Transcript Solute Transport in the Vadose Zone

Solute Transport in
the Vadose Zone
Quantification of oozing,
spreading and smearing
1
Overview
Much of the attention in vadose zone and
groundwater in general results from interest in
contaminant transport.
We will review
basic formulations of sorption and degradation
Plug flow (piston flow) modeling approach
Convective/Dispersive approach
Remember:
Any errors in you solution to water flow will be
propagated in your solute transport estimates
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Partitioning between phases: Sorption
The total concentration C (in mass per volume) is
the sum of sorbed and aqueous
C  b cs  cl
b = bulk density of the porous media [mass dry
media per total volume]
cs = concentration adsorbed to media [mass of
solute adsorbed per mass of dry media]
 = volumetric water content [volume of water per
total volume],
cl = solute concentration liquid phase [mass of
solute in liquid phase per volume of water].
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A Brief Discussion of Sorption
An isotherm relates cs to cl in
a mathematical form
Typical Assumptions:
Each chemical species acts
independently
Rub: with limited number of
adsorption sites, this doesn’t work
Desorption and adsorption follow Solid with
the same isotherm
adsorbed
Rub: There is hysteresis between
adsorption and desorption AND
time dependence!
Concentration cs
Liquid with
concentration cl
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Isotherms
Three most popular relationships
Freundlich
Linear
cs
cs = kf cl1/n
cs = kd cl
Langmuir
cs = a cl Q/(1+acl)
cl
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Linear Isotherm
c s = Kd c l
What is so great about the linear isotherm?
Two things:
For low concentration (i.e., when most sorption sites
are unoccupied), the linear isotherm is a good
description.
It makes the math easy! (allows us to find solutions
that we can understand).
Problems:
If dealing with concentrated sources or limited
sorption sites.
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Langmuir Isotherm
Q = adsorption sites/mass
cs = a cl Q/(1+acl)
a = k1/k2 where
k1 = rate of adsorption
k2 =rate of desorption
Notes:
for acl <<1 this reduces to cs = aQcl (linear isotherm)
for acl >>1 this reduces to cs =Q Makes sense since
Q is the sorption capacity of the soil (recall CEC)
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What’s so great about Langmuir isotherm?
The high and low concentration behavior
makes intuitive sense
We can “derive” the Langmuir relationship
from a simplified model.
Consider a block of stuff with Q adsorption sites
per unit mass
at equilibrium the rate of sites
being filled (ra) will equal the
rate of sites being vacated (rd)
Assuming that each site acts
independently, the probability
of sorption will be proportional
to the probability of a solute
molecule hitting that site
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“Deriving” the Langmuir isotherm
So we estimate the adsorption rate as
Proportionality
Constant
Fraction of
sites filled
Concentration
in Liquid
 Q  cs 
ra  k ' 
cl
 Q 
Similarly, we may estimate the rate of
desorption as being proportional to the
number of sites filled
Fraction of
 cs 
rd  k  
Q 
sites filled
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Langmuir derivation...
At equilibrium, ra = rd. Equating these
 Q  cs
k ' 
 Q

 cs 
cl  k  

Q
letting k = k’/k” and multiplying each side by Q
cs  k(Q  c s )cl
Solving for the sorbed concentration
kQcl
cs 
1  kcl
as desired.
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Transport: Basic Processes
3 basic mechanisms by which solutes move:
advection
 diffusion
dispersion
Advection (A.K.A. convection)
movement of the solute with the bulk water in a
macroscopic sense.
Advective transport ignores the microscopic
processes, but simply follows the bulk Darcian flow
vectors.
The crowd metaphor: in a march with thousands, a
small group will still stay together
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Basic Processes 2: Diffusion
Diffusion:
the spreading of a compound through the effects of
molecular motion
Governed by Fick’s law
jdi ff  D 3 c
 mass

time  area 
tends to mix areas of high concentration with areas
of lower concentration.
the rapidity of diffusive spreading linked to molecular
velocities and path length between collisions.
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Diffusion Cont.
For a given temperature, any given molecule has a
particular energy, and thus velocity.
Since kinetic energy is related to the square of
velocity, diffusion rates changes with the square-root
of temperature (as measured in degrees K),
 varies little over typical groundwater temperature
ranges.
Summarized by the diffusion coefficients
are on the order of 0.2 cm2/sec in gases
0.00002 cm2/sec in liquids
a factor of 10,000 higher in gases due to the lower rate of
molecular collisions.
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Diffusion, cont.
The Crowd Scene metaphor:
diffusion corresponds to the movement that
happens when they put the dance music on as
darkness falls at the end of the march.
People start bouncing around
Slowly you and your buddies spread out in the
crowd, making your designated driver very anxious
about how you will all ever be brought together
again.
Right to worry in that she is working in direct
opposition to the aggressive force of entropy, a
tough foe.
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Basic Processes 3: Dispersion
Dispersion is:
Mixing which occurs due to differences in velocities
of neighboring parcels of fluid.
Occurs at many scales (compared to diffusion which
is strictly a molecular-scale process).
The crowd scene metaphor:
they have turned the music off, and your chaperone has
reassembled the group to leave.
 Some members get stripped as the crowd moves past
obstructions, others caught up quick moving groups
2 problems:
 (1) People hitting poles get left behind
 (2) people in the center of the crowd exit too quickly.
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Dispersion in Groundwater
Start at the scale of the intergranular channels which
the fluid moves through.
In these channels the fluid velocity is proportional to the
square of the distance from the local surfaces, leading to
separation of particles across these areas
A
B
C
D
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Tortuosity and beyond
The tortuousity of the intergranular space also
smears solutes.
At a larger scale (say the 1 m scale), there is
typically heterogeneity between materials of
differing permeability, which will again lead to
areas of higher and lower flow velocity, and
therefore dispersion.
Dispersion increases with
increasing scale as each
new dispersive process is
added to those which occur
at all of the lower scales.
A
B
C
D
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Plug or Piston Flow models
Movement is taken to be only due to advection
Processes of sorption and degradation still may be
included
How could this assumption be reasonable?
Typically don’t have data on the magnitude of dispersion
for media.
May argue that it is better to be explicit with lack of
knowledge rather than making a wild guess
If the solute is distributed relatively uniformly (as in
nitrogen), then dispersion and diffusion are not big players
If we don’t care about position, but just about final loading
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Plug Flow model
The notion is that all water molecules move
in lock-step.
Visualize marbles moving down a rubber tube
Push one in the top, and one comes out the bottom
The order of the marbles never changes (no mixing)
Solutes move in proportion to the fraction in
the liquid state
If non-adsorbed, solutes move with the water
For sorbed solutes it makes sense to use linear
partition, which does not cause dispersion
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Concentration (C/C o )
Plug flow description of processes
vel ed
a
r
t
nc e
a
t
s
i
D
Pure
Advecti on
Advecti on
with sorption
Advecti on
with decay
Advecti on
with sorption
and decay
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Example: Plug Flow Transport (Mills et al., 1985)
50,000 g/ha of naphthalene spilled
sandy loam soil with bulk density of 1.5 g/cm3
 = 0.22 cm3/cm3
water table at 1.5 meters
mean annual percolation of 40 cm.
first order partition coefficient kd =11,
half life of 1,700 days (decay rate  = 0.149 yr-1)
We want to know:
the quantity of naphthalene that will reach the
aquifer.
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Plug flow example (cont.)
Computing the
ml
cl 
q 


plug flow velocity v s  vw



m

m

cs   cl 
s
is simply a matter
l
of computing the
 q 
ratio of the water

= 
kd    
to solute velocity
(retardation factor)
The water velocity
is the flux divided
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by the moisture
0.22  1.5(11)
=
content.
= 2.3 cm/yr.
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Plug flow example (completed)
At 2.3 cm/yr, it takes 65 yr. to go 1.5 m
The half life is 4.66yrs
From the definition of half life we find the decay
rate 
c/co = 0.5 = exp(-t1/2) = exp(-  4.66)
 = 0.149 yr-1
Thus the final mass is
M = M0 exp(-0.149 x 65)
=50,000gr/ha x exp(-0.149 x 65) = 3.1 gr/ha
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What was so great about that?
Advantages of the plug flow approach:
No hidden steps or highly uncertain parameters
obtain expression which allows direct assessment
of uncertainty in key transport parameters (sorption,
percolation velocity, decay rate)
Disadvantages:
Not conservative in terms of the leading edge of the
plume which will get to the aquifer perhaps years
before the center of mass through
diffusion/dispersion
Reinforces a false sense of deterministic
knowledge of the outcome.
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The Advective/Dispersive Equation (ADE)
Also called the Convection-Dispersion
Equation (CDE)
Most widely used approach to describe solute
transport in porous media.
Derived by imposing the conservation of mass upon
transport which includes convection, diffusion, and
dispersion.
Scale dependent dispersion! In general requires
numerical methods for solution.
There are some very useful analytical solutions to
the ADE for special cases which give insight into
many real world problems.
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Scope of Application of ADE
Applicable in contexts as varied as
riverine discharges
atmospheric plumes
groundwater transport
In the vadose zone, the equation may be used to
describe any contaminant which does not move a as
a free phase (e.g., not NAPLs)
ADE assumes the solutes are hydrodynamically
inactive.
concentrations are so small that density induced flow is
ignored.
 Flow field must be known a priori. Any error in the
flow modeling will cause errors in solute modeling 26
Derivation of the ADE
Road map of our approach
(1) use a mass balance on an REV to obtain
solute mass conservation equation
(2) look at the flux term at a microscopic and
macroscopic level to identify transport
processes which are added to the solute
conservation equation
(3) add in chemical reactions (decay and
absorption) to obtain the ADE
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Mass Balance about an REV
Take an arbitrary volume
and compute the total
solute flux into the
volume, accounting for
source/sink terms.
S
j3
n3
j3 n3
C
  j 3 n3 dS    dV  
dV
t
S
V
V
rate of
delivery
through
surface
rate of
contribution change of
of sources
mass in the
or sinks
volume
j3
n3
j3 n3
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C
  j 3 n3 dS    dV  
dV
t
S
V
V
Recall derivation of Richards equation.
Transform the surface integral into a volume
integral using the Divergence Theorem
(k
n)dS

(
k)dV


S
V
Which gives us
C
  ( 3  j3 )dV    dV  
dV
t
V
V
V
gathering the integrals
C


V (3  j3 )  t   dV  0
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C


V (3  j3 )  t   dV  0
Since the volume V is completely arbitrary, we could
choose this to be any given point. The integrand
must be zero everywhere. So we have
C
 (3  j3 )  
t
which can be summarized as
Rate of Change = Fluxes in/out + sources/sinks
storage
a.k.a.: conservation of mass
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Now what about that Flux term?
We will now discuss in more detail
1. Advection
joint movement of the water/solute ensemble
2. Diffusion
Purely microscopic molecular solute movement
3. Dispersion
Scale dependent
Intrinsically anisotropic: tensor property
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Advection
The advected flux is computed through an area dA
with unit normal vector n in a local flow with vector
velocity u
total flux = (u•n c) dA [mass/time]
= jc•ndA
where jc = uc is the convective flux vector with
units [mass/(area•time)]
u
jc
n
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Diffusive Transport
Fick’s Law states that the net rate of diffusive
mass transport is proportional through the
diffusion coefficient D to the negative gradient
of concentration normal to the area, dA:
c
diffusive mass transport through
dA= -D
dA
n 3
mass
= D( 3 c  n3 )dA
t ime 
In flux notation
 D( 3 c n 3 )dA  (jdi ff n3 )dA
mass 
time 
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Advective/Diffusive Transport
the diffusive mass flux, jdiff, is defined
 mass 
jdi ff  D 3 c
time  area 
Combining this with the advective results we
have the net local (micro-scale) flux
j3  jc  jdi ff  u 3 c  D3 c
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Macroscopic Phenomena: Dispersion
The rub: how to deal with the variability in velocity in
a macroscopic sense?
Taylor’s approach of mean and deviations
Consider the local velocity to be composed of a sum
of the average local velocity with a deviation term
accounting for the departure of the local velocity
from the average
u 3  u 3  u 3
We may do the same for the concentration
c  c  c
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Macroscopic flux
Now we may put these mean/deviation expressions
into our flux equation
j3  (u3  u3 )(c  c) D3 (c  c)
Carrying out the products we obtain
j3  u3 c  u3c  u3 c  u3c  D 3c  D 3c
To obtain volume averaged flux, multiply by the
fraction of the volume taking part in the flow () and
take averages of all terms
j3   (u3 c  u3c  u3 c  u3 c  D3 c  D3c)
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j3   (u3 c  u3c  u3 c  u3 c  D3 c  D3c)
the average of a deviation is zero, so any constant
time the average of a deviation is also zero. Thus
u 3c  u3 c  D3c  0
and so our total flux becomes
j3   (u3 c  u 3c  D 3 c)
diffusive flux
convect ive flux
jconv   u 3 c
dispersive flux
jdi sp  u3 c
jdi ff   D 3 c
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dispersive flux: bjdi sp  u 3c
Dispersion is due to correlations between
variations in solute concentration and fluid velocity
High concentrations in
areas of low flux
c > 0 where u < 0, so
jdisp = c u < 0
High concentration in
areas of high flux
Center of Moving
Plume
c > 0 where u > 0, so
jdisp = c u > 0
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Great, how are we going to handle this?
For mathematical convenience we will take dispersion
to follow a pseudo-Fickian form:
jdi sp  u 3c  D 33c
D3 is the dispersion coefficient (second rank tensor)
Watch out: D3 is always anisotropic even if flow
is isotropic.
Dispersion in the longitudinal direction (in the
direction of flow) is always much greater than in
the transverse direction.
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Back to the ADE ...
Putting this form of the dispersion into the flux
j3   u3 c   (D + D3 )3 c
Putting this into the conservation of mass Eq.
C
  3  u 3 c   ( D + D 3 ) 3c   
t
we obtain the governing equation for solute
transport, the Advection Dispersion Equation!
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The dispersion tensor
The dispersion is a 3 x 3 tensor. If D3 is aligned with
the velocity field the off-diagonal terms go to zero
D3
 D xx

  D xy
 D xz

D yx
D yy
D yz
D zx 

D zy 
D zz 
z
y
x
u
D3
D x

 0
 0
0
Dy
0
0 

0 
D z 
if the media is uniform lateral to the direction of flow,
say in the y and z directions, then Dy = Dz, and we
may write this as a 2 x 2 tensor
D L
D2  
 0
0  DL = longitudinal dispersion

D T  DT = transverse dispersion
typically DL 10 DT
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About those dispersion Coefficients
We have two basic relationships to look at:
jdi sp  u 3c
jdi sp  D33c
We need to look particularly at the velocity deviation
Remember that the flow is laminar AND non-inertial
So if your double the flow rate, you double the
velocity everywhere
This doubles the mean velocity as well as the
deviation velocity
SO D3 is linear with velocity
If D3 dominates, then velocity*time and position of the
position of the solutes are related in a 1-to-1 manner
42
More on velocity and dispersion
The upshot:
IF DISPERSION DOMINATES
plume spreading will yield the same shape of
plume with consistent values of u * t
1
u = 4, t = 1
u = 1, t = 4
C/C0
0
Position
43
Dispersivity
D3 is a function of the porous media and the velocity
of the flow field.
We need a parameter which is a function of the
media alone
Define dispersivities:
D L   L u and DT   T u
.
As before, L 10 T
Intrinsic permeability and pore-scale dispersivity are
properties of the porous media, they are related
L
k
 25 to 50
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Scale Dependence of Dispersion
Consider the various scales at which velocities
will be regionally distributed
Micro: at no-slip boundaries compared to channel
core
Meso: Along structural elements (fissures/cracks/
bedding planes.
Macro: between units of differing properties (e.g.,
soil horizons)
Field: Pinch-outs, low permeability lenses etc.
Point: Dispersion increases monotonically with
scale due to additive processes
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46
Combining dispersion and diffusion
C
  3  u 3 c   ( D + D 3 ) 3c   
t
Wouldn’t it be nice if we could simply add the D’s?
Define the “Hydrodynamic Dispersion” D3’
D 3 ’  D + D3
To assign vales to D3’ we need to assess the
relative importance of diffusion and dispersion:
the dimensionless Peclet number, Pe, the ratio of
dispersion effects to diffusion effects
where d is the mean grain size
ud
Pe =
D
47
Hydrodynamic Dispersion vs Peclet Number (after Bear, 1972)
48
Hydrodynamic Dispersion Zones
Zone I: 0< Pe <0.4
Diffusion dominates
Zone II: 0.4 < Pe < 5 Mixed dispersion/diffusion
Zone III: 5 < Pe < 10 Dispersion dominates in
longitudinal, combined effects in
transverse
Zone IV: Pe > 10 and Re < 1Dispersion
dominates: laminar, non-inertial flow
Zone V: Pe > 10 and Re > 1Dispersion
dominates,but now D3 is a function of v49
Typical values of Pe
For dispersion to dominate we need Pe > 5
For a sandy soil, with a mean grain diameter,
d = 10-3 m and D = 10-11 m2/sec
 Pore water velocity needs to be greater than about
5 x 10-8 m/sec (1.6 m/year) to neglect the effects of
diffusion on the longitudinal spreading of the solute.
Had we considered a finer texture of soil this velocity
would increase linearly.
In the vadose zone we are typically in the tough
regions II and III.
Critical to correctly identify the values of d and u that
apply to your problem to determine the relative
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importance of diffusion and dispersion.
A brief note on decay
For mathematical convenience, and
because it is a reasonable approximation,
we use first order decay
A first order decay reaction is where
solute gain or loss is proportional to its
concentration
dC
 C    source or sink
dt
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