Finite Element Implicit 3D Subsurface Structural Modeling. (August 2022) - Irakarama, Modeste; Thierry-Coudon, Morgan; Zakari, Mustapha; Caumon, Guillaume
Abstract: We introduce a method for 3D implicit geological structural modeling from sparse sample points, where several conformable geological surfaces are represented by one single scalar field. Laplacian and Hessian regularization energies are discretized on a tetrahedral mesh using finite elements. This scheme is believed to offer some geometrical flexibility as it is readily implemented on both structured and unstructured grids. While implicit modeling on unstructured grids is not new, methods based on finite elements have received little attention. The finite element method is routinely used to solve boundary value problems. However, because boundary conditions are typically unknown in implicit subsurface structural modeling, the traditional finite element method requires some adjustments. To this end, we present boundary free discretizations of the Laplacian and Hessian energies that do not assume vanishing Neumann boundary conditions, thereby eliminating the boundary artifacts usually associated with that assumption. Furthermore, we argue that while an appropriate discretization of the Laplacian can be used to minimize the curvature of a function on triangulated meshes, it may fail to do so on tetrahedral meshes. Graphical abstract: Highlights: A finite element discretization of the Laplacian without imposing any boundary condition. A finite element discretization of the Hessian without imposing any boundary condition. Building 3D subsurface numerical models using finite elements. The Laplacian may not be as adequate for interpolation on tetrahedra as it is on triangles.