Øyvind Hjelle

Parallel solutions of static Hamilton-Jacobi equations for simulations of geological folds

Two new algorithms for numerical solution of static Hamilton-Jacobi equations are presented. These algorithms are designed to work efficiently on different parallel computing architectures, and numerical results for multicore CPU and GPU implementations are reported and discussed. The numerical experiments show that the proposed solution strategies scale well with the computational power of the hardware. The performance of the new methods are investigate for tow types of static Hamilton-Jacobi formulations are investigated, the isotropic eikonal equation and an anisotropic formulation used to simulate different types of geological folding. Simulations of geological folding is a key component in an industrial software used in oil and gas exploration. In particular, our experiments indicate that the new algorithms would be capable of accelerating an existing industrial simulator substantially. Direct comparison with the current industry code shows that computing times can be reduced from several minutes to a few seconds. The new methods are now being migrated to the industrial software.

A Numerical Framework for Modeling Folds in Structural Geology

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.

Accuracy and efficiency of stencils for the eikonal equation in earth modelling

Motivated by the needs for creating fast and accurate models of complex geological scenarios, accuracy and efficiency of three stencils for the isotropic eikonal equation on rectangular grids are evaluated using a fast marching implementation. The stencils are derived by direct modelling of the wave front, resulting in new and valuable insight in terms of improved upwind and causality conditions. After introducing a method for generalising first-order upwind stencils to higher order, a new second-order diagonal stencil is presented. Similarly to the multistencil fast marching approach, the diagonal stencil makes use of nodes in the diagonal directions, whereas the traditional Godunov stencil uses solely edge-connected neighbours. The diagonal stencil uses nodes close to each other, reaching upwind, to get a more accurate estimate of the angle of incidence of the arriving wave front. Although the stencils are evaluated in a fast marching setting, they can be adapted to other efficient eikonal solvers. All first- and second-order stencils are evaluated in a range of tests. The first test case models a folded structure from the Zagros fold belt in Iran. The other test cases are constructed to investigate specific properties of the examined stencils. The numerical investigation considers convergence rates and CPU times for non-constant and constant speed first-arrival computations. In conclusion, the diagonal stencil is the most efficient and accurate of the three alternatives.