TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractThe two-point flux, finite volume method (FVM-2P) is the most widely used method for solving the flow equation in reservoir simulations. For FVM-2P to be consistent, the simulation grid needs to be orthogonal (or K-orthogonal if the permeability field is anisotropic). It is well known that cornerpoint grids can introduce large errors in the flow solutions due to the lack of orthogonality in general. Multi-point flux formulations that do not rely on grid orthogonality have been proposed, but these methods add significant computational cost to solving the flow equation.Recently, 2.5D unstructured grids that combine 2D Voronoi areal grids with vertical projections along deviated coordinate lines have become an attractive alternative to corner point gridding. The Voronoi grid helps to maintain orthogonality areally and can mitigate grid orientation effects. However, experience with these grids is limited.In this paper, we present an analytical and numerical study of these 2.5D unstructured grids. We focus on the effect of grid deviation on flow solutions in homogeneous but anisotropic permeability fields. In particular, we consider the grid deviation that results from gridding to sloping faults. We show that FVM-2P does not converge to the correct solution as grid refines. We further quantify the errors for some simple flow scenarios using a technique that combines numerical analysis and asymptotic expansions.Analytical error estimates are obtained. We find that the errors are highly flow dependent and they can be global with no strong correlation with local non-orthogonality measures. Numerical tests are presented to confirm the analytical findings and to show the applicability of our conclusions to more general flow scenarios.

The two-point flux finite-volume method (2P-FVM) is the most widely used method for solving the flow equation in reservoir simulations. For 2P-FVM to be consistent, the simulation grid needs to be orthogonal (or k-orthogonal if the permeability field is anisotropic). It is well known that corner-point grids can introduce large errors in the flow solutions because of the lack of orthogonality in general. Multipoint flux formulations that do not rely on grid orthogonality have been proposed, but these methods add significant computational cost to solving the flow equation.Recently, 2.5D unstructured grids that combine 2D Voronoi areal grids with vertical projections along deviated coordinate lines have become an attractive alternative to corner-point gridding. The Voronoi grid helps maintain orthogonality areally and can mitigate grid orientation effects. However, experience with these grids is limited.In this paper, we present an analytical and numerical study of these 2.5D unstructured grids. We focus on the effect of grid deviation on flow solutions in homogeneous, but anisotropic, permeability fields. In particular, we consider the grid deviation that results from gridding to sloping faults. We show that 2P-FVM does not converge to the correct solution as the grid refines. We further quantify the errors for some simple flow scenarios using a technique that combines numerical analysis and asymptotic expansions. Analytical error estimates are obtained. We find that the errors are highly flow dependent and that they can be global with no strong correlation with local nonorthogonality measures. Numerical tests are presented to confirm the analytical findings and to show the applicability of our conclusions to more-general flow scenarios.

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We present a new global scale-up technology for calculating effective permeability and/or transmissibility. Using this innovative technology, we apply global flow solutions to improve scale-up accuracy. Global scale-up was proposed in the 1980s, and its benefits are well-described in the literature. However, global scale-up has not been adopted by the industry due to significant technical challenges in its application to real reservoir models. These models are often characterized by complex features like faults, pinch-outs, and isolated geobodies. Here, we overview several novel technologies that we have developed to overcome these difficulties. Numerical examples applying global scale-up to several reservoir models are presented. Comparisons with local scale-up methods currently used in the industry are made. The examples demonstrate that our new global scale-up technology leads to significant improvements in scale-up accuracy. In particular, when applied to challenging models with complicated fine-scale connectivity, the global scale-up method preserves the fine-scale connectivity more accurately than local scale-up methods. Sometimes, dramatic differences are seen. Moreover, we note that global scale-up can be especially effective when used in conjunction with unstructured grids. Accurate scale-up is a critical link between fine-scale geologic descriptions and coarse-scale reservoir simulation models used for development planning and reservoir management. Predictive reservoir models that are consistent with geologic and production data gathered at different scales are critical for these tasks. Global scale-up is a promising technique for building more accurate reservoir models. Introduction Reservoir modeling is a critical component in development planning and production management of oil and gas fields. The ultimate goal of reservoir modeling is to aid the decision making process throughout all stages of field life. During early field development, reservoir models are used to assess the risk and uncertainty in the field performance based on limited data. Once production begins, reservoir models are periodically refined or updated based on reservoir surveillance data. The updated models are then used for making field management decisions, such as in-fill drilling. For mature fields, accurate reservoir models are required to evaluate opportunities in enhanced oil recovery. A significant challenge in building predictive reservoir models is ensuring that the models are consistent with data collected at multiple scales. Reservoir models built at different scales for different purposes need to be consistent with each other and all available data. Such consistency is important to assess uncertainty and to understand field geology. The latter is typically achieved through history matching reservoir models using both field production and 4D seismic data along with other data. Although models that match field performance are non-unique, those that are consistent with both the underlying geology and the measured data provide better predictability [1]. This paper focuses on the consistent modeling of permeability at different scales. It is well-known that permeability measured from core analysis, well logs, and well tests can be very different (e.g., [2]), because each measurement probes different spatial and time scales. The scale difference can be orders of magnitude. For example, core plug measurements are conducted at centimeter scale, whereas well tests measure permeability at a spatial scale of 10 - 103 meters. Consistently incorporating all these data into a reservoir model with cell size of 50 to 100s of meters is non-trivial. Unlike porosity and other volumetric properties, permeabilities at different scales are not related using simple averaging equations. Single-phase flow-based scale-up is commonly used in the industry to link permeability data and models at different scales. Within this methodology, coarse scale permeability is calculated from numerical flow solutions on fine scale reservoir models; see [3–7] for comprehensive reviews. In this paper, we present a single-phase scale-up technology using global flow solutions and its application to reservoir modeling.

Dedicated to Olof B. Widlund on the occasion of his 60th birthday. SUMMARYWe prove extension theorems in the norms described by Stokes and Lamà e operators for the threedimensional case with periodic boundary conditions. For the Lamà e equations, we show that the extension theorem holds for nearly incompressible media, but may fail in the opposite limit, i.e. for case of absolutely compressible media. We study carefully the latter case and associate it with the Cosserat problem.Extension theorems serve as an important tool in many applications, e.g. in domain decomposition and ÿctitious domain methods, and in analysis of ÿnite element methods. We consider an application of established extension theorems to an e cient iterative solution technique for the isotropic linear elasticity equations for nearly incompressible media and for the Stokes equations with highly discontinuous coe cients. The iterative method involves a special choice for an initial guess and a preconditioner based on solving a constant coe cient problem. Such preconditioner allows the use of well-known fast algorithms for preconditioning.Under some natural assumptions on smoothness and topological properties of subdomains with small coe cients, we prove convergence of the simplest Richardson method uniform in the jump of coecients. For the Lamà e equations, the convergence is also uniform in the incompressible limit.Our preliminary numerical results for two-dimensional di usion problems show fast convergence uniform in the jump and in the mesh size parameter.