Solute Transport in the Vadose Zone

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Solute Transport in the Vadose Zone

• Quantification of oozing, spreading and smearing

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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 your 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
• ?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 doesnt work
• Desorption and adsorption follow the same
  isotherm
• Rub There is hysteresis between adsorption and
  desorption AND time dependence!

Solid with adsorbed Concentration cs
Liquid with concentration cl
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Isotherms
Three most popular relationships
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Linear Isotherm

• cs Kd cl
• 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

• cs a cl Q/(1acl)

Q adsorption sites/mass
a k1/k2 where k1 rate of adsorption k2
rate of desorption

• Notes
• for acl ltlt1 this reduces to cs aQcl (linear
  isotherm)
• for acl gtgt1 this reduces to cs Q Makes sense
  since Q is the sorption capacity of the soil
  (recall CEC)

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Whats 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 sitesbeing filled
  (ra) will equal therate of sites being vacated
  (rd)
• Assuming that each site acts independently, the
  probabilityof sorption will be proportionalto
  the probability of a solutemolecule hitting that
  site

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Deriving the Langmuir isotherm

• So we estimate the adsorption rate
  asSimilarly, we may estimate the rate of
  desorption as being proportional to the number of
  sites filled

Proportionality Constant
Concentration in Liquid
Fraction of sites unfilled
Fraction of sites filled
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Langmuir derivation...

• At equilibrium, ra rd. Equating
  theseletting k k/k and multiplying each
  side by QSolving for the sorbed concentration
• 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 Ficks law
• 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

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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.

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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 dont 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 dont 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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Plug flow description of processes
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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
• We want to know the quantity of naphthalene
  that will reach the aquifer.

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Plug flow example (cont.)

• Computing the plug flow velocity is simply a
  matter of computing the ratio of the water to
  solute velocity (retardation factor)
• The water velocity is the flux divided by the
  moisture content.

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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, can describe contaminant that
  is not a free phase (e.g., not NAPLs)
• ADE solutes are hydrodynamically inactive.
• concentrations small so density induced flow is
  ignored.
• Flow field must be known a priori. Any error in
  the flow modeling will cause errors in solute
  modeling

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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 flux at a microscopic and macroscopic
  level to identify processes to include in 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.

rate of change of mass in the volume
contribution of sources or sinks
rate of delivery through surface
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• Recall derivation of Richards equation. Transform
  the surface integral into a volume integral using
  the Divergence TheoremWhich gives
  usgathering the integrals

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• 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
• 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 (un c) dA mass/time
• jcndA
• where jc uc is the convective flux vector with
  units mass/(areatime)

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Diffusive Transport

• Ficks 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, dAIn
  flux notation

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Advective/Diffusive Transport

• the diffusive mass flux, jdiff, is
  definedCombining this with the advective
  results we have the net local (micro-scale) flux

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Macroscopic Phenomena Dispersion

• The rub how to deal with the variability in
  velocity in a macroscopic sense?
• Taylors 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
• We may do the same for the concentration

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Macroscopic flux

• Now we may put these mean/deviation expressions
  into our flux equationCarrying out the
  products we obtain To obtain volume averaged
  flux, multiply by the fraction of the volume
  taking part in the flow (?) and take averages of
  all terms

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• the average of a deviation is zero, so any
  constant time the average of a deviation is also
  zero. Thusand so our total flux becomes

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• Dispersion is due to correlations between
  variations in solute concentration and fluid
  velocity

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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
• 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
  fluxPutting this into the conservation of mass
  Eq.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 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
• DL longitudinal dispersion
• DT transverse dispersion
• typically DL 10 DT

z
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About those dispersion Coefficients

• We have two basic relationships to look atWe
  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 velocitytime and position
  of the position of the solutes are related in a
  1-to-1 manner

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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
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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
• As before, ?L 10 ?T
• Intrinsic permeability and pore-scale
  dispersivity are properties of the porous media,
  they are related

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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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Combining dispersion and diffusion

• Wouldnt it be nice if we could simply add the
  Ds?
• Define the Hydrodynamic Dispersion D3 D3
  ? 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
  effectswhere d is the mean grain size

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Hydrodynamic Dispersion vs Peclet Number (after
Bear, 1972)
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Hydrodynamic Dispersion Zones

• Zone I 0lt Pe lt0.4 Diffusion dominates
• Zone II 0.4 lt Pe lt 5 Mixed dispersion/diffusion
• Zone III 5 lt Pe lt 10 Dispersion dominates in
  longitudinal, combined effects in
  transverse
• Zone IV Pe gt 10 and Re lt 1Dispersion
  dominates laminar, non-inertial flow
• Zone V Pe gt 10 and Re gt 1Dispersion
  dominates,but now D3 is a function of v

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Typical values of Pe

• For dispersion to dominate we need Pe gt 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 decrease 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 importance of diffusion and dispersion.

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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