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Assessment of petrophysical quantities inspired by joint multifractal approach

In this paper joint multifractal random walk approach is carried out to analyze some petrophysical quantities for characterizing the petroleum reservoir. These quantities include Gamma emission (GR), sonic transient time (DT) and Neutron porosity (NPHI) which are collected from four wells of a reservoir. To quantify mutual interaction of petrophysical quantities, joint multifractal random walk is implemented. This approach is based on the mutual multiplicative cascade notion in the multifractal formalism and in this approach $L_0$ represents a benchmark to describe the nature of cross-correlation between two series. The analysis of the petrophysical quantities revealed that GR for all wells has strongly multifractal nature due to the considerable abundance of large fluctuations in various scales. The variance of probability distribution function, $λ_{\ell}^2$, at scale $\ell$ and its intercept determine the multifractal properties of the data sets sourced by probability density function. The value of $λ_0 ^2$ for NPHI data set is less than GR's, however, DT shows a nearly monofractal behavior, namely $λ_0 ^2\rightarrow 0$, so we find that $λ_0^2({\rm GR})>λ_0^2({\rm NPHI})\ggλ_0^2({\rm DT})$. While, the value of Hurst exponents can not discriminate between series GR, NPHI and DT. Joint analysis of the petrophysical quantities for considered wells demonstrates that $L_0$ has negative value for GR-NPHI confirming that finding shaly layers is in competition with finding porous medium while it takes positive value for GR-DT determining that continuum medium can be detectable by evaluating the statistical properties of GR and its cross-correlation to DT signal.