Page 77 - Reservoir Formation Damage

60 Reservoir Formation Damage

where D denotes the diameter, f\(D) and / 2(^) are the distribution
functions for the fine and coarse fractions, and w is the fraction of the
fine fractions.
Popplewell et al. (1988, 1989) used the p-distribution function to
represent the skewed size distribution, because the diameters of the
smallest and the largest particles are finite in realistic porous media. For
convenience, they expressed the P-distribution function in the following
modified from:

m
m
f(x) = x am (l-x) / \x am (\-x} dx (3-19)
I J
in which jc denotes a normalized diameter defined by:

(3-20)

and are the smallest and the largest diameters, respectively, a
£> min £> max
and m are some empirical power coefficients. The mode, x m, and the
2
spread, a , for Equation 3-19 are given, respectively, by:
x m=a/(a + (3-21)

and

(am + l)(m +1)
(3-22)

Chang and Civan (1991, 1992, 1997) used this approach successfully in
a model for chemically induced formation damage.

Fractal Distribution. Fractal is a concept used for convenient mathe-
matical description of irregular shapes or patterns, such as the pores of
rocks, assuming self-similarity. The pore size distributions measured at
different scales of resolution have been shown to be adequately described
by empirically determined power law functions of the pore sizes (Garrison
et al., 1993; Verrecchia, 1995; Karacan and Okandan, 1995; Perrier et al.,
1996). The expression given by Perrier et al. (1996) for the differential
pore size distribution can be written in terms of the pore diameter as:

dV_ l
, 0<d<e (3-23)
dD