Ensemble-Based Assimilation of Nonlinearly Related Dynamic Data in Reservoir Models Exhibiting Non-Gaussian Characteristics

Inverse modeling techniques for estimating reservoir parameters (e.g., transmissivity, permeability) utilize some dynamic (secondary) information (e.g., hydraulic head or production data at well locations). Ensemble based data assimilation methods are one such class of inverse modeling techniques. Ensemble Kalman filter (EnKF) in particular is built around the basic framework of linearly updating a state vector ψf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\psi ^f}$$\end{document} to ψa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\psi ^a}$$\end{document}, based on observed dynamic data. Components of the state vector are reservoir model parameters such as transmissivity, permeability, storativity, porosity, hydraulic head, and phase-saturation. Although EnKF is able to update a large number of parameters successively as data become available, it has some shortcomings. It is optimal only in the cases where multivariate joint distribution describing the relationship between components in the state vector is Gaussian. The relationship between the model parameters and the observed data space is quantified in terms of a covariance function, which is adequate to describe a multi-Gaussian distribution. These assumptions and simplifications may result in updated models that are inconsistent with the prior geology (exhibiting non-Gaussian characteristics) and that, in turn, may yield inaccurate predictions of reservoir performance. The aim of this research work is to propose a novel method for data assimilation which is free from the Gaussian assumptions. This new method can be used to assimilate dynamic data into reservoir models using an ensemble-based approach. Model updates are performed after transforming the model parameters into indicator variables. The indicator transform is insensitive to non-linear operations, and it thus allows us to overcome some of the limitations of the traditional EnKF.