Mineral Content Conversion by Piecewise Polynomial Truncated Singular Value Decomposition Algorithm in the Context of Formation Element Logging

Abstract

In spite of diverse types of elements and minerals in the stratum, there are only a few kinds they are mainly concentrated, thus the key to element-mineral conversion lies in the quantitative relationship between main elements and main minerals, which task, however, is challenged in practical logging. Firstly, the element types detected by logging tools are fixed, while the mineral types vary in different regions, which makes it difficult to determine the mineral type merely from logging data accurately. Secondly, the mineral types are generally more than the element types detected in the stratum, leading to a multiplicity of the conversion results. Therefore aiming to solve such challenges, this paper establishes a reasonable conversion relationship between the element and mineral, and ultimately attains relatively precise results through certain mathematical method, i.e. piecewise polynomial truncated singular value decomposition (PP-TSVD) algorithm that eliminates “small” singular values of the ill-conditioned conversion coefficients matrix, and meanwhile makes a superimposition of the effective components in the “small” singular value spectrum in the solution of the TSVD method, which greatly reduces the mean square error of the solution, so as to improve the accuracy and reliability of the solution. In addition, PP-TSVD method can determine either the mineral content of single mineral or diversified types, on condition that the known element types are not less than the mineral types to seek. When mineral types are numerous, the integration of some types into one mineral is imperative in order to establish suitable conversion coefficients. In this paper, based on previous studies and the #JD1 well data measured by ECS logging tool, the PP-TSVD method is employed to study the calculation and conversion between the element content and mineral content.