Knowledge of the types, quantities, and conditions of the minerals forming the petroleum-bearing rocks is important for assessment of their formation damage potential and design preventive and stimulation techniques to alleviate formation damage. As stated by Chakrabarty and Longo (1997), "Minerals are usually quantified using mineral properties available from published data and rock properties measured in the laboratory used cored samples or in the field using geochemical well logs." In the literature, several approaches have been proposed for this purpose.
For example, the rapid mineral quantification method developed by Chakrabarty and Longo (1997) can be used for quantification of minerals both in the laboratory and downhole. They begin by expressing each measured rock property, Ү¡, as a mass fraction, ƒ¡, weighted sum of the properties, χ¡j, of the minerals present in the rock by the property balance equation:
where n and p denote the number of rock properties measured and the number of different mineral phases present in the rock, respectively, and £, denotes the measurement error. Because, the mineral properties, χ¡j, can be obtained from the literature, the mineral compositions, ƒʲ :1,2,...,n, can be calculated by solving the set of linear algebraic equations formed by Eq. 6-7.
The system, represented by Eq. 6-8, is under-determined when the number of measured rock properties is less than the number of 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:
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:
The superscripts "7" and "-1" refer to the transpose and inverse of the matrices, respectively. V(e) and V(u) are diagonal matrices, whose elements 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:
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:
Then, Chakrabarty and Longo (1997) determine the goodness of the estimates of the mineral fractions by the following sum of the squares of the deviations:
As stated by Chakrabarty and Longo (1997), the chemical laboratories measuring the rock properties usually also provide the standard deviations of the measured properties. Similarly, the service companies running the wireline logs can provide the standard deviations of the geochemical well logging. They suggest that the XRD pattern can provide the initial estimates of the mineral fractions. Otherwise, a reasonable initial guess, such as evenly distributed mineral fractions can be used. Chakrabarty and Longo (1997) demonstrated their method, called the modified matrix algebra-based method, by several examples. Chakrabarty and Longo (1997) show satisfactory comparisons of the measured and predicted properties of the various rock samples.
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