Scale-Dependent Dispersivities and Scale-Dependent Dispersivities and The Fractional Convection - Dispersion The Fractional Convection - Dispersion EquationEquation

Mike Sukop/FIU

Primary Source:Ph.D. DissertationDavid BensonUniversity of Nevada Reno, 1998

2

OutlineOutline

MotivationPorous Media and

ModelsDispersion ProcessesRepresentative

Elementary VolumeConvection-

Dispersion Equation

Scale DependenceSolute TransportConventional and

Fractional Derivatives-Stable Probability

DensitiesLevy Flights ApplicationConclusions

3

MotivationMotivation

Scale Effects Need for Independent Estimation

Scale Effects Need for Independent Estimation

4

DispersionDispersion

Simulated Toxaphene Concentrations 50 Years After Recharge Begins

0

50

100

150

200

250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Concentration (ug/l)

Dep

th (

feet

)

General Simulation Conditionsq: 0.5 ft/d: 0.4

b:1.58 kg/lfoc: 0.00018No Degradation

Initial Mass: 6.51 lb/acKoc: 100,000Retardation Factor: 72

AD

EQ

Toxaphene Health-B

ased Guidance Level

Dispersivity = 10 m

Dispersivity = 1 m

5

Soil/Aquifer MaterialSoil/Aquifer Material

6

Real Soil MeasurementsReal Soil Measurements

X-Ray Tomography

7

What is Dispersion?What is Dispersion?

Spreading of dissolved constituent in space and time

Three processes operate in porous media: Diffusion (random Brownian motion) Convection (going with the flow) Mechanical mixing (the tough part)

8

Solute DispersionSolute Dispersion

Diffusion OnlyDiffusion Only

Time = 0Time = 0

Modified from Serrano, 1997

9

Solute DispersionSolute Dispersion

Diffusion OnlyDiffusion Only

Time > 0 Modified from Serrano, 1997

10

Solute DispersionSolute Dispersion

Advection OnlyAdvection Only

Average Pore Water Velocity Average Pore Water Velocity

Time > 0x > x0

Time > 0x > x0

Time = 0x = x0

Time = 0x = x0

Modified from Serrano, 1997

11

Solute DispersionSolute Dispersion

Water Velocities Vary on sub-Pore Scale

Mechanical Mixing in Pore Network

Mixing in K Zones

Water Velocities Vary on sub-Pore Scale

Mechanical Mixing in Pore Network

Mixing in K Zones

Modified from Serrano, 1997

12

Solute DispersionSolute Dispersion

Mechanical Dispersion, Diffusion, Advection Mechanical Dispersion, Diffusion, Advection

Average Pore Water Velocity Average Pore Water Velocity

Time = 0x = x0

Time = 0x = x0

Time > 0x > x0

Time > 0x > x0

Modified from Serrano, 1997

13

Representative Elementary Representative Elementary Volume (REV)Volume (REV)

From Jacob Bear

14

Representative Elementary Representative Elementary Volume (REV)Volume (REV)

General notion for all continuum mechanical problems

Size cut-offs usually arbitrary for natural media (At what scale can we afford to treat medium as deterministically variable?)

15

Soil Blocks (0.3 m)Soil Blocks (0.3 m)

Phillips, et al, 1992

16

Aquifer (10’s m)Aquifer (10’s m)

17

Laboratory and Field ScalesLaboratory and Field Scales

18

Problems with the CDEProblems with the CDE Problems with the CDEProblems with the CDE

x

cv

x

cD

t

c

2

2

Macroscopic, REV, Scale dependence,Brownian Motion/Gaussian distribution

19

Scale Dependence of DispersivityScale Dependence of Dispersivity

Gelhar, et al, 1992

20

Scale Dependence of DispersivityScale Dependence of Dispersivity

Neuman, 1995

21

Scale Dependence of DispersivityScale Dependence of Dispersivity

Pachepsky, et al, 1999 (in review)

22

Scale DependenceScale Dependence

Power law growth Deff = Dxs

Perturbation/Stochastic DEsStatistical approaches

23

Scale DependenceScale Dependence

Serrano, 1996

tDtD uxx2

2

222

l

tmDtD uyy

22

222

hn

ATu

24

Conventional DerivativesConventional Derivatives

1 rr

rxdx

xd

From Benson, 1998

25

Conventional DerivativesConventional Derivatives

1 rr

rxdx

xd

From Benson, 1998

26

Fractional DerivativesFractional Derivatives

The gamma function interpolates the factorial function. For integer n, gamma(n+1) = n!

0

1)( dtetx tx

27

quuq xuq

uxD

)1(

)1(

Fractional DerivativesFractional Derivatives

From Benson, 1998

28

Another Look at DivergenceAnother Look at Divergence

For integer order divergence, the ratio of surface flux to volume is forced to be a constant over different volume ranges

29

Another Look at DivergenceAnother Look at Divergence

From Benson, 1998

30

Another Look at DivergenceAnother Look at Divergence

From Benson, 1998

31

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-5 -4 -3 -2 -1 0 1 2 3 4 5

x

f(x

) = 2 (Normal)

= 1.8

= 1.5

Standard Symmetric Standard Symmetric -Stable -Stable Probability DensitiesProbability Densities

32

Standard Symmetric Standard Symmetric -Stable -Stable Probability DensitiesProbability Densities

0.0001

0.0010

0.0100

0.1000

1.0000

-5 -4 -3 -2 -1 0 1 2 3 4 5

x

f(x)

= 2 (Normal)

= 1.8

= 1.5

33

Standard Symmetric Standard Symmetric -Stable -Stable Probability DensitiesProbability Densities

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100

x

f(x)

= 2 (Normal)

= 1.8

= 1.5

= 1.2

34

Brownian Motion and Levy Brownian Motion and Levy FlightsFlights

DuU

D

eu

uDuU

uuU

uuU

Prln

lnPrln

1,1Pr

Pr

35

Monte-Carlo Simulation of Monte-Carlo Simulation of Levy FlightsLevy Flights

Power Law Probability Distribution

00.10.20.30.40.50.60.70.80.9

1

0 5 10 15

u

Pr(

U>

u)

D=1.7D=1.2

Uniform Probability Density

0

0.2

0.4

0.6

0.8

1

Pr(x)

x

36

MATLAB Movie/MATLAB Movie/Turbulence AnalogyTurbulence Analogy

FADE (Levy Flights)

100 ‘flights’, 1000 time steps each

50500

37

Dt

vtxerf

CC

21

20

Ogata and Banks (1961)Ogata and Banks (1961)

Semi-infinite, initially solute-free medium

Plane source at x = 0Step change in concentration at t =

0

38

ADE/FADEADE/FADE

Dt

vtxerf

CC

21

20

1

0 12 Dt

vtxserf

CC

39

Error FunctionError Function

dxxfzerfz

0

2

2xexf

40

-Stable Error Function-Stable Error Function

dxxfzserfz

0

2

k

k

k

xk

kxf 2

0

)112

()!12(

)1(1

41

Scaling and TailingScaling and Tailing

=0.12

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100 120 140

Time (min)

C/C

0

Data

FADE Fit

ADE Fit

11 cm 17 cm 23 cm

After Pachepsky Y, Benson DA, and Timlin D (2001) Transport of water and solutes in soils as in fractal porous media. In Physical and Chemical Processes of Water and Solute Transport/Retention in Soils. D. Sparks and M. Selim. Eds. Soil Sci. Soc. Am. Special Pub. 56, 51-77 with permission.

42

Scaling and TailingScaling and Tailing

Depth Dispersion Coefficient

(cm) CDE(cm2/hr)

FADE(cm1.6/hr)

11 0.035 0.030

17 0.038 0.029

23 0.042 0.028

43

ConclusionsConclusions

Fractional calculus may be more appropriate for divergence theorem application in solute transport

Levy distributions generalize the normal distribution and may more accurately reflect solute transport processes

FADE appears to provide a superior fit to solute transport data and account for scale-dependence