Page 140 - Reservoir Formation Damage

122 Reservoir Formation Damage

different minerals present in the rock, and over-determined, otherwise, and
determined when they are equal. In order to handle both of these cases
and alleviate any instability problems associated with the solution of
Eq. 6-8, Chakrabarty and Longo (1997) supplemented the property
balance equation (Eq. 6-8) with the following constraining equation:

c = C' f + u (6-9)

This equation incorporates any prior information available or initial
guesses on the fractional compositions of the minerals present in rocks,
in which c is a vector of the initial guesses of the mineral fractions, C is
a unit diagonal matrix, and u is a vector of errors associated with the
initial guesses of the fractions of the various minerals.
Chakrabarty and Longo (1997), then, combines Eqs. 6-8 and 9 as:

X'f
(6-10)
C'f

from which, the estimates of the mineral fractions is expressed by:

T
= [X T • {V(e)}~ 1 * X + C ' {V(u)}
(6-11)

The superscripts "7" and "-1" refer to the transpose and inverse of the
matrices, respectively. V(e) and V(u) are diagonal matrices, whose ele-
ments are the error variances (standard deviations) of the measured rock
properties and the initial guesses of the mineral fractions, respectively.
Chakrabarty and Longo (1997) expressed the variances of the mineral
fractions by:

v(f) = [x -X+C - c (6-12)

which is the same as the first part of Eq. 6-11.
Using Eq. 6-8 without the error term and Eq. 6-11, the rock properties
are estimated by:

y = (6-13)