Mathematical and Numerical Properties of Control-Volume, Finite-Element Scheme for Reservoir Simulation

Abstract

Summary This paper presents the mathematical properties of a control-volume, finite-element (CVFE) scheme. We show that appropriate constraints on finite-element grids and convenient definitions for volumes and transmissibilities lead to a convergence property for the CVFE scheme. This convergence property proves that use of the CVFE scheme is mathematically correct for reservoir simulation. With the control-volume concept, the local balances of each component are fully satisfied. Because the discretized equations resulting from the scheme can be solved by classic methods, the CVFE scheme can be implemented easily in a general-purpose simulator. In the implementations in this paper, we use prismatic finite elements for 3D full-field simulations and triangles for 2D simulations. The grid is generated with automatic techniques, popular in structural engineering, resulting in numerical diffusion that is as isotropic as possible; therefore, grid-orientation effects are controlled and accuracy is easily improved through natural grid refinement. Examples show that, for the same accuracy, computational costs are lower for results obtained with the CVFE scheme than for those obtained with finite-difference grids. The CVFE scheme is an excellent alternative to flexible gridding techniques used in finite-difference simulators because the entire reservoir can be gridded as required, without use of special techniques for local grid modifications. Introduction Because finite-element methods are naturally convenient for gridding areas with irregular boundaries and local refinements, several attempts have been made to use finite elements for reservoir simulation. Douglas1 and Chavent et al.2 used discontinuous and mixed finite elements for two-phase incompressible flows; Eymard et al.3 used mixed hybrid finite elements, which lead to smaller linear systems, an advantage over previous methods; and Lemonier4 reported the first attempt with the CVFE method. In all these attempts, finite-element methods are used to solve a "pressure equation." Then, the conservation equations of all the components except one are sequentially solved by various techniques, all of which lead to control-volume methods. None of the methods used in these attempts has been generalized successfully to model realistic black-oil cases. This can be explained by the following:no finite-element method exists for the Buckley-Leverett saturation equation,use of different schemes for the pressure equation and the component conservation equation excludes correct balances for all the components, andonly schemes that keep all the balances correct have been shown to handle compressibility and thermodynamic phenomena. On the other hand, the usual finite-difference methods5–7 have been improved for more-complex physical processes (compositional, thermal, and chemical flood simulation) by use of fully implicit and implicit-pressure/explicit-saturation (IMPES)-coupled methods that provide perfect balances for all the components. Note that these usual finite-difference schemes are, in fact, control-volume, finite-difference (CVFD) schemes. Until recently, CVFE schemes have been shown to present the advantages of finite-element grids and fully coupled schemes. Examples of such applications to thermal and compositional simulation have been presented.8–10 However, these papers have shown applications with no proof that the CVFE scheme is mathematically correct and convergent. The purpose of this paper is two-fold. First, we present the mathematical and numerical features of the CVFE scheme where some convergence properties are helpful for the definition of the fundamental concepts and the elimination of unsuitable considerations. Then, we present some applications to demonstrate the advantages of the CVFE scheme compared with classic CVFD methods. The Appendix summarizes the mathematical proof of the CVFE convergence property. CVFE Scheme Refs. 5 through 7 give a complete description of reservoir equations in black-oil, thermal, and compositional cases, and Refs. 8 through II give detailed presentations of CVFE schemes. Because the CVFE scheme is presented in this paper from a conceptual point of view, the general conservation equation is expressed asEquation 1 In the most general case, the accumulation functions, A; the transport functions, B; and the potential functions, C, are nonlinear functions of pressure and of all the convective variables. Scheme Formulation. Formulation of the scheme involves four steps.The reservoir is gridded by elements, such as triangles and parallelograms (if a 2D model is available), or other polyhedral volumes (for 3D simulation). If the reservoir includes several rock types, each element must be homogeneous.Test functions, Xj(x), of any Point x of the reservoir are defined for each Node j of the grid (Fig. 1). These functions are, in the case of triangles in 2D or tetrahedrals in 3D, piecewise continuous linear functions ("chapeau" functions) for every Point x of the reservoir that meet the essential propertyEquation 2 Control volumes are located at the nodes of the grid withEquation 3 Note that this definition can be made without any description of the boundaries of each control volume. However, such a description9,10 does not yield any additional properties for the convergence of the scheme. Because of Eq. 2, the local and global volumes are naturally respected. In the case of triangles or tetrahedrals, each node of an element is respectively given one-third or one-quarter of the volume of this element. p. 283-289 Scheme Formulation. Formulation of the scheme involves four steps.The reservoir is gridded by elements, such as triangles and parallelograms (if a 2D model is available), or other polyhedral volumes (for 3D simulation). If the reservoir includes several rock types, each element must be homogeneous.Test functions, Xj(x), of any Point x of the reservoir are defined for each Node j of the grid (Fig. 1). These functions are, in the case of triangles in 2D or tetrahedrals in 3D, piecewise continuous linear functions ("chapeau" functions) for every Point x of the reservoir that meet the essential propertyEquation 2 Control volumes are located at the nodes of the grid withEquation 3 Note that this definition can be made without any description of the boundaries of each control volume. However, such a description9,10 does not yield any additional properties for the convergence of the scheme. Because of Eq. 2, the local and global volumes are naturally respected. In the case of triangles or tetrahedrals, each node of an element is respectively given one-third or one-quarter of the volume of this element. p. 283–289

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