Calculus Review
Partial Derivatives
3
2.5
2
1.5
1
0.5
0
-0.5
S19
-1
S13
19
13
16
X
10
S7
7
4
1
-1.5
Y
S1
• Functions of more than one variable
• Example: h(x,y) = x4 + y3 + xy
2.5-3
2-2.5
1.5-2
1-1.5
0.5-1
0-0.5
-0.5-0
-1--0.5
-1.5--1
Partial Derivatives
• Partial derivative of h with respect to x at a
y location y0
• Notation dh/dx|y=y0
• Treat ys as constants
• If these constants stand alone, they drop
out of the result
• If they are in multiplicative terms involving
x, they are retained as constants
Partial Derivatives
3
2.5
2
1.5
1
0.5
0
-0.5
S19
-1
S13
19
13
16
X
10
S7
7
4
1
-1.5
• Example:
• h(x,y)
= x4 + y3 + xy
• dh/dx|y=y0 = 4x3
+ y0
S1
Y
2.5-3
2-2.5
1.5-2
1-1.5
0.5-1
0-0.5
-0.5-0
-1--0.5
-1.5--1
Gradients
• del C (or grad C)
C
C
C  i
j
x
y
• Diffusion (Fick’s 1st Law):
J   DC
Numerical Derivatives
• slope between points
I  P  O  L   A
Poisson Equation
h
q  K
x


h
h
 K x   K x
 yb  Rxy  0
x
x  x 

 h
 x

 h
R

2
x
T
2
h 


x
R
x  x
x

x
T

Analytical Solution to Poisson
Equation
h
x   R
x
T

h
R
  x    T x
•Incorporate flux BCs (including zero
flux) here!
• h/x|0 = 0; i.e., a no flow
groundwater divide
h
R
  x  c1
x
T
 R


h


x

c
   T 1 x
R 2
h
x  c1 x  c 2
2T
Laplace Equation
 h
0
2
x
2
Poisson Equation
h
R


2
x
T
2
Heat/Diffusion Equation Derivation
z
Jx|x
y
x + x
x
J
x x
 Jx
x  x

C
yz 
xyz
t
C
J  D
x
Heat/Diffusion Equation Derivation

C
C
  D
D
x x
x


C
yz 
xyz
t
x  x 
 C C
D 2 
t
x
2
Fully explicit FD solution to Heat
Equation
t-t
x +x
x -x
C/t|t-t/2
Estimate here
C|x, t
t
x
Fully explicit FD solution to Heat
Equation
C x ,t  C x ,t  t

Dt
 2 C x  x ,t  t  2C x ,t  t  C x  x ,t  t
x
• Need IC and BCs

No diffusive flux BC
•
Fick’s law
•
If ∂C/∂x = 0, there is no flux
•
Finite difference expression for ∂C/∂x is
C
J  D
x
C
x

x  x / 2
C x  x  C x
x
C x  x  C x
•
Setting this to 0 is equivalent to
•
‘Ghost’ points outside the domain at x + x
•
Then, if we make the concentration at the ghost points equal to the
concentration inside the domain, there will be no flux
•
Often the boundary conditions are constant in time, but they need not be
Closed Box
• Finite system:
C0 n 
h  2nl  x
h  2nl  x 
C
erf
 erf



2 n 
4Dt
4Dt 
Superposition of original
process and reflections
C
0
x 0
500
t = 1000
450
400
C (mu lu -2)
350
300
t = 2000
250
200
t =11000
150
100
50
C
x
0
-50
-30
-10
10
x (lu)
30
50
0
xmax
Basic Fluid Dynamics
Viscosity
•
•
•
•
Resistance to flow; momentum diffusion
Low viscosity: Air
High viscosity: Honey
Kinematic viscosity
Reynolds Number
• The Reynolds Number (Re) is a non-dimensional
number that reflects the balance between viscous and
inertial forces and hence relates to flow instability (i.e.,
the onset of turbulence)
• Re = v L/n
• L is a characteristic length in the system
• Dominance of viscous force leads to laminar flow (low
velocity, high viscosity, confined fluid)
• Dominance of inertial force leads to turbulent flow (high
velocity, low viscosity, unconfined fluid)
Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Separation
Eddies and Cylinder Wakes
S.Gokaltun
Florida International University
Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)
Eddies and Cylinder Wakes
Re = 30
Re = 40
Re = 47
Re = 55
Re = 67
Re = 100
Re = 41
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Poiseuille Flow
Jean Léonard Marie Poiseuille; 1797 – 1869.
From Sutera and Skalak, 1993. Annu. Rev. Fluid
Mech. 25:1-19
Poiseuille Flow
• In a slit or pipe, the velocities at the walls are 0
(no-slip boundaries) and the velocity reaches its
maximum in the middle
• The velocity profile in a slit is parabolic and
given by:
G 2
2
u( x ) 
(a  x )
2
• G can be gravitational pressure gradient
(rg for example in a vertical slit) or
(linear) pressure gradient (Pin – Pout)/L
u(x)
x=0
x=a
Dispersion
• Mixing induced by velocity variations
• No velocity, no dispersion
Geoffrey Ingram Taylor; 1886 - 1975.
http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Taylor_Geoffrey.html
Taylor Dispersion
Taylor Dispersion
C
C
 2C
 v
D 2
t
x
x
1.2
1
C/C0
0.8
0.6
LBM Result
Analytical Solution
0.4
0.2
0
0
50000
100000
150000
Time (time steps)
2
2
U
W
D  Dm 
210Dm
200000
250000
Taylor/Aris Dispersion
Dispersion Coefficient (lu2 ts-1)
0.07
Analytical Solution:
2
2
D = U W /(210 Dm) + Dm
0.06
0.05
0.04
0.03
0.02
Distance of measured
breakthrough curve from source:
50 lu
0.01
150 lu
300 lu
0.00
0
5
10
15
20
25
Width (lattice units)
Stockman, H.W., A lattice-gas study of retardation and dispersion in
fractures: assessment of errors from desorption kinetics and
buoyancy, Wat. Resour. Res. 33, 1823 - 1831, 1997.
30
35
40
Diffusion in Poiseuille Flow
Pore Volume
Breakthrough Curves
Influent Solution:
Concentration C0
• ‘Piston’ Flow – no
dispersion
• Dispersed Flow
• Retarded/
Dispersed Flow
Effluent Solution:
Concentration C
Breakthrough Curve
q = 1 m/y
100
Initial and Boundary
Conditions:
C (mg/l)
10 m
80
C(x,0) = 0
C(0,0<t<1) = 100
C(0,t>1) = 0
60
General Conditions:
q = 1 m/year
 = 0.5
40
 = 0.1 m
20
0
0
2
4
6
Time (years)
8
10
Continuous Source
Pulse Source
Peclet Number
• Inside a pore, the dimensionless Peclet
number (Pe ≡ vl/Dm, with l a characteristic
length) indicates the relative importance of
diffusion and convection;
– large values of Pe indicate a convection
dominated process
– small values of Pe indicate the dominance of
diffusion
Dimensionless Diffusion-Dispersion
Coefficient
• The dimensionless diffusion-dispersion
coefficient D* ≡ Dd/Dm reflects the relative
importance of hydrodynamic dispersion and
diffusion
• For porous media with well-defined
characteristic lengths (i.e., bead diameter in
packed beds of uniformly sized glass beads), D*
can be estimated from Pe based on empirical
data
Empirical relationship between dimensionless dispersion coefficient and Peclet number with data for
uniformly sized particle beds. Adapted from Fried, JJ and Combarnous MA (1971) Dispersion in porous
media. Adv. Hydroscience 7, 169-282.
Classes of Behavior
• Different classes of behavior proposed based on
the observed relationship between Pe and D*
– Class I: very slow flow, dominance of diffusion
– Class II: transitional with approximately equal and
additive hydrodynamic dispersion and diffusion
– Class III: hydrodynamic dispersion dominates, but the
role of diffusion is still non-negligible,
– Class IV: diffusion negligible
– Class V: velocity so high that the flow of many fluids is
turbulent
The process:
• Measure grain size l
• Look up Dm (10-5 cm2 s-1)
– http://www.hbcpnetbase.com/
• Know mean pore water velocity from v =
q/n
• Compute Pe (= vl/Dm)
• Take D* (=Dd/Dm) from graph
• Compute Dd = D* Dm
Ion Diffusion Coefficients in Water
Organic Molecule Diffusion
Coefficients in Water
Large-scale Dispersion
Neuman, 1995
Rule of Thumb:
Dispersivity = 0.1
Scale
Neuman, 1995
CDE
z
Jx|x
C
J x x  J x xx yz  t xyz
* C
J d  D
x
y
x + x
x

C
C
 vC x  D*
 vC x x  D*
x x
x

•Key difference from diffusion here!
• Convective flux

C
yz 
xyz

t
x  x 
 C 


*
C D  x  C
v


x 
x
t
1st Order Spatial Derivative
C
x

x  x / 2
C x  x  C x
x
• Worked for estimating second order
derivative (estimate ended up at x).
• Need centered derivative approximation
C xx  C xx
C

x x
2x
CDE Explicit Finite Difference
C x ,t  C x ,t t



Dt
vt
 2 C x x ,t t  2C x ,t t  C x x ,t t 
C x  x ,t t  C x x ,t t
2x
x
• Grid Pe = vL/D, where L is the grid
spacing
• Pe < 1, 4, 10

Isotherms
• Linear: Cs = Kd Cw
• Freundlich: Cs = Kf Cw1/n
• Langmuir: Cs = Keq Cst Cw/(1 + Keq Cw)
Koc Values
• Kd = Koc foc
Organic Carbon Partitioning Coefficients for Nonionizable Organic Compounds. Adapted
from USEPA, Soil Screening Guidance: Technical Background Document.
http://www.epa.gov/superfund/resources/soil/introtbd.htm
Compound
Acenaphthene
mean Koc (L/kg)
5,028
Compound
1,4-Dichlorobenzene(p)
mean Koc (L/kg)
687
Compound
Methoxychlor
mean Koc (L/kg)
80,000
Aldrin
48,686
1,1-Dichloroethane
54
Methyl bromide
9
Anthracene
24,362
1,2-Dichloroethane
44
Methyl chloride
6
1,1-Dichloroethylene
65
Methylene chloride
trans-1,2-Dichloroethylene
38
Naphthalene
1,231
1,166,733
1,2-Dichloropropane
47
Nitrobenzene
141
76
1,3-Dichloropropene
27
Pentachlorobenzene
36,114
25,604
Pyrene
70,808
84
Styrene
912
Benz(a)anthracene
Benzene
Benzo(a)pyrene
Bis(2-chloroethyl)ether
Bis(2-ethylhexyl)phthalate
Bromoform
Butyl benzyl phthalate
Carbon tetrachloride
Chlordane
Chlorobenzene
Chloroform
459,882
66
114,337
126
14,055
158
51,798
Dieldrin
Diethylphthalate
10
Di-n-butylphthalate
1,580
1,1,2,2-Tetrachloroethane
Endosulfan
2,040
Tetrachloroethylene
272
Toluene
145
Endrin
11,422
260
Ethylbenzene
207
57
Fluoranthene
49,433
DDD
45,800
Fluorene
DDE
86,405
DDT
792,158
Toxaphene
1,2,4-Trichlorobenzene
79
95,816
1,783
8,906
1,1,1-Trichloroethane
139
Heptachlor
10,070
1,1,2-Trichloroethane
77
Hexachlorobenzene
80,000
Trichloroethylene
97
Dibenz(a,h)anthracene
2,029,435
-HCH (-BHC)
1,835
o-Xylene
241
1,2-Dichlorobenzene(o)
390
b-HCH (b-BHC)
2,241
m-Xylene
204
g-HCH (Lindane)
1,477
p-Xylene
313
Retardation
• Incorporate adsorbed solute mass
r b Kd
R  1

V
R
Vs
Retardation
C
C
C
v
D 2 R
x
x
t
2
Kinetics
• dC/dt = constant: zero order
• dC/dt = -kC: first order
C ( T ) C
• Integrate:

C (0)
C
T
   kt
0
C (T )
ln
 kT
C (0)
1
ln
2 T
1/ 2
k
Two-Site Conceptual Model
Instantaneous Adsorption Sites
Mobile Water
Air

Kinetic Adsorption Sites
dC s 2
  1  F KdC  Cs 2 
dt
Two-site model
• Selim et al., 1976; Cameron and Klute,
1977; and many more
• Instantaneous equilibrium and kineticallylimited adsorption sites
• Different constituents:
• “Soil minerals, organic matter, Fe/Al
oxides”
• ‘F’ = Fraction of instantaneous sites
• ‘’ = First-order rate constant
Batch Sorption Kinetics
2600
Mean column
 = 0.06 hr-1
Concentration (dpm/ml)
2400
2200
First Order Model
for All Data
2000
1800
1600
1400
First Order Model
for t > 1 hour:  = 0.11 hr-1
1200
1000
0
5
10
15
Time (hours)
20
25
Two-Region Conceptual Model
Dynamic Soil Region
Mobile Water

Air
Immobile Water
Stagnant Soil Region
dCim
im  1  F rb Kd    Cm  Cim 
dt
STANMOD
• CXTFIT Toride et al.[1995]
• For estimating solute transport parameters using a
nonlinear least-squares parameter optimization method
• Inverse problem by fitting a variety of analytical solutions
of theoretical transport models, based upon the onedimensional convection-dispersion equation (CDE), to
experimental results
• Three different one-dimensional transport models are
considered:
– (i) the conventional equilibrium CDE;
– (ii) the chemical and physical nonequilibrium CDEs; and
– (iii) a stochastic stream tube model based upon the local-scale
equilibrium or nonequilibrium CDE
http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM
STANMOD
• CHAIN van Genuchten [1985]
• For analyzing the convective-dispersive
transport of solutes involved in sequential firstorder decay reactions.
• Examples:
– Migration of radionuclides in which the chain
members form first-order decay reactions, and
– Simultaneous movement of various interacting
nitrogen or organic species
http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM
STANMOD
• 3DADE Leij and Bradford [1994]
• For evaluating analytical solutions for two- and three-dimensional
equilibrium solute transport in the subsurface.
• The analytical solutions assume steady unidirectional water flow in
porous media having uniform flow and transport properties.
• The transport equation contains terms accounting for
–
–
–
–
solute movement by convection and dispersion, as well as for
solute retardation,
first-order decay, and
zero-order production.
• The 3DADE code can be used to solve the direct problem and the
indirect (inverse) problem in which the program estimates selected
transport parameters by fitting one of the analytical solutions to
specified experimental data
http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM
STANMOD
• N3DADE Leij and Toride [1997]
• For evaluating analytical solutions of two- and three-dimensional
nonequilibrium solute transport in porous media.
• The analytical solutions pertain to multi-dimensional solute transport
during steady unidirectional water flow in porous media in systems
of semi-infinite length in the longitudinal direction, and of infinite
length in the transverse direction.
• Nonequilibrium solute transfer can occur between two domains in
either the liquid phase (physical nonequilibrium) or the absorbed
phase (chemical nonequilibrium).
• The transport equation contains terms accounting
–
–
–
–
solute movement by advection and dispersion,
solute retardation,
first-order decay
zero-order production
http://www.ussl.ars.usda.gov/models/stanmod/stanmod.HTM
2- and 3-D Analytical
Solutions to CDE
Equation Solved:
C
C
C
C
C
v
 Dxx 2  Dyy 2  Dzz 2  R
x
x
y
z
t
2
2
2
• Constant mean velocity in x direction only!
•Hunt, B., 1978, Dispersive sources in
uniform ground-water flow, ASCE
Journal of the Hydraulics Division, 104
(HY1) 75-85.
‘Instantaneous’ Source
• Solute mass only
– M1, M2, M3
• Injection at origin of coordinate system (a
point!) at t = 0
‘Continuous’ Source
• Solute mass flux
– M1, M2, M3 = dM1,2,3/dt
• Injection at origin of coordinate system (a
point!)
Instantaneous and Continuous
Sources
• 1-D
2

M1
  x  vt  
Ci 
exp

2 Dxxt
 4 Dxxt 

 xv  exp  x v erfc  x  vt 

M 1 exp
 2D 
 2 D t
xx

2
D


xx 

xx 

Cc 

2v

xv 
x  vt


 
erfc 

exp

 2 D t
2
D
xx


xx










2-D Instantaneous Source
2
2 

M2
 x  vt 
y
Ci 
exp


4 D yyt 
4t Dxx Dyy
 4 Dxxt
2-D Instantaneous Source Solution
Dyy
Dxx
t = 25
t=1
Back dispersion
Extreme
concentration
t = 51
3-D Instantaneous Source
2
2
2 

M3
 x  vt 
y
z
Ci 
exp



3 3
4 Dyyt 4 Dzz t 
8  t Dxx Dyy Dzz
 4 Dxxt
3-D Instantaneous Source Solution
Dzz
Dyy
Dxx
t=1
t = 25
Back dispersion
t = 51
Extreme
concentration
3-D Continuous Source

 xv  exp  Rv erfc  R  vt

M 3 exp
 2D 
 2 D t
xx 

2 Dxx  
xx


Cc 

8R D yy Dzz 
 R  vt 
 Rv 





exp
erfc
 2D 

2 D t 
xx 

xx 


R
Dxx
2 D xx
x y
z
D yy
Dzz
2
2








StAnMod (3DADE)
• Same equation (mean x velocity only)
• Better boundary and initial conditions
• Leij, F.J., T.H. Skaggs, and M.Th. Van
Genuchten, 1991. Analytical solutions for
solute transport in three-dimensional semiinfinite porous media, Water Resources
Research 20 (10) 2719-2733.
Coordinate systems
• x increasing downward
z
z
r
y
y
x
x
Boundary Conditions
• Semi-infinite source
z
y
-∞
-∞
x
Boundary Conditions
• Finite rectangular source
z
b
-a
a
-b
x
y
Boundary Conditions
• Finite Circular Source
z
r=a
y
x
Initial Conditions
• Finite Cylindrical Source
z
y
r=a
x1
x2
x
Initial Conditions
• Finite Parallelepipedal Source
z
b
y
a
x1
x2
x
x2
x1
x
r=a
z
y
Comparing with Hunt
• M3 = r2 (x1 – x2) Co
(=1, small, high C)
• Co = 1/[r2 (x1 – x2)] = 106  for r = x= 0.01
Wells?
• Finite Parallelepipedal Source
Pathlines
Scale-Dependent Dispersivities and
The Fractional Convection - Dispersion
Equation
Primary Source:
Ph.D. Dissertation
David Benson
University of Nevada Reno, 1998
Representative Elementary
Volume (REV)
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?)
Conventional Derivatives
 
r
dx
 rx r 1
dx
From Benson, 1998
Conventional Derivatives
 
r
dx
 rx r 1
dx
From Benson, 1998
Fractional Derivatives
The gamma function interpolates the factorial
function. For integer n, gamma(n+1) = n!

( x)   t e dt
0
x 1 t
Fractional Derivatives
(u  1) u q
D x 
x
(q  u  1)
q
u
From Benson, 1998
Standard Symmetric -Stable
Probability Densities
0.35
0.30
f(x)
0.25
0.20
 = 2 (Normal)
 = 1.5
0.15
0.10
 = 1.8
0.05
0.00
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
Standard Symmetric -Stable
Probability Densities
1.0000
f(x)
0.1000
 = 1.5
0.0100
 = 1.8
0.0010
 = 2 (Normal)
0.0001
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
Standard Symmetric -Stable
Probability Densities
1
0.1
f(x)
0.01
 = 1.2
0.001
 = 1.5
0.0001
0.00001
 = 1.8
 = 2 (Normal)
0.000001
0.0000001
1
10
x
100
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 

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
u
15
FADE (Levy Flights)
MATLAB Movie/
Turbulence Analogy
500
50
100 ‘flights’, 1000 time steps each
Scaling and Tailing
1.0
C/C0
0.8
=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
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
lbm
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
Continuous Time Random
Walk Model
Primary Sources:
Berkowitz, B, G. Kosakowski, G. Margolin, and H. Scher,
Application of continuuous time random walk theory to tracer test
measurents in fractured and heterogeneous porous media, Ground
Water 39, 593 - 604, 2001.
Berkowitz, B. and H. Scher, On characterization of anomalous
dispersion in porous and fractured media, Wat. Resour. Res. 31,
1461 - 1466, 1995.
Mike Sukop/FIU
Introduction
• Continuous Time Random Walk (CTRW)
models
– Semiconductors [Scher and Lax, 1973]
– Solute transport problems [Berkowitz and
Scher, 1995]
Introduction
• Like FADE, CTRW solute particles move
along various paths and encounter
spatially varying velocities
• The particle spatial transitions (direction
and distance given by displacement vector
s) in time t represented by a joint
probability density function y(s,t)
• Estimation of this function is central to
application of the CTRW model
Introduction
• The functional form y(s,t) ~ t-1-b (b > 0) is of
particular interest [Berkowitz et al, 2001]
 b characterizes the nature and magnitude of
the dispersive processes
Ranges of b
 b ≥ 2 is reported to be “…equivalent to the
ADE…”
– For b ≥ 2, the link between the dispersivity (
= D/v) in the ADE and CTRW dimensionless
bb is bb = /L
 b between 1 and 2 reflects moderate nonFickian behavior
• 0 < b < 1 indicates strong ‘anomalous’
behavior
Fits
1.0
Data
CTRW Fit
ADE Fit
C/C0
0.8
0.6
11 cm
17 cm
23 cm
0.4
0.2
0.0
0
20
40
60
80
100
Time (min)
120
140
Gas Phase Transport
Principal Sources:
VLEACH, A One-Dimensional Finite
Difference Vadose Zone Leaching
Model, Version 2.2 – 1997. United States
Environmental Protection Agency, Office
of Research and Development, National
Risk Management Research Laboratory,
Subsurface Protection and Remediation
Division, Ada, Oklahoma.
Šimůnek, J., M. Šejna, and M.T. van
Genuchten. 1998. The HYDRUS-1D
software package for simulating the
one-dimensional movement of water,
heat, and multiple solutes in variablysaturated media. Version 2.0,
IGWMC - TPS - 70, International
Ground Water Modeling Center,
Colorado School of Mines, Golden,
Colorado, 202pp., 1998.
Effective Diffusion
• Tortuosity (T = Lpath/L) and percolation (2D)
Macroscopic Gas Diffusion
x
D  J
C
dC
J  D
dx
2
D  a  4/3
    a
D0   
0.5
Maxwell (1873)
Buckingham (1904)
0.4
Penman (1940)
Marshall (1959)
0.3
D/Do
Millington (1959)
Wesseling (1962)
0.2
Currie (1965)
WLR(Marshall):
Moldrup et al (2000)
0.1
0
0
0.1
0.2
0.3
Volumetric Air Content
0.4
0.5
Total Mass
• At Equilibrium:
Henry’s Law
Cg  KH Cw
• Dimensionless:
• Common:
atm m3 mol-1
Total Mass
• At Equilibrium:
MT ( z, t )  Cl  (   )KH Cl  rb Kd Cl 
VLEACH
• Processes are
conceptualized as
occurring in a
number of
distinct, userdefined polygons
that are vertically
divided into a
series of userdefined cells
Voronoi
Polygons/
Diagram
• Voronoi_polygons
– close('all')
– clear('all')
– axis equal
– x = rand(1,100); y
= rand(1,100);
– voronoi(x,y)
Chemical Parameters
• Organic Carbon Partition Coefficient (Koc)
= 100 ml/g
• Henry’s Law Constant (KH) = 0.4
(Dimensionless)
• Free Air Diffusion Coefficient (Dair) = 0.7
m2/day
• Aqueous Solubility Limit (Csol) = 1100
mg/l
Soil Parameters
•
•
•
•
Bulk Density (rb) = 1.6 g/ml
Porosity (f) = 0.4
Volumetric Water Content (q) = 0.3
Fraction Organic Carbon Content (foc) =
0.005
Environmental Parameters
• Recharge Rate (q) = 1 ft/yr
• Concentration of TCE in Recharge Water
= 0 mg/l
• Concentration of TCE in Atmospheric Air =
0 mg/l
• Concentration of TCE at the Water Table =
0 mg/l
Dispersion!
• Dispersivity is implicit in the cell size (l)
and equal to l/2 (Bear 1972)
• Numerical dispersion but can be used
appropriately
Dispersion
100
VLEACH 0.1 m cells
Initial and Boundary
Conditions:
VLEACH 1 m cells
C(x,0) = 100 mg/l
C(0,t) = 0 mg/l
 = 0.05 m
 = 0.5 m
C (mg/l)
80
General Conditions:
 =5m
60
VLEACH 10 m cell
q = 1 m/year
 = 0.5
CDE Flux-averaged
concentrations
(Dispersivity as
shown)
40
VLEACH time step:
0.01 years
20
0
0
5
10
Time (years)
15
20
M.C. Sukop. 2001. Dispersion in VLEACH and similar models. Ground Water 39, No. 6, 953-954.
Hydrus
Hydrus
• Solves
– Richards’ Equation
– Fickian solute transport
– Sequential first order decay reactions
Governing Equation
w ,k 1, g ,k 1, and s,k 1
Provide linkage with preceding
members of the chain
Density-Dependent Flows
Primary source:
User’s Guide to SEAWAT: A Computer Program for
Simulation of Three-Dimensional Variable-Density GroundWater Flow
By Weixing Guo and Christian D. Langevin
U.S. Geological Survey
Techniques of Water-Resources Investigations 6-A7,
Tallahassee, Florida2002
Sources of density variation
• Solute concentration
• Pressure
• Temperature
Freshwater Head
• SEAWAT is based on the
concept of equivalent
freshwater head in a saline
ground-water environment
• Piezometer A contains
freshwater
• Piezometer B contains water
identical to that present in
the saline aquifer
• The height of the water level
in piezometer A is the
freshwater head
Converting between:
Mass Balance
• (with sink term)
Density
(and soon T!)
Densities
•
•
•
•
Freshwater: 1000 kg m-3
Seawater: 1025 kg m-3
Freshwater: 0 mg L-1
Seawater: 35,000 mg L-1
dr 1025 1000kg m

3
dC
35 kg m
3
 0.714