Contaminant transport in the subsurface

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Contaminant transport in the subsurface

• Two phase medium VOIDS SOLIDS
• The voids typically filled with some fluid
  liquid or gas, sometimes both, thus giving a
  three phase medium

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Darcys law
h L K L/T
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Constant Head Apparatus (in lab)
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Continuity equation, incompressible fluid
Dimensions 1/T, multiplying by density gives
M/L3T
Dimensions 1/L
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Velocities in two phase media
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• Fig 6.2 BR

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Dispersive flux in the subsurface

• For a one dimensional flow characterized by a
  uniform seepage velocity vx, we might expect the
  dispersion coefficients D to be proportional to
  velocity
• Dispersivity is a characteristic of the system
  that will need to be determined experimentally
  (recall determination of dispersion coefficients
  in the atmosphere and surface waters)

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Diffusive and dispersive fluxes in the subsurface

• Since the diffusive and dispersive fluxes are
  defined similarly, if we let Dx represent the sum
  of diffusivity and dispersion coefficient both
  fluxes will have been included in the
  advection-diffusion equation. Thus
• We have independent means for measuring
  diffusivity (Dd ) which, is determined by
  temperature, pressure, and molecular properties
  of the diffusing molecule as well as the mixture
  in which it is diffusing
• (recall the theoretical, empirical correlations
  for gases and liquids)

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Tortuosity

• The tortuous path taken in the x direction by
  diffusing molecules means that the diffusive flux
  is less than the flux in the case of an available
  straight path.
• We can define tortuosity as
• see (Fig 6.2 BRN)
• Then our dispersion coefficient expression
  becomes
• t lt 1, 0.56 0.8 for granular media (Bear,
  1972)
• Not easy to measure independently

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Adsorption

• The adhesion of a component in the fluid to the
  solid surface.
• Depends on
• Temperature
• Solute
• Solid
• C mass of solute per volume of fluid
• S mass of solute per mass of dry solid

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S
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Accumulation in two phases
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• Figure 6.1 BR

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Advection-dispersion in two phases(no decay)
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• Recall the case examined before
• With initial and boundary conditions
• S(0,t) So for all tgt0
• ?S/?x 0 at xL
• S(x,0) 0 at t0
• And analytical solution

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• In BR notation for subsurface contaminant
  transport, assuming constant Dx
• Initial and boundary conditions
• C(0,t) Co for all tgt0
• ?C/?x 0, at xL
• C(x,0) 0 at t0
• Analytical solution (Eqn 6.17 BR) at xL

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• The previous formulation describes a tracer (dye)
  test (step input) in a column of porous material.
• The amount of liquid passed through the column is
  usually measured in number U of pore volumes (Eqn
  6.39 BR)
• When the solution is expressed in terms of U and
  plotted on appropriate coordinates, the
  dispersion coefficient can be obtained from the
  slope. (See Fig 6.10a BR)

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• Figure 6.10a

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Dispersion in a sand column, Example 6.4
BRN(see Ex6_4BR.xls)
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PARTICLE SIZE DISTRIBUTION

• Gaussian, or normal

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ALTERNATE FORM FOR GAUSSIAN DISTRIBUTION
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GAUSSIAN DISTRIBUTION, INTEGRATED FORM

• (Table 8.3 de Nevers)
• Probability scale a scale linear in z
• Gaussian distribution gives linear plot on
• normal (y axis) vs probability (x axis)
  coordinates
• Slope standard deviation
• mean value is at z 0

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Table 8.3 de Nevers

• Values of the cumulative frequency integral

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Figure 8.8 de Nevers

• Graph with (log) probability scale