Elliptical Mathematical Scholium of Petrophysical Modulation of Volume of Shale Paradigm from Linear and Non-Linear Confluxibilities

Abstract: Shale characterization and volume prognosis using only the linear confluxibilities creates quantifiable petrophysical constraints on the volumetric illation of oil and gas reserves since it is an inverse homology to formation porosity. Although, linear correlations are quicker and most often used but are most often degraded by quantificational errors due to the degree of formation heterogeneity complexities, while its degree of linearity varies with severity of complex sandstone reservoirs such as shaly-sands and sandy-shales noticeable in the X-Field, Niger Delta, Nigeria. The non existence of non-linear correlations encapsulated in most logging software increases the degree of intricacies of the linear rubric and puts enormous strains on the magnitude of petrophysical based uncertainties. The aim of this study is to carry out a detailed comparative study of linear and non-linear homologies for astute prognosis of volume of shales from gamma ray well logs in a Field in the Niger Delta. The main objective is to develop new mathematical models/correlations for quicker quantitative reservoir elucidation and effective management of both linear and non-linear models of volume of shales from gamma ray logs. Quantitative and qualitative petrophysical interpretation techniques were comprehensively used for both the linear and 4 non linear correlations in 5 oil wells with 25 sand zones in the X-Field. Results of volume of shale showed collective convergent declining curves of all the non-linear models with increasing gamma ray content in contrast to declining straight line of linear model. Results of modified volume of shale for this field gave Vshmod = -0.46In(GRlog) + 1.997 with square regression (R2) of 0.996. This perhaps shows increased volume of shale profile with depth. Model validation results showed good matches with R2 > 0.9 with Steiber 2 given as R2 = 0.999, Steiber 3 given as R2 = 0.991, average Steiber given as R2 = 0.999, Larinov (older rocks) given as R2 = 0.997, Larinov (tertiary rocks) given as R2 = 0.999, Clavier given as R2 = 0.999 and IGR (gamma ray index) given as R2 = 0.999. Detailed petrophysical re-evaluation of the non-linear models was achieved with perhaps appreciable square regression and a clear departure from the conventional linear model initially perceived as the most accurate. It is recommended that more studies be done with large data volume from regional Field studies to buttress the few points highlighted in this research.

Keywords: Shales, correlations, linear, non-linear, evaluation, petrophysics, sandstone reservoirs.