# A New Differentiable Parameterization Based on Principal Component Analysis for the Low-Dimensional Representation of Complex Geological Models

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## Abstract

A new approach based on principal component analysis (PCA) for the representation of complex geological models in terms of a small number of parameters is presented. The basis matrix required by the method is constructed from a set of prior geological realizations generated using a geostatistical algorithm. Unlike standard PCA-based methods, in which the high-dimensional model is constructed from a (small) set of parameters by simply performing a multiplication using the basis matrix, in this method the mapping is formulated as an optimization problem. This enables the inclusion of bound constraints and regularization, which are shown to be useful for capturing highly connected geological features and binary/bimodal (rather than Gaussian) property distributions. The approach, referred to as optimization-based PCA (O-PCA), is applied here mainly for binary-facies systems, in which case the requisite optimization problem is separable and convex. The analytical solution of the optimization problem, as well as the derivative of the model with respect to the parameters, is obtained analytically. It is shown that the O-PCA mapping can also be viewed as a post-processing of the standard PCA model. The O-PCA procedure is applied both to generate new (random) realizations and for gradient-based history matching. For the latter, two- and three-dimensional systems, involving channelized and deltaic-fan geological models, are considered. The O-PCA method is shown to perform very well for these history matching problems, and to provide models that capture the key sand–sand and sand–shale connectivities evident in the true model. Finally, the approach is extended to generate bimodal systems in which the properties of both facies are characterized by Gaussian distributions. MATLAB code with the O-PCA implementation, and examples demonstrating its use are provided online as Supplementary Materials.

## Keywords

Non-Gaussian parameterization Geological modeling History matching Inverse problem Data assimilation Regularization Soft-thresholding Histogram transform Oil reservoir simulation## Notes

### Acknowledgments

We thank the industrial affiliates of the Stanford University Reservoir Simulation Research (SUPRI-B) and Smart Fields Consortia for partial funding of this work. We are grateful to Vladislav Bukshtynov and Oleg Volkov for implementing the adjoint formulation in AD-GPRS and for their assistance with its use. We also acknowledge Andre Journel, Albert Reynolds and Pallav Sarma for useful discussions and suggestions.

## Supplementary material

## References

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