A Discrete Imaging Formulation for History Matching Complex Geologic Facies

Abstract Estimation of geologic facies with complex connectivity patterns from limited direct and indirect measurements is facilitated by exploiting recent advances in discrete imaging methods. Classical model calibration techniques have difficulty in honoring solution discreteness and preserving facies connectivity. The existing methods for calibration of facies models either focus on preserving the facies connectivity and incorporate discreteness as a post-processing step, or they attempt to generate conditional samples from a discrete prior model (training image), which can be computationally demanding. In this work, we propose a novel framework for discrete geologic facies reconstruction from dynamic production data by combining connectivity-preserving parameterizations with discrete regularization techniques such as well-potentials that are inspired by recent advances in discrete tomography. For calibration of discrete geologic facies against flow data, we propose a method to promote solution discreteness and incorporate geologic connectivity information. To obtain discrete solutions we invoke well-potential regularization functions that penalize continuous solutions. The regularization penalty function is minimized along with the mismatch between model predictions and observed production data. To incorporate the geologic connectivity patterns, we learn plausible geologic patterns from available prior (training) models. This is done by learning parametric representations of facies connectivity such as the truncated singular value decomposition (TSVD) or learned sparse geologic dictionaries. We solve the resulting regularized minimization problem by implementing an efficient gradient-based algorithm known as the alternating direction method of multipliers (ADMM). Through several numerical experiments, we show that the proposed formulation offers a flexible facies model calibration approach that can be applied to problems with multiple facies types. An important aspect of this method is that it incorporates the discreteness of the underlying structure as a soft constraint in the inversion process, without a requirement for post-processing of the solution, which can potentially violate data match requirements. The implementation is amenable to iterative gradient-based algorithms and allows gradual, systematic, and plausible morphing of a given facies model to match the observed data. We present several case studies that illustrate the superiority of the proposed method to existing approaches in the literature for calibration of discrete facies distribution against production data.

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