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Truncated Gaussian/pluri-Gaussian representations and level set representations are two methodologies for implicit representation that can be applied to large-scale subsurface structures. Identification of facies in a petroleum reservoir from dynamic well data is a common field of application for these representation methodologies. No comparison of the methodologies has appeared in the literature to date. The paper seeks to improve on this situation by comparing selected level set and truncated Gaussian/pluri-Gaussian representations in detail. Strong similarities are found, in the sense that every truncated Gaussian/pluri-Gaussian representation considered has a level set counterpart. Furthermore, the transition from a truncated Gaussian/pluri-Gaussian representation to the corresponding level set representation is easily accessible. In addition to the comparison aspect, this paper also introduces a novel level set representation-the hierarchical level set representation-that removes a difficulty present in existing level set representations in a shape estimation setting. It is shown that the hierarchical level set representation corresponds to a well-known truncated pluri-Gaussian representation. [ABSTRACT FROM AUTHOR]

We consider the impact of using time-lapse seismic data in addition to production data for permeability estimation in a porous medium with multiphase fluid flows, such as a petroleum reservoir under water-assisted production. Since modeling seismic wave propagation in addition to modeling fluid flows in the reservoir is quite involved, it is assumed that the time-lapse seismic data have already been inverted into fluid saturation differences (pseudoseismic data). Because an inversion process often leads to considerable error growth, we will consider pseudoseismic data with large uncertainties. The impact of pseudoseismic data is assessed through permeability estimation with and without such data and through application of some uncertainty measures for the estimated parameters. A multiscale algorithm is used for the parameter estimations, so that potential differences in attainable permeability resolution will be easily revealed. The numerical examples clearly indicate that the permeability estimation problem is stabilized at a higher level of resolution when pseudoseismic data are applied in addition to production data, even if the pseudoseismic data have large associated uncertainties. Use of the parameter uncertainty measures confirm these results. [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]

An extension of a methodology for identifying fluid conductivity in porous media flow, i.e. adaptive multiscale estimation, is presented. Adaptive multiscale estimation has been successful in identifying coarse scale structures of the spatially dependent fluid conductivity by introduction of a low number of parameters. The parameterization choices allowed in the existing methodology experience some problems in identifying fluid conductivity with spatial dependency in a slanting manner, i.e. the spatial dividing lines between different fluid conductivity values do not follow the grid lines in a quadrilateral grid. In this work it is investigated whether more parameterization choices allow the adaptive multiscale estimation methodology to resolve slanting fluid conductivities or not. It is found that when adaptive multiscale estimation is allowed to choose parameterization from a richer parameterization selection, it is capable of resolving structures in slanting fluid conductivities without increasing the CPU time too much. Examples utilizing the new parameterization choices are shown. [ABSTRACT FROM AUTHOR]

Adaptive multiscale estimation of a spatially dependent diffusion function is considered. When utilizing pressure data, the method has been successful in identifying coarse-scale structures of the diffusion function. In this article the method is used with pressure and fluid rate data as information sources. The estimate is expected to be improved when additional data are included. However, this expectation is not always met, and in some cases the simultaneous use of these data types in the estimation might even cause problems. The reason for the lack of improvement is investigated together with methods for successful estimation utilizing pressure and fluid rates. [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]