Novel Ensemble Data Assimilation Algorithms Derived from A Class of Generalized Cost Functions

Ensemble data assimilation algorithms are among the state-of-the-art history matching methods. From an optimization-theoretic point of view, these algorithms can be derived by solving certain stochastic nonlinear-leastsquares problems. In a broader picture, history matching is essentially an inverse problem, which is often nonlinear and ill-posed, and may not possess any unique solution. To mitigate these noticed issues, in the course of solving an inverse problem, domain knowledge and prior experience are often incorporated into a suitable cost function within a respective optimization problem. This helps to constrain the solution path and promote certain desired properties (e.g., sparsity, smoothness) in the solution. Whereas in the inverse problem theory there is a rich class of inversion algorithms resulting from various choices of cost functions, there are few ensemble data assimilation algorithms which in their practical uses are implemented in a form beyond nonlinear-least-squares. This work aims to narrow this noticed gap. Specifically, we consider a class of generalized cost functions, and derive a unified formula to construct a corresponding class of novel ensemble data assimilation algorithms, which aim to promote certain desired properties that are chosen by the users, but may not be achieved by using the conventional ensemble-based algorithms. As an example, we consider a channelized reservoir characterization problem, and formulate history matching as some minimum-average-cost problems with two new cost functions. In one of them, our objective is to restrict the changes of total variations of reservoir models during model updates. While in the other, our goal is instead to curb the modifications of histograms of reservoir models. While these two cost functions may appear unconventional in the context of ensemble data assimilation, the corresponding assimilation algorithms derived from our proposed formula are very similar to the conventional iterative ensemble smoother (IES). As such, our previous experience with the IES can be smoothly transferred into the implementations and applications of these new algorithms. In addition, the experiment results indicate that using either of these two new algorithms leads to better history matching performance, in comparison to the original IES.