INTERFACIAL 3.3 TENSION, CONTACT ANGLE, AND DENSITY
Interfacial tension (ab , a is one phase and b is another) can be thought of
as the force per unit length (or energy per unit area) acting at a liquidliquid or liquid-solid interface.
Interfacial tension - arises because of unbalanced cohesional forces on
molecules at the interface. The tension causes the interface between the
two fluids to contract and form an area that is as small as possible.
If one phase is liquid and the other vapor, then we use the term surface
tension denoted by .
e.g.
CEE 440
Water bugs, rain drops, Goretex, oil slick
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
1
Consider a soap film stretched across a wire frame with one movable side.
L
The work done in extending the movable member dx is:
Work = E = dA= L dx
(3.58)
E = change in energy
= surface tension (force/length = N/m; energy/area = J/m2)
L = length (m)
dA = change in area (m2)
dx = change in length
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
2
Alternatively we could think of an expanding soap bubble:
dR
R
the surface tension of the soap film results in stable,
spherical bubble
The free energy of the soap bubble surface is initially:
Einit = ______
4 R2
(3.59)
If the radius of the soap bubble is increased by dR, then the final free
energy is:
Efinal = 4 (R+dR)2
CEE 440
(3.60)
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
3
It follows that the change in surface free energy or the work required to
expand the soap bubble is:
Work = E = Efinal Einit =
(4 R2 + 8 R dR + 4 dR2) 4 R2
(3.61)
E = 8 R dR + 4 dR2 ~ 8 R dR
(3.62)
Since increasing the soap bubble radius increases the surface free energy,
the tendency to do so must be balanced by a pressure difference across the
film P. Hence, fluid on the concave side of the interface (i.e., inside the
soap bubble) is at a higher pressure.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
4
The work against this pressure difference is given as follows:
Work = E = P * Area * Distance
Distance = Rfinal Rinit = R + dR R = dR
(3.63)
E = P 4 R2 dR
(3.64)
This work equals the work required to expand the bubble by dR.
P 4 R2 dR = 8 R dR
(3.65)
Rearranging:
P = 2 / R
(Young-Laplace Equation)
(3.66)
So as the bubble gets larger the pressure difference decreases and vice
versa.
This example also demonstrates that interfacial tension results in curved
surfaces between phases.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
5
Consider the following configuration:
Patm =
R3
R2
Valve 1
Valve 2
What happens when
valve 1 and valve 2 are
opened?
R2=R3
R3
Valve 3
P2>P3 so the bubble on
the left will stop
shrinking when R2=R3
CEE 440
R2
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
6
In general surfaces are not perfectly spherical:
R2
R1
As a result, we use the more general form of the Young-Laplace equation:
P = (1/R1 + 1/R2)
CEE 440
(3.67)
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
7
What about if we have three phases present, say a solid surface, and two
fluids, or three immiscible fluids:
gas
air
liquid
solid
oil
water
Interfacial tension between the gas-liquid (gl), between the gas-solid (gs),
and between the liquid-solid (ls) can cause fluids to form a curved interface
with a contact angle () between them. The contact angle indicates which
fluid has a higher affinity for the surface.
If <90o, liquid wets and has greater affinity for surface
(=0o for perfectly liquid wetting)
If >90 o, gas wets and has greater affinity for surface
(=180o for perfectly gas wetting)
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
8
In some cases the interfacial tensions of the spreading liquid are not great
enough to allow a contact angle to form; therefore, the liquid spreads
without limit. This can be assessed with the spreading coefficient:
Fs = gs - (gl + ls)
g=gas, l=liquid, s=solid
OR
Fs = aw - (ao + ow)
(3.68)
a=air, o=organic, w=water
If Fs>0, the liquid spreads. If Fs<0, the liquid forms a pool.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
9
How would changes
in the spreading
coefficient affect
NAPL migration?
As Fs increases
there is less
residual NAPL, and
more free NAPL
available for
migration
ow
ao
aw
pentane
hexane
heptane
octane
decane
dodecane
tetradecane
hexadecane
49
51.1
50.2
50.81
52
52.8
52.2
53.3
15.5
18.4
19.66
21.62
23.83
24.91
26.56
26.95
72.21
72.4
72.59
72.66
72.74
72.74
72.7
71.83
halogenated aliphatics
carbon tetrachloride
bromoform
chloroform
methylene chloride
perchloroethylene
trichloroethylene
45
40.85
32.8
28.31
47.48
34.5
27.04
45.53
27.32
27.84
31.74
28.8
72.74
0.7
72.75 -13.63
72.56 12.44
72.12 15.97
72.75 -6.47
72.69
9.39
alkanes
Fs
7.71
2.9
2.73
0.23
-3.09
-4.97
-6.06
-8.42
Demond and Lindner, ES&T, 1993
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
10
The interfacial forces between two fluids and a solid also give rise to a
phenomenon known as capillarity. Capillarity causes mercury to rise in a
thermometer.
gas
r
Consider a circular capillary
tube inserted into liquid:
P=0
liq u i d
h
We want to calculate how far in the tube the liquid will rise. Upward
capillary forces exert pressure on the liquid; downward gravity is exerting
pressure on the liquid. These two pressures (or forces) must be equal
when the liquid interface is static.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
11
gas
r
P=0
Capillary pressure acting on the liquid is expressed with
the Young-Laplace equation:
Pc = P = 2 / R =
2 cos () / r
liq u i d
h
(3.69)
Gravity pressure acting on the liquid is:
P = g h
= l g
(3.70)
= density
g= gravity
Setting these pressures equal to each other:
R
r
P = Pg Pl = 2 cos / r = g h (3.71)
h = 2 cos / (r g)
(3.72)
cos()=r/R
R=r/cos()
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
12
P=0
What happens if we replace the wetting
liquid with oil, which has a contact angle
greater than 90o with the capillary wall.
h = 2 cos / (r g)
gas
r
h
liquid
What happens if we have a sloped capillary tube shown below:
gas
r
P=0
gas
liq u i d
+
R
r
h
P=0
liq u i d
h
For the case of a sloped capillary tube we simply include the slope of the
wall in the capillary calculations:
P = Pg Pl = 2 cos (+) / r = g h
CEE 440
(3.73)
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
13
3.4. EXTENDING CAPILLARY PHENOMENON TO NAPL ENTRAPMENT
Non-aqueous phase liquids
(NAPLs) migrate into the
subsurface and move down
through the soil matrix toward the
water table.
As they move through the soil
matrix small liquid globules
become trapped inside small pore
spaces between soil grains and
are left behind the main infiltrating mass of NAPL.
Migration continues until all NAPL has become trapped, until lighter than water
NAPLs (LNAPLs) reach the water table and float, and/or until denser than
water NAPLs (DNAPLs) reach an impermeable barrier.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
14
In 1975, Friedrich Schwille at the Bundesanstalt fr Gewsserkunde started
with some bench-scale lab experiments to investigate the behavior of
chlorinated solvents in the subsurface
Dichloromethane released into water saturated glass beads
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
15
Infiltration of chlorinated hydrocarbons into a sandy media
<10 min
30 min
1 hour
2 hour
Infiltration of 23 liters of PCE (dyed in red) over two hours in a tank with a height of
1.60 m.
(simulation of a leaking pipe)
Two layers of sand with a K of 8 x 10-4 (m/s) (upper part) and 32 x 10-4 (m/s) (lower
part)
Experiment by Schwille (1984)
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
16
Water Outlet
Inject DNAPL
Residual NAPL is trapped in individual pore
spaces so the individual blobs are much
smaller than the main NAPL body.
Glass Wool
Imaging Region
I.D.= 1cm
Length=1cm
For a NAPL to penetrate a porous matrix
the sum of viscous and buoyant pressures
must exceed capillary pressure.
Pc < Pv + Pb
(3.74)
Th
e
im
ag
e
ca
nn
ot
be
di
sp
la
ye
d.
Yo
ur
co
m
pu
ter
m
ay
DNAPL, water
and silica sand
Glass Wool
Water Inlet
The extent of residual NAPL can be inferred
from several key equilibrium parameters.
dgrain=1
mm
CEE 440
dgrain=0.5mm
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
17
The magnitude of the capillary
pressure can be defined by the
Laplace equation (Note:
equation shown is for idealized
spherical interface):
Pc = Po P w
= 2 ow cos () / r
(3.75)
This equation can be used to approximate the threshold value of the capillary
pressure that must be exceeded for DNAPL to pass through a pore throat of
radius r.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
18
Pore Diameter
E.g., In the pore below, describe the change in capillary pressure as the NAPL
migrates from left to right.
dmin
1.6
advancing receeding
y
water front dmax
0.8 NAPL
front
1.2
x
0.4
0
0
0.5
1
1.5
2
PORE LENGTH
This illustration indicates there should exist a relationship between capillary
pressure and the relative amounts of NAPL and water in a porous medium.
This is the case, but the complex geometry and wettability of natural pore
spaces precludes direct computation of the relationship - ERGO, it must be
measured.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
19
The measured relationship is the capillary pressure-saturation function or
capillary pressure curve illustrated in the figure below (Note: this macroscopic
function is defined only for volume elements of a porous medium that are
large relative to the volume of individual grains):
PCE
H2O
N2
Source: Pankow and Cherry
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
20
BYPASSING AND DEAD-END PORE WATER TRAPPING MECHANISMS
WATER SATURATED
NAPL
FLOW
CEE 440
NAPL SATURATED
WITH RESIDUAL
WATER
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
21
In the previous figure, PCE DNAPL advanced into previously water
saturated porous media. This is known as drainage. Once the PCE source
is exhausted, the PCE DNAPL will continue to migrate away from the
source and it will be replaced by water. Now we get increasing water
saturation but decreasing capillary pressure. This is known as wetting.
Shown below is a draining-wetting capillary pressure curve:
Note that water
saturation at any
particular capillary
pressure is less
during wetting than
during drainage.
This is known as
hysteresis.
Source: Pankow and Cherry?
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
22
SNAP-OFF AND BYPASSING NAPL TRAPPING MECHANISMS
WATER SATURATED
WITH RESIDUAL NAPL
NAPL SATURATED
WATER
FLOW
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
23
Definitions
Sw = Vw/Vv = wetting phase saturation
Snw = Vnw/Vv = nonwetting phase
saturation
Vw = volume of wetting phase
Vnw = volume of nonwetting phase
Vv = volume of void (or pore) space
1 = Sw + Snw
Swr = residual wetting phase
Snwr = residual nonwetting phase
Pd = displacement pressure = minimum
capillary pressure required to initiate
invasion of a water saturated porous
media by DNAPL, also known as
bubbling pressure Pcb
DNAPL
What causes hysteresis?
-Variable pore dimensions
-Contact angle aging
Water
DNAPL
Water
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
24
Several relationships empirical are available to calculate the relationship
between capillary pressure and saturation. Two of these are shown
below:
Brooks-Corey (1966) equation
Se = (Pc/Pd)-,
Pc>=Pd
(3.76)
Se = 1,
(3.77)
Se = normalized wetting-uid
saturation or effective saturation
Se=(Sw-Swr)/(Sm-Swr)
Sm = maximum water saturation is
unity when DNAPL is invading
water and 1-Snwr when water
displaces DNAPL
Swr = wetting uid saturation
Sw = saturation of wetting uid
= pore size distribution index
Pd and are t from the data
CEE 440
14
12
decreases
10
Pc/Pd
Pc<Pd
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
Se
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
25
van-Genuchten equation
Se = (1+(Pc/(g))n)-m
(3.78)
hc = Pc/(g)
(3.79)
, n, and m are tting parameters, typically it is assumed that m=1-(1/n)
70
70
60
60
50
alpha decreases
40
hc
hc
50
30
30
20
20
10
10
0
0
0
0.5
Se
CEE 440
n decreases
40
1
0
0.5
1
Se
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
26
ii. Frictional Pressure
Water moving past the particle (i.e. moving groundwater) imparts a pressure
through frictional force which serves to displace the globule:
Pv w hw L
w = density of water
hw = hydraulic gradient in the
displacing water (dhw/dz)
L = length of globule
(function of grain size)
CEE 440
(3.80)
L
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
27
iii. Buoyant Pressure
Buoyancy pressure arises from density differences between NAPL and water:
Pb g L sin()
-For DNAPLs in groundwater,
buoyancy hinders the
upward displacement and
aids downward
displacement.
-For LNAPLs in groundwater,
buoyancy hinders the
downward displacement and
aids the upward
displacement.
CEE 440
(3.81)
L
g
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
28
EXAMPLE OF BUOYANCY VERSUS CAPILLARY PRESSURE FOR NAPL POOL
Consider a DNAPL release from storage containers. The DNAPL has leaked
into the subsurface, penetrated the groundwater table, and is perched on an
aquitard (low permeability layer) as a pool. How can we determine if the
DNAPL has penetrated the aquitard.
Z=vertical coordinate
measured upward
W=distance between bottom
residual
of pooled DNAPL and
DNAPL
saturated zone
water table
W
Z
T=distance between bottom
T
pooled DNAPL
of pooled DNAPL and
top of pooled DNAPL
aquitard
unsaturated zone
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
29
Z=vertical c
measure
W=distance
residual
of poole
DNAPL
saturated zone
water tab
W
Z
T=distance b
T
pooled DNAPL
of poole
top of po
aquitard
unsaturated zone
From basic hydrology we know that water
pressure increases with depth below the water
table according to the following equation:
Pw = - w g Z + C1
(3.82)
(note: we could evaluate C1 by setting Pw=0 at Z=W)
We know that the pressure of the nonwetting phase increases similarly with
depth below the top of the pooled DNAPL:
Pnw = - nw g Z + C2
(3.83)
Subtracting these two equations we can solve for the buoyancy pressure:
Pb = Pnw Pw = - (nw - w) g Z + (C2 C1)
= - g Z + C
CEE 440
(3.84)
(3.85)
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
30
Z=vertical coo
measured
W=distance b
residual
of pooled
DNAPL
saturated zone
water tabl
ZW
T=distance be
T
pooled DNAPL
of pooled
top of poo
aquitard
unsaturated zone
Since Pb=0 at Z=T we can evaluate for C to get:
Pb = g (T-Z),
for Z<=T
(3.86)
Z
Graphically we can illustrate this as:
Pw
W
How might soil properties vary such
that the capillary pressure-saturation
curve changes and NAPL more easily
penetrates the aquitard?
CEE 440
Pb
pressure
Dra ina ge ca pilla ry pre ssure -sa tura tion curve s
hc (cm water)
Given the capillary pressure-saturation
drainage curve for the aquitard, what
conditions are necessary for the NAPL
to migrate through this low
permeability barrier?
Pb > Pd = hd g
Pnw
T
70
60
50
40
30
20
10
0
0
0.5
1
Se
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
31
Example:
A monitoring well which spans both the unsaturated and saturated zone
detects high concentrations (near the solubility limit) of two different organic
chemicals, perchloroethylene and jet fuel (PCE was used to spray the grease
and dirt off jets) The specific gravity of the perchloroethylene is ~1.50. The
specific gravity of the jet fuel is ~0.89 (Note: specific gravity is the density of
the chemical divided by the density of water). In separate measurements,
the interfacial tension of pechloroethylene/water and jet fuel/water is
determined to be 36 dyne/cm and 35 dyne/cm, respectively (not statistically
different). The organic/water contact angles are also determined to be not
statistically different. Which chemical do you expect to find at higher
residual saturation in the vadose zone? Which chemical do you expect to
find at higher residual saturation in the saturated zone?
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
32
Solution
The two organics have similar properties excluding density. Hence, lets
consider buoyancy pressure:
Pb g L sin()
In the unsaturated zone the density difference between jet fuel and air is
less than between PCE and air so more jet fuel will remain trapped.
Jet fuel is less dense than water so the buoyancy force will prevent jet fuel
from entering the saturated zone; only PCE will be found below the
water table. This excludes, of course, the occurrence of water table
fluctuations.
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
33
Soil Type
Contaminant Spill for 7 years
Permeability
(mean value)
6.8e-11 m2
Hanford Coarse
(12 m)
Hanford Fine
(13 m)
6.6e-12 m2
Upper Plio-Pleistocene
(4.5 m)
1.7e-13 m2
Lower Plio-Pleistocene
(6 m)
7.8e-12 m2
Ringold Formation
(20 m)
1.3e-11 m2
1 order of magnitude lower than mean
1 order of magnitude higher than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
Mean
CEE 440
34
Simulation Scenario
CCl4 Infiltration Rates
Water Infiltration Rates
1955 (6 mo) 0.062 m/yr
1956 0.277 m/yr
1957 0.277 m/yr
1958 0.264 m/yr
1959 0.283 m/yr
1960 0.286 m/yr
1961 0.379 m/yr
1962 (6 mo) 0.178 m/yr
1955 (6 mo) 3.04 m/yr
1956 2.47 m/yr
1957 2.95 m/yr
1958 3.92 m/yr
1959 3.07 m/yr
1960 4.42 m/yr
1961 4.23 m/yr
1962 (6 mo) 1.98 m/yr
1962 1993 over all areas
Nov Feb 7.5 cm/yr
Mar Oct 0.5 cm/yr
CCl4 infiltration occurs one week per four weeks because of sporadic disposals
Water infiltration occurs continuously
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
35
1 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
36
2 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
37
3 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
38
4 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
39
5 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
40
6 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
41
7 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
42
8 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
43
9 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
44
10 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
45
11 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
46
12 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
47
13 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
48
14 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
49
15 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
50
16 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
51
17 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
52
18 yr
Contaminant Spill for 7 years
Mean
CEE 440
Mea1norder of magnitude lower than mean
f a gi t o g
1 or1doerrdeorf ommganntiuuddee lhiwheerr than mean
than mean
1 order of magnitude higher
than mean
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
53