The fate of fracturing water: A field and simulation study

This paper examines the properties of the Iterated Ensemble Smoother (IES) and the Multiple Data Assimilation Ensemble Smoother (ES–MDA) for solving the history matching problem. The iterative methods are compared with the standard Ensemble Smoother (ES) to improve the understanding of the similarities and differences between them. We derive the three smoothers from Bayes’ theorem for a scalar case which allows us to compare the equations solved by the three methods, and we can better understand which assumptions are applied and their consequences. When working with a scalar model, it is possible to use a vast ensemble size, and we can construct the sample distributions for both priors and posteriors, as well as intermediate iterates. For a linear model, all three methods give the same result. For a nonlinear model, the iterative methods improve on the ES result, but the two iterative methods converge to different solutions, and it is not clear which should be the preferred choice. It is clear that the ensemble of cost functions used to define the IES solution does not represent an exact sampling of the posterior-Bayes’ probability density function. Also, the use of an ensemble representation for the gradient in IES introduces an additional approximation compared to using an exact analytic gradient. For ES–MDA, the convergence, as a function of increasing number of uniform update steps, is studied for a huge ensemble size. We illustrate that ES–MDA converges to a solution that differs from the Bayesian posterior. The convergence is also examined using a realistic sample size to study the impact of the number of realizations relative to the number of update steps. We have run multiple ES–MDA experiments to examine the impact of using different schemes for choosing the lengths of the update steps, and we have tried to understand which properties of the inverse problem imply that a non-uniform update step length is beneficial. Finally, we have examined the smoother methods with a highly nonlinear model to examine their properties and limitations in more extreme situations.

/pdf/analysis-of-iterative-ensemble-smoothers-for-solving-inverse-3akizxd9jv.pdf

Analysis of iterative ensemble smoothers for solving inverse problems

Geologic CO2 sequestration monitoring design: A machine learning and uncertainty quantification based approach

Analysis of the performance of ensemble-based assimilation of production and seismic data

Towards a robust parameterization for conditioning facies models using deep variational autoencoders and ensemble smoother

An Adaptive Ensemble Smoother With Multiple Data Assimilation for Assisted History Matching

Impact of Capillary Suction on Fracture Face Skin Evolution in Waterblocked Wells

A History Matching Procedure for Non-Gaussian Facies Based on ES-MDA

We consider the problem of choosing among a suite of potential reservoir surveillance operations. We frame the problem in terms of two questions: (1) Which surveillance operation is the most useful? (2) What is the expected value of the reduction in uncertainty in the reservoir variable J (e.g., cumulative oil production) that would be achieved if we were to conduct each surveillance operation to collect and history match the data obtained? Note that the objective is to answer these questions with an uncertain reservoir description and without any actual measurements. We propose a procedure based on information theory to answer these questions. We apply the proposed method to two simple problems, a nonlinear toy problem and a simple waterflooding problem. The results are verified by an exhaustive history-matching procedure, which is reasonably rigorous but very computationally demanding. We find that the mutual information approach is a fast and reliable alternative to the history-matching approach.

Optimal choice of a surveillance operation using information theory

Estimation of Mutual Information and Conditional Entropy for Surveillance Optimization

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