Building the carbonate pore-type classifier for well logging via the blended training dataset

Abstract

The exploration of carbonate rocks has outstanding economic benefits, as well as facing the extreme challenge of reservoir characterization. This article has proposed a data-based description scheme generalizing carbonate pore-type characteristics from both laboratory measurements and theoretical predictions to the well logging dataset. Firstly, in the feature space of elastic properties, we employed the supervised machine learning (ML) algorithm to convert this pore-type classification process from a typical nonlinear inversion to sample label allocation problem. Secondly, to alleviate the inherent scale gaps between data sources, virtual samples were randomly mixed into the laboratory measured dataset. Through inheriting or mimicking statistical elastic features of limited core samples, the new built training dataset could improve the overall sample richness and thus help the ML algorithms making better identification decisions. On the one hand, this scheme was verified by 74 carbonate samples. In the feature space of high dimensions, the blended dataset trained radial basis function support vector machine accurately separated different carbonate pore systems. Moreover, using logging curves of a carbonate gas field, we verified the generalization capability of this scheme over unbalanced data scales. Searching skills were used to optimize model and classifier setups according logging curves of a specific interval. Finally, with the help of the vertical label distributions, logging elastic response modes and historical pore evolution footprints were further studied.

Appendix: Equivalent elastic modulus of multiple pore-type materials

Assume n = K_m/G_m and m = n(P^*i− Q^*i). Polarization factors P and Q are decomposed as primary and secondary terms P₀(n), Q₀(n) and P₁(m, n) depending on n and (m, n). For penny-shaped pores, Li and Zhang (2010) proofed parameter m satisfying

$$ P^{{{*}i}} - Q^{{{*}i}} = - \frac{1}{5} + \frac{{{{K^{*} } \mathord{\left/ {\vphantom {{K^{*} } {G^{*} \left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 4} \right)}}} \right. \kern-\nulldelimiterspace} {G^{*} \left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 4} \right)}}}}{{\pi \alpha \left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 1} \right)}} - \frac{{4\left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 4} \right)}}{{15\pi \alpha \left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 1} \right)}} - \frac{{8\left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 4} \right)}}{{15\pi \alpha \left( {{{3K^{*} } \mathord{\left/ {\vphantom {{3K^{*} } {G^{*} }}} \right. \kern-\nulldelimiterspace} {G^{*} }} + 2} \right)}} $$

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Hence, decoupled DEM solution of dry rock matrix bulk and shear moduli is

$$ \left\{ \begin{gathered} K_{{\text{d}}} = K_{{\text{m}}} (1 - \varphi )^{{P_{{^{0} }} + P_{{^{1} }} }} e^{{P_{{^{1} }} \varphi }} \hfill \\ G_{{\text{d}}} = G_{{\text{m}}} (1 - \varphi )^{{Q_{{^{0} }} + Q_{{^{1} }} }} e^{{Q_{{^{1} }} \varphi }} \hfill \\ \end{gathered} \right. $$

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Transforming Eq. 13 to a first order Taylor expansion, we get

$$ \left\{ \begin{aligned} K_{{\text{d}}} &= K_{{\text{m}}} (1 - \varphi )^{{P_{{^{0} }} + P_{{^{1} }} }} e^{{P_{{^{1} }} \varphi }}\\& \approx K_{{\text{m}}} (1 - \varphi )^{{P_{{^{0} }} }} (1 - \varphi )^{{P_{{^{1} }} }} (1 - \varphi )^{{{ - }P_{{^{1} }} }} \\&= K_{{\text{m}}} (1 - \varphi )^{{P_{{^{0} }} }} \hfill \\ G_{{\text{d}}} &= G_{{\text{m}}} (1 - \varphi )^{{Q_{{^{0} }} + Q_{{^{1} }} }} e^{{Q_{{^{1} }} \varphi }} \\&\approx G_{{\text{m}}} (1 - \varphi )^{{Q_{{^{0} }} }} (1 - \varphi )^{{Q_{{^{1} }} }} (1 - \varphi )^{{{ - }Q_{{^{1} }} }} \\&= G_{{\text{m}}} (1 - \varphi )^{{Q_{{^{0} }} }} \hfill \\ \end{aligned} \right. $$

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Now, considering N components pore system, we use a continuous division process replacing host modulus by last porous matrix modulus from i = N to 1. The bulk modulus K^*i is derived from K^*i−1,

$$ \begin{gathered} K_{{_{{\text{d}}} }}^{ * i} = K^{ * i - 1} (1 - v^{{_{i} }} \varphi )^{{P_{0}^{ * i} + P_{1}^{ * i} }} e^{{P_{0}^{ * i} }} \hfill \\ K_{{_{{\text{d}}} }}^{ * i} \approx K^{ * i - 1} (1 - \varphi )^{{v^{{_{i} }} (P_{0}^{ * i} + P_{1}^{ * i} )}} e^{{P_{0}^{i} }} = K_{{\text{m}}} (1 - \varphi )^{{\sum\limits_{i = 1}^{N} {v^{{_{i} }} (P_{0}^{ * i} + P_{1}^{ * i} )} }} e^{{\sum\limits_{i = 1}^{N} {v^{{_{i} }} P_{1}^{ * i} } }} \hfill \\ \end{gathered} $$

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Shear modulus can be derived in the same way. Therefore, we get a decoupled DEM solution for the multiple pore-type materials (i.e., carbonate rocks)

$$ \left\{ \begin{gathered} K_{{_{{{\text{dry}}}} }}^{ * i} = K_{{\text{m}}} (1 - \varphi )^{{\sum\limits_{i = 1}^{N} {v^{{_{i} }} (P_{0}^{i} + P_{1}^{i} )} }} e^{{\varphi \sum\limits_{i = 1}^{N} {v^{{_{i} }} P_{1}^{i} } }} \hfill \\ G_{{_{{{\text{dry}}}} }}^{ * i} = G_{{\text{m}}} (1 - \varphi )^{{\sum\limits_{i = 1}^{N} {v^{{_{i} }} (Q_{0}^{i} + Q_{1}^{i} )} }} e^{{\varphi \sum\limits_{i = 1}^{N} {v^{{_{i} }} Q_{1}^{i} } }} \hfill \\ \end{gathered} \right. $$

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