Scale-Dependent Dispersivities and
The Fractional Convection Dispersion Equation
Primary Source:
Ph.D. Dissertation
David Benson
University of Nevada Reno, 1998
Mike Sukop/FIU
Outline
Motivation
Porous Media and
Models
Dispersion Processes
Representative
Elementary Volume
Convection-Dispersion
Equation
Scale Dependence
Solute Transport
Conventional and
Fractional Derivatives
a-Stable Probability
Densities
Levy Flights
Application
Conclusions
2
Motivation
Scale Effects
Need for Independent Estimation
3
Dispersion
Simulated Toxaphene Concentrations 50 Years After
Recharge Begins
0
ADEQ Toxaphene Health-Based Guidance Level
50
100
Depth (feet)
150
200
250
300
350
400
Initial Mass: 6.51 lb/ac
Koc: 100,000
Retardation Factor: 72
General Simulation Conditions
q: 0.5 ft/d
q: 0.4
rb:1.58 kg/l
foc: 0.00018
No Degradation
Dispersivity = 1 m
450
Dispersivity = 10 m
500
0
0.1
0.2
0.3
0.4
0.5
Concentration (ug/l)
0.6
0.7
0.8
4
Soil/Aquifer Material
5
Real Soil Measurements
X-Ray
Tomography
6
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)
7
Solute Dispersion
Diffusion Only
Time = 0
Modified from
8
Serrano, 1997
Solute Dispersion
Diffusion Only
Time > 0
Modified from
9
Serrano, 1997
Solute Dispersion
Advection Only
Average Pore
Water Velocity
Time = 0
x = x0
Time > 0
x > x0
Modified 10
from
Serrano, 1997
Solute Dispersion
Water Velocities
Vary on sub-Pore
Scale
Mechanical Mixing
in Pore Network
Modified from
Serrano, 1997
Mixing in K Zones
11
Solute Dispersion
Mechanical Dispersion, Diffusion, Advection
Average Pore
Water Velocity
Time = 0
x = x0
Time > 0
x > x0
Modified 12
from
Serrano, 1997
Representative Elementary
Volume (REV)
13
From Jacob Bear
Representative Elementary
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?)
14
Soil Blocks (0.3 m)
15
Phillips, et al, 1992
Aquifer (10’s m)
16
Laboratory and Field
Scales
17
Problems with the CDE
c
c
c
 D 2 v
t
x
x
2
Macroscopic, REV, Scale dependence,
Brownian Motion/Gaussian distribution
18
Scale Dependence of Dispersivity
19
Gelhar, et al, 1992
Scale Dependence of Dispersivity
20
Neuman, 1995
Scale Dependence of Dispersivity
21
Pachepsky, et al, 1999 (in review)
Scale Dependence
Power law growth Deff = Dxs
Perturbation/Stochastic DEs
Statistical approaches
22
Scale Dependence
Serrano, 1996
Dx t   Dx   t
2
u
2m  t
Dy t   Dy 
2
l
2
 
2
u
 A
2
T
2
nh
2
u
2
2
23
Conventional Derivatives
 
r
dx
 rx r 1
dx
24
From Benson, 1998
Conventional Derivatives
 
r
dx
 rx r 1
dx
25
From Benson, 1998
Fractional Derivatives
The gamma function interpolates the factorial
function. For integer n, gamma(n+1) = n!

( x)   t e dt
x 1 t
0
26
Fractional Derivatives
(u  1) u q
D x 
x
(q  u  1)
q
u
27
From Benson, 1998
Another Look at
Divergence
For integer order divergence, the ratio of
surface flux to volume is forced to be a
constant over different volume ranges
28
Another Look at
Divergence
29
From Benson, 1998
Another Look at
Divergence
30
From Benson, 1998
Standard Symmetric a-Stable
Probability Densities
0.35
0.30
f(x)
0.25
0.20
a = 2 (Normal)
a = 1.5
0.15
0.10
a = 1.8
0.05
0.00
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
31
Standard Symmetric a-Stable
Probability Densities
1.0000
f(x)
0.1000
a = 1.5
0.0100
a = 1.8
0.0010
a = 2 (Normal)
0.0001
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
32
Standard Symmetric a-Stable
Probability Densities
1
0.1
f(x)
0.01
a = 1.2
0.001
a = 1.5
0.0001
0.00001
a = 1.8
a = 2 (Normal)
0.000001
0.0000001
1
10
x
100
33
Brownian Motion and Levy
Flights
Pr U  u   u
D
Pr U  u   1, u  1
ln Pr U  u    D ln u
ue
 ln  PrU u 

D 

34
Monte-Carlo Simulation of
Levy Flights
Power Law Probability Distribution
Uniform Probability Density
1
Pr(U>u)
0.8
x
0.6
0.4
0.2
0
Pr(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
D=1.7
D=1.2
0
5
10
15
u
35
FADE (Levy Flights)
MATLAB Movie/
Turbulence Analogy
500
50
100 ‘flights’, 1000 time steps each
36
Ogata and Banks (1961)
Semi-infinite, initially solute-free medium
Plane source at x = 0
Step change in concentration at t = 0
C0
C
2

 x  vt 

1  erf 
 2 Dt 

37
ADE/FADE

 x  vt 

1  erf 
 2 Dt 

 x  vt 
C0 

C
1  serf 
1 

a 
2 


Dt



C0
C
2
38
Error Function
erf  z  
2
z


f  x dx
0
f x   e
x
2
39
a-Stable Error Function
z
serf a  z   2 fa  x dx
0
(1)
2k  1
2k
fa  x   
(
 1) x
 k 0 (2k  1)!
a
1

k
40
Scaling and Tailing
1.0
C/C0
0.8
q=0.12
Data
FADE Fit
ADE Fit
0.6
11 cm
17 cm
23 cm
0.4
0.2
0.0
0
20
40
60
80
100
120
140
Time (min)
After Pachepsky Y, Benson DA, and Timlin D (2001) Transport of water and solutes in soils as in fractal porous media. In Phys ical and Chemical
41
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.
Scaling and Tailing
Depth
(cm)
Dispersion Coefficient
11
CDE
2
(cm /hr)
0.035
FADE
1.6
(cm /hr)
0.030
17
0.038
0.029
23
0.042
0.028
42
Conclusions
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
43