Page 371 - Petrophysics 2E

CENTRIFUGE MEASUREMENT OF CAPILLARY PRESSURE 339

length with the radial distances between the inlet and end of the
core [21].

(5.47)

which is the average saturation, 5, between r and re.
The corresponding capillary pressure is:

(P,) = 0.5ApN2(rz - r2) (5.48)

Solving for r:

2 2 -1 112
r = re[l - Pc(0.5ApN re) I (5.49)

Differentiating Equation 5.48 with respect to r and solving for dr:

dr = -(ApN2r)-'dPc (5.50)

Substituting Equations 5.49 and 5.50 into Equation 5.47 and recognizing
that the following conditions exist:

(P,) = 0.5ApN2(r: - rf)
P, = Oatr = re
Pc = (Pc)i at r = ri

algebraic reduction yields:
1 -112

5 = * (SdP,) [ 1 - (&) - It2)] (5.51)
(1
2Pc)i

This basic equation (Equation 5.5 1) gives the exact relationship
between the average saturation, 5, the saturation at any point in the
core, S, and the inlet capillary pressure, (Pc)i. The inlet face saturation,
Si, corresponding to (Pc)i is obtained by solving Equation 5.51. Van
Domselaar [21] attempted to derive a general solution for Equation 5.51,
but, as shown by Rajan, it also involved an approximation [22].
Rajan developed a general solution for Equation 5.5 1 without using the
Hassler and Brunner simplifying assumptions [22]. Calculation of Si using
Raja's expression, however, also requires evaluation of the derivative,
dS/d(P,)i, which can be obtained from the least-squares fit of the data