The Fractional Advection--Dispersion Equation: Development and ...

University of Nevada Reno

The Fractional Advection--Dispersion Equation: Development and Application

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Hydrogeology

by

David Andrew Benson

Stephen W. Wheatcraft, Dissertation Advisor

May 1998

E 1997, 1998 David Andrew Benson All Rights Reserved

i

The dissertation of David Andrew Benson is approved:

Dissertation Advisor

Department Chair

Dean, Graduate School

University of Nevada Reno

May 1998

ii

Dedicated to my father, who taught me how to think about the world, and to my mother, who taught me how to live in it.

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ACKNOWLEDGEMENTS No metaphysician ever felt the deficiency of language so much as the grateful. - Charles Caleb Colton, Lacon Oddly, I must first thank my Master’s advisor, David Huntley. When I told him I was considering a Ph.D., without hesitation he told me to go to UNR and talk to Steve Wheatcraft. I have never received more 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.

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Many of my old friends kept me in--touch (and in cheap digs!) during this long process, so I thank Tom, Chris, Greggy and Strato, Don (thanks for the deal on the scooter, yeah right!), Laura, and John and Laura. My mother is the smartest person I know. She should have been the U.S. ambassador to the U.N., but 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.

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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.

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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

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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

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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, 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