Abstract
The multipoint-flux approximation (MPFA) methods have been popular for reservoir simulation, since they handle permeability anisotropy and heterogeneity on irregular grids, they are locally mass conservative and have an explicit expression for the flux. However, they are in general not coercive, and hence, only conditionally convergent. Further, the matrix is in general nonsymmetric, and the cell stencil may become somewhat wide in some cases, increasing the run time of the linear solvers.
The control-volume finite element (CVFEM) method was introduced for reservoir simulation in the beginning of the 1990ies as a locally mass conservative alternative to the Galerkin finite element (FEM) method. Recently a vertex-centered finite element (VAG) method that can be applied on general grids and that has an explicit flux have been developed [R. Eymard et al., ESAIM: Mathematical Modelling and Numerical Analysis, 46(2), 2012]. The method is symmetric and coercive, and produces a small cell stencil.
We show that the VAG, CVFEM, and FEM are, in fact, identical for single phase flow on tetrahedral grids. Next, the VAG, CVFEM and MPFA methods are compared for hexahedral grids. We observe, that the VAG and CVFEM methods produce similar results; further, it is found that the VAG and CVFEM scheme converge for a wider range of problems than the MPFA methods, however when the MPFA-methods converge, the convergence rate in flux is better than for the other methods.
At last, the VAG flux expression is applied for tracer flow and simple two-phase flow simulations. We investigate the limit case when the volume of the vertex control volumes goes to zero.
