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It is well known that inverse problem theory using the
Gauss-Newton method with restrictions on step length can be applied to obtain realizations
of rock property fields (simulator gridblock log-permeabilities and porosities). If one
wishes to simulate permeability and porosity values at thousands of gridblocks by
conditioning to large amounts of production data, then computation of sensitivity
coefficients, solution of the Gauss-Newton matrix problem and related matrix
multiplications become computationally expensive. In order to reduce the computational
effort required for large-scale inversion with large amounts of data, it is necessary to
use a reparameterization technique. The purpose of this study is then to develop an
effective method of reparameterization for solving large-scale single-phase reservoir
inverse problems with large amounts of pressure data using the subspace methodology. Unlike other methods such as pilot point, the proposed parameterization in terms of gradients of sub-objective functions preserves the high rate of convergence of the conventional Gauss-Newton method (with the full parameterization) while keeping the features of the resulting realizations. However, the results show that appropriate choice of subspace vectors is required for the subspace method to be successful. A poor choice of subspace vectors results in slow convergence. An efficient implementation that uses an adjoint method to compute the subspace vectors and the “gradient simulator” method to compute sensitivities to subspace vectors is described. These techniques eliminate the need of forming the entire sensitivity matrix directly and more importantly make the proposed subspace method to be applicable to multiphase problems.
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