The most important and difficult aspect of modeling a naturally fractured reservoir is the correct calculation of the exchange of fluids between the matrix rock and the surrounding fractures. Several authors have published alternative techniques for handling this problem over the past few years. However, because each of these alternatives has some limitations, a new and more general technique has been developed. This new technique is used to simulate matrix/fracture exchange with special emphasis on the gravity forces included in the exchange terms. The exchange terms and the gravity forces within the exchange terms simulate the behavior of a single matrix block surrounded .by fractures that may contain several different fluids. The gravity forces are internally calculated as functions of saturation. This technique has been incorporated into a new three-dimensional (3D), threephase, fully implicit model for simulating fluid flow in a naturally fractured reservoir. The description of the porous medium might include highly fractured, microfractured, and nonfractured regions. Several examples explain the use of a new naturally fractured reservoir model and the essential differences between the new approach and those used in earlier naturally fractured reservoir models. IntroductionThe numerical simulation of naturally fractured reservoirs is a subject that has been described extensively in the literature of the oil industry for the last 25 years.In the initial work done, the primary goal of the simulation of fractured systems was to study pressure behavior during well tests. This was based primarily on the analytical solutions obtained by Barenblatt et at. l and Warren and Root Z for describing singlephase flow near a wellbore in a naturally fractured reservoir. Their work was based on the concept of a fractured continuum filled with noncontinuous matrix blocks. Warren and Root also delivered a one-dimensional (ID) radial model, which Kazemi 3 later extended to study two-dimensional (2D) flow during a well test. Further developments were made by Iffly et at.,4 Yamamoto et at.,5 Kleppe and Morse, 6 and others, but these models were used either to match laboratory results or to study the behavior of a single matrix block.More recently, more complex problems have been studied with full-field reservoir models. Our interest in this paper is the study of fractured reservoirs with full-field models.Examples are presented that show some of the differences between a new fractured reservoir model and other models described in the literature. Comparisons are also made that point out some of the complexities of fractured reservoir simulation and the potential problems with the use of a single-porosity approach to study various types of fractured reservoir behavior.One of the first papers to describe a fractured reservoir simulator that could be applied to full-field model studies was by Saidi 7 and was concerned with the simulation of the very highly fractured reservoirs in Iran. This model represented the fracture system as a ...

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractA proper modeling of tertiary recovery processes such as gas injection or WAG (Water Alternating Gas) requires an adequate three-phase flow model. This allows to better predict the recovery efficiency, gas storage reservoir performance as well as the well injectivity.For gas drainage, a previous paper [25] presented a new threephase flow model based on a theoretical analysis and validated through experimental approach. For WAG injection, there is an additional complexity due to the need to model the imbibition that occurs when gas saturation decreases. To tackle the modeling of hysteresis problem, a comprehensive approach was followed. First, successive drainage and imbibition experiments were conducted under various conditions of initial saturations. A new three-phase model taking into account the hysteresis is presented and validated on the experiments.Indeed, as shown in previous experimental studies, hysteresis was found to depend not only on the drainage/imbibition process (saturation history) but also on the cycle considered (displacement history) where cycle names the association of two consecutive displacements (drainage and imbibition). In this study, a relevant analytical expression of the hysteresis is proposed avoiding any negative effect of numerical instabilities. The new formulation was implemented in a reservoir simulator and WAG experiments have been successfully simulated. The impact on breakthrough time, overall recovery efficiency was tested through large scale reservoir simulations.

A new model has been developed specifically to study very large, heterogeneous oil and gas reservoirs. By using unique approaches to the simulation of fluid properties and dual porosity/permeability systems, the model is able to accurately simUlate complex reservoir flow performance that was previously difficult or not feasible to model. Descriptions of the fluid flow characteristics of the model are included in the Appendix. Potential applications to several real field simulation problems are discussed to show the advantages of the model as compared to more traditional approaches. Additionally, several actual simulations of hypothetical reservoirs are shown and the results are compared with both standard single porosity and dual porosity black oil models currently available. The enhanced representation of the physical system compared to classical black oil models is achieved in a way which is efficient in the use of both the memory and processor resources of the computer, whilst the modular nature of the simulator, coupled with advanced programming and documentation standards, will facilitate its further development and maintenance. 305

A new model has been developed specifically to study very large, heterogeneous oil and gas reservoirs. By using unique approaches to the simulation of fluid properties and dual porosity/permeability systems, the model is able to accurately simUlate complex reservoir flow performance that was previously difficult or not feasible to model. Descriptions of the fluid flow characteristics of the model are included in the Appendix. Potential applications to several real field simulation problems are discussed to show the advantages of the model as compared to more traditional approaches. Additionally, several actual simulations of hypothetical reservoirs are shown and the results are compared with both standard single porosity and dual porosity black oil models currently available. The enhanced representation of the physical system compared to classical black oil models is achieved in a way which is efficient in the use of both the memory and processor resources of the computer, whilst the modular nature of the simulator, coupled with advanced programming and documentation standards, will facilitate its further development and maintenance. 305

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