Capillary Pressure Permeability Relations

1
Capillary Pressure / Permeability Relations
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Single Capillary Tube

• Previously discussion showed that for a single
  capillary
• consistent units required
• Pc dyne/cm2
• s dyne/cm
• r cm

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Bundle of Capillary Tubes Model

• Purcell (ABW page 169) developed a model which
  considers the porous media to be a bundle of
  capillary tubes of varying sizes
• This model was used to develop a relationship for
  predicting permeability from the drainage
  capillary pressure curve
• for a discrete set of capillary sizes, in
  consistent units
• k cm2
• s dyne/cm
• Pc dyne/cm2

4
Purcells Equation

• For a continuous distribution of capillary sizes,
  determined from the Pc(Sw) function (consistent
  units)

• a is an dimensionless, empirical lithology
  factor (Purcell used symbol, l)
• The function 1/Pc2 could be integrated
  numerically
• OR, we could use the Sw model for Pc
• Note that the function 1/Pc2 has value of zero if
  SwltSwi

5
Purcells Equation

• For oilfield units,
• k md
• s dyne/cm
• Pc psia
• Note Handwritten notes in margin of ABW
  photocopy is missing a squared symbol on
  conversion from dyne/cm2 per atmosphere
• Self Study Be able to derive this conversion
  constant (and its units), 10.66

6
Purcells Equation

• Using an average lithology factor, a0.216,
  Purcell showed that permeability can be estimated
  from the drainage capillary pressure curve
• For permeability, estimates that are the correct
  order of magnitude are considered fairly accurate

Perfect Correlation Trend
Actual Correlation Trend
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Application of Sw Model

• To integrate the function 1/Pc2, we consider
  piecewise integration

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Application of Sw Model

• From our definition of Sw
• Integral requires dSw
• Sw model for Pc

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Application of Sw Model

• Substituting dSw for dSw in integrand changes
  limits

10
Application of Sw Model

• Substituting Sw model for Pc in integrand

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Purcells Equation with Sw Model

• Finally, for oilfield units