Development of meshless computational algorithms for seismic exploration

Abstract

Summary Radial basis function-generated finite differences (RBFFD), a high-order mesh-less method for scientific computing, offer an entirely new window of opportunity for numerically handling domains with irregular boundaries and discontinuous material interfaces. This is critical for the accurate solution of PDEs that are pertinent to seismic exploration, where wave reflections from vast numbers of curved material interfaces produce the information used for inversion and thus for effective hydrocarbon discovery/recovery. Present state-of-the-art in the industry focus on lattice-type grid methods, suggesting that reduction to first-order near boundaries will be difficult to avoid. Although some more accurate interface treatment methods have been proposed, these are too complex to generalize to 3-D. In contrast, the mesh-free nature of RBFFD allow for the freedom of optimal node placement. The interface approach we propose starts by using RBF-FD node sets that align with the interfaces on a local scale, but with nodes that are otherwise ‘evenly scattered' in the domain between interfaces. There is therefore no constraint on the location or curvature of other nearby interfaces. Across the interface, a 'fictitious point' approach is employed, allowing for smooth continuation of the PDE's dependent variables. Preliminary tests on both the 2-D acoustic and elastic wave equations show that this approach indeed combines the best features of its different components, high order of accuracy away from interfaces and no significant degradation when interfaces are encountered. With regard to seismic exploration, the RBFFD method may offer a new level of accuracy and cost effectiveness for elastic wave simulations in heavily stratified and fractured media, such as those holding hydrocarbon deposits.