A Numerical Framework for Modeling Folds in Structural Geology
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A Numerical Framework for Modeling Folds in Structural Geology Øyvind Hjelle · Steen A. Petersen · Are Magnus Bruaset
Received: 30 December 2012 / Accepted: 25 February 2013 / Published online: 19 March 2013 © International Association for Mathematical Geosciences 2013
Abstract A numerical framework for modeling folds in structural geology is presented. This framework is based on a novel and recently published Hamilton–Jacobi formulation by which a continuum of layer boundaries of a fold is modeled as a propagating front. All the fold classes from the classical literature (parallel folds, similar folds, and other fold types with convergent and divergent dip isogons) are modeled in two and three dimensions as continua defined on a finite difference grid. The propagating front describing the fold geometry is governed by a static Hamilton–Jacobi equation, which is discretized by upwind finite differences and a dynamic stencil construction. This forms the basis of numerical solution by finite difference solvers such as fast marching and fast sweeping methods. A new robust and accurate scheme for initialization of finite difference solvers for the static Hamilton–Jacobi equation is also derived. The framework has been integrated in simulation software, and a numerical example is presented based on seismic data collected from the Karama Block in the North Makassar Strait outside Sulawesi. Keywords Folding · Dip isogons · Front propagation · Hamilton–Jacobi · Anisotropy · Upwind finite differences · Fast marching Ø. Hjelle () Kalkulo AS, Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway e-mail: [email protected] S.A. Petersen Statoil ASA, P.O. Box 7200, 5020 Bergen, Norway A.M. Bruaset Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway A.M. Bruaset Department of Informatics, University of Oslo, P.O. Box 1080, 0316 Oslo, Norway
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Math Geosci (2013) 45:255–276
1 Introduction When investigating the geological evolution of a given region, concurrent theories will arise and compete as geologists try to fit together the puzzle pieces provided by collected data. Traditionally, geology has been centered on a qualitative understanding of physical processes spanning tens and hundreds of millions of years. However, through innovative use of different data sources, such as seismic profiles, electromagnetic recordings, and bore samples, computer-based models have become quantitative supplements and corrections to the qualitative interpretation. Such models are of particular value to oil and gas companies searching for hydrocarbon reserves across wide ranges of geological settings, often being of very high complexity. In particular, computer-based models of geological folds are important when mapping out paleogeographical features, such as when identifying geological structures that have supported accumulation of sediments in a basin. In a recent work, Hjelle and Petersen (2011) presented a new mathematical framework for fold modeling. Using this framework, the main classes of folding described by Ramsa
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