New Geological Parameterizations for History Matching Complex Models

Advisors

Louis Durlofsky, primary advisor
Tapan Mukerji, advisor
Pallav Sarma, advisor

Abstract

In this thesis, we introduce and apply several new approaches to represent complex geological models in terms of a relatively small number of parameters. These concise representations are then used for history matching, in which the geological parameters characterizing the system are varied in order to generate models that provide predictions in agreement with observed production data. We first introduce a parameterization procedure based on principal component analysis (PCA). 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 non-Gaussian property distributions. The approach, referred to as optimization-based PCA (O-PCA), is applied here for binary-facies, three-facies and bimodal systems, including nonstationary models. These different types of geological scenarios are represented by varying the form and parameters in the O-PCA regularization term. The O-PCA procedure is applied both to generate new (random) realizations and for data assimilation, though our emphasis in this work is on its use for gradient-based history matching. The gradient of the O-PCA mapping, which is required for gradient-based history matching methods, is determined analytically or semi-analytically, depending on the form of the regularization term. O-PCA is implemented within a Bayesian history-matching framework to provide both the maximum a posteriori (MAP) estimate and to generate multiple history-matched models for uncertainty assessment. For the latter, O-PCA is combined with the randomized maximum likelihood (RML) method. The O-PCA method is shown to perform well for history matching problems, and to provide models that honor hard data, retain the large-scale connectivity features of the geological system, match historical production data and, when used with RML, provide an estimate of prediction uncertainty. O-PCA is also used to directly parameterize upscaled models. In this case the geological parameters represented in O-PCA are directional transmissibilities. Use of this representation should enable history matching to be performed using upscaled models, which could lead to significant computational savings. Finally, we consider kernel PCA (KPCA) methods, which entail the application of PCA in a high-dimensional 'feature space.' In order to map the model back to input (physical) space, KPCA requires the solution of the so-called pre-image problem. The pre-image determination is, in general, a challenging optimization problem. A robust pre-image method, which is based on approaches from the machine learning community, is proposed for KPCA representations of complex geological systems. Consistent with the treatment used for O-PCA, we also introduce a regularized version of KPCA, referred to as R-KPCA. The impact of the pre-image method and regularization is demonstrated by generating new (random) realizations and through flow assessments. R-KPCA is also applied for history matching (MAP estimate) and is shown to provide accurate models. Our overall R-KPCA procedure may offer some slight benefit over O-PCA, but the relative simplicity of O-PCA is a key advantage.