1. Capillary Pressure / Permeability Relations.

2. Single Capillary Tube
Previously discussion showed that for a single capillary:
consistent units required
Pc dyne/cm2
s dyne/cm
r cm

3. 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. Purcell’s 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 Sw<Swi

5. Purcell’s 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. Purcell’s Equation
Using an average lithology factor, a=0.216, 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
Actual Correlation Trend
“Perfect” Correlation Trend
Note that the range of values for SwD is from 0 to 1

7. Application of Sw* Model
To integrate the function 1/Pc2, we consider piecewise integration

8. Application of Sw* Model
From our definition of Sw*:
Integral requires dSw:
Sw* model for Pc:

9. Application of Sw* Model
Substituting dSw* for dSw in integrand changes limits

10. Application of Sw* Model
Substituting Sw* model for Pc in integrand

11. Purcell’s Equation with Sw* Model
Finally, for oilfield units: