Lecture 03

Lecture 3 Contaminant Transport Mechanisms and Principles

BASIC DEFINITIONS Ground surface Vadose zone, unsaturated zone

Capillary fringe Water table

Saturated zone Confining bed

Below ground surface (BGS) Water-table, phreatic, or unconfined aquifer Confined aquifer or artesian aquifer

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

MICRO VIEW OF UNSATURATED ZONE Contaminant concentrations: water

Cw, mg/L concentration in water

air Cg, mg/L or ppmv concentration in gas solid

Cs, gm/kg concentration in solids

PARTITIONING RELATIONSHIPS

Solid ↔ water

Cs mg/kg solid = Kd = Cw mg/L water

Kd = partition coefficient

Water ↔ vapor

Cg

mol/m3 air =H= 3 Cw mg/m water

H = Henry’s Law constant

HENRY’S LAW CONSTANT H has dimensions: atm m3 / mol H’ is dimensionless H’ = H/RT R = gas constant = 8.20575 x 10-5 atm m3/mol °K T = temperature in °K

NOTE ON SOIL GAS CONCENTRATION Soil gas is usually reported as: ppmv = parts per million by volume

Cg (ppmv) =

Cg (mg/L) × 24,000 mL/mole molecular weight g/mole

VOLUME REPRESENTATION

Gas volume, Vg Void volume, VV

Water volume, VW

Solid volume, VS

Total volume, VT

VOLUME-RELATED PROPERTIES Bulk density = ρb = mass of solids total volume Porosity = 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 CONCENTRATION IN SOIL Total mass in unit volume of soil: CT = ρb Cs + θw CW + θg Cg If soil is saturated, θg = 0 and θW = n CT = ρb Cs + n CW

NOMENCLATURE FOR DARCY’S LAW Q = KiA K = hydraulic conductivity i = hydraulic gradient = dh/dL A = cross-sectional area

Velocity of ground-water movement u = Q / n A = q / n = K i / n = average linear velocity n A = area through which ground water flows q = 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

∂C JD = −DO ∂x

in surface water !!!

DO is molecular diffusion coefficient [L2/T]

DIFFUSIVE FLUX In porous medium, geometry imposes constraints:

∂C ∂C JD = − τ DO n = −D*n ∂x ∂x τ = tortuosity factor D* = effective diffusion coefficient Factor n must be included since diffusion is only in pores

TORTUOSITY Solute must travel a tortuous path, winding through pores and around solid grains Common empirical expression: L = straight-line distance Le = actual (effective) path τ ≈ 0.7 for sand

⎛L ⎞ τ = ⎜⎜ ⎟⎟ ⎝ Le ⎠

2

NOTES ON DIFFUSION Diffusion is not a big factor in saturated groundwater flow – dispersion dominates diffusion Diffusion can be important (even dominant) in vapor transport in unsaturated zone

MECHANICAL DISPERSION

C B

C B A

A

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:

∂C JM = −DM n ∂x Dispersion coefficient, DM = αL u αL = longitudinal dispersivity (units of length)

TRADITIONAL VIEW OF HYDRODYNAMIC DISPERSION

ACTUAL OBSERVATIONS OF PLUMES

USGS Cape Cod Research Site

Source: NOAA Coastal Services Center, http://www.csc.noaa.gov/crs/tcm/98fall_status.html Accessed 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 – HORIZONTAL VIEW

Significant longitudinal dispersion, but limited lateral dispersion

OBSERVED BROMIDE PLUME – VERTICAL VIEW

Limited vertical dispersion

LONGITUDINAL DISPERSION VS. LENGTH SCALE

Lateral and vertical dispersivity

TRANSPORT EQUATION Combined transport from advection, diffusion, and dispersion (in one dimension): J =JA +JD +JM ∂C ∂C J = nuC − D * n − DM n ∂x ∂x ∂C J = nuC − D H ∂x

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

TRANSPORT EQUATION Consider conservation of mass over control volume (REV) of aquifer. REV = Representative Elementary Volume REV must contain enough pores to get a meaningful representation (statistical average or model)

TRANSPORT EQUATION Change in contaminant mass with time

∂C T ∂t ∂C T ∂t

Flux in less flux out of REV

Sources and sinks due to reactions

=

−∇•J

±

S/S

(1)

=

∂J − ∂x

±

S/S

(2)

TRANSPORT EQUATION CT = =

total mass (dissolved mass plus mass adsorbed to solid) per unit volume ρb CS + n CW = ρb CS + n C

(3)

Note: W subscript dropped for convenience and for Consistency with conventional notation Substitute Equation 3 into Equation 2:

∂ (ρbCS ) ∂ (nC) ∂ ⎛ ∂C ⎞ + =− ⎜ nuC − DHn ⎟ ± S/S ∂t ∂t ∂x ⎝ ∂x ⎠ ↑ no solid phase in flux term

(4)

TRANSPORT EQUATION CS = Kd C by definition of Kd Assume spatially uniform n, ρb, Kd, u, and DH and no S/S

∂C ∂C ∂ C (ρbK d + n) = −nu + nDH 2 ∂t ∂x ∂x 2 DH u ∂ C ∂C ∂C + =− 2 K n K n ρ + ρ + ∂t ⎞ ∂x ⎛ b d ⎞ ∂x ⎛ b d ⎜ ⎟ ⎜ ⎟ n n ⎝ ⎠ ⎝ ⎠ 2

(5)

(6)

TRANSPORT EQUATION “Retardation factor”, Rd

ρbK d + n n

=

ρbK d 1+ n

=

Rd

(7)

Substituting Equation 7 into Equation 6:

u ∂ C DH ∂ C ∂C =− + ∂t R d ∂x R d ∂x 2 2

(8)

Effect of adsorption to solids is an apparent slowing of transport of dissolved contaminants Both 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

x

-

[ [

x-a x+a

Relative Concentration C/C0

1.0 0.8 0.6 0.4 0.2 0.0 t1

t2

t0

Spreading of a solute slug with time due to diffusion. A slug of solute was injected into the aquifer at time t with a resulting initial concentration of C0 . 0

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

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:

⎛ R d x − ut ⎞ Co ⎟ erfc⎜ C(x, t ) = ⎜ 4R D t ⎟ 2 d H ⎠ ⎝

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

1.0 0.9 0.8 0.7 0.6 0.50 0.4 0.3 0.16

0.2

Mean

Relative Concentration C/C0

0.84

0.1

x

0.0

+s

s

x = ut/Rd The profile of a diffusing front as predicted by the complementary error function. Adapted from Fetter, C. W. Contaminant Hydrogeology. New York: Macmillan Publishing Company, 1992.

Moving front of contaminant from constant source 10

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

9

Concentration, C(x,t)

8 7

t=1 ut = 1

6

t=3 ut = 3

t=5 ut = 5

5 4 3 2 1 0 0

1

2

3

4

5

Distance, x Moving front of contaminant from constant source

6

7

8

9

Effect of dispersion coefficient

Effect of Rd on moving front of contaminant Effect of retardation

10 9

Concentration, C(x,t)

t=3 ut = 3 Rd = 1

t=3 ut = 3 Rd = 2

8 7

C0 = 10 u=1 DH = 0.1

6 5 4 3 2 1 0 0

1

2

3

4

5

Distance, x

6

7

8

9

1-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources

.

Continuous Input of Mass Per Unit Time M Starting at Time t = 0

Dimensions

Pulse Input of Mass M

1-D

M exp - (x-vt) C= 1/2 1/2 4Dxt 2np t D x

.

M, M are instantaneous or continuous plane sources M

.

[ [ 2

x=0

v

.

(

x-vt C = M erfc 2nv 2 Dx t

x=0

v

(

.

Continuous Input of Mass Per Unit Time M in Steady State

.

C= M nv

( for x > 0 (

x=0

M L2

M M 2 L T

v to ∞

t=0

t = t1

Mass Front at input here time t

Mass input here

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

2-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources

Dimensions 2-D

.

M, M are instantaneous or continuous line sources

[[ . [ [

M M

M L

M L-T

Continuous . Input of Mass Per Unit Time M Starting at Time t = 0

Pulse Input of Mass M

[

(x-vt)2 y2 M exp + C= 4Dx t 4Dy t 4np t Dx Dy v t = t1 t=0 y

x

[

.

M

C= 4np 1/2 (vr)1/2

[ [ ( (

exp (x-r)v erfc r-vt 2D x 2 Dx t Dy

v

Continuous Input . of Mass Per Unit Time M in Steady State

[ [

.

exp (x-r)v C= 1/2 1/2 2D x 2np (vr) Dy M

v

Plume at time t y

y x

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

x

to ∞

3-D SOLUTIONS Transport of a Conservative Substance from Pulse and Continuous Sources

Dimensions 3-D

.

M, M are instantaneous or continuous point sources

[[ . [ [

M M

M L M T

Continuous . Input of Mass Per Unit Time M Starting at Time t = 0

Pulse Input of Mass M

exp -

.

M

C=

[

8np

3/2 3/2

t

(x-vt)2 4Dx t

+

Dx Dy D z y2

4Dy t

+

v

z y

z2 4Dz t

[

M C= 8np r DyD z

z y

[ [ ( (

exp (x-r)v erfc r-vt 2D x 2 Dx t

v

Continuous Input . of Mass Per Unit Time M in Steady State

.

M C= 4np r DyD z

z y

x t=0

[ [

exp (x-r)v 2D x

v

to ∞ t = t1

Plume at time t

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