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
If we only consider advection and
start with a "point" of material with Co=1000mg/l
A point has no volume so can it have a concentration? So why do we say a “point”?
K = 0.1 cm/sec
dh = 10 cm
dl = 100 cm
φ = 0.2
How long will it take for the material to move 50cm?
What will the concentration be at that location at that time?
1
However concentration will decreasse due to
DIFFUSION and DISPERSION
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
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
2
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
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
3
Dispersivities at various scales & measured by various
methods as compiled by Stan Davis et al.
Table B1 in the book, "Ground Water Tracers"
Table B1 CONTINUED Dispersivities at various scales &
measured by various methods from "Ground Water Tracers"
4
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
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
5
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
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
6
Error Function
Tables are listed in
the back of ground
water hydrology
books
WARNING
EXCEL DOES NOT
APPROXIMATE
THIS
ACCURATELY AT
EXTREME VALUES
OF BETA
Suppose that source enters the up gradient end of a column
At a continuous concentration of Co=1000mg/l
K = 0.1 cm/sec
dh = 10 cm
dl = 100 cm
φ = 0.2
02
Dispersivity αx = 5 cm
What will the concentration be at 50 cm after 1000sec?
cm
sec 10cm = 0.05 cm
average linear velocity
0.2 100cm
sec
cm
distance traveled in 1000sec? d = v t = 0.05
1000 sec = 50cm
sec
Kdh
v=
=
φdl
0 .1
By inspection we know that the concentration should be 0.5*Co=500mg/l
But let’s carry out the calculation
7
Experiment with the spreadsheet
http://inside.mines.edu/~epoeter/_GW/22ContamTrans/C1d.xls
http://inside.mines.edu/~epoeter/_GW/22ContamTrans/C1d.xls
Note the values of C using only the first term and then both terms at
times and locations where y
your intutition allows y
you to know the
concentration.
When is use of the second term important? When does excel cause it
to be in error?
Try
x = 50, 49, 51, 0, 100
Consider other times
times.
Where and when can you know the correct C?
The second term is important for calculating C near the
source.
Analytical Solution for transport in 1-D flow field
slug source
3D spreading
without chemical reaction
8
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
Suppose a slug source
enters a uniform flow field
With an initial mass of Mo=1000mg
K = 0.1
0 1 cm/sec
/
dh = 10 cm
dl = 100 cm
φ = 0.2
dispersivity αx = 5 cm
dispersivity αy = 1/5 αx
dispersivity αz = 1/10 αx
What will the concentration be at 50 cm
directly down gradient after 1000sec?
9
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
10
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
⎠
⎝
11
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
Suppose a source continuously enters that uniform flow field
With an initial concentration of Co=1,000mg/l
pause to consider relationship of mass and concentration
Mass = Conc * Volume
Mass/Time = Conc * Velocity * Area = Conc * Q (Q is discharge)
Mass = Conc * Q * Time
Envision the source is submerged and
emanates from a 0.5cm high x 1cm wide zone
pause to consider the character of the source geometry
v = 0.05 cm/sec
d spe s ty αx = 5 c
dispersivity
cm
dispersivity αy = 1/5 αx
dispersivity αz = 1/10 αx
What will the concentration be at 50 cm
directly down gradient after 1000sec?
pause to consider the coordinate system
12
What do you make of the concentration relative to the C we obtained for
the slug source?
How much mass enters the system in 1000sec?
M = CQT = CAVDT
How would you go about developing a contour map of the plume?
If you did not know the dispersivities, how could you use this equation
to estimate them?
How might you set up the problem if 8g/d arrived at the water table over
a 1m2 area in an aquifer with the properties and conditions used for the
example?
Analytical Solutions for transport provide smoothed
representations of plumes
Be sure to practice using this topic’s exercises
View an animation of contaminant transport
Consider how what you see will affect:
1) the predictions you make using the analytical solutions
2) the concentrations you obtain in samples from field sites
View DVD
NOW CONSIDER THE NON-MECHANICAL ASPECTS OF
CONTAMINANT TRANSPORT
13
Decay
dN
= −λN
dt
or N = N oe (− λt )
where
h
λ=
decay
constant
0.693
T1
2
0.693 is the
natural log of 0.5
For a material with a half-life of 12 yrs, how much is
left after 40 yrs? (Hint figure it as a % of initial mass)
N = N o e ( − λt )
λ=
0.693
T1
2
14
It is often said that material is essentially gone after
7 half-lives. How much is left then?
N = N o e ( − λt )
λ=
0.693
T1
2
Retardation - Adsorption
Units
Kd
ml
mg
R=
Vwater
Vcontaminant
⎛ ρb
⎞
= ⎜⎜ 1 + K d ⎟⎟
φe
⎝
⎠
15
What is the Retardation Coefficient for a site with
⎞
⎛ ρ
Vwater
Kd = 0.01ml
R=
= ⎜⎜1 + b K d ⎟⎟
Vcontaminant ⎝ φe
mg
⎠
effective porosity of 0.3
particle density of 2.65 g/cc
What is the Retardation Coefficient for a site with
Ground water velocity = 0.05 cm/sec
Contaminant velocity = 0.0009 cm/sec
R=
⎞
⎛ ρ
Vwater
= ⎜1 + b K d ⎟⎟
Vcontaminant ⎜⎝ φe
⎠
16
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
=
−
2
∂t
R ∂x
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
Analytical Solution for transport in
uniform 1D flow
continuous source
3D spreading
With Decay
17
C( x, y , z , t ) =
⎛ x − v xt ⎞ ⎞
Co ⎛⎜
⎟⎟
erfc⎜
⎜ 2 D t ⎟⎟
8 ⎜
x ⎠⎠
⎝
⎝
C ( x, y , z , t ) =
⎛ ⎛
⎞
⎛
⎞⎞
⎜ ⎜ y+ Y ⎟
⎜ y − Y ⎟⎟
⎜ erf ⎜
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ ⎜
⎜
Analytical
x ⎟ Solution
x ⎟⎟
Analytical
⎜⎜ ⎜ 2 Dy ⎟ Solution
⎜ 2 Dy ⎟ ⎟⎟
v⎠
⎝ in v ⎠ ⎠
⎝for⎝ transport
⎛
4 λα x
⎜1 − 1 +
⎜
v
⎝
⎛
⎛
⎛
⎜
⎜ x − v t ⎜ 1 + 4 λα x
⎜
⎜
⎜
v
⎝
⎜ erfc ⎜
2 α xv t
⎜
⎜
⎜
⎜
⎝
⎝
for transport in
⎛ ⎛
⎞1D flow
⎛
⎞⎞
uniform
⎜ ⎜ z + Z ⎟1D
⎜ z − Z ⎟⎟
uniform
flow
⎜ erf ⎜
2 ⎟⎟
2 ⎟ − erf ⎜
continuous
source
⎜ ⎜
⎜
continuous
x ⎟⎟
x ⎟ source
⎜⎜ ⎜ 2 D ⎟
⎜ 2 D ⎟ ⎟⎟
3D
spreading
v ⎠⎠
v
⎝ spreading
⎠
⎝
⎝ 3D
with
Decay
with Decay
z
⎛ x
Co
exp ⎜
⎜ 2α x
8
⎝
z
⎛
⎜
⎜ 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
Y and
and Z
Z
Are
the source
source
A th
Are
the
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
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
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
⎠⎠
ON
ONTHE
THE
CENTER
CENTERLINE
LINE
i.e.
y=z=0
If R>1
Divide v by R
⎛ x
Co
exp⎜
⎜ 2α x
2
⎝
⎛
4λα x
⎜1 − 1 +
⎜
v
⎝
⎛
⎛
⎛
⎜
⎜ x − v t ⎜1 + 4λα x
⎜
⎜
v
⎜
⎝
⎜ erfc ⎜
2 α xv t
⎜
⎜
⎜
⎜
⎝
⎝
⎞⎞
⎟⎟
⎟⎟
⎠⎠
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟⎟
⎠
⎟⎟
⎟⎟
⎟⎟
⎠⎠
⎛ ⎛ Y ⎞ ⎞⎛ ⎛ Z ⎞ ⎞
⎜ erf ⎜
⎟ ⎟⎜ erf ⎜
⎟⎟
⎜ ⎜ 2 α x ⎟ ⎟⎜ ⎜ 2 α x ⎟ ⎟
y ⎠ ⎝
z ⎠⎠
⎝
⎝ ⎝
⎠
18
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
⎜
⎜
⎝
⎝
⎞
⎟
⎟ − 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
⎠⎠
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
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
⎞ ⎞ ⎞⎟
⎟⎟
⎟ ⎟⎟
⎠
⎟⎟
⎟⎟
⎟⎟
⎠⎠
3D spreading
with Decay
⎛
Y
⎛
⎜
⎜ y+
2
⎜ erf ⎜
⎜
⎜ 2 αyx
⎜
⎜
⎝
⎝
⎞
⎟
Y ⎞⎞
⎛
⎟⎟
⎜ y−
2 ⎟⎟
⎜
⎟ − erf
Upper
⎜ 2 α x ⎟⎟
⎟ case
⎟⎟
⎜
⎟
⎠⎠
⎝
⎠
Y and
Z
⎛
Z ⎞
Z ⎞⎞
⎛
⎛
⎜Are
⎜ z +the
⎟ source
⎜ z−
⎟⎟
2 ⎟ − erf ⎜
2 ⎟⎟
⎜ erf ⎜
⎜
x⎟
⎜ 2 α and
⎜ 2 α x ⎟⎟
width
height
⎜
⎟
⎜
⎟⎟
⎜
⎝
⎝
z
y
⎠
⎝
z
⎠⎠
x
⎞⎞
⎟⎟
⎟⎟
⎠⎠
C ( x, y , z , steadystat e) =
⎛ x
Co exp⎜
⎜ 2α x
⎝
⎛
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
19
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?)
A transport model for your exploration:
http://inside.mines.edu/~epoeter/_GW/22ContamTrans/TransportModel/tdpf1.0web/pflow/pflow.html
Explore plume spreading as a function of heterogeneity as represented by K variation AND
local heterogeneity as represented by the input dispersivity
Create grid. Make sure you understand the size of the system you are working with.
P
Properties:
ti
Run
R att least
l
t1h
homogeneous and
d 1 heterogeneous
h t
model
d l
Calculate heads
Choose particle movement for flow (this is by random walk … advecting based on Ks and gradient
then randomly displacing each particle based on dispersivity)
Be aware of the number of particles you use given spacing, grid size and your drawn area
Use the same particles for the above comparison of 1 homogeneous and 1 heterogeneous model
Choose # days per second such that you will get transport across your grid in a matter of a minute
g estimate of travel time given
g
gradient,
g
, K,, porosity
p
y and distance))
or so ((make a rough
Always choose to show center of mass, std deviation bars of particles and plot the variance of
particle locations.
Run both your homogeneous and heterogeneous models with and without local dispersion. When
you use local dispersion make sure it is a reasonable value. Try varying the value.
For all cases note the spatial variance of the particles. Explain the results.
When does it stop plotting spatial variance? Why?
20