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Abstract:
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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 . |