calculus review. partial derivatives functions of more than one variable example: h(x,y) = x 4 + y 3...
TRANSCRIPT
Calculus Review
Partial Derivatives
• Functions of more than one variable• Example: h(x,y) = x4 + y3 + xy
1 4 7
10 13 16 19S1
S7
S13
S19
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
X
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
Partial Derivatives
• Partial derivative of h with respect to x at a y location y0
• Notation h/x|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
• Example: • h(x,y) = x4 + y3 + xy
• h/x|y=y0 = 4x3 + y0
1 4 7
10 13 16 19S1
S7
S13
S19
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
X
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)
• Diffusion (Fick’s 1st Law):
y
C
x
CC
ji
CDJ
Numerical Derivatives
• slope between points
ALOPI
Poisson Equation
x
hKq
0
yxRybx
hK
x
hK
xxx
T
R
x
xh
xh
xxx
T
R
x
h
2
2
Analytical Solution to Poisson Equation
T
R
xx
h
xT
R
x
h
1cxT
R
x
h
xcx
T
Rh 1
212
2cxcx
T
Rh
•Incorporate flux BCs (including zero flux) here!
• h/x|0 = 0; i.e., a no flow groundwater divide
Laplace Equation
02
2
x
h
Poisson Equation
T
R
x
h
2
2
Heat/Diffusion Equation Derivation
x + x
y
z
x
Jx|x
zyxt
CzyJJ
xxxxx
x
CDJ
Heat/Diffusion Equation Derivation
t
C
x
CD
2
2
zyxt
Czy
x
CD
x
CD
xxx
Fully explicit FD solution to Heat Equation
C|x, t
x
x +x
C/t|t-t/2 Estimate here
t-t
t
x -x
Fully explicit FD solution to Heat Equation
• Need IC and BCs
ttxxttxttxxttxtx CCC
x
tCC
,,,2,, 2
D
No diffusive flux BC• Fick’s law
• If ∂C/∂x = 0, there is no flux • Finite difference expression for ∂C/∂x is
• 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
x
CDJ
x
CC
x
C xxx
xx
2/
xxxCC
Closed Box• Finite system:
• Standard ‘Bounce-back’ from solids boundary works for diffusion
n
n Dt
xnlherf
Dt
xnlherf
CC
4
2
4
2
20
0max
xx
C
00
x
C
Superposition of original process and reflections
0
50
100
150
200
250
300
350
400
450
500
-50 -30 -10 10 30 50x (lu)
C (
mu
lu
-2)
t = 1000
t = 2000
t =11000
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/• 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 WakesS
.Go
kaltu
n
Flo
rida
Inte
rna
tion
al U
nive
rsity
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 = 41Tritton, 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:
)(2
)( 22 xaG
xu
x = 0 x = a
u(x)
• G can be gravitational pressure gradient (g for example in a vertical slit) or (linear) pressure gradient (Pin – Pout)/L
Dispersion
• Mixing induced by velocity variations
• No velocity, no dispersion
Taylor Dispersion
Geoffrey Ingram Taylor; 1886 - 1975.
htt
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/ww
w-h
isto
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cs.s
t-a
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rew
s.a
c.u
k/P
ictD
isp
lay/
Ta
ylo
r_G
eo
ffre
y.h
tml
Taylor Dispersion
0
0.2
0.4
0.6
0.8
1
1.2
0 50000 100000 150000 200000 250000
Time (time steps)C
/C0
LBM ResultAnalytical Solution
2
2
x
CD
x
Cv
t
C
mm D
WUDD 21022
Taylor/Aris Dispersion
Analytical Solution:
D = U2 W2 /(210 Dm) + Dm
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0 5 10 15 20 25 30 35 40
Width (lattice units)
Dis
pe
rsio
n C
oe
ffic
ien
t (l
u2 ts
-1)
50 lu 150 lu 300 lu
Distance of measured breakthrough curve from source:
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.
Diffusion in Poiseuille Flow
Pore Volume
Breakthrough Curves
• ‘‘Piston’ Flow – no Piston’ Flow – no dispersiondispersion
• Dispersed FlowDispersed Flow• Retarded/ Retarded/
Dispersed FlowDispersed Flow
Influent Solution: Concentration C0
Effluent Solution: Concentration C
Breakthrough Curve
0
20
40
60
80
100
0 2 4 6 8 10Time (years)
C (
mg
/l)
Initial and Boundary Conditions:
C(x,0) = 0C(0,0<t<1) = 100C(0,t>1) = 0
General Conditions:
q = 1 m/year = 0.5
= 0.1 m
10 m
q = 1 m/y
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
Neuman, 1995
Rule of Thumb:
Dispersivity = 0.1 Scale
CDE
x + x
y
z
x
Jx|x
zyxt
CzyJJ
xxxxx
x
CDJ d
*
zyxt
Czy
x
CDCv
x
CDCv
xxxx
xx
**
t
C
xx
CD
x
Cv
*•Key difference from diffusion here!
• Convective flux
1st Order Spatial Derivative
x
CC
x
C xxx
xx
2/
x
CC
x
C xxxx
x
2
• Worked for estimating second order derivative (estimate ended up at x).
• Need centered derivative approximation
CDE Explicit Finite Difference
• Grid Pe = vL/D, where L is the grid spacing
• Pe < 1, 4, 10
ttxxttxxttxxttxttxxttxtx CC
x
tCCC
x
tCC
,,,,,2,, 2
v2
D
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 mean Koc (L/kg) Compound mean Koc (L/kg) Compound mean Koc (L/kg)
Acenaphthene 5,028 1,4-Dichlorobenzene(p) 687 Methoxychlor 80,000
Aldrin 48,686 1,1-Dichloroethane 54 Methyl bromide 9
Anthracene 24,362 1,2-Dichloroethane 44 Methyl chloride 6
Benz(a)anthracene 459,882 1,1-Dichloroethylene 65 Methylene chloride 10
Benzene 66 trans-1,2-Dichloroethylene 38 Naphthalene 1,231
Benzo(a)pyrene 1,166,733 1,2-Dichloropropane 47 Nitrobenzene 141
Bis(2-chloroethyl)ether 76 1,3-Dichloropropene 27 Pentachlorobenzene 36,114
Bis(2-ethylhexyl)phthalate 114,337 Dieldrin 25,604 Pyrene 70,808
Bromoform 126 Diethylphthalate 84 Styrene 912
Butyl benzyl phthalate 14,055 Di-n-butylphthalate 1,580 1,1,2,2-Tetrachloroethane 79
Carbon tetrachloride 158 Endosulfan 2,040 Tetrachloroethylene 272
Chlordane 51,798 Endrin 11,422 Toluene 145
Chlorobenzene 260 Ethylbenzene 207 Toxaphene 95,816
Chloroform 57 Fluoranthene 49,433 1,2,4-Trichlorobenzene 1,783
DDD 45,800 Fluorene 8,906 1,1,1-Trichloroethane 139
DDE 86,405 Heptachlor 10,070 1,1,2-Trichloroethane 77
DDT 792,158 Hexachlorobenzene 80,000 Trichloroethylene 97
Dibenz(a,h)anthracene 2,029,435 -HCH (-BHC) 1,835 o-Xylene 241
1,2-Dichlorobenzene(o) 390 -HCH (-BHC) 2,241 m-Xylene 204
-HCH (Lindane) 1,477 p-Xylene 313
Retardation
• Incorporate adsorbed solute mass
Kd
R b1
Vs
VR
Retardation
t
CR
x
CD
x
Cv
2
2
Kinetics• dC/dt = constant: zero order
• dC/dt = -kC: first order
• Integrate:
TTC
Ctk
C
C0
)(
)0(
kTC
TC
)0(
)(ln
2/121
lnT
k
Two-Site Conceptual ModelTwo-Site Conceptual Model
Instantaneous Adsorption Sites
Mobile WaterAirAir
Kinetic Adsorption Sites
22 1 s
s CKdCFdt
dC
Two-site model
• Selim et al., 1976; Cameron and Klute, 1977; and many more
• Instantaneous equilibrium and kinetically-limited adsorption sites
• Different constituents:• “Soil minerals, organic matter, Fe/Al
oxides”• ‘F’ = Fraction of instantaneous sites• ‘’ = First-order rate constant
Batch Sorption KineticsBatch Sorption Kinetics
1000
1200
1400
1600
1800
2000
2200
2400
2600
0 5 10 15 20 25Time (hours)
Co
nc
en
tra
tio
n (
dp
m/m
l)
First Order Model for All Data
First Order Model for t > 1 hour: = 0.11 hr-1
Mean column = 0.06 hr-1
Two-Region Conceptual ModelTwo-Region Conceptual Model
Dynamic Soil Region
Mobile Water
AirAir
Immobile Water
Stagnant Soil Region
immbimim CCKdF
dt
dC 1
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 one-dimensional 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 first-order 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:
• Constant mean velocity in x direction only!
t
CR
z
CD
y
CD
x
CD
x
Cv zzyyxx
2
2
2
2
2
2
•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
tD
vtx
tD
MC
xxxx
i 4exp
2
21
tD
vtxerfc
D
vx
tD
vtxerfc
D
vx
v
Dxv
M
C
xxxx
xxxxxxc
22exp
22exp
2
2exp1
2-D Instantaneous Source
tD
y
tD
vtx
DDt
MC
yyxxyyxx
i 44exp
4
222
2-D Instantaneous Source Solution
Dyy
Dxx
Back dispersion Extreme concentration
t = 1t = 25
t = 51
3-D Instantaneous Source
tD
z
tD
y
tD
vtx
DDDt
MC
zzyyxxzzyyxx
i 444exp
8
222
33
3
3-D Instantaneous Source SolutionDzz
Dxx
Back dispersion
Extreme concentration
t = 1t = 25
t = 51
Dyy
3-D Continuous Source
tD
vtRerfc
D
Rv
tD
vtRerfc
D
Rv
DDR
Dxv
M
C
xxxx
xxxx
zzyy
xxc
22exp
22exp
8
2exp3
zz
xx
yy
xx
D
Dz
D
DyxR 222
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 semi-infinite porous media, Water Resources Research 20 (10) 2719-2733.
Coordinate systems
• x increasing downward
x
z
y
x
z
y
r
Boundary Conditions
• Semi-infinite source
x
z
y
-∞
-∞
Boundary Conditions
• Finite rectangular source
x
z
y
-b
-a
b
a
Boundary Conditions
• Finite Circular Source
x
z
y
r = a
Initial Conditions
• Finite Cylindrical Source
x
z
y
r = a
x1
x2
Initial Conditions
• Finite Parallelepipedal Source
x
z
y
x1
x2
b
a
Comparing with Hunt
• M3 = r2 (x1 – x2) Co (=1, small, high C)
• Co = 1/[r2 (x1 – x2)] = 106 for r = x= 0.01 x
z
y
r =
a
x 1 x 2
Wells?• Finite Parallelepipedal Source
x
z
y
x1
x2
b
a
Pathlines
Scale-Dependent Dispersivities and Scale-Dependent Dispersivities and The Fractional Convection - Dispersion The Fractional Convection - Dispersion
EquationEquation
Primary Source:Ph.D. DissertationDavid BensonUniversity of Nevada Reno, 1998
Representative Elementary Representative Elementary Volume (REV)Volume (REV)
From Jacob Bear
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?)
Conventional DerivativesConventional Derivatives
1 rr
rxdx
xd
From Benson, 1998
Conventional DerivativesConventional Derivatives
1 rr
rxdx
xd
From Benson, 1998
Fractional DerivativesFractional Derivatives
The gamma function interpolates the factorial function. For integer n, gamma(n+1) = n!
0
1)( dtetx tx
quuq xuq
uxD
)1(
)1(
Fractional DerivativesFractional Derivatives
From Benson, 1998
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
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
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
Brownian Motion and Levy Brownian Motion and Levy FlightsFlights
DuU
D
eu
uDuU
uuU
uuU
Prln
lnPrln
1,1Pr
Pr
Monte-Carlo Simulation of Levy Monte-Carlo Simulation of Levy FlightsFlights
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
MATLAB Movie/MATLAB Movie/Turbulence AnalogyTurbulence Analogy
FADE (Levy Flights)
100 ‘flights’, 1000 time steps each
50500
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.
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
lbm
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
Continuous Time Random Walk Model
Mike Sukop/FIU
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.
IntroductionIntroduction
• Continuous Time Random Walk (CTRW) models – Semiconductors [Scher and Lax, 1973]– Solute transport problems [Berkowitz and
Scher, 1995]
IntroductionIntroduction
• 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 (s,t)
• Estimation of this function is central to application of the CTRW model
IntroductionIntroduction
• The functional form (s,t) ~ t-1- ( > 0) is of particular interest [Berkowitz et al, 2001]
characterizes the nature and magnitude of the dispersive processes
Ranges of
≥ 2 is reported to be “…equivalent to the ADE…” – For ≥ 2, the link between the dispersivity (
= D/v) in the ADE and CTRW dimensionless b is b = /L
between 1 and 2 reflects moderate non-Fickian behavior
• 0 < < 1 indicates strong ‘anomalous’ behavior
Fits
0.0
0.2
0.4
0.6
0.8
1.0
0 20 40 60 80 100 120 140Time (min)
C/C
0
DataCTRW Fit
ADE Fit
11 cm 17 cm 23 cm
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 variably-saturated 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
3/4
2
0a
a
D
D
dx
dCDJ
C
xJD
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5Volumetric Air Content
D/D
o
Maxwell (1873)
Buckingham (1904)
Penman (1940)
Marshall (1959)
Millington (1959)
Wesseling (1962)
Currie (1965)
WLR(Marshall):Moldrup et al (2000)
Total Mass
• At Equilibrium:
Henry’s Law
• Dimensionless:
• Common:
wHg CKC
atm m3 mol-1
Total Mass
• At Equilibrium:
ldblHlT CKCKCtzM )(),(
VLEACH
• Processes are conceptualized as occurring in a number of distinct, user-defined polygons that are vertically divided into a series of user-defined 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
0
20
40
60
80
100
0 5 10 15 20Time (years)
C (
mg
/l)
VLEACH 0.1 m cells
VLEACH 1 m cells
VLEACH 10 m cell
CDE Flux-averagedconcentrations (Dispersivity asshown)
Initial and Boundary Conditions:
C(x,0) = 100 mg/lC(0,t) = 0 mg/l
General Conditions:
q = 1 m/year = 0.5
VLEACH time step:
0.01 years
= 0.05 m
= 0.5 m
= 5 m
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
1,1,1, and ,, kskgkw 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 Ground-Water Flow
By Weixing Guo and Christian D. LangevinU.S. Geological SurveyTechniques 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
714.0m kg 35
m kg 100010253
3
dC
d