Longitudinal Dispersion in Porous Media

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Longitudinal Dispersion in Porous Media

• A. G. Hunt and T. E. Skinner
• Department of Physics
• Wright State University

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What is longitudinal dispersion?

• It is what causes contaminant plumes to spread
  out in the direction of flow.
• The spreading is due to heterogeneity in the
  medium.
• Consider a (saturated) medium that is, at the
  pore scale, homogeneous in the mean.
• Heterogeneity comes from distribution of pore
  sizes (and shapes).

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For such systems we know already

• K (the hydraulic conductivity)
• ? (the electrical conductivity)
• ka (the air permeability)
• D (the diffusion constants, solute and gas)
• Electromechanical properties
• Thermal conductivity (?)
• as functions of saturation.

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In fact we have developed a systematic approach
which yields all of these properties in terms of
either critical path analysis (from percolation
theory), percolation scaling, or both. How would
one incorporate dispersion into such a mix?
except thermal conductivity
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ANSWER
Find the pdf W(gmin) that a particular
path through the system is limited by a
minimum conductance gmin, find the time it takes
for a particle to traverse the path with gmin,
and use the probabilistic transformation W(gmin)
dgmin W(t) dt.
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Important Percolation Input

• Saturated K
• K (S)
• ? (S)
• kA (S)
• dispersion

• Competition
• Geometry (except very near threshold)
• Topology (almost all)
• Topology exclusively
• both

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For K, competition between pore-size distribution
and topology puts controlling g enough past
critical percolation that paths are not fractal.
Dispersion is still influenced by fractal path
distribution.
Exactly this contrast Is built into our
calculation framework
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Two inputs to t(g)

• Pore-size distribution (geometry) and flow rates
  (streamlines)
• Tortuosity (topology of connections)

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For any pore of radius r, cross-sectional area A
t ? r /v ? r A / Q
For a path with minimum pore radius r
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Topological complication (tortuosity)
Note that we choose d for optimal path along
backbone cluster
Product of two
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Approximate distribution of controlling
conductances (from previous publications),
giving streamline probabilities actual
distribution defined in terms of the exponential
integral, but also with logarithmic singularity.
Distribution comes from cluster statistics of
percolation theory with sharp cut-off at maximum
departure from critical percolation and minimum
cluster size equal to system length, x.
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Work left to do

• Tortuosity cannot actually diverge in a finite
  system.
• Compare and contrast with Stanleys group, which
  uses a distribution of arrival times other than
  the optimal one, but does not use distribution of
  controlling conductances.
• Other?

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Conclusions

• We have made an interesting start on a
  complicated problem.
• It is widely believed that the saturated
  hydraulic conductivity and dispersion are closely
  related note that g value for which t has a
  local minimum is closely related to the optimal g
  value which defines the saturated hydraulic
  conductivity it also produces a spike in solute
  arrival
• Whether specific results at this point are
  compatible with experiment is a little doubtful
  the tail is probably not fat enough and it is not
  clear if such a spike as predicted has been
  observed.