Search

We consider calculation of the Bayesian model evidence, which is an essential component in realistic uncertainty quantification. The main motivation is large-scale porous-media-flow problems, where plain Monte-Carlo (MC) integration is computationally infeasible. We propose a simplistic multilevel (ML) estimator and argue why this estimator is expected to perform well within a certain problem class, and point out that many important problems, including many porous-media-flow problems, belong to this class. Four test cases are utilized to assess the performance of the proposed ML estimator. Test cases I and II contain two strongly linked toy models, but only Test case II belongs to the problem class where the proposed ML estimator is expected to perform well. Test cases III and IV are concerned with single-phase and two-phase porous-media flows, respectively. The results show that the ML estimator clearly and consistently outperforms MC integration in Test cases II, III, and IV, while this is not the case for Test case I. • Motivate the novel multilevel method for calculation of the Bayesian model evidence within a certain problem class. • Demonstrate good performance of this method on problems with O(1000) parameters belonging to this class. • Demonstrate poor performance of this method on problems outside this class. [ABSTRACT FROM AUTHOR]

The interrelation (denoted SNS) between sensitivity, nonlinearity and scale, associated with the inverse problem of permeability (fluid conductivity) identification from fluid pressure observations in porous-media flow, is considered. The family of models considered includes both single-phase and two-phase flows, with applications to groundwater flow/primary recovery in petroleum reservoirs and to water flooding of petroleum reservoirs, respectively. SNS is important for regularization of both of these ill-posed inverse problems, but so far, SNS has been shown to exist only for single-phase flow. Several multiscale/multiresolution estimation techniques, explicitly or implicitly based on SNS, have, however, been developed and applied to practical permeability estimation both for single- and two-phase flows. In this paper, SNS is shown to exist for one-dimensional, two-phase flow. Moreover, very similar approaches are applied to show the existence of SNS both for single- and two-phase flows. To convey some insight into the role of SNS for two-dimensional, two-phase flow, a discussion of a more speculative nature concerning this issue is also included. [ABSTRACT FROM AUTHOR]

We propose a solution strategy for parameter estimation, where we combine adaptive multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the nonlinear inverse problem of recovering a coefficient function in a system of differential equations from spatially sparsely distributed measurement data. The specific equations considered in this paper describe two-phase porous-media flow where a coefficient function defining absolute permeability (fluid conductivity) is estimated based on fluid pressure observations in wells. This inverse problem is known to be ill-posed. The spatial variability of the sought coefficient function is unknown and will typically vary within the porous medium. Due to limited information in the available data, mainly coarse-scale features of the existing variability in the coefficient function will be attainable. In AME, one starts out with a single parameter representation of the sought function, whereafter the domain is successively divided into finer rectangular zones, each representing a constant parameter value of the coefficient function. The strong restrictions on the zone geometry may lead to overparameterization. LSE is a method for moving curves and enables adjustments of the zone structure into more general geometries. In order to perform well, LSE requires a reasonable starting point. In this paper, we have developed a methodology to combine AME and LSE, where at each step either refinements or deformation of the zone structure may be conducted, depending on which method promises the better result. The combined approach seems promising with respect to recovering coarse-scale features of the sought coefficient functions with a low number of parameters. [ABSTRACT FROM AUTHOR]

Abstract: We consider identification of absolute permeability (hydraulic conductivity) based on time series of pressure data in sparsely distributed wells for two-phase porous-media flow. For this problem, it is impossible to recover all details of the parameter function. On the other hand, a coarser, approximate recovery may be sufficient for many applications. We propose a novel solution approach, based on reparametrization, for such approximate identification of the parameter function. We use a nonlinear, composite representation, which is detached from the computational grid, allowing for a flexible representation of the parameter function at many resolution levels. This is utilized in a sequential multi-level estimation of the parameter function, starting at a coarse resolution, which is then gradually refined. The composite representation is designed to allow for smooth as well as sharp transitions between regions of nearly constant parameter value. Moreover, it facilitates the estimation also of the structure and smoothness of the parameter function itself. As a limiting case, the chosen representation is reduced to a zonation with implicit representation of the interior boundaries that is equivalent to a level-set representation. A motivation for the selected representation and the multi-level estimation is presented in terms of an analysis of sensitivity and nonlinearity. Numerical examples demonstrate identification of coarse-scale features of reference permeability distributions with varying degree of smoothness. Comparisons show how the multi-level strategy stabilize the identification and avoid local minima of the objective function compared to a single-level strategy. [Copyright &y& Elsevier]