Abstract
This work investigates an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints in reservoir data assimilation problems. The proposed workflow is built upon a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES), and inherits the benefits of ensemble-based data assimilation algorithms in geoscience applications. Unlike the traditional ensemble assimilation algorithms, the proposed workflow admits cost functions beyond the form of nonlinear-least-squares, and has the potential to develop an infinite number of constrained assimilation algorithms. In the proposed workflow, we treat data assimilation with constraints as a constrained optimization problem. Instead of relying on a general-purpose numerical optimization algorithm to solve the constrained optimization problem, we derive an (approximate) closed form to iteratively update model variables, but without the need to explicitly linearize the constraint systems. The established model update formula bears similarities to that of an iterative ensemble smoother (IES). Therefore, in terms of theoretical analysis, it becomes relatively easy to transit from an ordinary IES to the proposed constrained assimilation algorithms, and in terms of practical implementation, it is also relatively straightforward to implement the proposed workflow for users who are familiar with the IES, or other conventional ensemble data assimilation algorithms like the ensemble Kalman filter (EnKF). Apart from the aforementioned features, we also develop efficient methods to handle two noticed issues that would be of practical importance for ensemble-based constrained assimilation algorithms. These issues include localization in the presence of constraints, and the (possible) high dimensionality induced by the constraint systems. We use one 2D and one 3D case studies to demonstrate the performance of the proposed workflow. In particular, the 3D example contains experiment settings close to those of real field case studies. In both case studies, the proposed workflow achieves better data assimilation performance in comparison to the choice of using an original IES algorithm. As such, the proposed workflow has the potential to further improve the efficacy of ensemble-based data assimilation in practical reservoir data assimilation problems.
Highlights
Data assimilation aims to estimate the quantities of interest (QoI), e.g., model variables, based on certain sources of information
Within the framework of Bayesian data assimilation, where the QoI are probability density functions (PDFs) of model variables, one more option to account for the constraints is to impose certain restrictions on the shapes of the prior PDFs, e.g., in terms of truncated PDFs when dealing with inequality constraints [20]
Our contributions in this work include the following aspects: We develop an ensemble-based workflow to simultaneously handle generic, nonlinear equality and inequality constraints based on a recently proposed umbrella algorithm, called the generalized iterative ensemble smoother (GIES) [23]
Summary
Data assimilation aims to estimate the quantities of interest (QoI), e.g., model variables (states and/or parameters), based on certain sources of information. Our proposed DASC workflow (through GIES) contains the following four features: (1) The ability to simultaneously handle generic, nonlinear equality and inequality constraints; (2) No need for explicitly evaluating the gradients of the constraint systems with respect to the QoI; (3) The ability to handle high-dimensional constraint systems; (4) A tailored adaptive localization scheme to tackle the adverse effects of a small ensemble size, while taking into account the presence of constraint systems These features can be considered as desirable practical benefits that help promote the applicability of the proposed DASC workflow to generic constrained data assimilation problems. We show that with the efficient methods to handle the issues of localization in the presence of constraints and high-dimensional data induced by the constraints, the proposed constrained assimilation algorithm works well in this particular case study This observation implies that the proposed algorithm has the potential to handle real-field reservoir characterization problems with practical constraints. We conclude the whole work with some technical discussions and future research directions
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