Page 348 - Petrophysics 2E

3 1 6 PETROPHYSICS: RESERVOIR ROCK PROPERTIES

Neglecting the small term dz2/R~R2, the increase in the area is equal to:

A’B’C’D‘ - ABCD = LlLz x dz X - + - (5.4)
:2)

The isothermal work [(N/m) x m2 = Nm] required to expand the area
against the surface tension is:

Work (1) = o(L1L2 x dz) - + - (5.5)
(il i2)

The isothermal work done by the increase of pressure to advance the
surface a distance dz is equal to:

Work(2) = p(LlL2 x dz) (5.6)

Equating the two work quantities and cancelling common terms yields
the capillary pressure as a function of interfacial tension and the radii of
curvature [2]:
Pc = CT ($ + &) (5.7)

When a porous medium is considered, R2 in Equation 5.7 may
be negative; therefore, the more general equation for the capillary
pressure is:
Pc = o (& * &) (5.8)

If the radii of curvature are equal (in a capillary tube, for example),
Equation 5.8 reduces to:

20
pc = -K (5.9)

The special case of the Plateau equation (Equation 5.9) may be used
to derive a relation from the interfacial geometry of a wetting fluid in a
capillary. Figure 5.3 is an exaggerated view of a capillary tube containing
water as the wetting phase in contact with a non-wetting fluid (gas or
oil). The radius of the spherical interface is larger than the radius of the