Chapter

# Benchmark 3D: the VAG scheme

DOI: 10.1007/978-3-642-20671-9_99

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**ABSTRACT:**We present here a number of test cases and meshes which were designed to form a benchmark for finite volume schemes and give a summary of some of the results which were presented by the participants to this benchmark. We address a two-dimensional anisotropic diffusion problem, which is discretized on general, possibly non-conforming meshes. In most cases, the diffusion tensor is taken to be anisotropic, and at times heterogeneous and/or dis-continuous. The meshes are either triangular or quadrangular, and sometimes quite distorted. Several methods were tested, among which finite element, discontinous Galerkin, cell centred and vertex centred finite volume methods, discrete duality finite volume methods, mimetic meth-ods. The results given by the participants to the benchmark range from the number of unknowns, the errors on the fluxes or the minimum and maximum values and energy, to the order of con-vergence (when available). - [Show abstract] [Hide abstract]

**ABSTRACT:**We present the use of the Vertex Approximate Gradient scheme for the simulation of multiphase flow in porous media. The porous volume is distributed to the natural grid blocks and to the vertices, hence leading to a new finite volume mesh. Then the unknowns in the control volumes may be eliminated, and a 27-point scheme results on the vertices unknowns for a hexahedral structured mesh. Numerical results show the efficiency of the scheme in various situations, including miscible gas injection. Keywordstwo-phase flow in porous media-vertex approximate gradient scheme-reservoir simulation12/2010: pages 409-417; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper concerns the discretisation on general 3D meshes of multiphase compositional Darcy flows in heterogeneous anisotropic porous media. Extending Coats’ formulation [15] to an arbitrary number of phases, the model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium and dynamically manages phase appearance and disappearance. The spatial discretisation of the multiphase compositional Darcy flows is based on a generalisation of the Vertex Approximate Gradient scheme, already introduced for single-phase diffusive problems in [24]. It leads to an unconditionally coercive scheme for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes, and the number of unknowns on tetrahedral meshes is considerably reduced, compared with the usual cell-centred approaches. The efficiency of our approach is exhibited on several examples, including the nearwell injection of miscible CO2 in a saline aquifer taking into account the vaporisation of H2O in the gas phase as well as the precipitation of salt.Computational Geosciences 01/2012; 16:987-1005. · 1.42 Impact Factor

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