1,nip
gam=neg+dgam1*dble(j-1)
u=big
if(j.eq.1)then
if(b.gt.zero)u=zero
if(abs(b-one).lt.small)u=one/enat
elseif(j.eq.nip)then
if(b.lt.zero)u=zero
if(abs(b-neg).lt.small)u=one/enat
else
u1=tan(halfpi*gam)*halfpi*(gam+one/b)
u2=halfpi*(one+b*gam)/cos(halfpi*gam)
if(u2.gt.big)u2=big
if(u1.gt.500.d0)then
u=big
else
u=u2*exp(u1)
endif
endif
arg=u*neg*exp(neg*x/b)
if((arg.gt.500.d0).or.(abs(u).lt.1.0d-190))then
f=zero
else
f=u*exp(arg)
endif
df=df+f+temp
temp=f
continue
df=half*dgam1*df*exp(neg*x/b)/(pi*abs(b))
df=half*dgam1*df*exp(neg*x/b)*pi/(4.0d0*abs(b))
return
end
114
I.7 Program CVX.F -- calculates the concnetration versus distance profiles for 1--D Levy
walks governed by the fractional ADE. It calls the subroutines DFASTD and CFASTD
listed above.
c
c
c
c
c
c
c
c
c
This program computes the analytic solutions for plumes undergoing
Levy walks with the fractional ADE governing equation. In this
case the program solves concentration versus distance at a single
user-specified time.
The program prompts for the time of interest, the constant
velocity,(x) range in which to compute C(x), the order of the
space derivative (1<alpha<=2), the skewness (-1<beta<1), the
constant dispersion coefficient (diff), the initial contaminant
”mass” (c0 times x0) and the number of integration points.
C
COPYRIGHT DAVID A. BENSON 1997 - 1998
implicit real*8 (a-h,o-z)
implicit integer (i-n)
parameter (nrp=50)
real*8 neg,k,lambda
character*40 fileout
dimension posdf(nrp+1),poscf(nrp+1),posxt(nrp+1)
data pi/3.1415926535d0/
one=1.0d0
two=2.0d0
neg=-1.0d0
half=0.5d0
zero=0.0d0
enat=2.718281828459045d0
time=zero
write(*,*)’ Enter name of output file: ’
read(*,’(a)’)fileout
open(14,file=fileout)
write(*,*)’ Enter time, velocity, and x-range (x1 to x2):’
read(*,*)time,v,x1,x2
write(*,*)’Enter alpha (2.0=Fickian), beta, the disp. coeff.,’
write(*,*)’the exponent on time, # of int. points (>500) and ini
1tial mass (C0):’
read(*,*)a,b,diff,texp,np,co
c----- If this is C v. x then I need to loop through and figure out the
c
scaling parameter (c) at each x-position. If it is C v. t then a
c
single scale factor works and I figure out a single density.
c
Remember that: a=alpha, b=beta, c=scale, d=shift
c
phi(k) = exp(idk + (ck)^a (1-ibtan(pi*a/2)))
c
F(t,a,b,c,d)=F((t-d)/c,a,b,1,0) (McCulloch max skew, page 2), and
c
f(t,a,b,c,d)=1/c*f((t-d)/c,a,b,1,0)
c
Remember also that for Laplace forms, beta=1, see Zolotarev, p.114.
nxp=100
c=one
d=v*time
dx=(x2-x1)/dble(nxp)
c---- Use an initial scale parameter c.
c---- This won’t change for C|t but will for C|x.
c--- loop thru time on the outside, x on the inside
115
c
c
c
do 10 j=1,nrp+1
time=rt*dble(j-1)/dble(nrp)
if(abs(time).le.1.0d-2)time=1.0d-2
do 20 i=1,nxp+1
x=x1+dble(i-1)*dx
if(abs(x).le.1.0d-2)then
if(x.lt.0.0)x=-1.0d-2
if(x.ge.0.0)x=1.0d-2
endif
write(6,*)x
c=(diff*time)**(texp/a)
c---- scale the argument (t) sent to the standard generators
t=(x-d)/c
c---- Get a standard cumulative distribution function at point x
call cfastd(cf,a,b,t,np)
c---- next the density
call dfastd(df,a,b,t,np)
c---- Correct the df according to the scaling constant (c) and alpha
c---- calc a temp df for a check
c
tdf=tdf+dx*df
df=df*(one/c)
if(df.lt.1.0d-50)df=1.0d-20
if(cf.lt.1.0d-50)cf=1.0d-20
write(14,500)x,co*df,co*(1.0-cf)
c
write(14,500)x,c*df,1.0-cf
20
continue
c 10 continue
stop
500 format(1p,4e14.4)
end
116
I.8 Program CVT.F -- calculates the concentration versus time graph for a fixed point in
1--D space for a plume undergoing Levy walks governed by the fractional ADE. It calls the
subroutines CFASTD and DFASTD listed above.
c
c
c
c
c
c
c
c
c
This program computes the analytic solutions for plumes undergoing
Levy walks with the fractional ADE governing equation. In this
case the program solves concentration versus time at a single
user-specified time (i.e. a breakthrough curve).
The program prompts for the distance, the constant velocity,
time range in which to compute C(t), the order of the space
derivative (1<alpha<=2), the skewness (-1<beta<1), the constant
dispersion coefficient (diff), and the number of
integration points.
C
COPYRIGHT DAVID A. BENSON 1997 - 1998
implicit real*8 (a-h,o-z)
implicit integer (i-n)
parameter (nrp=50)
real*8 neg,k,lambda
character*40 fileout
dimension posdf(nrp+1),poscf(nrp+1),posxt(nrp+1)
data pi/3.1415926535d0/
one=1.0d0
two=2.0d0
neg=-1.0d0
half=0.5d0
zero=0.0d0
enat=2.718281828459045d0
time=zero
write(*,*)’ Enter name of output file: ’
read(*,’(a)’)fileout
open(14,file=fileout)
write(*,*)’ Enter position (x), v, and time-range (t1 to t2):’
read(*,*)y,v,t1,t2
write(*,*)’Enter alpha (2.0=Fickian), beta, the disp. coeff.,’
write(*,*)’the exponent on time, and # of int. points (>500) :’
read(*,*)a,b,diff,texp,np
c----- If this is C v. x then I need to loop through and figure out the
c
scaling parameter (c) at each x-position. If it is C v. t then a
c
single scale factor works and I figure out a single density.
c
Remember that: a=alpha, b=beta, c=scale, d=shift
c
phi(k) = exp(idk + (ck)^a (1-ibtan(pi*a/2)))
c
F(t,a,b,c,d)=F((t-d)/c,a,b,1,0) (McCulloch max skew, page 2), and
c
f(t,a,b,c,d)=1/c*f((t-d)/c,a,b,1,0)
c
Remember also that for Laplace forms, beta=1, see Zolotarev,
c
p.114.
ntp=100
c=one
d=zero
dt=(t2-t1)/dble(ntp)
c---- Use an initial scale parameter c.
c---- This won’t change for C|t but will for C|x.
do 10 j=1,ntp+1
117
time=t1+dble(j-1)*dt
if(abs(time).le.1.0d-2)time=1.0d-2
c---- at time t the fixed point y is somewhere on the snapshot curve:
x=y-v*time
if(abs(x).le.1.0d-3*y)then
if(x.lt.0.0)x=-1.0d-3*y
if(x.ge.0.0)x=1.0d-3*y
time=(y-x)/v
endif
write(6,*)time,time**(1./a)
c=(diff*time)**(texp/a)
c---- scale the argument (t) sent to the standard generators
t=(x-d)/c
c---- Get a standard cumulative distribution function at point x
call cfastd(cf,a,b,t,np)
c---- next the density
call dfastd(df,a,b,t,np)
c---- Correct the pdf according to the scaling constant (c) and alpha
c
10
500
df=df*(one/c)
if(df.lt.1.0d-50)df=1.0d-20
if(cf.lt.1.0d-50)cf=1.0d-20
write(14,500)time,1.0-cf
write(14,500)(v*time/y-1.)/time**(1./a),c*df,cf
continue
stop
format(1p,4e14.4)
end
118
I.9 Program FRACDISP.F -- A 1--D numerical approximation of the symmetric fractional
ADE. The program calculates the concentration versus distance graph for any number of
fixed times for an arbitrary initial plume undergoing Levy walks.
c---c---c---c---c---c---c---c--c---c---c---c---c---c---c----
This program calculates the 1-D spread of an initial slug of
of contaminant size x0 with concentration C0 in a constant
velocity field. Since this is a numerical solution, all of the
IC’s, BC’s, and parameters can be changed by the user. THis
program is meant to be validated against the delta function
analytic solution and simulate the Cape Cod experiment, where
velocty (v) and D (diff) are relatively constant is space and
time.
The fractional calculus parameters are alpha (the order of the
space derivative) and beta (the order of the time derivative). If
alpha=2 and beta=1, the classical ADE is recovered. The user is
advised to keep the product r=diff*dt**beta/dx**alpha as low as
possible by specifying small timesteps (dt) or large space between
nodes (dx). Note that the continuous-release (step function)
solution is simultaneously calculated.
c---- copyright David A. Benson 02/016/98
parameter (nodes=400)
dimension cdel(nodes),ccont(nodes),prttime(100)
data cdel,ccont/nodes*0,nodes*0/
open (unit=12, file=’fracxt.prn’)
write(*,*)’enter velocity, x0, C0, diff. coeff., alpha, beta:’
read(*,*)v,x0,c0,diff,alpha,beta
write(*,*)’enter dx, dt, number of printouts, and time of each:’
read(*,*)dx,dt,npts
do 5 i=1,npts
read(*,*)prttime(i)
write(*,*)prttime(i)
5
continue
c---- place initial conditions
tmass=c0*x0
ninit=max(1,int(x0/dx))
c0=tmass/(dx*float(ninit))
write(*,*)’initial contaminant adjusted to (c,x):’,c0,dx*ninit
ninit1=nodes/2-ninit/2
write(*,*)ninit,ninit1,nodes/2
10
do 10 i=1,nodes/2
ccont(i)=1.0
11
do 11 i=ninit1,ninit1+ninit-1
cdel(i)=c0
rstd=diff*dt**beta/dx**alpha
t=0.
npt=1
nts=int(prttime(npts)/dt)+npts
c---- time loop
do 110 j=1,nts
if((t+dt).gt.prttime(npt))then
dtreal=prttime(npt)-t
r=diff*dtreal**beta/dx**alpha
t=t+dtreal
119
else
t=t+dt
r=rstd
endif
cdelold=0.0
ccontold=1.0
tm1=0.0
tm2=0.0
do 100 i=2,nodes-1
ct1=cdel(i)
ct2=ccont(i)
grdown=ccont(i)-ccontold
grup=ccont(i+1)-ccont(i)
sign=1.0
if(grdown.lt.0.0)sign=-1.0
grdown=sign*(abs(grdown))**(alpha-1.0)
sign=1.0
if(grup.lt.0.0)sign=-1.0
grup=sign*(abs(grup))**(alpha-1.0)
temp2=r*(grup-grdown)
sign=1.0
if(temp2.lt.0.0)sign=-1.0
ccont(i)=ccont(i)+sign*(abs(temp2))**(1./beta)
grdown=cdel(i)-cdelold
grup=cdel(i+1)-cdel(i)
sign=1.0
if(grdown.lt.0.0)sign=-1.0
grdown=sign*(abs(grdown))**(alpha-1.0)
sign=1.0
if(grup.lt.0.0)sign=-1.0
grup=sign*(abs(grup))**(alpha-1.0)
temp2=r*(grup-grdown)
sign=1.0
if(temp2.lt.0.0)sign=-1.0
cdel(i)=cdel(i)+sign*(abs(temp2))**(1./beta)
100
200
c
cdelold=ct1
ccontold=ct2
tm1=tm1+dx*cdel(i)
tm2=tm2+dx*ccont(i)
continue
if(abs(t-prttime(npt)).lt.1.e-10)then
npt=npt+1
write(*,*)t,tm1,tm2
scale=(diff*t)**(beta/alpha)
nfirst=max(2,(nodes/2+1-int(v*t/dx)))
do 200 i=nfirst,nodes-1
if(cdel(i).gt.1e-15)write(12,*)dx*(float(i-nodes/2)-0.5)/scale,
1
ccont(i),
2
v*t+dx*(float(i-nodes/2)+0.5),
3
scale*cdel(i)
write(12,*)
if(npt.gt.npts)stop
120
110
endif
continue
stop
end
121
APPENDIX II
STABLE LÉVY MOTION CALCULATIONS
II.1 The Green’s Function Chapman--Kolmogorov Equation for random walks of random
duration
Given the intermediate density
q(x, t) =
t
∞

x′=−∞
 q(x′, τ)p(x − x′; t − τ)dτ + δ(x − 0)δ(t − 0)
(A2.1)
0
And the expression for the propagator:
t
P(x, t) =
 q(x′, t − τ)Φ(τ)dτ
(A2.2)
0
Taking Laplace transforms of the last two convolutions gives
∞

q(x, s) =
q(x′, s)p(x − x′, s) + δ(x − 0)
(A2.3)
x′=−∞
P(x, s) = q(x, s)Φ(s)
(A2.4)
where the Laplace transform is denoted by a change of variable t → s. The last equality can be applied at
any point including x′:
q(x′, s) = P(x′, s)∕Φ(s)
(A2.5)
Substituting (A2.5) into (A2.3)
∞

q(x, s) =
P(x′, s)p(x − x′, s)∕Φ(s) + δ(x − 0)
(A2.6)
x′=−∞
∞

q(x, s)Φ(s) =
P(x′, s)p(x − x′, s) + δ(x − 0)Φ(s)
(A2.7)
x′=−∞
Using (A2.4),
∞
P(x, s) =

P(x′, s)p(x − x′, s) + δ(x − 0)Φ(s)
(A2.8)
x′=−∞
The inverse Laplace transform gives
P(x, t) =
∞

x′=−∞
t
 P(x′, τ)p(x − x′, t − τ) + δ(x − 0)Φ(t)
0
(A2.9)
122
II.2 Exact Solutions for the transformed α--stable densities
We have for any integer jumps of size r
∞
p~ (k, s) =
 e dt  δ(r − t )p(r)e
μ
−st
−ikr
dr
(A2.10)
0
The second integral is a convolution, so the product of the Fourier transforms of p(r) and the delta function
gives:
p~ (k, s) =
e
−ikt μ−st ~
p~ (k, s) = p~ (k)e −ik
e
p(k)dt
(A2.11)
−tμ−st
(A2.12)
dt
The integral can be expanded as a sum of Laplace transforms of power functions, and the expression of the
characteristic function of p(r) is known explicitly since it is α--stable:
~
p(k, s) = e
C|k|α−ik
∞
+ 1)
 (− 1)j!sjΓ(μj
≡ e C|k| −ikC s
μj+1
α
(A2.13)
j=0
where Cs signifies the power series of s which converges for μ > 0. One should note that for μ = 1, the Laplace
transform reduces to
p~ (k, s) = e
(C|k|α−ik)
(A2.14)
s+1
Now the remaining transform of interest follows
∞
p(s) =
∞
 e dt  p(r, t)dr = p(k = 0, s)
−st
0
~
(A2.15)
0
∞
(− 1) jΓ(μj + 1)
= Cs
p(s) = 
j!s μj+1
(A2.16)
j=0
which, for μ = 1, equals
p(s) =
1
s+1
(A2.17)
Now the propagator is given by
~
P(k, s) =
1 − exp(− s μα)
s − sC sexp(|k| α − ik)
(A2.18)
f(s)
1 − C sexp(|k| α − ik)
(A2.19)
which can be written simply
~
P(k, s) =
123
which does not likely have a convenient inverse Laplace transform. The second derivative of (A2.14) with
respect to k is
p~ kk(k, s) = C sCα(α − 1)|k| α−2 + (Cα|k| α−1sign(k) − i) 2e C|k|
α−ik

(A2.20)
Evaluated at k=0, the expression is infinite for α < 2. For α ≥ 2, the expression is simply
p~ kk(k, s) = − C s
(A2.21)
The behavior of the propagator and its variance can be discerned by the case μ = 1, since exact representations
are easily gained.
The series are converging power series of s, so the order of the Laplace transform suggests that the variance
is finite. These complete expressions for the propagator and the variance have not, to my knowledge, been
analytically solved. An open question is whether numerical inversion of the expressions yields more useful
or accurate estimates than those given by simplification of the densities, as will be shown immediately.
II.3 Calculation of power--law transition density Fourier/Laplace transforms
For walks on a lattice with spacing Δ, denote r = jΔ. Using the Zipf (power--law) walk density and assuming
symmetric walks, we have for the Laplace--Fourier transform at the point k=0:
∞
p(s) = p(k = 0, s) = C
e
∞
dt  ei0Δjδ(Δj − t ν)(Δj) −λ
−st
(A2.22)
j=1
0
∞
p(s) = C
e
−st −νλ
t
(A2.23)
dt
Ò
where the cutoff is now the time required for the smallest jump (Ò = Δ1/ν). Making the substitution y = st
gives
∞
e
p(s) = Cs νλ−1
−y −νλ
y
dy
(A2.24)
sÒ
Note the error in Eq. 3.381--3 of Gradshteyn and Ryzhik (1994) which should give:
p(s) = Cs νλ−1Γ(1 − νλ, sÒ)
(A2.25)
The incomplete gamma function has a series representation (Eq. 8.354--2, Gradshteyn and Ryzhik [1994])
of
Γ(a, b) = Γ(a) −
∞
 (−n!(a1)+nba+n
n)
(A2.26)
n=0
so the waiting time transform has a series representation of
(1−νλ)
p(s)
= s νλ−1Γ(1 − νλ) − Ò
−
C
(1 − νλ)
∞
sn
 (−n!(n1)n+Ò(n+1−νλ)
1 − νλ)
n=1
(A2.27)
124
To satisfy the requirement of being a density, p(s=0) = 1. Therefore C = --(1--νλ)/Ò(1-- νλ) and νλ>1.
∞

s νλ−1Γ(2 − νλ)
(− 1) n(1 − νλ)Ò ns n
+
p(s) = 1 −
n!(n + 1 − νλ)
Ò 1−νλ
n=1
(A2.28)
We will be interested in 2 ≤ λ ≤ 3 (between ballistic and Brownian motion); therefore, ν will be no smaller
than 1/3. For small s (large time), the dominant term in the summation will depend on the magnitude of νλ.
Define the constant C Ò = Γ(2 − νλ)∕Ò 1−νλ. To first order,
p(s) ≈

1 − C Òs νλ−1
1 − τs
1 < νλ < 2
2 < νλ
(A2.29)
in agreement with Blumen et al. (1989). In the second case, the constant τ is simply the mean waiting time
per step, which is finite only for 2 < νλ.
As previously stated, two different approximations are needed for p(k,s): one for very small k and one valid
over a large range of k. In the first instance, a somewhat simpler integral to manipulate uses the combined
distributions:
~
p(k, s) − p(s) = C
∞

(e−ikr − 1)
r=−∞
e
δ(r − tν)r −λdt
−st
(A2.30)
Since the jump probability is symmetric (p(r) = p(--r)) and identity 2cos(x) = eix -- e- ix, the exponential can
be simplified to:
∞
p(k, s) − p(s) = C  (cos(kr) − 1)e −sr
~
1∕ν
r −λ
(A2.31)
r=1
For very small values of k, the cosine can be expanded as 1--(rk)2.
p~ (k, s) − p(s) = − k 2C
∞
 e−sr
1∕ν
r 2−λ
(A2.32)
r=1
The summation converges for ν(λ--2)>1 and the right hand side is reasonably approximated by --k2(C1--sC2)
where the constants are zeta functions: C1 = ζ(ν(λ--2)) and C2 = ζ(ν(λ--2)--1). Blumen et al. (1989) ignore
the dependance of k2 on s, which causes different short--time behavior of the variance (Figure A2.1). For
simplicity, the long--time behavior can ignore the mixed sk2 term. Within the range 0<ν(λ--2)<1, the summation diverges. The summation can be approximated by an integral after substituting x = (jΔ)1/ν = r1/ν:
∞
p~ (k, s) − p(s) = − k 2C
e
−sx −ν(λ−2)
x
dx
Δ 1∕ν
which leads to a power series similar to p(s) above:
p~ (k, s) − p(s) = − k 2Cνs ν(λ−2)−1Γ(1 − ν(λ − 2), sΔ 1∕ν)
(A2.33)
125
106
105
104
r 2 
103
100
10
ν(λ--2) = 1.7
1
0.1
1
10
100
103
time
Figure A2.1 Particle travel distance variance when mixed sk2 terms are
included in the small--k approximation of the Laplace--Fourier transformed
conditional Lévy walk probability p(r,t). The dashed line indicates results from
simplified form used by Blumen et al. (1989).
∞


(− sΔ 1∕ν) n
ν(λ−2)−1

= k Cνs
Γ(1 − ν(λ − 2)) −
n!(n + 1 − ν(λ − 2))


0
2
(A2.34)
For small s (or small Δ), the first term will be large and dominate the series, which diverges for s = 0. The
constants can be consolidated leaving to first order:
ν(λ − 2) > 1
C1

2
−
≈
−
⋅
p(k, s) p(s)
k  ν(λ−2)−1
ν(λ − 2) < 1
C3s
~
(A2.35)
Recalling eq. (4.48), the space--time probability accurate as k→0 used in the calculation of variance is approximated by 4 distinct cases. The first two cases listed below pertain to walks with a finite mean residence time
(2 < νλ), while the second two are for infinite mean residence time (1 < νλ < 2):
126
1 < ν(λ − 2)
 1 − τs − C1k2
 1 − τs − C3k2sν(λ−2)−1
ν(λ − 2) < 1
~
p(k, s) ≈ 
1 < ν(λ − 2)
1 − C Òs νλ−1 − C 1k 2

νλ−1
ν(λ−2)−1
2
ν(λ − 2) < 1
1 − CÒs − C3k s
(A2.36)
For values of λ < 4, the third density cannot exist, since ν(λ--2) > 1 and νλ < 2 are mutually exclusive (Figure
4.5).
In order to calculate the propagator, the transform p(k,s) needs to be accurate over a large range of k. One
can assume for simplicity that the transformed density p(k,s) is dominated by two terms, one for small k and
another for small s. Several methods can be employed to evaluate the cosine Fourier transform of the walk
density, the easiest following the path already taken twice above, which relies on an integral of mixed exponential and algebraic form. We will denoting a walk of j positions on a lattice with spacing Δ by r (i.e. r =
jΔ). The smallest allowable jump is therefore Δ, since the power law diverges at zero. For symmetric walks
and s=0, the sum is approximated by an integral:
p~ (k, s = 0) = C
∞
∞
 e−ikjΔ  e0tδ(t − (jΔ)1∕ν)(jΔ)–λdt ≈ C  e−ikrr–λdr
∞
j=1
(A2.37)
Δ
0
For unknown reasons, Klafter et al. (1987) and Blumen et al. (1989) have the quantity (--λ--1+1/ν) as the exponent on r, which carries throughout many of their subsequent calculations. The transformed densities presented by those authors would yield erroneous propagators. The substitution y = ikr in the previous equation
gives
∞
p~ (k, s = 0) = C|ik| λ−1
e
−y
(A2.38)
|y| –λdy
ikΔ
The integral is expanded in the series:

p~ (k, s = 0) = C|ik|λ−1Γ(1 − λ, ikΔ) = C|ik| λ−1 Γ(1 − λ) −
 (−n!(11)−(ikΔ)

λ + n)
∞
n
1−λ+n
(A2.39)
n=0
∞
(− 1) n(ikΔ)n
 n!(1
− λ + n)
(A2.40)
1) n(1 − λ)(ikΔ)n
 (−n!(1
− λ + n)
(A2.41)
p~ (k, s = 0)
= |ik| λ−1Γ(1 − λ) −
C
n=0
Evaluation of the constant at k = 0 gives
p~ (k, s = 0) = |ik| λ−1Γ(2 − λ) −
∞
n=0
which has separable real and imaginary components. Using the identity ix = eiÕx/2 = cos(Õx/2) + i⋅sin(Õx/2)
and realizing that a symmetric density has a purely real Fourier transform, hence the imaginary component
can be disregarded, leaves to first order (depending on whether λ > 3):
127
(1 − λ)k 2Δ 2
p~ (k, s = 0) = 1 − |k| λ−1Γ(2 − λ) cos(Õ (λ − 1)) −
2
2(3 − λ)
(A2.42)
Since there are two classes of mean waiting time (finite and infinite) and two classes of excursion length (also
finite and infinite), there are four combination of these. We will not continue to develop the density for finite
waiting time and excursion length variance, since this leads to the well--known Brownian motion. The last
term in the previous equation disappears when Δ is small compared to k. This expression can be shortened
and combined with the approximation for p(k=0,s) to form the three densities of interest:
 1 − τs − C2|k|λ−1

νλ−1 −
C 2|k| λ−1
p~ (k, s) ≈ 1 − C Òs
 1 − C sνλ−1 − C1k2
Ò

2 < νλ
(A2.43)
1 < νλ < 2
1 < νλ < 2
where the first two densities are for 2 < λ < 3 and the third is for λ ≤ 3.
Blumen et al. (1989) compared similar expressions with a more complete numerical calculation of the full
density expression (A2.37) over a large range of the parameters k and s. The simplified formulas provide
reasonably good approximations except in the crossover region in which both terms are comparable in magnitude. The simplified expressions should best represent the behavior of a random walker at relatively large
distance with respect to time and vice--versa. The utility of the simplified transition density transform will
become apparent in the sequel, when analytic expressions are inverted from the Fourier--Laplace space. An
open question is whether better representations of the density and numerical inversion would yield appreciably better results.
128
APPENDIX III
VELOCITY AUTOCOVARIANCE OF LÉVY WALKS
III.1 Velocity Autocovariance for Lévy Walks with Lower Cutoff
Let Ri be a random variable representing the magnitude of a Lévy walk taking values from (0,1). The probability density of Ri, denoted fRi(r), is approximated by the Pareto density for r larger than some lower cutoff
(Ò), since the density fRi(r) = Cr- 1-- α diverges at r=0. The density is generally set to a constant below Ò. A
logical choice is CÒ- 1-- α (Figure A3.1). To be a density, the integral from r=0 to infinity must be equal to
p(r)
p(r) = r- 1-- α
Ò
r
Figure A3.1 Pareto distribution with lower cutoff.
unity, so for the shaded are in Figure A4.1, the density is:

α
Ò −1−α
f Ri(r) = αÒ ⋅
α+1
r −1−α
0<r<Ò
Ò<r
(A3.2)
We now define a random variable Rx, which is the value of the walk length at location x within a trajectory
(sequence) of iid walks of length Ri. Since the longer walks occupy more of the total distance along the sequence, the probability of Rx is weighted by the individual walk lengths, or fRx = C¡r¡fRi(r), where the constant
assures that a density results. The constant evaluates to the expected value of Ri, so the probability density
associated with the value of Rx is
f Rx(r) = C ⋅
rÒr
−1−α
−α
0<r<Ò
Ò<r
(A3.3)
129
To calculate the spatial autocovariance of velocity, we are interested in the probability that the particle is travelling at some speed at a point along the trajectory. The only way that this can be calculated is if the velocity
is a function of the walk size at that point in space (Rx). Using the power function introduced by Klafter et
al. (1987) we have now a random variable V = Rx1-- 1/ν. Using the transformation y = r1-- 1/ν and consolidating
constants into C, the probability density of V is calculated from fRx(r) (c.f., Ross [1988]) as
f V(y) = f Rx(r) dr
dy
Ò−1−αy
f V(a) = C ⋅ 
 y
ν+1
ν−1
(A3.4)
0 < y < Ò 1−1∕ν
1−να
ν−1
(A3.5)
Ò 1−1∕ν < y
Now the definition of autocovariance is invoked:
∞
R vv(ξ) ≡ V xVx+ξ =
∞
  yb f
(A3.6)
V x,V x+xi(y, b)dydb
–∞ –∞
where fVx,Vx+ξ(y,b) is the joint density of the velocity at points x and x+ξ in the trajectory. The joint density
is related to the marginal density by conditioning:
f Vx,V x+ξ(y, b) = f Vx(y)f V x+ξ(b|V x = y)
(A3.7)
If two points are within the same walk, they have the same velocity. The conditional probability is the probability that the two points are in the same walk times the Dirac delta function δ(y--b). The probability that
the two points are within the same walk is merely (1--ξ/r). If the two points in the trajectory are in different
walks, the values are independent and the marginal densities are multiplied by the probability that the two
walks are in different walks (ξ/r). Finally, if the lag is larger than a walk of magnitude r, then the velocities
at the two endpoints are independent:
f Vx,Vx+ξ(y, b) =
fV (y)(1 − ξ∕r)δ(y − b) + fV (y)fV

f V (y)f V (b)

x
x
x
(b)ξ∕r
x+ξ
r≥ξ
(A3.8)
r<ξ
x+ξ
Now the densities can be evaluated in terms of one variable, since velocity is a function of the individual walk
distance (r):
fV (y)(1 − ξ∕y )δ(y − b) + fV (y)fV
ξ(y, b) = 
f V (y)f V (b)

x
f Vx,V
ν
ν−1
x
x+
x
x+ξ
ν
(b)ξ∕y ν−1
x+ξ
y ≥ ξ 1−1∕ν
(A3.9)
y < ξ 1−1∕ν
The delta function that is used when the velocity is constant within a single walk reduces the autocovariance
double integral for the first joint density term to a single integral. The condition of independence for each
walk reduces the integral for the other two terms to:
130
∞
R vv(ξ) =
bf
2
∞
V x(b)(1 − ξ∕b
ν
ν−1
)db + V
ξ1−1∕ν
 yf
V x(y)(ξ∕y
ν
ν−1
)dy + V
2
(A3.10)
ξ 1−1∕ν
For walks that are equally likely in the positive and negative directions, the last two terms are zero, leaving
for the spatial velocity autocovariance
∞
bf
R vv(ξ) =
2
V x(b)(1 − ξ∕b
ν
ν−1
(A3.11)
)db
ξ1−1∕ν
One can return this integral to variables that depend on distance, not velocity, by making the transformation
r=bν/(ν- 1) and following the rules for density transforms, resulting in:
∞
R vv(ξ) =
r
2−2∕ν
f Rx(r)(1 − ξ∕r)dr
(A3.12)
ξ
which can be couched in a more general form using the functional relationship for velocity V=g(R):
∞
R vv(ξ) =
 g (r) ⋅ r ⋅ f (r)(1 − ξ∕r)dr
2
(A3.13)
Ri
ξ
Since the density of the walks lengths is Paretian or α--stable, the integral will diverge for certain velocity
functions. In particular, divergence is assured when α < 3--2/ν, or when ν > 2/(3--α). We will impose a finite
largest jump size (M) in this case obtain converging solutions. This largest jump size is akin to a “correlation
length” since it describes the largest continuous excursion of a particle.
Different functions are used for the marginal velocity density when the lag (ξ) is greater or less than the walk
cutoff length (Ò). This results in two different expressions for the autocovariance, depending on the relative
size of the lag versus the cutoff. For ξ<Ò the integral (A3.12) can be split according to the different marginal
density functions for r greater or less than Ò:
Ò
R VV(ξ) = CÒ −1−α
r
M
3−2∕ν(1 − ξ∕r)dr
+C
ξ
r
2−2∕ν−α(1
− ξ∕r)db
(A3.14)
Ò
Evaluation of the integrals is straightforward and yields

R VV(ξ) = C
(1 + α)ξÒ 2−2∕ν−α
(− 1 − α)Ò 3−2∕ν−α
+
+
(4 − 2∕ν)(3 − 2∕ν − α) (3 − 2∕ν)(2 − 2∕ν − α)
3−2∕ν−α
ξ 1−2∕νÒ −1−α
ξM 2−2∕ν−α
+ M
−
(4 − 2∕ν)(3 − 2∕ν) 3 − 2∕ν − α 2 − 2∕ν − α

(A3.15)
131
With a velocity variance of

3−2∕ν−α
(− 1 − α)Ò 3−2∕ν−α
+ M
(4 − 2∕ν)(3 − 2∕ν − α) 3 − 2∕ν − α
VAR(V) = R VV(0) = C

(A3.16)
The autocorrelation function and semivariogram are:
à V(ξ) = 1 +
ξ 4−2∕νÒ −1−a
ξM 2−2∕ν−α
(1+α)ξÒ 2−2∕ν−α
+ (4−2∕ν)(3−2∕ν) − 2−2∕ν−α
(3−2∕ν)(2−2∕ν−α)
(A3.17)
(−1−α)Ò 3−2∕ν−α
M 3−2∕ν−α
+ 3−2∕ν−α
(4−2∕ν)(3−2∕ν−α)
γ V(ξ) =
ξ 4−2∕νÒ −1−a
ξM 2−2∕ν−α
(−1−α)ξÒ 2−2∕ν−α
−
+
(3−2∕ν)(2−2∕ν−α)
(4−2∕ν)(3−2∕ν)
2−2∕ν−α
(A3.18)
(−1−α)Ò 3−2∕ν−α
M 3−2∕ν−α
+ 3−2∕ν−α
(4−2∕ν)(3−2∕ν−α)
Which, despite the ugliness is very nearly linear with respect to ξ, if one neglects the small second term in
the numerator. The linear autocorrelation is expected for lags smaller than the cutoff, since the velocity is
a constant value. The correction becomes important as the lag and cutoff sizes are comparable.
For the case ξ>Ò one has
M
R VV(ξ) = C
r
2−2∕ν−α(1 − ξ∕r)dr
(A3.19)
ξ

R VV(ξ) = C
ξ 3−2∕ν−α
M 3−2∕ν−α − ξM 2−2∕ν−α +
3 − 2∕ν − α 2 − 2∕ν − α (3 − 2∕ν − α)(2 − 2∕ν − α)
à V(ξ) =

ξM2−2∕ν−α
ξ 3−2∕ν−α
M3−2∕ν−α
−
+
3−2∕ν−α
2−2∕ν−α
(3−2∕ν−α)(2−2∕ν−α)
(A3.20)
(A3.21)
(−1−a)Ò 3−2∕ν−α
M 3−2∕ν−α
+ 3−2∕ν−α
(4−2∕ν)(3−2∕ν−α)
Several simplifications present themselves immediately.
III.2 Lévy Walks with Converging Autocovariance
If the walks are specified with an velocity function such that the longest walks are sufficiently slowed, i.e.
ν > 2/(3--α) then the integrals converge at infinity. Each term above containing a β is zero, leaving for ξ<Ò
ξ
Ò
(1 + α)ξÒ
1 − α)Ò
(4 −(−2∕ν)(3

+
+
− 2∕ν − α) (3 − 2∕ν)(2 − 2∕ν − α) (4 − 2∕ν)(3 − 2∕ν)
R VV(ξ) = C
3−2∕ν−α
VAR(V) =
à V(ξ) = 1 +
2−2∕ν−α
C(− 1 − α)Ò 3−2∕ν−α
(4 − 2∕ν)(3 − 2∕ν − α)
ξ 4−2∕νÒ −1−α
(1+α)ξÒ 2−2∕ν−α
+
(3−2∕ν)(2−2∕ν−α)
(4−2∕ν)(3−2∕ν)
(−1−a)Ò 3−2∕ν−α
(4−2∕ν)(3−2∕ν−α)
1−2∕ν
−1−α
(A3.22)
(A3.23)
(A3.24)
132
(4 − 2∕ν)(3 − 2∕ν − α) ξ
(3 − 2∕ν)(2 − 2∕ν − α) Ò
γ V(ξ) ≈
(A3.25)
And more importantly, for the case of larger lags ξ>Ò one has
R VV(ξ) = C
ξ 3−2∕ν−α
(3 − 2∕ν − α)(2 − 2∕ν − α)

4 − 2∕ν
ξ
à V(ξ) =
(− 1 − α)(2 − 2∕ν − α) Ò
(A3.26)
3−2∕ν−α
(A3.27)

4 − 2∕ν
ξ
γ V(ξ) = 1 −
(− 1 − α)(2 − 2∕ν − α) Ò
3−2∕ν−α
(A3.28)
III.3 Autocovariance with Velocity proportional to Lévy Walk Size
The spatial velocity autocovariance function can be simplified greatly by assuming that the velocity is roughly proportional to the individual walk size within a trajectory. Then the quantity 2/ν becomes very small and
the velocity probability density is simply fV(y) = C fRx(r). The autocovariance function requires an upper
cutoff (M) for convergence and for ξ<Ò reduces to:
R VV(ξ) = C
1 − α)Ò
(− 4(3
− α)
3−α
+
(1 + α)ξÒ 2−α ξÒ −1−α M 3−α ξM 2−α
+
+
−
(3)(2 − α)
12
3−α
2−α

(A3.29)
With a velocity variance of
1 − α)Ò
(−(4(3
− α)

(A3.30)
ξ 4Ò −1−a
ξM 2−α
(1+a)ξÒ 2−α
+ 12 − 2−α
3(2−α)
1+
3−α
(−1−a)Ò 3−α
+M
4(3−α)
3−α
(A3.31)
3−α
VAR(V) = C
3−α
+M
3−α
The autocorrelation function and semivariogram are:
à V(ξ) =

(A3.32)
2−α(1 − ξ∕r)dr
(A3.33)
ξ
γ V(ξ) ≈ 3 − α
2−α M
For the case ξ>Ò one has
M
R VV(ξ) = C
r
ξ

3−α
ξM 2−α
ξ 3−α
−
+
R VV(ξ) = C M
3−α
2−α
(3 − α)(2 − α)
à V(ξ) =
M3−α
3−α
ξM 2−α
ξ3−α
+ (3−α)(2−α)
2−α
3−α
(−1−a)Ò 3−α
+M
4(3−α)
3−α
−

(A3.34)
(A3.35)
133


ξ
ξ
− 1
γ V(ξ) ≈ 3 − α
2−α M
2−α M
3−α
(A3.36)
III.4 Full α--stable density.
This approach uses an exact series expansion of fRx(r) for 1≤α≤2:
1
f Rx(r) = Õ
∞
k
+ 1 + 1r 2k
 (2k(−+1)1)!
Γ2k α
(A3.37)
k=0
For simplicity, group all of the expressions that do not involve r into f(k):
f Rx(r) =
∞
 f(k)r2k
(A3.38)
k=0
We derive the autocovariance for the simplest case of velocity proportional to jump size (2/μ = 0). The spatial
autocovariance of V becomes
∞
R VV
(ξ) = C  (r − r ξ)  f(k)r
3
∞
2
2k
dr
(A3.39)
k=0
ξ
The summation converges, so the integral can be moved inside:
R VV(ξ) = C
∞
 f(k)  (r3 − r2ξ)r2k dr
∞
k=0
(A3.40)
ξ
Once again the integral evaluates to infinity, but we are interested in the ratio of this expression to the total
variance, so an upper jump size (β) will be imposed. Evaluation of the integral gives:
R VV(ξ) = C
3+2k
ξ4+2k
 f(k)4β+4+2k2k − ξβ

+
3 + 2k (3 + 2k)(4 + 2k)
∞
(A3.41)
k=0
The total variance of velocity with an upper limited jump size is given by the expression
M
VAR(V) = C lim
M→∞

∞
r3
 f(k)r2k dr
(A3.42)
k=0
0
∞
4+2k

f(k) M
4 + 2k
M→∞
VAR(V) = C lim
(A3.43)
k=0
The autocorrelation function and semivariogram for 0≤ξ≤M are therefore:
∞
ξ

 f(k)− ξ M3+2k + (3+2k)(4+2k)
Ã(ξ) = 1 −
4+2k
3+2k
k=0
∞
(A3.44)
 f(k) M4+2k
k=0
4+2k
134
∞
ξ

 f(k)− ξ M3+2k + (3+2k)(4+2k)
γ(ξ) =
4+2k
3+2k
k=0
∞
(A3.45)
 f(k) 4+2k
k=0
M 4+2k
135
APPENDIX IV
FRACTIONAL DERIVATIVES AND THEIR PROPERTIES
A number of excellent texts describe the long history and analytic properties of fractional derivatives and
fractional differential equations (Miller and Ross [1993]; Samko et al. [1993]). Analysis of fractional derivatives is finding exposure in more mainstream texts as well (Debnath [1995]) Perhaps the most natural development is given by Anton Grunwald and is concisely summarized by Lavoie et al. (1976). Grunwald started
with the finite--difference formulations of any order:
F(x) − F(x − h)
,
h
F(x) − 2F(x − h) + F(x − 2h)
,
h2
F(x) − 3F(x − h) + 3F(x − 2h) − F(x − 3h)
h3
(A4.1)
These quotients give the familiar first, second, and third differences of F(x). The formulas in (A4.1) can be
generalized to differences of arbitrary order. The order can be integer, real, or complex; here we use a rational
symbol q:
Δ qhF(x)
=
 (− 1)kqkF(x − (q − k)h)
∞
(A4.2)
k=0
where h
Δ qh
=
(x--a)/n
=
the qth --order difference operator with dependence on the
discretization h,

q
(− 1) k k
is the binomial coefficient.
The relationship between the fractional derivative and the fractional difference is given by:
Δ qF(x)
hq
h→0
D qF(x) = lim
(A4.3)
Another definition is based on the idea of “partial integration.” In short, a fractional derivative is an integer
derivative of a partial integral that extends the traditional Cauchy formula for iterated integrals. Starting with
the fundamental theorem of calculus (n=1):
x
x
 f(t)dt = n!1 dxd (x − t)
a
Integrating both sides gives
a
n−1f(t)dt
(A4.4)
136
x1
x
x
 dx  f(x )dx = n!1 (x − t)
1
0
0
a
a
n−1
(A4.5)
f(t)dt
a
Integrating n times gives Cauchy’s formula:
x n−1
x
 dx  dx
n−1
x1
n−2
a
a

x
 f(x )dx = (n −1 1)! (x − t)
0
0
a
n−1
f(t)dt
(A4.6)
a
which can be identified as an operator:
x
I nf(x) = 1
n!
(x − t)
(A4.7)
n−1f(t)dt
a
The integer--order (n) in Cauchy’s formula can be generalized to rational--order integration. This rational--order integration is commonly called the Riemann--Liouville integral for values of q greater than zero:
x
I qa+F(x) = 1
Γ(q)
(x − ζ)
q−1
(A4.8)
F(ζ)dζ
a
where Γ(q) is the Gamma function, which extends the factorial function to real numbers:
∞
Γ(z) =
x
z−1e −xdx
(A4.9)
0
The composition rule and the Riemann--Liouville integral give an equivalence between the nearest integer
and rational order derivatives (Oldham and Spanier [1974]; Miller and Ross [1993]; Samko, et al. [1993]).
If n is the smallest integer larger than q, we have
n
d qF(x)
≡ D qa+F(x) = d n I n−q
F(x)
q
d(x − a)
dx a+
=
 1 x

n−q−1
F(ζ)dζ
Γ(n − q) (x − ζ)


a
dn
dx n

(A4.10)
The fractional derivative can be thought of as partially integrating back from the next highest derivative. For
values of r between 0 and 1, 1/Γ(r) is approximately equal to r (Figure A4.1). The lower limit of integration
is commonly set to zero or minus infinity. When infinite bounds are used, the limit is eliminated from the
subscript and only the direction of integration (+ or --) is noted:
x
d q f(x)
dx q
= D q+f(x) =
dn
1
Γ(n − q) dx n
 (x − ζ)
−∞
n−q−1
f(ζ)dζ
(A4.11)
137
10.0
5.0
Γ(x)
0.00.0
1.0
1/Γ(x)
3.0
2.0
4.0
x
Figure A4.1 Plots of the functions Γ(x) and 1/Γ(x) for 0 < x < 4. Note that n! = Γ(n+1).
Since this is a convolution, one could integrate from the other side of (x), defining another related type of
fractional differentiation (Samko et al. [1993]) denoted
∞
dq
d(− x) q
f(x) = D q−f(x) =
(− 1) n
dn
Γ(n − q) dx n
 (ζ − x)
n−q−1
f(ζ)dζ
(A4.12)
x
Both of these formulas reduce to the single shorthand representation (note the change of (+) to (--) and vice-versa inside the integral):
∞
D qf(x) =
1) n
(
dn
Γ(n − q) dx n
t
n−q−1
f(x  t)dt
(A4.13)
0
These fractional differo--integral operators (FDOs) have many of the properties that ordinary derivatives
have. They are linear, i.e. for β % 9:
D q(βf(x) + g(x)) = βD qf(x) + D qg(x)
(A4.14)
However, the fractional derivative of a constant is no longer zero. Specifically:
D qa+(1) =
(x − a) q
Γ(1 − q)
(A4.10)
Additionally, FDOs do not necessarily scale functions as do the integer derivatives. For example, consider
a scale transform to X of x with respect to the lower limit a: βX = βx -- βa + a. Then:
138
D qa+f(βX) =
d qf(βX)
d q(βX)
q
=
β
d(x − a) q
d(βX − a) q
(A4.11)
The scaling is more easily seen when a = 0:
d qf(βx)
d q(βx)
= βq
q
d(x)
d(βx) q
(A4.12)
Like integer derivatives, a fractional derivative of a power function of x reduces the exponent by the order
of the differentiation:
D qa(x − c) u =
Γ(u + 1)
(x − c) u−q
Γ(q − u + 1)
(A4.15)
Γ(u + 1)
x u−q
Γ(q − u + 1)
(A4.16)
or simply
D qx u =
We use the fractional derivative of the Dirac delta function δ(x--a) with b ≤ c ≤ d, defined by
d
 δ(x − c)f(x)dx = f(c)
(A4.17)
b
A fractional derivative with order 0 < q is directly obtained:
x
D q+δ(x
1
− c) =
Γ(n −
dn
q) dx n
 δ(t − c)(x − t)
n−q−1
−∞
(x 0− c)−q−1x < c
dt = 
x≥c
 Γ(− q)
(A4.18)
and
∞
(− 1) n d n
D q−δ(x − c) =
Γ(n − q) dx n
 δ(t − c)(x − t)
n−q−1
x
(c − x)−q−1 x ≤ c
dt =  Γ(− q)
x>c
 0
(A4.19)
We will make use of the formula for L2 inner products, (f(x),g(x)) = sf(x)g(x)dx. When one of the functions
is a fractional derivative, Samko et al. (1993) and Debnath (1995) show that the (+) and (--) direction FDOs
are adjoint operators:
∞
(g(x), D q+f(x)) =
x

1
Γ(1 − q)
g(x) d
dx
x=−∞
∞
=
1
Γ(1 − q)

x=−∞
 (x − t) f(t)dt dx
q
t=−∞
x
d
dx

t=−∞
g(x)(x − t) qf(t)dtdx
139
∞
=
∞
 dxd  g(x)(x − t) f(t)dx dt
1
Γ(1 − q)
q
t=−∞
x=t
∞
=
1
Γ(1 − q)

∞
 g(x)(x − t) dx
f(t)dt (− 1) d
dt
t=−∞
q
x=t
= (D q−g(x), f(x))
(A4.20)
Finally, by definition, every FDO is a convolution in Laplace space, so the Laplace transforms of FDOs lead
to natural results. This is perhaps the most useful property of the FDOs with respect to solving partial differential equations. The convolution (*) of functions F and G is given by:
x
F*G=
 G(x − ξ)F(x)d(ξ)
(A4.13)
−∞
The Laplace transform (L) of a convolution is the product of the Laplace transforms of the two functions,
i.e.
L(F * G) = L(F)L(G)
(A4.14)
Equation (A4.13) is based on the convolution of the functions F and xq-- 1, leading to a Laplace transform with
parameter s of (c.f., Oldham and Spanier [1974]):
n−1
L(D q0+f(x))
q
= s L(f) −
 skDq−1−k
f(x)
0+
(A4.21)
k=0
If the function and its derivatives are zero at the origin, this reduces to:
L(D q0+f(x)) = s qL(f(s))
(A4.22)
Using Mellin transforms as an intermediate step, Debnath (1995) calculates the Fourier transform of a fractional integral. Samko et al. (1993) also give a generalization of the operation of Fourier transforms on fractional derivatives. Note the change of (+) to (--) and vice--versa:
F(D qg(x)) = ( ik) qg~ (k)
(A4.23)
where
Õ
( ik) q = |k| qe iq2sign(k)
(A4.24)
140
APPENDIX V
FINITE DIFFERENCE APPROXIMATION
OF THE FRACTIONAL DERIVATIVE
Arguments for the definitive choice of a two--point finite difference approximation of the fractional derivative
are beyond the scope of this study. Several logical choices present themselves immediately, but some will
not deliver the salient features of the derivatives they approximate, especially the scaling property. This is
principally due to the fact that the fractional derivative is a convolution that relies on past values or appears
to have spatial “memory.” By definition, the finite difference is a local operator. In order to accurately calculate the fractional space derivative, an implicit scheme that can solve the linear integral operator (such as Galerkin finite elements) will likely be needed. However, the plots in Chapter 7 indicate that a simple difference
operator performs reasonably well. Several operators that seem likely candidates do not fare so well. For
example, the “local” Taylor series given by Kolwankar and Gangal (1996):
f(x + Δx) = f(x) + D qf(x)
Δx q + R
Γ(q + 1)
(A5.1)
where R is a remainder containing higher fractional derivatives, and the authors have defined the local derivative Dq by making the lower limit of the Riemann fractional integral (Chapter 5) approach the upper:
d q(f(x) − f(y))
d(x − y) q
y→x
(A5.2)
D qf(x) = lim
x
d
1
y→x Γ(1 − q) dx
D qf(x) = lim
 f(t)(x −− f(y)
dt
t)
q
(A5.3)
y
Rearranging (A5.1) gives
D qf(x) ≈ Γ(q + 1)
f(x + Δx) − f(x)
Δx q
(A5.4)
Denoting f(x) = Ci and setting the order of differentiation q to (α--1), this expression clearly will not give the
scaling behavior desired in the FADE, since the approximation of the entire PDE becomes:
≈
C ti − C t+Δt
i
Γ(q + 1)DΔt
(C i+1 − C i) − (Ci − C i−1)
Δx α
(A5.5)
which is a second--order diffusion equation approximation with a larger “effective” dispersion coefficient.
A plume modeled by this equation will have a greater rate of spreading then its second--order counterpart,
but the plume will be invariant after scaling by t1/2.
Another candidate arises from the Marchaud definition of fractional derivation (c.f., Miller and Ross [1993])
using a lower bound of x--Δx:
141
x
d
1
Γ(1 − q) dx
x

q
f(x)
f(t)
+
dt =
q
q
(x − t)
Γ(1 − q)Δx
Γ(1 − q)
x−Δx
 (xf(x)−−t) f(t) dt
1+q
(A5.6)
x−Δx
The mean value theorem could be used to approximate the last integral at x--Δx/2:
x
f(x) − f(x − zΔx∕2)
− f(x − zΔx))
 (xf(x)−−t) f(t) dt = Δx(f(x)(zΔx∕2)
=C
Δx
1+α
α
1+α
(A5.7)
x−Δx
giving, for the whole derivative:
f(x)
α f(x) − f(x − zΔx) = C 1f(x) − C 2f(x − zΔx)
+
Δx α z 1+α
Δx α
Δx α
Γ(1 − α)D α−1 =
(A5.8)
The value of the function at x--zΔx must be interpolated from the known locations x and x--Δx. If a linear
interpolation using f(x) and f(x--Δx) is employed, one is left with another version of (A5.4) that is simply
variable upstream weighting. This obviously results in Gaussian spreading and the well--known scaling by
t1/2.
One might also try the series definitions of the Riemann--Liouville operator given originally by Grunwald
(see Miller and Ross [1993]). This definition is exactly equivalent to the integral definition using zero as
the lower bound. Since we ultimately seek a local approximation, we will use this definition to approximate
a lower bound of --∞ (indicating infinite spatial dependence):
D q0f(x)
1 Δx −q
= lim
Γ(−
q)
n→∞
n−1
− q)
 Γ(k
f(x − kΔx)
Γ(k + 1)
(A5.9)
k=0
So that the fractional derivative at x+Δx is approximately
1 Δx −q
Γ(−
q)
n→∞
D q0f(x + Δx) = lim
n−1
− q)
 Γ(k
f(x − (k − 1)Δx)
Γ(k + 1)
(A5.10)
k=0
and the fractional derivative at x--Δx is
1 Δx −q
n→∞ Γ(− q)
D q0f(x − Δx) = lim
n−1
− q)
 Γ(k
f(x − (k + 1)Δx)
Γ(k + 1)
(A5.11)
k=0
Taking the first difference of these gives an approximation of the q+1th derivative approaching the point x
from the negative side:
D q+1
f(x) ≈
+
1
2Γ(− q)Δx q+1

Γ(− q)f(x + Δx) + Γ(1 − q)f(x) +
 Γ(kΓ(k++1 −2) q) − Γ(k −Γ(k)1 − q)f(x − kΔx)
n−1
k=1

(A5.12)
142
For a symmetric approximation, we need the derivative approaching from the positive side, which will be
approximated by
D q+1
− f(x) ≈
1
2Γ(− q)Δx q+1

Γ(− q)f(x − Δx) + Γ(1 − q)f(x) +
 Γ(kΓ(k++1 −2) q) − Γ(k −Γ(k)1 − q)f(x + kΔx)
n−1
(A5.13)

k=1
So that the total symmetric finite difference approximation is
D q+1
f(x) ≈
+

Γ(2 − q)
Γ(2 − q)
1
f(x − Δx) + 2Γ(1 − q)f(x) +
f(x + Δx) +
2
2
2Γ(− q)Δx q+1

Γ(kΓ(k++1 −2) q) − Γ(k −Γ(k)1 − q)(f(x + kΔx) + f(x − kΔx))
n−1
(A5.14)
k=2
One should note that the coefficients on the terms in the summation stay large for several terms. For example,
when k=3, the coefficient is roughly the same size as the coefficients on f(xΔx). This suggests that an accurate finite difference scheme based on this definition will require 7 or more nodes in each dimension, rendering the solution unmanageable. An open question is whether this approximation converges to the analytic
solutions.
Returning to the simpler, local approximation (A5.4) we see that the value of the function at x--zΔx must be
interpolated from the known locations x and x--Δx. If a linear interpolation using f(x) and f(x--Δx) is
employed, one is left with another version of (A5.4) that is simply variable upstream weighting. Choosing
a different interpolation scheme (perhaps with some prior knowledge of the function) will presumably give
a better estimation and the scaling property. Since the finite difference “localizes” the transport history of
the contamination, one should opt for an interpolation that can recover the global character of a plume. In
the case of α--stable transport, the flux can be weighted according to the gradient. Low gradients, implying
the plume edges, are given higher flux (relative to Fickian flux) by a simple power function (Figure A5.1):
J=D
(C(x + Δx) − C(x)) α−1
(Δx) α−1
(A5.15)
This immediately suggests a heuristic, ad hoc approximation of the qth fractional derivative as:
D qf(x) ≈ C
(f(x + Δx) − f(x)) q
(Δx) q
(A5.16)
We can now take a cursory look at the convergence of the numerical operators to their analytic counterparts.
In an analytic sense, one can examine how the operator acts upon a known function, say integer powers of
x (f(x) = xp):
143
10
1
.1
FLUX
.01
α = 1.3
α = 1.5
α = 1.7
α=2
.001
0.001
0.01
0.1
GRADIENT
1
10
Figure A5.1 Flux as a power function of gradient. This is the basis for the numerical approximation
implemented in Chapter 7.
(f(x + Δx)
Δx q
Δx→0
D q ≈ lim
− f(x)) q


p
x p − x p + 1 x p−1Δx + O(Δx 2)
=
= p qx qp−q

q
Δx q
(A5.17)
which is different than the desired analytic result for 0 < q < 1 and q < --p (Miller and Ross, 1993):
D q−x p =
Γ(q − p) p−q
x
Γ(− p)
(A5.18)
This result can be extended to real values of p by analytic extension. This creates the discrepancy noticed
in the slope of the power tails in the simulations presented in Chapter 7. For example, if the function is a
power function with p = --α--1 (the expected slope of an α--stable density), the numerical approximation of
the (α--1) derivative should result in a function of power (--α--2)(α--1). Analytically, the (α--1) derivative of
x- α- 1 is x- 2α, so the numerical approximation is exact only when α=2 (Table A5.1). The approximation appears to give more weight to the tails than the analytic solution (see also Figure 7.2). This simplistic analysis
also suggests that a constant be placed before the numerical approximation. The constant is on the order:
p- qΓ(q--p)/Γ(--p). For the case of fractional dispersion of order α, this constant should be
(1+α)1-- αΓ(2α)/(Γ(1+α)). Note that when α = 2 (the classical case), the constant reduces to unity and the first
forward difference is recovered.
144
We can also analyze the local derivative (A5.4) of Kolwankar and Gangal (1996) in a similar manner. Once
again a power function is used where f(x) = xp. The approximation is:
f(x + Δx) − f(x)
=
D ≈ lim
Δx q
Δx→0
q

p
x p − x p + 1 x p−1Δx + O(Δx 2)
Δx q
= lim px p−1Δx 1−q = 0
Δx→0
(A5.19)
This approximation clearly does not recover the analytic function. As shown above, this function also does
not fulfill the required scaling characteristic.
Three approximations were examined in this Appendix. The two--point approximation of the local derivative
of Kolwankar and Gangal (1996) does not scale correctly, nor does it converge to an analytic power function.
The series expansion from the Grunwald definition yields a 7-- (or more) point finite difference equation
which may prove too computationally costly to implement. A convergence analysis of the Grunwald approximation was not undertaken. An ad hoc definition scales properly (see Chapter 7) and converges to a power
function, but the exponent is different than the analytic solution for derivative orders different than unity (α
≠ 2).
α
Analytic Exponent
(--2α)
Numeric Exponent
(--α--2)(α--1)
2.0
- 4.
- 4.
1.8
- 3.6
- 3.04
1.6
- 3.2
- 2.16
1.4
- 2.8
- 1.36
1.2
- 2.4
- 0.64
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.