Bayesian Method for Rapid Multi-Well Interpretation

It is standard practice to estimate porosity, water saturation, and mineralogy of formations with complex solid and fluid compositions by minimizing the error between well logs and their numerical simulations, subject to realistic petrophysical and material-balance constraints. For unconventional organic-shale formations, the minimization is usually performed using Bayesian inversion, which incorporates field-specific a priori correlations and uncertainty quantification. However, no efficient multi-well application of this method is yet available because of the high computational cost associated with Markov chain Monte Carlo (McMC) sampling. Additionally, in many cases, the available well logs/core data are not sufficient to obtain accurate petrophysical estimations times connection.

We introduce and successfully verify a new workflow for rapid multi-well interpretation of wireline logs and core data. First, well logs are corrected for tool and borehole effects (e.g. shoulder beds, laterolog vs. induction resistivity tools, LWD measurements, etc.) by running separate inversions. This step outputs layer-by-layer physical properties (e.g. resistivity, gamma ray, density, and neutron migration length) that are subsequently combined to estimate compositional/petrophysical properties via Bayesian joint inversion. As a necessary step in the preparation for joint inversion, we establish petrophysical models and statistical relationships among different components in key wells where core data and spectroscopy measurements are available. Finally, we apply our McMC Bayesian joint inversion to the same formation penetrated by other nearby wells with limited measurements.

Our interpretation workflow circumvents multiple difficulties for multi-well interpretation. First, performing separate inversions ensures that the input layer-by-layer physical properties are free of any uncertainty caused by shoulder beds and different borehole instruments or drilling conditions. Second, to speed up calculations, we implement the quasi-Newton McMC sampling technique and use a pre-computed surrogate model for nuclear properties. This combination reduces the computational time by a factor of 30 and 100, respectively. Third, we adopt an extended Bayesian framework that automatically implements different petrophysical models for each rock type.

The proposed method is validated with a synthetic example and wireline measurements acquired across two wells in the Wolfcamp shale formation. Results show that 80% of the core data are within the 80% confidence intervals of the estimations.

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