Data assimilation with soft constraints (DASC) through a generalized iterative ensemble smoother

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.

Hyper-Parameter Optimization for Improving the Performance of Localization in an Iterative Ensemble Smoother

“…( 15), both criteria to estimate threshold values for the Au-toAdaLoc scheme are based on certain statistical analysis methods. Although our experience shows that the localization scheme established in this way appears to work reasonably well in a number of studies (Luo and Bhakta, 2020;Luo, 2021;Luo and Cruz, 2022;Ranazzi et al, 2022), it is possible to further improve its performance. Our main idea is that, instead of relying on a statistical analysis method to determine the threshold values, we parameterize the threshold values instead, and then use an ensemble-based CHOP procedure to optimize the parameterized localization scheme.…”

“…Nevertheless, should a large number of field datasets, e.g., 4D seismic (Lorentzen et al, 2019;Luo et al, 2017;Soares et al, 2020), be assimilated into a big reservoir model, then it could be computationally challenging to construct a big tapering matrix for the update of each reservoir model. In this regard, we expect that a few strategies can be adopted to mitigate the consumption of computer memory, which include: (1) sparse model/data representation (Canchumuni et al, 2019;Lorentzen et al, 2019;Luo et al, 2017;Soares et al, 2020) to reduce the size(s) of reservoir model and/or observation data; (2) projection of observation data onto the ensemble subspace and then using the projected data as the effective observations (Luo et al, 2019;Luo and Cruz, 2022); (3) local analysis in which each update focuses on a small group of model variables and observation data points (Chen and Oliver, 2017;Soares et al, 2021). In future work, we will test some of these strategies in relevant data assimilation problems.…”

“…An alternative to distance-based localization is correlation-based localization (Lacerda et al, 2019;Luo et al, 2018;Luo and Bhakta, 2020;Ranazzi et al, 2022), in which a tapering matrix is formulated based on sample correlations -rather than physical distances -between ensembles of model variables and simulated observation data points. As elaborated in Luo et al (2018Luo et al ( , 2019, correlation-based localization is able to overcome or mitigate the aforementioned long-standing issues arising in distance-based localization, and can be used in certain situations, e.g., when model variables and/or observations do not possess physical locations (Luo et al, 2018) or when localization needs to be done for an ensemble of gradients projected onto ensemble subspaces (Luo and Cruz, 2022), where the conventional distance-based localization does not seem to be straightforwardly applicable.…”