History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm

For large-scale history matching problems, applying the Gauss–Newton (GN) or the Levenberg–Marquardt (LM) algorithm is computationally expensive. However, these algorithms can be efficiently applied with parameterization based on a truncated singular value decomposition (SVD) of a dimensionless sensitivity matrix, where a truncated SVD is computed by using the Lanczos method. The SVD parameterization algorithm has been previously combined with randomized maximum likelihood (RML) to simultaneously generate multiple realizations of the reservoir model. The resulting algorithm, called SVD-EnRML, has been applied for simulation of permeability fields of 2D synthetic reservoirs. In this work, the SVD-EnRML algorithm is extended for the simulation of both porosity and permeability fields of 3D reservoirs. In the proposed extension, a dimensionless sensitivity matrix is defined for each set of correlated model parameters. A limitation of the original algorithm is due to the fact that a square root of the covariance matrix is required as a transformation from the original space to a dimensionless space. In this work, this limitation is resolved by introducing ensemble-based regularization based on utilizing an ensemble of unconditional realizations of the reservoir model. Although the proposed extension fits well within the original algorithm, a modified SVD-EnRML algorithm is introduced to mainly improve the computational efficiency. Computational results, composed of two different examples, show that the algorithm can be efficiently applied for the simulation of rock property fields and performance predictions of 3D reservoirs.