CONTAMINANT TRANSPORT
MECHANICAL ASPECTS
ADVECTION
v=
Kdh
= average linear velocity
φ edl
DISPERSION/DIFFUSION
due to variable advection
that occurs in the transition zone between
two domains of the fluid with different compositions
(diffusion is caused by chemical gradients)
Later we will look at some fundamental
Some NONMECHANICAL ASPECTS : Decay & Sorption
In the direction of flow we consider
LONGITUDINAL DISPERSION:
Velocity variation
within pores:
Velocity variation
between pores:
Variation of
flow path lengths
TRANSVERSE DISPERSION (normal to the flow path):
Splitting of
flow paths
1
These physical mixing processes are combined and referred to as
"Mechanical Dispersion"
Mechanical dispersion is related to average pore velocity by
dispersivity (α)
Mechanical Dispersion = D = α v
dispersivity (α)
units of length
increases with increased heterogeneity
and thus with travel distance
Diffusion:
Movement of dissolved species from areas of high
concentration to low concentration
Fick's Law:
Flux = F = − D
∂C
∂l
D in open water for common groundwater ions
~1x10-9 to 2x10-9 m2/sec
D* represents D in porous media, and is reduced due to
tortuosity
y and effective p
porosity
y
D* ~ 2x10-11 to 5x10-10 m2/sec
some suggest D* = D
τ=
φe
τ
actual path
direct path
2
Transport Equations
The combined mechanical and chemical diffusion process
is treated with a Fick's Law approach
F = − Dl
∂C
∂l
But here D is
Hydrodynamic Dispersion expressed as
Dl = α l v l + D*
Studies indicate scale dependence of dispersivity, α
Dispersivities at various scales & measured by various
methods as compiled by Stan Davis et al.
Table B1 in the book, "Ground Water Tracers"
3
Table B1 CONTINUED Dispersivities at various scales &
measured by various methods from "Ground Water Tracers"
Break through Curves
Co continuous
source starting
at t=0
C outflow
initially fresh water
x=L
x=0
Note t’ would be average travel
time to this point. Why?
C=0
x=0
C
Co
1.0
0.5
x=L
along the column at various times
t1
t2
0
C
Co
Contour C/Co versus x
location at one time
0.3
0.1
0.0
1.0
0.9
0.7
Co
0.5
schematic C/Co for t=t’
t3
x
1.0
Graph C/Co at one x
location as a
function of time
0.5
0first arrival t
Graph C/Co versus x
for 3 different times
1
average arrival time t2
t3 constant C=Co
4
Mechanical Transport Equations can be derived by considering
an elemental volume as we did for the flow equations
We leave the derivation to a later course & consider the practical analytical forms
∂C
∂ 2C
∂ 2C
∂ 2C
∂C
= Dl 2 + Dt 2 + Dv 2 − vl
∂l
∂t
∂ll
∂lt
∂lv
C
t
l
D
v
ll
lt
lv
concentration in fluid
Note differing form of
time
flow equations
spatial coordinate
∂h T ⎡ ∂ 2 h ∂ 2 h ⎤
dispersion tensor
= ⎢
+
⎥
interstitial velocity
∂t S ⎣ ∂x 2 ∂y 2 ⎦
reflects the flow direction
reflects the direction transverse laterally to flow
reflects the direction transverse vertically to flow
Equation for mechanical transport in 1-D
∂C
∂ 2C
∂C
= Dx 2 − v x
∂t
∂x
∂x
C
t
x
D
concentration in fluid
time
spatial coordinate
dispersion tensor
v
interstitial velocity
5
Analytical Solution for transport in 1-D flow field
continuous source
1D spreading
without chemical reaction
This is an appropriate model for transport along a sand column
It will over estimate C at x if applied to a case with spreading in the
transverse lateral or vertical directions
It will predict the break through curves we looked at earlier
⎛ x − v xt ⎞
⎛ x + v xt ⎞ ⎞
⎛ vxx ⎞
Co ⎛⎜
⎜
⎟
⎟⎟
⎟erfc
C=
erfc
f
+ exp⎜⎜
f ⎜
⎟
⎜2 D t⎟
⎜ 2 D t ⎟⎟
2 ⎜
⎝ Dx ⎠
x ⎠
x ⎠⎠
⎝
⎝
⎝
erfc is the complimentary error function
Error Function
Tables are listed in
the back of ground
water hydrology
books
6
Analytical Solution for transport in 1-D flow field
slug source
3D spreading
without chemical reaction
Analytical Solution for transport in 1-D flow field
slug source
3D spreading
without chemical reaction
C(x = vt + X, y = Y, z = Z ) =
M
8( πt )
3
2
⎛ X2
Y2
Z 2 ⎞⎟
⎜−
−
−
⎜ 4 Dx t 4 Dy t 4 Dz t ⎟
⎝
⎠
exp
p
Dx Dy Dz
IMPORTANT! X Y Z = distance from center of mass
Maximum concentration will occur at
the center of mass
Where X=Y=Z=0
M
Cmax =
8( πt )
3
2
Dx Dy Dz
7
So we just considered an Analytical Solution for transport in
1-D flow field
slug source
3D spreading
without chemical reaction
C(x = vt + X, y = Y, z = Z ) =
M
8( πt )
3
2
⎛ X2
Y2
Z 2 ⎞⎟
⎜−
−
−
⎜ 4 Dx t 4 Dy t 4 Dz t ⎟
⎝
⎠
exp
Dx Dy Dz
X Y Z = distance from center of mass in each direction
NEXT Analytical Solution for transport in
1D flow field
continuous source
3D spreading
without chemical reaction
8
Analytical Solution
for transport in
uniform 1D flow
continuous source
3D spreading
without chemical
reaction
see previous
graphic
g
ap c
Upper case
Y and Z
Are the source
width and height
Analytical Solution
for transport in
uniform 1D flow
continuous source
3D spreading
without chemical
reaction
If source is on the
water table such that
spreading is only
downward
Omit (/2) on Z terms
C( x, y , z , t ) =
⎛ x − v xt ⎞ ⎞
Co ⎛⎜
⎟⎟
erfc⎜
⎜ 2 D t ⎟⎟
8 ⎜
x ⎠⎠
⎝
⎝
⎛ ⎛
⎞
⎛
⎞⎞
⎜ ⎜ y+ Y ⎟
⎜ y − Y ⎟⎟
⎜ erf ⎜
2 ⎟⎟
2 ⎟ − erf ⎜
⎜ ⎜
⎜
x ⎟⎟
x⎟
⎜⎜ ⎜ 2 Dy ⎟
⎜ 2 Dy ⎟ ⎟⎟
v ⎠⎠
v
⎝
⎠
⎝
⎝
⎛ ⎛
⎞
⎛
⎞⎞
⎜ ⎜ z+ Z ⎟
⎜ z − Z ⎟⎟
⎜ erf ⎜
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ ⎜
⎜
x⎟
x ⎟⎟
⎜⎜ ⎜ 2 Dz ⎟
⎜ 2 Dz ⎟ ⎟⎟
v
v ⎠⎠
⎠
⎝
⎝ ⎝
⎛ x − v xt ⎞ ⎞
Co ⎛⎜
⎟⎟
C( x, y , z , t ) =
erfc⎜
⎜ 2 D t ⎟⎟
8 ⎜
x ⎠⎠
⎝
⎝
⎛ ⎛
⎜ ⎜ y+ Y
⎜ erf ⎜
2
⎜ ⎜
x
⎜⎜ ⎜ 2 Dy
v
⎝ ⎝
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟ ⎟⎟
⎠⎠
⎛ ⎛
⎞
⎛
⎞⎞
⎜ ⎜
⎟
⎜
⎟⎟
⎜ erf ⎜ z + Z ⎟ − erf ⎜ z − Z ⎟ ⎟
⎜ ⎜
⎜
x ⎟⎟
x⎟
⎜⎜ ⎜ 2 Dz ⎟
⎜ 2 Dz ⎟ ⎟⎟
v ⎠⎠
v
⎠
⎝
⎝ ⎝
⎞
⎛
⎟
⎜ y− Y
⎟ − erf ⎜
2
⎜
⎟
x
⎜ 2 Dy
⎟
v
⎠
⎝
9
Analytical Solution
for transport in
uniform 1D flow
continuous source
3D spreading
without chemical
reaction
If source is of full
vertical extent in a
confined aquifer
OR
if you are far from a
limited extent
source in a confined
aquifer
⎛ x − v xt ⎞ ⎞
Co ⎛⎜
⎟⎟
erfc⎜
C( x, y , z , t ) =
⎜ 2 D t ⎟⎟
4 ⎜
x ⎠⎠
⎝
⎝
⎛ ⎛
⎜ ⎜ y+ Y
⎜ erf ⎜
2
⎜ ⎜
x
⎜⎜ ⎜ 2 Dy v
⎝ ⎝
⎞
⎛
⎟
⎜ y− Y
⎟ − erf ⎜
2
⎟
⎜
x
⎟
⎜ 2 Dy
v
⎠
⎝
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟ ⎟⎟
⎠⎠
Change Co/8 to Co/4
Omit z terms
Decay
dN
= −λN
dt
or N = N oe (− λt )
0.693
where
h
λ=
T1
decay
constant
2
0.693 is the
natural log of 0.5
10
Retardation - Adsorption
Units
Kd
ml
mg
R=
Vwater
Vcontaminant
⎛ ρb
⎞
⎜
= ⎜ 1 + K d ⎟⎟
φe
⎝
⎠
Equation for transport in 1-D with
Decay, Retardation, Reaction, Source
Divide D's and V's by R
∂C D x ∂ 2C v x ∂C W(C − C' ) CHEM
+
+
− λC
=
−
∂t
R ∂x 2 R ∂ x
Rφb
φ
C
t
b
x
D
R
concentration in fluid
time
aquifer thickness
spatial coordinate
dispersion tensor
retardation
t d ti
coefficient
ffi i t
interstitial velocity
v
W
source fluid flux
φ
porosity
C'
concentration of source fluid
CHEM chemical reaction source/sink per unit volume of aquifer
lambda
decay constant
11
Analytical Solution for transport in
uniform 1D flow
continuous source
3D spreading
With Decay
C( x, y , z , t ) =
⎛ x − v xt ⎞ ⎞
Co ⎛⎜
⎟⎟
erfc⎜
⎜ 2 D t ⎟⎟
8 ⎜
x ⎠⎠
⎝
⎝
⎛ ⎛
⎞
⎛
⎞⎞
⎜ ⎜ y+ Y ⎟
⎜ y − Y ⎟⎟
⎜ erf ⎜
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ ⎜
⎜
Analytical
x ⎟ Solution
x ⎟⎟
Analytical
⎜⎜ ⎜ 2 Dy ⎟ Solution
⎜ 2 Dy ⎟ ⎟⎟
v⎠
⎝ in v ⎠ ⎠
⎝for⎝ transport
for transport in
⎛ ⎛
⎞1D flow
⎛
⎞⎞
uniform
⎜ ⎜ z + Z ⎟1D
⎜ z − Z ⎟⎟
uniform
flow
⎜ erf ⎜
2 ⎟ − erf ⎜
2 ⎟⎟
continuous
source
⎜ ⎜
⎜
continuous
x ⎟ source
x ⎟⎟
⎜⎜ ⎜ 2 D ⎟
⎜ 2 D ⎟ ⎟⎟
3D
spreading
v⎠
v ⎠⎠
⎝ spreading
⎝
⎝ 3D
with Decay
Decay
with
z
z
C ( x, y , z , t ) =
⎛ x
Co
exp ⎜
⎜ 2α x
8
⎝
⎛
4 λα x
⎜1 − 1 +
⎜
v
⎝
⎛
⎛
⎛
⎜
⎜ x − v t ⎜ 1 + 4 λα x
⎜
⎜
⎜
v
⎝
⎜ erfc ⎜
2 α xv t
⎜
⎜
⎜
⎜
⎝
⎝
⎛
⎜
⎜ erf
⎜
⎜
⎝
⎛
⎜
⎜ erf
⎜
⎜
⎝
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟⎟
⎠
⎟⎟
⎟⎟
⎟⎟
⎠⎠
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎠
Z ⎞
Z ⎞⎞
⎛
⎛
⎜ z+
⎜ z−
⎟⎟
⎟
2 ⎟ − erf ⎜
2 ⎟⎟
⎜
⎜ 2 αzx ⎟
⎜ 2 α zx ⎟⎟
⎜
⎜
⎟⎟
⎟
⎝
⎝
⎠⎠
⎠
Y
⎛
⎜ y+
2
⎜
⎜ 2 αyx
⎜
⎝
⎞
⎟
⎟ − erf
⎟
⎟
⎠
Y
⎛
⎜ y−
2
⎜
⎜ 2 α yx
⎜
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
If R>1
Divide v by R
Upper
Upper case
case
Y and
and Z
Z
Y
Are
the
source
A th
Are
the source
width
Note this
width and
and height
height
includes a
Same
Same
simplification
modifications
modifications apply
apply
of D=αv
for
for downward
downward &
&
no
x
x
no vertical
vertical
Dy if D* is ignored then equivalent to α y v which = α y x
spreading
spreading
v
v
12
C ( x, y , z , t ) =
Analytical
AnalyticalSolution
Solution
⎛ x ⎛ in
for
for
transport
transport
in4λα
C
⎜1 − 1 +
C ( x, y , z , t ) =
exp ⎜
⎜ 2α ⎜
8
⎝1D
uniform
uniform
1D⎝flow
flow v
⎛
⎛
⎛
⎞ ⎞ ⎞⎟
⎜
continuous
continuous
source
source
⎜ x − v t ⎜1 + 4 λα
⎟⎟
⎜
⎟ ⎟⎟
⎜
⎜
v
⎝
⎠
⎜ erfc
⎜ spreading
⎟⎟
3D
3D
spreading
2 α vt
⎜
⎜
⎟⎟
⎜ with
⎜ withDecay
Decay⎠⎟ ⎠⎟
⎝
⎝
o
x
x
⎛ x
Co
exp⎜
⎜ 2α x
2
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
x
⎛
⎛
⎛
⎜
⎜ x − v t ⎜1 + 4λα x
⎜
⎜
v
⎜
⎝
⎜ erfc ⎜
2 α xv t
⎜
⎜
⎜
⎜
⎝
⎝
x
⎛
Y
⎛
⎜
⎜ y+
2
⎜ erf ⎜
⎜
⎜ 2 αyx
⎜
⎜
⎝
⎝
⎞
⎟
⎟ − erf
⎟
⎟
⎠
Y
⎛
⎜ y−
2
⎜
⎜ 2 αyx
⎜
⎝
⎠
⎝
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎠
Upper
Uppercase
case
YYand
andZZ
⎛
Z ⎞
Z ⎞⎞
⎛
⎛
z +the
⎜Are
⎜the
⎟ source
⎜ z − width
⎟⎟
Are
source
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ erf ⎜
⎜
2
α
x
2
α
x
⎜
⎟
⎜
⎟⎟
width
and
height
⎜ and
⎟ height
⎜
⎟⎟
⎜
⎝
⎝
z
z
⎠⎠
ON
ONTHE
THE
CENTER
CENTERLINE
LINE
If R>1
Divide v by R
i.e.
y=z=0
⎛
4λα x
⎜1 − 1 +
⎜
v
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟⎟
⎠
⎟⎟
⎟⎟
⎟⎟
⎠⎠
⎛ ⎛ Y ⎞ ⎞⎛ ⎛ Z ⎞ ⎞
⎜ erf ⎜
⎟ ⎟⎜ erf ⎜
⎟⎟
⎜ ⎜ 2 α x ⎟ ⎟⎜ ⎜ 2 α x ⎟ ⎟
y ⎠ ⎝
z ⎠⎠
⎝
⎝ ⎝
⎠
C ( x , y , z , steadystat e ) =
Analytical
AnalyticalSolution
Solution
⎛ x ⎛ in
for
for
transport
transport
in4λα
C
⎜1 − 1 +
C ( x, y , z , t ) =
exp ⎜
⎜ 2α ⎜
8
⎝1D
uniform
uniform
1D⎝flow
flow v
⎛
⎛
⎛
⎞ ⎞ ⎞⎟
⎜
continuous
continuous
source
source
⎜ x − v t ⎜1 + 4 λα
⎟⎟
⎜
⎜
⎜
v ⎟⎠ ⎟ ⎟
⎝
⎜ erfc
⎜ spreading
⎟⎟
3D
3D
spreading
2 α vt
⎜
⎜
⎟⎟
⎜ with
⎜ withDecay
⎟⎟
Decay
⎝
⎠⎠
⎝
o
x
x
⎞⎞
⎟⎟
⎟⎟
⎠⎠
x
x
⎛
Y
⎛
⎜
⎜ y+
2
⎜ erf ⎜
⎜
⎜ 2 αyx
⎜
⎜
⎝
⎝
⎞
⎟
Y ⎞⎞
⎛
⎟⎟
⎜ y−
2 ⎟⎟
⎜
⎟ − erf
Upper
Upper
case
⎜ 2 α x ⎟⎟
⎟ case
⎟⎟
⎜
⎟
⎠⎠
⎝ Z
⎠andZ
YYand
⎛
Z ⎞
Z ⎞⎞
⎛
⎛
+
−
z
z
⎜
⎟
⎜
⎟
⎜
⎟
Are
Are
the
the
source
source
width
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ erf ⎜
⎜
⎟
2
α
x
2
α
x
⎜
⎟
⎜
⎟
width
and
height
⎜ and
⎟ height
⎜
⎟⎟
⎜
⎝
⎝
z
y
⎠
⎝
z
⎠⎠
AT
ATSTEADY
STEADYSTATE
STATE
i.e.
Mass is decaying as fast
as it is being supplied at
the source
⎛ x
Co
exp ⎜
⎜ 2α x
4
⎝
⎛
⎜
⎜ erf
⎜
⎜
⎝
⎛
⎜
⎜ erf
⎜
⎜
⎝
If R>1
⎛
4 λα x
⎜1 − 1 +
⎜
v
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎠
Z ⎞
Z ⎞⎞
⎛
⎛
⎜ z+
⎟
⎜ z−
⎟⎟
2
2
⎜
⎟⎟
⎟ − erf ⎜
⎜ 2 αzx ⎟
⎜ 2 α zx ⎟⎟
⎜
⎜
⎟⎟
⎟
⎝
⎠
⎝
⎠⎠
Y
⎛
⎜ y+
2
⎜
⎜ 2 αyx
⎜
⎝
⎞
⎟
⎟ − erf
⎟
⎟
⎠
Y
⎛
⎜ y−
2
⎜
⎜ 2 αyx
⎜
⎝
Divide v by R
13
Analytical Solution
for transport in
uniform 1D flow
⎛ x source
⎛
continuous
C
4 λα
⎜1 − 1 +
C ( x, y , z , t ) =
exp ⎜
o
⎜ 2α x ⎜
⎝
⎝
8
⎛
⎛
⎛
⎜
⎜ x − v t ⎜1 + 4 λα x
⎜
⎜
⎜
v
⎝
⎜ erfc ⎜
2 α xv t
⎜
⎜
⎜
⎜
⎝
⎝
v
x
⎞⎞
⎟⎟
⎟⎟
⎠⎠
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟⎟
⎠
⎟⎟
⎟⎟
⎟⎟
⎠⎠
3D spreading
⎛ x
Co exp⎜
⎜ 2α x
⎝
with Decay
⎛
Y
⎛
⎜
⎜ y+
2
⎜ erf ⎜
⎜
⎜ 2 αyx
⎜
⎜
⎝
⎝
⎞
⎟
⎟ − erf
⎟
⎟
⎠
Y
⎛
⎜ y−
2
⎜
⎜ 2 αyx
⎜
⎝
⎠
⎝
⎞⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠⎠
Upper case
Y and Z
⎛
Z ⎞
Z ⎞⎞
⎛
⎛
⎜Are
⎜ z +the
⎟ source
⎜ z−
⎟⎟
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ erf ⎜
⎜
2
α
x
2
α
x
⎜
⎟
⎜
⎟⎟
width
and
height
⎜
⎟
⎜
⎟⎟
⎜
⎝
⎝
z
z
C ( x, y , z , steadystat e) =
⎛
4λα x
⎜1 − 1 +
⎜
v
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
⎛ ⎛ Y ⎞ ⎞⎛ ⎛ Z ⎞ ⎞
⎜ erf ⎜
⎟ ⎟⎜ erf ⎜
⎟⎟
⎜ ⎜ 4 α x ⎟ ⎟⎜ ⎜ 4 α x ⎟ ⎟
y ⎠ ⎝
z ⎠⎠
⎝
⎝ ⎝
⎠
⎠⎠
STEADY STATE
ON THE
i.e.
CENTER LINE Mass is decaying as fast
as it is being supplied at
the source
i.e.
y=z=0
If R>1
Divide v by R
THINK IN TERMS OF ORGANIZING THE
ANALYTICAL SOLUTIONS IN TERMS OF
THE TYPE OF SOURCE:
SLUG OR CONTINUOUS
TYPE OF SPREADING:
1D, 2D, 3D
TYPE OF CONTAMINANT BEHAVIOR:
DECAYING, ADSORPING
(and if so steady-state? center-line?)
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