Sardar Afra

Heterogeneous reservoir characterization using efficient parameterization through higher order SVD (HOSVD)

Parameter estimation through reduced-order modeling play a pivotal role in designing real-time optimization schemes for the Oil and Gas upstream sector through the closed-loop reservoir management framework. Reservoir models are in general complex, nonlinear, and large-scale, i.e., large number of states and unknown parameters. Consequently, model reduction techniques are of great interest in reducing the computational burden in reservoir modeling and simulation. Furthermore, de-correlating system parameters in all history matching and reservoir characterization problems is an important task due to its effects on reducing ill-posedness of the system. In this paper, we utilize the higher order singular value decomposition (HOSVD) to reparameterize reservoir characteristics, e.g. permeability, and perform several forward reservoir simulations by the resulted reduced order map as an input. To acquire statistical consistency we repeat all experiments for a set of 1000 samples using both HOSVD and Proper orthogonal decomposition (POD). In addition, we provide RMSE analysis for a better understanding in process of comparing HOSVD and POD. Results show that HOSVD provide a better performance in a RMSE point of view.

Permeability Parametrization Using Higher Order Singular Value Decomposition (HOSVD)

Model reduction is of highly interest in many science and engineering fields where the order of original system is such high that makes it difficult to work with. In fact, model reduction or parametrization defined as reducing the dimensionality of original model to a lower one to make a costly efficient model. In addition, in all history matching problem, in order to reduce the ill-posed ness of the problem, it is necessary to de-correlate the parameters. Proper orthogonal decomposition (POD) as an optimal transformation is widely used in parameterization. To obtain the bases for POD, it is necessary to vectorize the original replicates. Therefore, the higher order statistical information is lost due to slicing the replicates. Another approach that deals with the replicates as they are, is high order singular value decomposition (HOSVD). In the present work permeability maps dimension is reduced using HOSVD image compression method. Unknown permeability maps are also estimated using HOSVD and results of both parts compared to those of SVD.

Studying the possibility of peaking phenomenon in linear support vector machines with non-separable data

Typically, it is common to observe peaking phenomenon in the classification error when the feature size increases. In this paper, we study linear support vector machine classifiers where the data is non-separable. A simulation based on synthetic data is implemented to study the possibility of observing peaking phenomenon. However, no peaking in the expected true error is observed. We also present the performance of three different error estimators as a function of feature and sample size. Based on our study, one might conclude that when using linear support vector machines, the size of feature set can increase safely.

Tensor based geology preserving reservoir parameterization with Higher Order Singular Value Decomposition (HOSVD)

Parameter estimation through robust parameterization techniques has been addressed in many works associated with history matching and inverse problems. Reservoir models are in general complex, nonlinear, and large-scale with respect to the large number of states and unknown parameters. Thus, having a practical approach to replace the original set of highly correlated unknown parameters with non-correlated set of lower dimensionality, that captures the most significant features comparing to the original set, is of high importance. Furthermore, de-correlating systems parameters while keeping the geological description intact is critical to control the ill-posedness nature of such problems. We introduce the advantages of a new low dimensional parameterization approach for reservoir characterization applications utilizing multilinear algebra based techniques like higher order singular value decomposition (HOSVD). In tensor based approaches like HOSVD, 2D permeability images are treated as they are, i.e., the data structure is kept as it is, whereas in conventional dimensionality reduction algorithms like SVD data has to be vectorized. Hence, compared to classical methods, higher redundancy reduction with less information loss can be achieved through decreasing present redundancies in all dimensions. In other words, HOSVD approximation results in a better compact data representation with respect to least square sense and geological consistency in comparison with classical algorithms. We examined the performance of the proposed parameterization technique against SVD approach on the SPE10 benchmark reservoir model as well as synthetic channelized permeability maps to demonstrate the capability of the proposed method. Moreover, to acquire statistical consistency, we repeat all experiments for a set of 1000 unknown geological samples and provide comparison using RMSE analysis. Results prove that, for a fixed compression ratio, the performance of the proposed approach outperforms that of conventional methods perceptually and in terms of least square measure. HighlightsIntroduced new permeability parameterization using High Order Singular Value Decomposition (HOSVD).HOSVD yield reduced computational time as compared to classical SVD based on the same compression ratio.The new methodology improves geological description by capturing all important features (spatial) as compared to SVD methods.The HOSVD method is general for any type of reservoir parameter.HOSVD can be applied in the optimization under the uncertainty paradigm.

Efficient Inference of Reservoir Parameter Distribution Utilizing Higher Order SVD Reparameterization

Reservoir parameter inference is a challenging problem to many of the reservoir simulation workflows, especially when it comes to real reservoirs with high degree of complexity and non-linearity, and high dimensionality. In a history matching problem that adapts the reservoir properties grid blocks, the inverse problem leads to an ill-posed and very costly optimization schemes. In this case, it is very important to perform geologically consistent reservoir parameter adjustments as data is being assimilated in the history matching process. Therefore, ways to reduce the number of reservoir parameters need to be sought after. In this paper, we introduce the advantages of a new parameterization method utilizing higher order singular value decomposition (HOSVD) which is not only computationally more efficient than other known dimensionality reduction methods such as, SVD and DCT, but also provides a consistent model in terms of reservoir geology. HOSVD power is due to its ability to supply a reliable low-dimensional reconstructed model while keeping higher order statistical information and geological characteristics of reservoir model. In HOSVD, we take the snapshots in a 2D or 3D approach, i.e., do not vectorize original replicates, and stack them up into a tensor form, i.e. a multi-way array in multilinear algebra which leads to implementing tensor decomposition. Technically, we performed HOSVD to find the best lower rank approximation of this tensor that is an optimization problem utilizing alternating least square method. This results in a more consistent reduced basis. We applied this novel parameterization method to the SPE10 benchmark reservoir model to show its promising parameterization performance. We illustrate its advantages by comparing its performance to the regular SVD (PCA) in a history matching framework using EnKF, as well as characterization performance of the ensemble-based history matching approaches along with HOSVD. Overall, HOSVD outperforms SVD in terms of reconstruction and estimation performance.