4. CHAPTER SITE PROPERTIES AFFECTING REMEDIATION
My teaching goals for this chapter are for you to:
1) learn how site specific properties affect contaminant transport in soil and
groundwater
2) Derive the advection-dispersion equation
3) Calculate travel times for contaminants in groundwater based on relevant
hydraulic properties
4) Evaluate how dispersion processes dilute contaminants in groundwater
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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TRANSPORT PROPERTIES: ADVECTION, DISPERSION, AND
SORPTION
Transport properties refer to all of the factors that influence the
movement of contaminants in the subsurface. The physics of transport
are a function of what phase the contaminants are in. In CEE440 we will
focus primarily on aqueous and vapor phase transport. Transport of pure
phase chemicals is a complex problem that is best left to a course which
focuses on multiphase flow.
In CEE440 we will derive partial differential equations that describe the
processes which affect transport. Analytical solutions to these equations
will be provided for you. You will work with these solutions to evaluate
how subsurface parameter affects contaminant transport.
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2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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Geologic/Hydrogeologic Heterogeneity
An extremely important factor that we will only consider qualitatively is
heterogeneity.
Heterogeneity exists at all scales (i.e. at both the pore and the Darcy scale)
and results in highly variable contaminant distributions in the subsurface.
Heterogeneity is one of the biggest obstacles facing Environmental
Remediation and the focus of much current research (including mine).
LOW PERMEABILITY CLAY & SILT
TILL
SAND AND GRAVEL
FRACTURED ROCK
BEDROCK
Figure 4.1. Illustration of subsurface heterogeneity
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A couple of definitions:
Water table - surface along which all pores are filled with water at atmospheric
pressure
Aquifer- porous medium capable of storing water and transport substantial
quantities of water under natural conditions; can be a uniform porous media
such as sand or a highly variable one such as fractured rock; can be either
confined or unconfined.
Classification
Clay
Silt
Sand
Gravel
CEE 440
of soil components based on particle size:
< 2 m
2 m 63 m
63 m 2 mm
> 2 mm
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4.1. Groundwater Flow Rate - ADVECTION
Flow through an aquifer is laminar (exceptions in fractured rock) and
described by Darcys law:
v = Q/A = - KH dh/dl
(4.1)
h1 = z1 + hp,1
hp,1
Q
h2 = z2 + hp,2
hp,2
h = h2 - h1
h = hydraulic head
hp = pressure head
z1
z2
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Q
z = elevation head
v =Darcy flux or specific
discharge [m/s]
Q = volumetric discharge
[m3/s]
A = cross-sectional area
[m2]
dh/dl = hydraulic
gradient varies from
10-5 to 10-1, typical
value is 10-3
KH = hydraulic
conductivity [m/s]
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KH depends primarily on:
the size of the pores or channels
fluid viscosity (for groundwaters of interest this changes less than 2X)
The hydraulic conductivity can span many orders of magnitude as
illustrated below:
10-12
10-10
10-8
KH [m/s]
10-6
104
10-2
1
gravel (1-10mm)
clean sand (0.1-1mm)
sitly sand (0.01-1mm)
silt (1-10m)
clay (<1m)
CEE 440
sandstone, limestone
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When calculating contaminant transport we usually want to know how
fast a chemical is moving in the ground. The Darcy flux does not give
us this value. Instead, we need to calculate the average linear velocity
as demonstrated below:
v
v
v = v n, n=porosity
CEE 440
(4.2)
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For our purposes we will often assume that dh/dl~-10-3, n=0.33, KH can
be chosen from the chart above depending on the solid material. With
these values we can calculate some typical travel values for different
porous media.
Examples:
Porous
media
KH
[m/s]
Gravel
v
10-3 to 1
3x10-6 to 3x10-3
100 to 100,000
km/yr
Sand
10-4 to 10-1
3x10-7 to 3x10-4
10 to 10,000
To
Silt
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Range
[m/s]
year]
[m/
10-7 to 10-3
3x10-10 to 3x10-6
0.01 to 100
mm/yr
v
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4.2 Groundwater Dispersion (Fetter, Contaminant Hydrology, 1993)
Groundwater moves at rates that are both greater than and less than the
average linear velocity. This is due to hydrodynamic dispersion, a
phenomenon caused by mechanical dispersion and molecular diffusion.
4.2.1. Mechanical dispersion is caused by three basic phenomena:
1) as fluid moves through the pores, it will move
faster in the center of the pores than along the
edges
2)
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some of the particles will travel along longer
flow paths in porous media than other particles
to go the same linear distance
3) some pores are larger than others, which
allows the fluid flowing through these pores to
move faster
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2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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4.2.2 Molecular Diffusion
-While advection is occurring, the contaminant is diffusing in all directions.
The contaminant can also diffuse into dead-end pores. Diffusion is very
slow relative to the rate of advection. However, diffusion causes additional
mixing and greater dilution of the solute at the advancing edge of flow.
-The process of molecular diffusion can not be separated from mechanical
dispersion in flowing groundwater. The two are combined to define a
parameter called the hydrodynamic dispersion coefficient, D.
D = Dmech + Dp
where the pore diffusion coefficient Dp=Dmol*n
D = v + Dp
where the dispersivity is
Hydrodynamic dispersion gives rise to both:
Longitudinal dispersion - mixing along the direction of flow path.
Transverse dispersion mixing normal to flow path ( T=1/20* L).
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4.2.3 Illustrations of the Effects
of Dispersion (Mackay et al.,
Water Resources Research, 22
(13), 2017-29, 1986.
In the mid 1980s tracer experiments
were conducted at the Borden aquifer
site in Canada. The conservative
tracer Chloride was used to evaluate
the effects of dispersion on transport.
The sorbing tracers carbon
tetrachloride and tetrachloroethene
were used to evaluate the effects of
sorption on transport. The field site
and sampling locations are shown on
this page.
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4.3 Modeling ADVECTION and DISPERSION of Solute (Freeze and
Cherry, Groundwater, Appx. X, 1979)
Now we can derive the advection-dispersion equation to describe solute
transport in flowing groundwater (ignoring sorption and reaction) [from Freeze
and Cherry, 1979]
Assumptions: porous media is homogeneous (n is constant) and isotropic (same
in all directions)
media is saturated with water
flow is steady state
Darcys law describes flow
Fick's groundwater first law describes steady state mass transfer
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In the same manner as we derived Fick's second law, we start with a mass
balance on a control volume
2dx
2dy
4 dy dz (Fx-Fx/x dx)
y
z
4 dy dz (Fx+Fx/x dx)
x
2dz
As before (note:this time we account for porosity in REV):
Rate In Rate Out = -8 dx dy dz Fx/ x
(4.3)
Rate of Accumulation = 8 dx dy dz n Caq/ t
(4.4)
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To close our mass balance we set these two equations equal to each other:
n Caq/ t = - Fx/ x
(4.5)
where:
Caq = concentration of solute in groundwater [g/m3]
n = (porosity) volume of flowing groundwater per volume of porous media in
REV [-]
nCaq = mass of solute per unit volume of porous media [g/m3]
x = length scale [m]
CEE 440
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This is where our derivation from the purely diffusive case differs. First,
we must consider advective flux in addition to dispersive or diffusive flux.
Second, we are going to consider the effect of the aquifer porosity on
both the advective and dispersive flux.
The mass of solute transported in the x direction by advection and
dispersion is:
Fx (advection) =
Fx (dispersive) =
Fx (total) =
[g /(m 2s)]
-n Dx
C aq
x
vx n C q - nDx
a
Caq
x
(4.6)
[g /(m 2s)]
vx n C q
a
(4.7)
[g /(m 2s)]
(4.8)
where:
Dx = dispersion coefficient [m2/s]
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Plugging in Fx(total) into our overall mass balance equation gives:
Caq
n Caq / t = - / x v x nCaq - nDx
x
Caq
Caq/ t = - / x ( v x C aq) + / x Dx
x
(4.9)
(4.10)
In a homogeneous medium in which the velocity is steady and uniform (i.e.
does not vary through time or space), and the dispersion coefficients do
not vary through space we obtain:
C aq / t = - v x C aq / x + D x
CEE 440
2 C aq
x
2
(4.11)
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This is known as the advection-dispersion equation and you will hear about it
and see it in abundance in this class and in many others. It is the primary
equation that governs mass transport through aquifers, through the
unsaturated zone, through activated carbon treatment columns, through
many treatment processes. Of course with each application there are
modifications to this equation.
The AD equation can be solved by either numerical or analytical methods.
Analytical methods involve the solution of PDEs using calculus based on initial
and boundary value conditions. They are limited to simple geometry and and
in general require that the aquifer be homogeneous. Numerical methods are
outside the scope of discussion. We are not concerned with how to solve this
equation. However, we are interested in examining available analytical
solutions to this equation and in observing how changes in the parameters
(Dx, v x , Caq, x, t) change the outcome of the solution.
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2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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4.3.1. Analytical Solutions to Advection-Dispersion Equation
In order to obtain a unique analytical solution to the PDE it is necessary to
define initial and boundary conditions. The initial conditions describe the value
of Caq at t=0. The boundary conditions specify the interaction between the area
under investigation and its external environment. There are three types of
boundary conditions for mass transport.
1) fixed concentration at one or more of the boundaries
2) fixed gradient at one or more of the boundaries
3) variable flux at one or more of the boundaries
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In class we will examine an analytical solution for only the first type of
boundary condition. In our homework we may examine solutions to other
boundary conditions and you may also want to consult the textbook by Fetter
(Contaminant Hydrogeology, 1993) for other solutions.
One Dimensional Step Change in Concentration (i.e. fixed concentration
boundary condition)
Caq(x,t=0)=0
(4.12)
Caq(x=0,t>0)=Caq,0
boundary condition
(4.13)
Caq(x= , t)=0
CEE 440
initial condition
boundary condition
(4.14)
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The solution to this problem is given by Ogata and Banks (U.S.
Geological Survey Professional Paper 411-A, 1961)
x - v x t
x + v x t
v x x
+ exp
C aq = 0.5 C aq ,0 erfc
D x erfc 2 D x t
2 D x t
(4.15)
Note: x is the length in the direction of flow and erfc is the
complementary error function (erfc(z) = 1- erf(z)).
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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Example #1
A drum of tritiated water (nonsorbing) is punctured and its contents slowly leak
into the ground (a silty sand). The first tritiated water reaches the initially clean
groundwater and mixes with it. As new tritiated water continues to reach the
groundwater, contaminated groundwater is transported down gradient. As a
simplification assume that the concentration of the tritiated groundwater at the
tritiated water point of entry is zero at t=0 and constant at t>0. Calculate the
relative concentration (Caq/Caq,0) of tritiated water at x=12m one year after the
tritiated water first reached the groundwater assuming: Dx=4x10-7 m2/s;
v x = 4.00x10-7 m/s.
What happens to Caq/C aq,0 if Dx changes to 5.00x10-8 m2/s, to 5.00x10-9 m2/s.
Why does it change the way it does?
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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Using the fixed concentration boundary condition solution to the advectiondispersion equation, we can calculate Caq/C aq,0 at x=12m one year after
contamination for each Dx value listed:
Dx = 4x10-7 m2/s; Caq/C aq,0 = 0.55
Dx = 5x10-8 m2/s; Caq/C aq,0 = 0.64
Dx = 5x10-9 m2/s; Caq/C aq,0 = 0.86
In order to understand why Caq/C aq,0 changes the way it does we need to plot
the breakthrough profiles. We can do this (via Excel) by calculating Caq/C aq,0
values at points in space between x=0.1 and 30 m and by plotting these points
on a graph. The results are shown below:
CEE 440
2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved.
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1
Tritia te d W a te r Front a t t= 1 ye a r
0.9
0.8
Caq/Caq,o
0.7
Dx =5e- 9 m2/s
0.6
Dx =5e- 8 m2/s
0.5
Dx =4e- 7 m2/s
0.4
0.3
0.2
0.1
0
0
5
10
15
x (m )
20
25
30
From the graph it is apparent that with increasing Dx, the profile becomes
more spread out. This is because with increasing Dx more mixing in the
direction of flow is occuring. As a result, the value of Caq/C aq,0 decreases
with increasing Dx.
What happens if have pulse input?
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D=0.5
D=1.5
D=2.5
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2D Plot of 3D dispersion
L=0.1
T=0.01
CEE 440
D=v+Dmol
L=0.1
T=0.01
L=0.4
T=0.02
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CEE 440
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