Diffusion, Electrical Conduction, and Flow in Porous Media

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Diffusion, Electrical Conduction, and Flow in
Porous Media

• Continuum Percolation Theory for Truncated Random
  Fractal Media

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Two Basic Applications of Percolation Theory to
Flow

• Near Perc. Threshold
• Gives scaling of hydraulic, electrical
  conductivity, air permeability with distance from
  threshold moisture content.
• Connectivity/ tortuosity

• Far from Perc. Threshold
• Gives dependence of same properties on moisture
  content through dependence of bottleneck pore
  size.
• Pore-size dependence

NOTE These two applications are not
independent (same critical percolation
probability for both)!
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And One to Accessibility (in Hysteresis)

• The probability that a given site (or volume) is
  connected to the infinite cluster above the
  percolation threshold is given by a universal
  function of percolation theory.

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Apply to Random (Truncated) Fractal Model of
Porous Media

• What is optimal form of percolation theory?
• Continuum.
• What are relevant variables?
• Critical volume fraction (moisture content,
  air-filled porosity).
• What about potential non-universal behavior?
• Much, though not all behavior is universal.

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Basic Results for Saturation Dependence of
Properties

• K(S) pore-size distribution dominates at high
  sat., connectivity/tortuosity at low sat.
• Electrical Conductivity similar
• Air permeability connectivity/tortuosity
  dominates throughout
• Solute and gas diffusion, connectivity/tortuosity
  dominate throughout

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Consider First Unsaturated K on Lattice of Tubes
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Appropriate Analogy in Continuum Percolation
Representation
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Result for K(?)
Note argument is not ? - ?t (unless ?1)!
Result from bottleneck pore.
Result from average resistance to flow over all
larger pores.
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Effects of Tortuosity, Connectivity
t1.88 (3D) 1.28 (2D)
In vicinity of ?t conductivity must scale as
Require both K and dK/d? to be continuous at
unknown ?x this generates prefactor and ?x.
Hydraulic Conductivity
Electrical Conductivity
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Pore-sizes
Connectivity/tortuosity
Hanford site data (Rockhold et al., 1988)
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Comparisons with Other Results
Balberg, Phil. Mag. 1987 if W(g) continues to g0
But if distribution continues to g0, this means
that smallest pore has zero radius. That means
that ? 1 in Rieu and Sposito model and,
yields Balberg result since, at ?1 (also for ?)
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Archies Law
Evaluate at saturation
Archies law
Use proportionality of critical volume
fraction to porosity (Hunt, AdWR)
m1.86 Thompson et al., 1987
Experiment finds
Kuentz et al., 2000, find (in 2-D simulations)
m1.28
To find limit of validity of Archies law set
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What happens if Archies law is not quite valid
(cross-over occurs before saturation)?
Electrical conductivity at saturation is enhanced
by
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Data from Katz and Thompson, Advances in Physics,
1987
R2 increases to over 0.3 if single point is
eliminated
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Application to Loma Prieta Earthquake Precursor
Signal

• Ultra-low frequency magnetic field effects were
  interpreted (Merzer and Klemperer) as due to
  increase in fault zone conductivity by factor 15.
  Authors suggested Archies law exponent changed
  from 2 to 1.
• Check A change from 1.88 to 1.28 is expected for
  a change in dimensionality from 3 to 2. This is
  consistent with development of 2-D network of
  interconnected micro-fractures in hours before
  earthquake.

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Theory has consequences for saturation dependence
of electrical conductivity
Data from Tusheng Ren for silt loams (two
adjustable parameters for entire family of
curves).
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Data from Tusheng Ren for sands. Power ca. 2.5,
not 1.88 Balberg non-universality?
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Additional data from Andrew Binley
Both cases one adjustable parameter, other
parameters from Cassiani
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Data from Jeffrey Roberts
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What About Air Permeability?
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Result observed by Moldrup et al., (2003) with
power equal to 1.84?0.54. Note that the expected
power in 2-dimensions is 1.28 (Derrida and
Vannimenus, 1982, J. Phys.) (cases with power
near 1 observed in clay-rich media, otherwise
near 2).
Two-dimensional configuration Steriotis et al.
(1999) compared with power of 1.28 one
adjustable parameter.
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Solute and Gas Diffusion
Typical definition
Results of numerical simulations of Ewing and
Horton xsystem length
Finite-size scaling (Fisher, 1970)
Value of universal exponent of percolation,
?0.88
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Final form substituting moisture content for
porosity
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How to Modify for Gas Diffusion?
Trivial modification
But not all air allowable pores have air, only
those accessible to the infinite percolation
cluster.
Extra factor is from percolation theory and
represents the fractional volume attached to the
infinite cluster (no adjustable parameters).
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Data compiled by Werner et al., 2004 (VZJ)
The Moldrup relationship, Dpm/Da?2.5/?,
originally proposed for sieved and repacked
soils, gave the best predictions of several
porosity-based relationships (Werner et al.,
2004)
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Repacked soils (presumably without structure)
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Rieu and Sposito WaterRetention Curves
Continuous truncated random fractal analogy to
Rieu and Sposito model
Porosity
Water retention curve
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Hydraulic Conductivity Limited Equilibration
Cross-over to regime of rapidly diminishing K
will have consequences for equilibration on
ceramic plates
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Hypothesis At Moisture Contents Approaching
Threshold Actual Water Removed is reduced by
ratio of
to
This is effectively a zero-dimension model with
correct constitutive relations rather than a
column model with inaccurate constitutive
relations.
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• The deviations in the fractal scaling of the
  water retention curves are predicted by the
  hydraulic conductivity derived from the fractal
  model using percolation theory for the hydraulic
  conductivity.
• This means that the fractal model actually
  predicts the deviations from the fractal model.
• Note that the deviations occurred at a moisture
  content related to the moisture content at which
  solute diffusion vanished in soils from another
  contintent (Moldrup et al., 2001).

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Hysteresis has Two Components

• Water removed from a pore must pass through a
  pore throat water imbibed must fill the entire
  pore body. Pressure ratio is ratio of two radii
  (typically about 2).
• All pores that allow water will contain that
  water during drainage, but only those connected
  to the infinite percolation cluster will contain
  water during imbibition.

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Percolation Component
Rieu and Sposito random fractal water retention
curve.
Same accessibility factor as for gas diffusion.
Product of two gives imbibition curve.
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Air entry pressure 11cm
Data from Bauters et al. (1998)
Characteristic pressure 5.5cm
Ratio of pressures consistent with
other experiments.
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Summary

• KS Competition
• K(S) Mainly CPA
• ?(S) Mainly Scaling
• ka(?0) Competition
• ka(?) Entirely Scaling
• D Entirely Scaling

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Conclusions

• Application of continuum percolation theory to
  fractal models yields results for saturation
  scaling of
• Hydraulic conductivity
• Air permeability
• Electrical conductivity
• Pressure-saturation curves including hysteresis
  and lack of equilibration
• Solute diffusion
• Gas diffusion (worst comparison with expt., still
  best theory)
• in agreement or accord with experiment.
• Theory yields consistent interpretations,
  parameters, analysis of relevance of pore size
  distributions compared with, e.g.,
  connectivity/tortuosity issues.
• Theory reinterprets fundamental soil physics.

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Relevant References

• Hunt, A. G., and Gee, G. W., 2002, Application of
  Critical Path Analysis to Fractal Porous Media
  Comparison with Examples from the Hanford Site,
  Advances in Water Resources, 25, 129-146.
• Hunt, A. G., and Gee, G. W., 2002, Water
  Retention of Fractal Soil Models Using Continuum
  Percolation Theory Tests of Hanford Site Soils,
  Vadose Zone Journal, 1, 252-260.
• Hunt, A. G., 2003, Percolative Transport and
  Fractal Porous Media, Chaos, Solitons, and
  Fractals, 19, 309-325.
• Hunt, A. G., and Ewing, R. P., 2003, On The
  Vanishing of Solute Diffusion in Porous Media at
  a Threshold Moisture Content, Soil Science
  Society of America Journal, 67, 1701-1702, 2003.
• Hunt, A. G., and Gee, G. W., 2003, Wet-End
  Deviations from Scaling of the Water Retention
  Characteristics of Fractal Porous Media, Vadose
  Zone Journal, 2, 759-765.
• Hunt, A. G., 2004, Continuum Percolation Theory
  for Water Retention and Hydraulic Conductivity of
  Fractal Soils 1. Estimation of the Critical
  Volume Fraction for Percolation, Advances in
  Water Resources, 27, 175-183.
• Hunt, A. G., 2004, Continuum Percolation Theory
  for Water Retention and Hydraulic Conductivity of
  Fractal Soils 2. Extension to Non-Equilibrium,
  Advances in Water Resources, 27, 245-257.
• Hunt, A. G., 2004, A note comparing van Genuchten
  and percolation theoretical formulations of the
  hydraulic properties of unsaturated media,
  accepted, June., 2004 to Vadose Zone Journal.
• Hunt, A. G., 2005, Continuum Percolation Theory
  for Saturation Dependence of Air Permeability,
  February issue of Vadose Zone Journal.
• Hunt, A. G., 2004, Continuum Percolation Theory
  and Archies Law Oct., 2004 issue of Geophysical
  Research Letters.
• Hunt, A. G., and T. E Skinner, 2005, Hydraulic
  Conductivity Limited Equilibration Effect on
  Water-Retention Characteristics, February issue
  of Vadose Zone Journal.
• Hunt, A. G., 2005, Comment on Ultra-low Frequency
  Magnetic Precursors to the Loma Prieta
  earthquake, by S. Klemperer and Merzer, Accepted
  to Pure and Applied Geophysics, January, 2005.