Implementation of streamline simulation based on finite element method in FEniCS

Abstract

Understanding flow and transport behaviors in subsurface porous rock is extremely important for various applications such as oil/gas reservoirs, groundwater, geothermal energy explorations, and CO2 sequestration. However, the relevant calculation is never an easy task because of the large areal extend and high heterogeneity of subsurface system. Although streamline-based simulations are suitable for solving large, geologically complex, and heterogeneous systems, they still present difficulties while working with the finite element method. This paper introduces a new algorithm to trace streamlines based on the underlying flow field in unstructured triangular mesh systems. The new solution procedure only requires velocities on each element vertex but not on cell facets, and therefore it does not rely on conservative fluxes. The algorithm is based on coupled ODEs derived from shape functions as well as the transformation formula between the actual and master elements. The analytical solution describing the trajectory of a streamline segment in the master element is derived so that the time-of-flight and exit coordinate (leaving the element) can be determined accurately and efficiently. Additionally, the presented algorithm allows the implementation of streamline-based methods in the programming framework, FEniCS, so as to extend the technique to other research areas.