Lecture 3

Contaminant Transport Mechanisms and Principles

BASIC DEFINITIONS

Ground surface

Belowgroundsurface(BGS)

Vadose zone,unsaturated zone

Capillary fringe

Water tableSaturated zone

Confining bedConfined aquifer orartesian aquifer

Water-table, phreatic,or unconfined aquifer

Capillary fringe may be >200 cm in fine siltIn capillary fringe water is nearly saturated, but held in tension in soil pores

MICRO VIEW OF UNSATURATED ZONE

air

water

solid

Contaminant concentrations:

Cw, mg/L concentration in water

Cg, mg/L or ppmvconcentration in gas

Cs, gm/kgconcentration in solids

PARTITIONING RELATIONSHIPS

Solid ↔ water

Kd = partition coefficient

Water ↔ vapor

H = Henry’s Law constant

watermg/Lsolid mg/kgK

CC

dw

s ==

watermg/mair mol/mH

CC

3

3

w

g ==

HENRY’S LAW CONSTANT

H has dimensions: atm m3 / mol

H’ is dimensionless

H’ = H/RTR = gas constant = 8.20575 x 10-5 atm m3/mol °KT = temperature in °K

NOTE ON SOIL GAS CONCENTRATION

Soil gas is usually reported as:ppmv = parts per million by volume

g/mole weight molecularmL/mole 24,000 (mg/L) C

(ppmv) C gg

×=

VOLUME REPRESENTATION

Gas volume, Vg

Water volume, VW

Solid volume, VS

Total volume, VT

Void volume, VV

VOLUME-RELATED PROPERTIESBulk density = ρb = mass of solids

total volumePorosity = n = θ = VV/VT

Volumetric water content or water-filled porosity = θW = VW/VT

Saturation = S = VW/VV

Gas-filled porosity = θg (or θa) = Vg/VT

θW + θg = n

CONTAMINANT CONCENTRATIONIN SOIL

Total mass in unit volume of soil:CT = ρb Cs + θw CW + θg Cg

If soil is saturated, θg = 0 and θW = nCT = ρb Cs + n CW

NOMENCLATURE FOR DARCY’S LAW

Q = K i AK = hydraulic conductivityi = hydraulic gradient = dh/dLA = cross-sectional area

Velocity of ground-water movementu = Q / n A = q / n = K i / n = average linear velocityn A = area through which ground water flowsq = Q / A = Darcy seepage velocity = Specific discharge

For transport, n is ne, effective porosity

ADVECTIVE FLUX

Flowing ground water carries any dissolved material with it → Advective Flux

JA = n u C mass / area / time

= mass flux through unit cross section due to ground-water advection

n is needed since no flow except in pores

DIFFUSIVE FLUX

Movement of mass by molecular diffusion (Brownian motion) – proportional to concentration gradient

in surface water !!!

DO is molecular diffusion coefficient [L2/T]

xCDJ OD ∂

∂−=

DIFFUSIVE FLUX

In porous medium, geometry imposes constraints:

τ = tortuosity factorD* = effective diffusion coefficient

Factor n must be included since diffusion is only in pores

xCn*D

xCnDJ OD ∂

∂−=

∂∂

−= τ

TORTUOSITY

Solute must travel a tortuous path, winding through pores and around solid grains

Common empirical expression:

L = straight-line distanceLe = actual (effective) path

τ ≈ 0.7 for sand

2

eLL

⎟⎟⎠

⎞⎜⎜⎝

⎛=τ

NOTES ON DIFFUSION

Diffusion is not a big factor in saturated ground-water flow – dispersion dominates diffusion

Diffusion can be important (even dominant) in vapor transport in unsaturated zone

MECHANICAL DISPERSION

ABC

A

BC

A arrives first, then B, then C → mechanical dispersion

MECHANICAL DISPERSION

Viewed at micro-scale (i.e., pore scale) arrival times A, B, and C can be predicted

Averaging travel paths A, B, and C leads to apparent spreading of contaminant about the mean

Spatial averaging → dispersion

MECHANICAL DISPERSION

Dispersion can be effectively approximated by the same relationship as diffusion—i.e., that flux is proportional to concentration gradient:

Dispersion coefficient, DM = αL u

αL = longitudinal dispersivity (units of length)

xCnDJ MM ∂

∂−=

TRADITIONAL VIEW OF HYDRODYNAMIC DISPERSION

ACTUAL OBSERVATIONS OF PLUMES

USGS Cape CodResearch Site

Source: NOAA Coastal Services Center, http://www.csc.noaa.gov/crs/tcm/98fall_status.htmlAccessed May 14, 2004. Source: U.S. Geological Survey, Cape Cod Toxic

Substances Hydrology Research Site, http://ma.water.usgs.gov/CapeCodToxics/location.html. Accessed May 14, 2004.

MONITORING WELL ARRAY

USGS MONITORING NETWORK

Source: http://ma.water.usgs.gov/CapeCodToxics/photo-gallery.html Photo by D.R. LeBlanc.

OBSERVED BROMIDE PLUME –

HORIZONTALVIEW

Significant longitudinal dispersion, but limitedlateral dispersion

OBSERVED BROMIDE PLUME –VERTICAL VIEW

Limited vertical dispersion

LONGITUDINALDISPERSION VS. LENGTH SCALE

Lateral and vertical dispersivity

TRANSPORT EQUATION

Combined transport from advection, diffusion, and dispersion (in one dimension):

DH = D* + DM = τ DO + αL u = hydrodynamic dispersion

xCDnuCJ

xCnD

xCn*DnuCJ

JJJJ

H

M

MDA

∂∂

−=

∂∂

−∂∂

−=

++=

TRANSPORT EQUATION

Consider conservation of mass over control volume (REV) of aquifer.

REV = Representative Elementary VolumeREV must contain enough pores to get a meaningful representation (statistical average or model)

TRANSPORT EQUATION

S/SxJ

tC

S/SJt

C

T

T

±∂∂

−=∂

∂

±•∇−=∂

∂

Change incontaminantmass withtime

Flux in lessflux out ofREV

Sources and sinks due toreactions

(1)

(2)

TRANSPORT EQUATIONCT = total mass (dissolved mass plus mass adsorbed to

solid) per unit volume= ρb CS + n CW = ρb CS + n C (3)

Substitute Equation 3 into Equation 2:

( ) ( ) S/SxCnDnuC

xtnC

tC

HSb ±⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

−=∂

∂+

∂ρ∂ (4)

↑ no solid phase in flux term

Note: W subscript dropped for convenience and for Consistency with conventional notation

TRANSPORT EQUATION

( )

2

2

db

H

db

2

2

Hdb

xC

nnK

DxC

nnK

utC

xCnD

xCnu

tCnK

∂∂

⎟⎠⎞

⎜⎝⎛ +ρ

+∂∂

⎟⎠⎞

⎜⎝⎛ +ρ

−=∂∂

∂∂

+∂∂

−=∂∂

+ρ (5)

CS = Kd C by definition of Kd

Assume spatially uniform n, ρb, Kd, u, and DH and no S/S

(6)

TRANSPORT EQUATION

ddbdb R

nK1

nnK

=ρ

+=+ρ

“Retardation factor”, Rd

2

2

d

H

d xC

RD

xC

Ru

tC

∂∂

+∂∂

−=∂∂

Substituting Equation 7 into Equation 6:

(8)

(7)

Effect of adsorption to solids is an apparent slowing of transportof dissolved contaminantsBoth u and DH are slowed

SOLUTION OF TRANSPORT EQUATION

Equation 8 can be solved with a variety of boundary conditions

In general, equation predicts a spreading Gaussian cloud

[ [

0.0

t1 t2 t0

0.2

0.4

0.6

0.8

1.0

x - a

x-

x + a

Spreading of a solute slug with time due to diffusion. A slug of solute wasinjected into the aquifer at time t with a resulting initial concentration of C .00

Adapted from: Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992.

C/C

0R

elat

ive

Con

cent

ratio

n

1-D SOLUTION OF TRANSPORT EQUATION

For instantaneous placement of a long-lasting source (for example, a spill that leaves a residual in the soil), solution is:

Where Co = C(x=0, t) = constant concentration at source location x = 0Solution is a front moving with velocity u/Rd

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

tDR4utxRerfc

2Ct,xC

Hd

do

+

0.0

0.2

0.3

0.10.16

0.50

0.84

0.4

0.6

0.7

0.8

0.9

1.0

Mea

n

σ σ

x =

x

ut/Rd

Adapted from Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992.

C/C

0R

elat

ive

Con

cent

ratio

n

The profile of a diffusing front as predicted by the complementary error function.

Moving front of contaminant from constant source

Moving front of contaminant from constant source

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9

Distance, x

Con

cent

ratio

n, C

(x,t)

C0 = 10u = 1DH = 0.1Rd = 1

t = 1ut = 1

t = 5ut = 5

t = 3ut = 3

Effect of dispersion coefficient

Effect of retardationEffect of Rd on moving front of contaminant

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9

Distance, x

Con

cent

ratio

n, C

(x,t)

C0 = 10u = 1DH = 0.1

t = 3ut = 3Rd = 2

t = 3ut = 3Rd = 1

1-D SOLUTIONS

Mass input here

Front at time t

1-D

M

C =

ML2

MM

L2T

M exp-2n D

t = 0 t = t1

1/2 1/2

x = 0 v

to

x = 0x = 0 vv

π x xt [ [(x-vt)4D t

2

C = M erfc2nv Dx

( (x-vt t C = M

nvfor x > 0

∞

( (

M, M are instantaneous or continuous plane

sources

.

.

..2

Transport of a Conservative Substance from Pulse and Continuous Sources

Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0

. Continuous Input of Mass PerUnit Time M in Steady State

.

Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.

Mass input here

2-D SOLUTIONS

2-DM, M are instantaneous

or continuous line sources

MML [[

M ML-T [[

t = 0 t = t Plume at time t1

to

v vv

x xy y

∞

.

. xy

C= M exp4n Dπ x xDy y[ [(x-vt)

4D t t

2 y4D t

2+ C= M exp erfc

4n (vr)1/2 1/2π xDy D tx[ [(x-r)v2D ( (r-vt

2

Transport of a Conservative Substance from Pulse and Continuous Sources

Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.

Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0

Continuous Input of Mass PerUnit Time M in Steady State

.

.C= M exp

2n (vr)1/2 1/2π xDy[ [(x-r)v2D

.

.

3-D SOLUTIONS

3-DM, M are instantaneous

or continuous point sources

MML [[

M MT [[

v v

.

.

C= M exp erfc8n rπ xDy D tx[ [(x-r)v

2D ( (r-vt2

Transport of a Conservative Substance from Pulse and Continuous Sources

Adapted from: Hemond, H. F. and E. J. Fechner-Levy. Chemical Fate and Transport in the Environment.2nd ed. San Diego: Academic Press, 2000.

Dimensions Pulse Input of Mass M Continuous Input of Mass PerUnit Time M Starting at Time t = 0

Continuous Input of Mass PerUnit Time M in Steady State

.

.C= M exp

x[ [(x-r)v2D

.

.

v

C =M

exp-

8n D

t = 0 t = t1

3/2 3/2

to

π x Dy

x

t

[ [(x-vt)4D t

2 2

y

y 2z4D t

∞

Dz Dz 4n rπ DyDz

+ +z4D t

z y

x

z y

Plume at time t

z y