A Probability Conditioning Method (PCM) for Nonlinear Flow Data Integration into Multipoint Statistical Facies Simulation

We present a probability conditioning method (PCM) for constraining multipoint statistical (MPS) facies simulation to nonlinear flow data. MPS has recently been introduced for flexible grid-based simulation of spatial connectivity in formations containing discrete geologic objects (e.g., fluvial channels) that are not amenable to conventional two-point geostatistical modeling. Using the higher-order statistics in MPS, facies realizations are simulated from a conceptual geologic continuity model known as a training image (TI). As a result, the simulated realizations inherit the complex structural connectivity and multipoint spatial statistics conveyed-by the TI. While conditioning multipoint simulation results on static hard (e.g., core) and soft (e.g., three-dimensional seismic) measurements is relatively straightforward, conditioning the simulated facies on nonlinear flow data is a nontrivial task. On the other hand, inversion methods that directly update post-simulation facies distributions have difficulty in reproducing the spatial connectivity (or higher-order statistics) implied by a TI. Using the PCM approach, we first invert the flow data to obtain a probabilistic spatial description of facies distribution (i.e., a probability map) and use the resulting facies probability map to guide MPS facies simulation from a specified TI. Since the probability map contains important information about the flow measurements, the simulated facies distributions are more likely to reproduce the observed flow data. While the proposed PCM approach can be used with different inversion algorithms, we choose the ensemble Kalman filter (EnKF) to extract facies distribution probabilities from flow data. We make this choice because (i) the ensemble form of the EnKF is less sensitive to discontinuity and nonuniqueness (randomness) introduced in conditioning facies simulation on a probability map, and (ii) the EnKF has been established as an effective subsurface data assimilation approach. The PCM implementation with the EnKF results in an improved performance of the filter updates, namely through the preservation of the facies correlation structure and the introduction of additional ensemble variability (spread) due to the resampling of facies from the TI after each update step. We discuss the important properties of the proposed PCM method and illustrate its effectiveness using several two-dimensional waterflooding problems in reservoirs containing two and three facies types. We conclude that PCM effectively combines the existing information in the flow data and the TI; it does so by using the former to infer probabilistic knowledge about inter-well and near-well spatial connectivity and the latter to ensure consistent facies structure and connectivity, where the flow data are inconclusive (e.g., away from measurement locations).