Mixed finite element methods for discretization and streamline tracing

This thesis develops mixed finite-element (MFE) methods for the discretization and numerical simulation of the reservoir flow equations. MFE methods provide the proper framework for the interpretation and theoretical analysis of multipoint flux approximations (MPFA), which are the industry standard discretization methods for reservoir simulation on advanced grids. The key contributions of this research are: (1) a proof of convergence of MPFA methods on three-dimensional rectangular parallelepipedic grids, and (2) a new streamline tracing method that gives exact streamlines from MPFA discretizations.;On two-dimensional triangular, or quadrilateral grids, and on three-dimensional tetrahedral grids, Wheeler and Yotov employed the first-order Brezzi--Douglas--Marini (BDM1) space to provide the link between MPFA and MFE methods, which allowed them to prove convergence of MPFA on such grids. On hexahedral grids, however, a different velocity space must be introduced to establish the bridge between MFE and MPFA discretizations.;In this dissertation, we present a new velocity space on three-dimensional hexahedra. The new velocity space is defined using four degrees of freedom per face, which are the normal components of the velocity field at the vertices of each face. The new space is compatible in the sense of Babuska and Brezzi with a piecewise constant pressure discretization and therefore yields a consistent MFE method. An error analysis of the new MFE discretization proves its convergence and leads to error estimates for the scalar and vector variables.;The application of a vertex-based quadrature rule reduces the new MFE to the widely used MPFA O-method on three-dimensional hexahedra. This represents the first direct link between MFE and MPFA on hexahedral grids, which we exploit to provide a proof of convergence of the MPFA O-method, along with error estimates for the pressure and velocity fields.;In the context of streamline simulation, the quality of the velocity field is essential to the accuracy of the overall method. The streamlines are traced by integration of the velocity field, which is interpolated from the MPFA fluxes. Current streamline tracing algorithms rely on low-order velocity reconstruction techniques that do not preserve the accuracy of the fluxes computed by MPFA discretizations. On advanced grids, this can lead to O(1) numerical errors.;In this work, we exploit the links between MPFA and MFE methods to interpret the MPFA fluxes as MFE degrees of freedom. The MPFA velocity field is then reconstructed through interpolation of the fluxes by the MFE velocity shape functions. Therefore, the streamlines are traced on the velocity field corresponding exactly to the MPFA discretization. After a detailed description of the streamline tracing algorithm, we provide comparisons of low- and high-order accurate tracing methods by means of challenging numerical experiments.