Spatial uncertainty quantification of siliciclastic reservoirs, using the distance kernel method

Assessing the uncertainty in reservoir performance is a necessary step during the exploration phase. To examine the uncertainty in flow response, a large set of realizations must be processed. There are several stochastic geostatistical algorithms capable of simulating multiple equiprobable realizations. Although these can show us the possible realities highlighting the spatial uncertainty, their handling is time- and CPU-consuming during the later processes, such as flow simulations. Consequently, only a small number of realizations can be post-processed in industrial practice. The purpose of this work is to develop a method, which will reduce the huge number of realizations in a way that the remaining ones retain the spatial uncertainty of a reservoir’s flow behavior, as would be demonstrated by a larger set of realizations. To solve this problem, ranking methods can be applied. Traditional ranking techniques, such as probability selection, are highly dependent on the applied static properties. In this paper, an alternative selection method is parameterized for measuring the pairwise dissimilarity between geostatistical models, with a distance function based on the hydrodynamic properties of the hydrocarbon reservoirs. The effectivity of the method is highly dependent upon the selected criteria. Thus, the distance function refers to the flow responses and allows visualizing the space of uncertainty through multidimensional scaling. A kernel transformation of the MDS data set is required to obtain a feature space where the K-means algorithm can discover non-linear structures in the basic data set. The final step of the method is the selection of the Earth models closest to the cluster centers. This tool allows for the selection of a subset of representative realizations, containing similar properties to the larger set

Membership-set estimation using random scanning and principal component analysis

A set-theoretic approach to parameter estimation based on the bounded-error concept is an appropriate choice when incomplete knowledge of observation error statistics and unavoidable structural model error invalidate the presuppositions of stochastic methods. Within this class the estimation of non-linear-in-the-parameters models is examined. This situation frequently occurs in modelling natural systems. The output error method proposed is based on overall random scanning with iterative reduction of the size of the scanned region. In order to overcome the problem of computational inefficiency, which is particularly serious when there is interaction between the parameter estimates, two modifications to the basic method are introduced. The first involves the use of principal component transformations to provide a rotated parameter space in the random scanning because large areas of the initial parameter space are thus excluded from further examination. The second improvement involves the standardization of the parameters so as to obtain an initial space with equal size extension in all directions. This proves to largely increase the computational robustness of the method. The modified algorithm is demonstrated by application to a simple three-parameter model of diurnal dissolved oxygen patterns in a lake

Effects of a Maximal Energy Scale in Thermodynamics for Photon Gas and Construction of Path Integral

In this article, we discuss some well-known theoretical models where an observer-independent energy scale or a length scale is present. The presence of this invariant scale necessarily deforms the Lorentz symmetry. We study different aspects and features of such theories about how modifications arise due to this cutoff scale. First we study the formulation of energy-momentum tensor for a perfect fluid in doubly special relativity (DSR), where an energy scale is present. Then we go on to study modifications in thermodynamic properties of photon gas in DSR. Finally we discuss some models with generalized uncertainty principle (GUP).Comment: This is a review article based on our works arXiv:1002.0192, arXiv:0908.0413, arXiv:1205.3919; v2: text overlap with gr-qc/0207085 remove