scale-dependent dispersivities and the fractional convection - dispersion equation
DESCRIPTION
Scale-Dependent Dispersivities and The Fractional Convection - Dispersion Equation. Primary Source: Ph.D. Dissertation David Benson University of Nevada Reno, 1998. Mike Sukop/FIU. Motivation Porous Media and Models Dispersion Processes Representative Elementary Volume - PowerPoint PPT PresentationTRANSCRIPT
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