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

p:/

/ww

w-h

isto

ry.m

cs.s

t-a

nd

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