Abstract

ABSTRACT Rock typing is critical in deepwater reservoir characterization to construct stratigraphic models populated with static and dynamic petrophysical properties. However, rock typing based on multiple well logs is challenging because different logging-tool physics exhibit different volumes of investigation. Consequently, large uncertainty is typically associated with rock typing in thinly bedded or laminated reservoirs because true physical properties cannot be resolved due to shoulder-bed effects. To circumvent this problem, we introduce a new Bayesian approach that inherently adopts the scientific method of iterative hypothesis testing to perform rock typing by simultaneously honoring different logging-tool physics in a multi-layered earth model. In addition to estimating the vertical distribution of rock types with maximum likelihood, the Bayesian method quantifies the uncertainty of rock types and associated petrophysical properties layer-by-layer. Bayesian rock classification is performed with a fast sampling technique based on the Markov-Chain Monte Carlo method, thereby enabling an efficient search of rock types to obtain final results. We use a fast linear iterative refinement method to simulate nuclear logs and a 2D forward modeling code to simulate arrayinduction resistivity logs. A rock-type distribution hypothesis is considered acceptable only when all observed well logs are reproduced with forward modeling. Both synthetic and field cases are used to verify the effectiveness of the new rock typing method. In a field case of deltaic gas reservoirs from offshore Trinidad, the Bayesian method differentiates rock types that exhibit subtle petrophysical variations due to grain size change. We show that the new method provides more than 77% agreement between log-derived and corederived rock types while conventional deterministic methods only achieve 60% agreement due to presence of thin beds and laminations. Rock types are verified with independent data sources such as laser particle size measurements and mercury injection capillary pressure. Even though large uncertainty is observed in thinly bedded and laminated zones, the Bayesian rock-typing method still yields rock types and petrophysical properties that agree well with core-plug measurements acquired in these layers. As a result, the overall correlation between log-derived permeability and coremeasured permeability is improved by approximately 16% when compared to conventional deterministic methods. More importantly, the quantified petrophysical uncertainty provides critical information for estimating the uncertainty of reservoir storage and productivity to guide decision-making for later phases of reservoir development. INTRODUCTION Conventional petrophysical rock typing heavily relies on representative laboratory core measurements, including mineral concentrations, grain-size and poresize distribution, fluid saturation, and fluid distribution Forward physical modeling can predict unique well-log responses given a vertical distribution of rock types and their associated petrophysical properties. However, petrophysical interpretation, which aims to estimate BAYESIAN HYPOTHESIS METHOD AND WORKFLOW where G is the forward function, m is a vector of rock type distribution, and [ , , , ] is the vector of observed well logs based on the rock type distribution and their petrophysical properties. In this study, volumetric concentration, water saturation, total porosity, apparent electrical conductivity, bulk density, neutron porosity, and gamma ray are denoted by i C , Bayes' theorem relates a-priori and posterior distributions in a way that makes the computations of q(m|d) tractable (1) The actual iteration number reaches the maximal iteration number, I max , and, Probabilistic Interpretation of Hybrid Rock Classes. Petrophysical zones segmented by well logs normally have a thickness ranging from 1 ft to 5 ft. In heterogeneous reservoirs, those intervals typically include hybrid rock classes composed of different basis rock types defined at the core scale. Two different approaches are used to describe hybrid rock classes: volumetric basis (Xu and Torres-Verdín, 2013b) and probabilistic basis. Rock Type Validation. Contingency tables In a contingency table, the contingency coefficient, C, is defined as the ratio of diagonal elements to the total number of samples, and quantifies rock typing accuracy as whereas Cramer's V quantifies the strength of the dependence between two variables as where N is the total number of rock samples, 2 χ is the Pearson's chi-squared test, and k is the number of rock types under comparison. SYNTHETIC CASE: INTERBEDDED SAND-SHALE SEQUENCE We construct a synthetic earth model of interbedded sand-shale sequences to illustrate the standard workflow of Bayesian rock typing. Five different rock types are assumed to be present in the reservoir: sands A, B, C, shaly sand D (with dispersed clay), and pure shale (SH). Sands A, B, and C were deposited with different flow energy; therefore they exhibit different grain sizes and reservoir quality. Shaly sand D has poor reservoir quality due to cementation of dispersed clay. Pure shales are non-reservoir facies. FIELD CASE: DELTAIC GAS RESERVOIR, OFFSHORE TRINIDAD The formation under consideration is a sandstone unit deposited in a deltaic sedimentary system in the Columbus Basin, offshore Trinidad (Liu, 2007; Xu and Basis Rock Classes from Core Measurements. We performed rock typing with Leverett's reservoir quality index (1941) calculated from routine porositypermeability data Bayesian Rock Typing from Logs. After defining rock types and their associated petrophysical properties, we invoke Bayesian rock typing to infer the rock type distribution by iteratively testing hypothetical rock types. Table 4 also shows that, in general, good rock types are less prone to misclassification. This behavior is partly due to the large population of good rock types, and partly due to their larger bed thickness. Permeability Estimation and Uncertainty Analysis. Each inferred distribution of rock types generates a corresponding permeability distribution obtained by calculating the permeability with rock-type specific porosity-permeability relations established with core data. A statistical analysis of all permeability distributions quantifies the uncertainty of permeability in each layer. Track 6 in SPWLA 54 th Annual Logging Symposium, June 22-26, 2013 10 was established between petrophysical rock types and grain sizes. Therefore, a stack of vertical rock types provides useful information to geologists for interpretation of geological facies and to reduce uncertainty when constructing a stratigraphic reservoir model based on well logs. Computational Performance Analysis. A common disadvantage of stochastic estimation methods is the need to perform a multitude of forward calculations to sample the posterior probability function in model space. Yang and Torres-Verdín (2013) introduced two strategies to enhance the efficiency of the stochastic method, which reduce the CPU time to approximately 4 hours to obtain 100 realizations of hypothetical distributions of rock types (100 ft interval) on a desktop PC (3.4 GHz CPU) and Matlab platform. The computer time required by the calculations can be further reduced with the implementation of parallel-computing techniques. CONCLUSIONS We developed a new core-calibrated and log-based Bayesian rock typing method that employs fast numerical simulation of well logs for iterative hypothesis testing. The new method effectively reduces shoulder-bed effects on well logs which give rise to significant uncertainty in rock typing across thinly bedded formations. A probabilistic method was introduced and successfully verified for describing hybrid rock classes. The application of the new method to a field case indicated more than 77% agreement between log-and core-derived rock types. Overall, the correlation between predicted permeability and core-measured permeability was improved by approximately 16% compared to conventional deterministic methods. In addition, the method quantified the uncertainty associated with rock-type identification and permeability estimation. The final distribution of maximum-likelihood rock types was consistent with the geological framework and provided useful information for stratigraphic reservoir construction and modeling. Computational performance can be a limitation when implementing the Bayesian rock typing method in field studies because of the requirement of heavy data processing. However, the method remains accurate and ACKNOWLEDGEMENTS We would like to thank BP Trinidad-Tobago (BPTT) for providing the field data used in this study. The work reported in this paper was funded by ABOUT THE AUTHORS Chicheng Xu is currently a Ph.D. candidate studying integrated reservoir characterization through geologically and petrophysically consistent rock classification based mainly on well logs and core data. His research focuses on applying rock classification to fundamental reservoir characterization subjects such as permeability prediction, saturation-height modeling, uncertainty analysis, heterogeneity characterization, petrophysical upscaling, fluid substitution, and sedimentary facies interpretation. He had more than 10 technical papers published in SPWLA, SPE, SEG, SCA, and AAPG. He received his B.S. in