Data assimilation using bayesian filters and B-spline geological models

This paper proposes a new approach to problems of data assimilation, also known as history matching, of oilfield production data by adjustment of the location and sharpness of patterns of geological facies. Traditionally, this problem has been addressed using gradient based approaches with a level set parameterization of the geology. Gradient-based methods are robust, but computationally demanding with real-world reservoir problems and insufficient for reservoir management uncertainty assessment. Recently, the ensemble filter approach has been used to tackle this problem because of its high efficiency from the standpoint of implementation, computational cost, and performance. Incorporation of level set parameterization in this approach could further deal with the lack of differentiability with respect to facies type, but its practical implementation is based on some assumptions that are not easily satisfied in real problems. In this work, we propose to describe the geometry of the permeability field using B-spline curves. This transforms history matching of the discrete facies type to the estimation of continuous B-spline control points. As filtering scheme, we use the ensemble square-root filter (EnSRF). The efficacy of the EnSRF with the B-spline parameterization is investigated through three numerical experiments, in which the reservoir contains a curved channel, a disconnected channel or a 2-dimensional closed feature. It is found that the application of the proposed method to the problem of adjusting facies edges to match production data is relatively straightforward and provides statistical estimates of the distribution of geological facies and of the state of the reservoir

B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure

Manifold interpolation and model reduction

One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of model reduction. Moreover, the principal approaches to data interpolation and Taylor-like extrapolation on matrix manifolds are outlined and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model Order Reduction

Trajectory Reconstruction Techniques for Evaluation of ATC Systems

This paper is focused on trajectory reconstruction techniques for evaluating ATC systems, using real data of recorded opportunity traffic. We analyze different alternatives for this problem, from traditional interpolation approaches based on curve fitting to our proposed schemes based on modeling regular motion patterns with optimal smoothers. The extraction of trajectory features such as motion type (or mode of flight), maneuvers profile, geometric parameters, etc., allows a more accurate computation of the curve and the detailed evaluation of the data processors used in the ATC centre. Different alternatives will be compared with some performance results obtained with simulated and real data sets

Force-Control for the Automated Footwear Testing System

The Automated Footwear Testing System (AFTS) is a robotic system designed to replicated the movement and loading of a shoe as it contacts the ground during common human movements. By doing so, the AFTS can serve as a system for the functional testing of different footwear designs in a manner that is difficult to achieve by standard testing systems. The AFTS consists of four main components: a robotic Stewart platform, a rigid fixed frame, a load cell and a prosthetic foot. Motion of the foot relative to the ground is created by rigidly fixing the foot to the frame and moving the platform relative to the foot. The Stewart platform has six degrees of kinematic freedom and can reproduce the required complex three-dimensional motion path within the limitations of its range of motion. While the platform is in contact with the footwear, the six-axis load cell measures the three-dimensional forces and moments acting on the prosthetic foot. For the AFTS, a movement path is specified, translated into platform coordinates and executed on the machine. During the execution, the load cell measures the forces and moments that act on the prosthetic foot. We wish to find the particular movement path of the Stewart platform that will generate the target force profile. Thus, we are interested in solving an inverse problem. The main goal of the workshop was to investigate potential solution methods for this ‘force-control’ problem, including looking into its feasibility