A New Mixed Finite Element and its Related Finite Volume Discretization on General Hexahedral Grids

2008 ◽  
Author(s):  
Se´bastien F. Matringe ◽  
Ruben Juanes ◽  
Hamdi A. Tchelepi
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Volume ◽  
Control Volume ◽  
Mixed Finite Element ◽  
Quadrature Rule ◽  
Mathematical Framework ◽  
Flow Equations ◽  
Hexahedral Elements ◽  
Flux Approximation

Modern reservoir simulation grids are generally composed of distorted hexahedral elements populated with heterogeneous and possibly full-tensor coefficients. The numerical discretization of the reservoir flow equations on such grids is a challenging problem. Finite volume methods based on a two-point flux approximation (TPFA) do not properly account for grid distortion or permeability anisotropy that is misaligned with the grid. Multipoint flux approximation (MPFA) methods have been developed to overcome these shortcomings. Although implemented and used in virtually every commercial reservoir simulator, a proof of convergence for MPFA methods on three-dimensional hexahedral grids has remained elusive. Here, we present a link between MPFA and a new mixed finite element methods (MFEM) on hexahedral grids, which provides a powerful mathematical framework for the analysis of MPFA. First, we introduce a new mixed finite element on 3D hexahedra. The new element defines a velocity field with bilinear normal components through element faces. Thus, the new velocity field is defined by 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 with a piecewise constant pressure discretization and yields a convergent discretization. The application of a vertex-based quadrature rule reduces the new mixed finite element method to a multipoint flux control volume method. For Cartesian grids, this is in fact the classical MPFA O-method. This provides for the first time a direct link between MFEM and MPFA on hexahedral grids, which we use to establish convergence of MPFA for 3D rectangular grids.

Tracing Streamlines on Unstructured Grids From Finite Volume Discretizations

SPE Journal ◽  
2008 ◽  
Vol 13 (04) ◽  
pp. 423-431 ◽  
Author(s):  
Sebastien F. Matringe ◽  
Ruben Juanes ◽  
Hamdi A. Tchelepi
Keyword(s):  
Finite Element ◽  
Velocity Field ◽  
Finite Difference ◽  
Finite Volume ◽  
Unstructured Grids ◽  
Control Volume ◽  
Next Generation ◽  
Flux Approximation ◽  
Streamline Tracing ◽  
Tracing Method

Summary The accuracy of streamline reservoir simulations depends strongly on the quality of the velocity field and the accuracy of the streamline tracing method. For problems described on complex grids (e.g., corner-point geometry or fully unstructured grids) with full-tensor permeabilities, advanced discretization methods, such as the family of multipoint flux approximation (MPFA) schemes, are necessary to obtain an accurate representation of the fluxes across control volume faces. These fluxes are then interpolated to define the velocity field within each control volume, which is then used to trace the streamlines. Existing methods for the interpolation of the velocity field and integration of the streamlines do not preserve the accuracy of the fluxes computed by MPFA discretizations. Here we propose a method for the reconstruction of the velocity field with high-order accuracy from the fluxes provided by MPFA discretization schemes. This reconstruction relies on a correspondence between the MPFA fluxes and the degrees of freedom of a mixed finite-element method (MFEM) based on the first-order Brezzi-Douglas-Marini space. This link between the finite-volume and finite-element methods allows the use of flux reconstruction and streamline tracing techniques developed previously by the authors for mixed finite elements. After a detailed description of our streamline tracing method, we study its accuracy and efficiency using challenging test cases. Introduction The next-generation reservoir simulators will be unstructured. Several research groups throughout the industry are now developing a new breed of reservoir simulators to replace the current industry standards. One of the main advances offered by these next generation simulators is their ability to support unstructured or, at least, strongly distorted grids populated with full-tensor permeabilities. The constant evolution of reservoir modeling techniques provides an increasingly realistic description of the geological features of petroleum reservoirs. To discretize the complex geometries of geocellular models, unstructured grids seem to be a natural choice. Their inherent flexibility permits the precise description of faults, flow barriers, trapping structures, etc. Obtaining a similar accuracy with more traditional structured grids, if at all possible, would require an overwhelming number of gridblocks. However, the added flexibility of unstructured grids comes with a cost. To accurately resolve the full-tensor permeabilities or the grid distortion, a two-point flux approximation (TPFA) approach, such as that of classical finite difference (FD) methods is not sufficient. The size of the discretization stencil needs to be increased to include more pressure points in the computation of the fluxes through control volume edges. To this end, multipoint flux approximation (MPFA) methods have been developed and applied quite successfully (Aavatsmark et al. 1996; Verma and Aziz 1997; Edwards and Rogers 1998; Aavatsmark et al. 1998b; Aavatsmark et al. 1998c; Aavatsmark et al. 1998a; Edwards 2002; Lee et al. 2002a; Lee et al. 2002b). In this paper, we interpret finite volume discretizations as MFEM for which streamline tracing methods have already been developed (Matringe et al. 2006; Matringe et al. 2007b; Juanes and Matringe In Press). This approach provides a natural way of reconstructing velocity fields from TPFA or MPFA fluxes. For finite difference or TPFA discretizations, the proposed interpretation provides mathematical justification for Pollock's method (Pollock 1988) and some of its extensions to distorted grids (Cordes and Kinzelbach 1992; Prévost et al. 2002; Hægland et al. 2007; Jimenez et al. 2007). For MPFA, our approach provides a high-order streamline tracing algorithm that takes full advantage of the flux information from the MPFA discretization.

scholarly journals MIXED FINITE ELEMENT METHODS: IMPLEMENTATION WITH ONE UNKNOWN PER ELEMENT, LOCAL FLUX EXPRESSIONS, POSITIVITY, POLYGONAL MESHES, AND RELATIONS TO OTHER METHODS

2013 ◽  
Vol 23 (05) ◽  
pp. 803-838 ◽  
Author(s):  
MARTIN VOHRALÍK ◽  
BARBARA I. WOHLMUTH
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Mixed Finite Element Methods ◽  
Discrete Maximum Principle ◽  
Polygonal Meshes ◽  
Linear Diffusion ◽  
Flux Approximation

In this paper, we study the mixed finite element method for linear diffusion problems. We focus on the lowest-order Raviart–Thomas case. For simplicial meshes, we propose several new approaches to reduce the original indefinite saddle point systems for the flux and potential unknowns to (positive definite) systems for one potential unknown per element. Our construction principle is closely related to that of the so-called multi-point flux-approximation method and leads to local flux expressions. We present a set of numerical examples illustrating the influence of the elimination process on the structure and on the condition number of the reduced matrix. We also discuss different versions of the discrete maximum principle in the lowest-order Raviart–Thomas method. Finally, we recall mixed finite element methods on general polygonal meshes and show that they are a special type of the mimetic finite difference, mixed finite volume, and hybrid finite volume family.

Transport Schemes in GungHo

2021 ◽  
Author(s):  
James Kent
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Method Of Lines ◽  
Finite Volume Methods ◽  
Kutta Method ◽  
Finite Volume Schemes ◽  
Dynamical Core ◽  
Runge Kutta Method ◽  
The Stability

<p>GungHo is the mixed finite-element dynamical core under development by the Met Office. A key component of the dynamical core is the transport scheme, which advects density, temperature, moisture, and the winds, throughout the atmosphere. Transport in GungHo is performed by finite-volume methods, to ensure conservation of certain quantaties. There are a range of different finite-volume schemes being considered for transport, including the Runge-Kutta/method-of-lines and COSMIC/Lin-Rood schemes. Additional horizontal/vertical splitting approaches are also under consideration, to improve the stability aspects of the model. Here we discuss these transport options and present results from the GungHo framework, featuring both prescribed velocity advection tests and full dry dynamical core tests. </p>

scholarly journals Mixed finite element-finite volume methods

2010 ◽  
Vol 17 (3) ◽  
pp. 385-410
Author(s):  
Khadija Zine Dine ◽  
Naceur Achtaich ◽  
Mohamed Chagdali
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Finite Volume Methods
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