A New Uncertainty Quantification Method for Petrophysical Parameters Based on NMR Relaxation Spectra

    Reservoir porosity, permeability, and saturation are regarded as the core parameters in oil and gas exploration. The prediction of reservoir productivity and the analysis of fluid transport behavior in porous media are directly influenced by these parameters. Nuclear magnetic resonance (NMR) is recognized as a non-invasive, non-destructive, and highly quantitative technique. Typically, echo signals are inverted to obtain relaxation spectra, which is then used to calculate rock parameters.

    In the existing methods, reservoir petrophysical parameters are usually calculated based on a relaxation spectrum obtained from a single inversion. However, the inversion of NMR relaxation spectra is inherently an ill-posed problem. Its solutions are characterized by non-uniqueness and high sensitivity to noise. In the traditional interpretation workflows, the single relaxation spectrum obtained from inversion is often assumed to be accurate and deterministic. Consequently, the propagation effect of inversion errors in the calculation of petrophysical parameters is ignored. This leads to a lack of credibility assessment for results in evaluations involving low signal-to-noise ratio (SNR) or unconventional reservoirs.

    To address this challenge, an improved Bootstrap resampling method is proposed in this study. It aims to achieve uncertainty quantification from NMR relaxation spectra to petrophysical parameters. The traditional approach of seeking a single solution is abandoned. Instead, single-measurement echo data are resampled multiple times to generate statistically significant pseudo-sample sets by fully mining the statistical fluctuation information hidden within the single measurement. Subsequently, each set of samples is inverted independently to construct a distribution set of relaxation spectra.

    By performing independent parameter calculations on the relaxation spectrum set, the distribution range, central tendency, and dispersion of each parameter can be obtained. Thus, the impact of data noise and pore size distribution differences on parameter estimation is revealed. On this basis, an error propagation model from the spectral domain to the parameter domain is established. Confidence intervals (CI) and prediction intervals (PI) for porosity, permeability, and saturation are calculated simultaneously. Specifically, model uncertainty caused by the non-uniqueness of the inversion algorithm is primarily quantified by the CI. Meanwhile, data uncertainty resulting from measurement noise is further incorporated into the PI, which provides a broader parameter interval range. A transition from "point estimation" to "interval estimation" for key petrophysical parameters is achieved by this method. Consequently, the robustness and credibility of parameter evaluation under complex reservoir conditions are significantly enhanced.

    This work was supported by the National Natural Science Foundation of China (42304118), the Frontier Interdisciplinary Exploration Research Program of China University of Petroleum Beijing (2462024XKQY009), the Young Elite Scientist Sponsorship Program by BAST (BYESS2023027).