The use of subspace methods for efficient conditioning of reservoir models to production data

It has been shown previously that realizations of rock property fields (simulator gridblock log-permeabilities and porosities) conditional to production data can be obtained by minimizing an objective function which includes a sum of production data mismatch terms squared plus a regularization term obtained from a prior geostatistical model. It is also well known that minimization using the Gauss-Newton method with restrictions on step length can be applied to generate such realizations. 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 reservoir inverse problems with large amounts of production data using the subspace methodology.; Unlike other methods such as the pilot point method, 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. The product of the prior model covariance matrix with gradients of the data sub-objective functions provides a good set of subspace vectors for reservoir inverse problems. Partitioning the data by well and then by time interval is an effective method of choosing subspace vectors.; Although computation of the optimal number of subspace vectors to maintain the fast convergence of the standard Gauss-Newton method may be expensive, we show that it is desirable to start with a small number of subspace vectors and gradually increase the number at each Gauss-Newton iteration until an acceptable level of data mismatch is obtained.; An efficient implementation that uses an adjoint method to compute the subspace vectors and the “gradient simulator” method to compute sensitivities to coefficients of subspace vectors is described. These techniques eliminate the need of forming the entire sensitivity matrix directly and more importantly make the proposed subspace method applicable to multiphase problems.