Three-dimensional streamline tracing method over tetrahedral domains

Abstract

Getting a clear understanding of the fluid velocity field in underground porous media is critical to various engineering applications, such as oil/gas reservoirs, CO2 sequestration, groundwater, etc. As an effective visualization tool and efficient transport behaviors solution algorithm, the streamline-based method was improved significantly by numerous studies conducted in the last couple of decades. However, the implementation of streamline simulation is still challenging while working with Finite Element Method (FEM) over 3D tetrahedral domains, where the mass conservation is not guaranteed. Considering the increased computational cost to enforce mass conservation in FEM and additional complexity, a new three-dimensional streamline tracing algorithm is presented that only relies on the velocity vector of a flow field on each vertex of a tetrahedron in a 3D unstructured mesh system. Owning to the shape functions and transformation equations between the master element and actual element, the exit coordinate leaving a tetrahedral element can be determined effectively. As a result, Time of Flight (TOF), the coordinate variable along each streamline, can be calculated accurately and efficiently because that the analytical solution depicting the trajectory in Master Element is deduced. The presented streamline-based method is tested under FEniCS, a programming framework for FEM, which eases the implementation and further development of the presented method.