Evaluation of Gaussian approximations for data assimilation in reservoir models

Abstract

The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework consists mainly in characterizing the posterior distribution of geologic parameters given a prior distribution and data from the reservoir dynamics. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly-nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing-up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone.In this paper we propose to numerically assess the performance of standard Gaussian approximations to probe the Bayesian posterior distribution. In particular we assess the performance of (i) the linearization around the maximum a posterior estimate, (ii) the randomized maximum likelihood and (iii) standard ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution we implement

Identification of the degradation coefficient for an anomalous diffusion process in hydrology

“…The Laplace approximation (LA) in essence is a linearization around the MAP point q MAP for sampling the posterior distribution of the solution (refer to [62]). It consists of approximating the posterior measure(or distribution) by ω ≈ N(q MAP , C MAP ), here C MAP = (J ′′ B (q MAP )) −1 is the inverse of Hessian of J B (q MAP ).…”

Parameter estimation for Ginzburg-Landau equation via implicit sampling

“…If we choose to compute the Hessian with finite differences, it would be very expensive. So we propose to use an approximation form of Hessian [20] H ≈ I − S (SS + R) −1 S,…”

Well-posed Bayesian geometric inverse problems arising in subsurface flow

“…However, recent work [20] has shown the potential detrimental effect of directly applying standard MCMC methods to approximate finite-dimensional posteriors which arise from discretization of PDE based Bayesian inverse problems. For standard subsurface flow models, the forward (parameter-to-output) map is nonlinear, and so even if the prior distribution is Gaussian, the posterior is in general nonGaussian.…”

“…as a discretized function) but are only capable of characterizing posteriors from one-dimensional problems on a very coarse grids [15,30]. While the aforementioned strategies offer a significant insight into to the solution of Bayesian inverse problems in subsurface models, there remains substantial opportunity for the improvement and development of Bayesian data assimilation techniques capable of describing the posterior distributions accurately and efficiently, using the mesh-independent MCMC schemes overviewed in [8] and applied to subsurface applications in [20]. In particular we aim to do so in this paper in the context of geometrically defined models of the geologic properties.…”

“…Although the aforementioned implementations are computationally feasible and may recover the truth within credible intervals (Bayesian confidence intervals), the methods may provide uncontrolled approximations of the posterior. Even for simple Gaussian priors, in [20] numerical evidence has been provided of the poor characterization that ensemble methods may produce when compared to a fully resolved posterior.…”