An iterative representer-based scheme for data inversion in reservoir modeling
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With the recent development of smart-well technology, the reservoir community now faces the challenge of developing robust and efficient techniques for reservoir characterization by means of data inversion. Unfortunately, classical history-matching methodologies do not possess computational efficiency and robustness needed to assimilate data measured almost in real time. Therefore, the reservoir community has started to explore techniques previously applied in other disciplines. Such is the case of the representer method, a variational data assimilation technique that was first applied in physical oceanography. The representer method is an efficient technique for solving linear inverse problems when a finite number of measurements are available. To the best of our knowledge, a general representer-based methodology for nonlinear inverse problems has not been fully developed. We fill this gap by presenting a novel implementation of the representer method applied to the nonlinear inverse problem of identifying petrophysical properties in reservoir models. Given production data from wells and prior knowledge of the petrophysical properties, the goal of our formulation is to find improved parameters so that the reservoir model prediction fits the data within some error given a priori. We first define an abstract framework for parameter identification in nonlinear reservoir models. Then, we propose an iterative representer-based scheme (IRBS) to find a solution of the inverse problem. Sufficient conditions for convergence of the proposed algorithm are established. We apply the IRBS to the estimation of absolute permeability in single-phase Darcy flow through porous media. Additionally, we study an extension of the IRBS with Karhunen-Loeve (IRBS-KL) expansions to address the identification of petrophysical properties subject to linear geological constraints. The IRBS-KL approach is compared with a standard variational technique for history matching. Furthermore, we apply the IRBS-KL to the identification of porosity, absolute and relative permeabilities given production data from an oil-water reservoir. The general derivation of the IRBS-KL is provided for a reservoir whose dynamics are modeled by slightly compressible immiscible displacement of two-phase flow through porous media. Finally, we present an ad-hoc sequential implementation of the IRBS-KL and compare its performance with the ensemble Kalman filter.