Formulating the history matching problem with consistent error statistics

Abstract

It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

Tailoring data assimilation to discontinuous Galerkin models

“…Ideally, any off‐diagonal error covariances can be explicitly accommodated in the DA algorithm. The use of diagonal $$$ \mathbf{R} $$$ is common, as the off‐diagonal error covariances are often not a priori known and additional steps and assumptions are necessary to estimate them (Campbell et al ., 2017; Evensen, 2021; Fowler et al ., 2018; Michel, 2018; Miyoshi et al ., 2013; Rainwater et al ., 2015; Stewart et al ., 2013). Because of this, and the fact that it allows for the use of more efficient algorithms to minimise the cost function in Equation (), it is in practice common to work with diagonal observation error covariances.…”

Implementation of Adaptive Localization for Enhancing Ensemble-Based History Matching in Hydrocarbon Reservoir Management

“…Fig. 13 shows an example of production rate vs time plot, including simulated observables and observations for a study case developed by Evensen (2021). The green lines represent the initial ensemble, the magenta lines stand for the updated ensemble, the black dots are the observations, and the black vertical lines are the spread of the measurement errors.…”

Using the (Iterative) Ensemble Kalman Smoother to Estimate the Time Correlation in Model Error

“…The impact of model error caused by unresolved processes on the forecast and DA results can last for several model time steps. Bennett (1992), typically way ahead of his time, extensively discussed the use of correlated model errors and solution of the problem using the representer method, Amezcua and Van Leeuwen (2018) formulated the time-correlated problem for ensemble smoothers, and Evensen (2021) extended this to iterative ensemble smoothers. An obstacle in this endeavor, however, is that it is hard to describe a prior on the model errors, especially if one is to include nontrivial probabilistic elements in both space and time.…”

“…An obstacle in this endeavor, however, is that it is hard to describe a prior on the model errors, especially if one is to include nontrivial probabilistic elements in both space and time. As a result, there has been interest in estimating model errors in DA schemes in the last two decades (Brasseur et al, 2005;Crommelin and Vanden-Eijnden, 2008;Zhu, Van Leeuwen, and Zhang, 2018;Lucini, Leeuwen, and Pulido, 2019;Bonavita and Laloyaux, 2020;Brajard et al, 2021;Pathiraja and Leeuwen, 2022;Evensen, 2021).…”