The Leverett J-Function

Part 11, Chapter 11: Capillary Pressure and Saturation-Height

Learning objectives

State the dimensionless Leverett J-function and its terms
Explain how dividing out the square root of k over phi cancels pore size
Collapse many rock types' capillary curves onto one J-Sw curve
Apply one cored well's capillary data across the whole field

One Curve for Many Rocks

Every rock type draws its own capillary-pressure curve, and they are not gentle variations on one shape. A tight rock needs many times the pressure of a clean sand to reach the same water saturation, so a field of mixed rock is a tangle of curves with no single one to carry into a saturation-height model. The question Leverett answered in 1941 is how to fold all of them into one.

Cancelling the Pore Size

The trick is that two things in the rock both track the pore-throat radius. Capillary pressure runs as $P_c \sim 1/r$ by Young-Laplace, while the quantity $\sqrt{k/\phi}$ , the reservoir quality index from the permeability chapter, runs as $r$ . Multiply one by the other and the radius cancels, leaving a dimensionless number that depends only on the shape of the pore system, not its size:

J(S_w) = 0.21645\,\frac{P_c}{\sigma \cos\theta}\sqrt{\frac{k}{\phi}}.

The constant 0.21645 makes $J$ dimensionless with $P_c$ in psi, the interfacial tension $\sigma$ in dyne/cm, and $k$ in millidarcies. Rocks that share a pore-system shape, however far apart their raw $P_c$ curves sit, collapse onto one master $J$ - $S_w$ curve.

The Payoff and the Limit

This is what makes capillary pressure usable at field scale. Measure $P_c$ on a handful of cores, fit one $J$ curve, and apply it to every porosity-permeability pair the logs deliver, turning a few laboratory samples into a saturation-height model for the whole reservoir. The limit is honest: real rocks differ in pore geometry, not only in $k/\phi$ , so the collapse is never quite perfect. The cure is to group the rock into a small number of types and fit one $J$ curve per type, which the next sections turn into the saturation-height function itself.

References

Leverett, M. C. (1941). Capillary behavior in porous solids. Transactions of the AIME, 142, 152-169.
Tiab, D. and Donaldson, E. C. (2015). Petrophysics, 4th ed. Gulf Professional Publishing.