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The windowing technique, introduced by Deimbacher and Heinemann (Ref. 3), allows a time-dependent replacement of grids for a defined area during a simulation run. A window can represent any kind of well with a gridded wellbore and an appropriate grid pattern around the well. Such an approach makes the generally used Peaceman well model superfluous. As the gridded wellbore and the grid blocks around it are small (some cu-ft) the computational stability requires small timesteps and a greater number of Newton-Raphson iterations. It is obvious that this is not feasible if the solution for the full-scale model and the well windows must be performed simultaneously. Therefore in a first step the fully implicit solution for the full-scale model will be calculated but the inner blocks of the windows are solved for the pressure only, without updating the saturations and mole fractions. This solution provides the boundary influx for the windows. In a second step the windows are calculated for the same overall time step with up to 1000 small local steps. This paper presents the general and practical applicability of this method. Windows, constructed by the PEBI (k-PEBI) method, are introduced automatically for all of the wells (vertical, horizontal and slanted). For testing purposes a real case field model was used. It will be shown that the quality of the results obtained for the model calculated with integrated radial grids around the wells and small overall timesteps are equal to those obtained for the same model using the windowing and local timestepping technique. Further it will be shown that solving the window model with large timestep lengths for the first and small lengths for the second solution step results in equal or smaller CPU times and less NR iterations in comparison to the conventional model. Introduction Adequate well representation in hydrocarbon reservoir simulation has been one of the central problems in the petroleum industry and topic of extensive research over the past decades. In 1978 Peaceman (Ref. 6) presented the concept of the equivalent wellblock radius, allowing to relate the computed pressure of a perforated block with the well flowing pressure for different block geometries. This approach was frequently discussed and adapted to fit different requirements, such as deliberate locations in a square rectangular block (Ref. 8). In 1986 Aziz and Pedrosa introduced the concept of "hybrid grids" in a Cartesian block system (Ref. 1), trying to overcome the insufficiencies of the Peaceman model. Gridding the near wellbore area by hybrid grids honors the radial nature of flow, small block sizes help handling large saturation changes caused by high production rates. The paper showed that the solution of the equations in the radially gridded well region can be decoupled from the solution of the equations of the conventional grid system by either applying an iterative or a direct solution method. Aziz and Pedrosa gave preference to the direct approach, finding it to converge faster. In 1996 Ding (Ref. 7) proposed a numerical well model, applicable for numerous geometries of vertical, deviated and horizontal wells in flexible grids. Heinemann et al. introduced the windowing technique, i.e. a time dependent replacement of grids or parameters in a limited area within the original block model, combined with local timestepping (Ref. 3, 4) that is highly suitable to model the near wellbore area. A direct solution approach is used, but fundamentally different from the approach used by Pedrosa and Aziz. A window can contain all kinds of wells, constructed by gridding techniques such as 3D or 3D k-PEBI gridding (Ref. 11), and can be solved using local timestep lengths different to the overall timestep length. Today the Peaceman approach is still commonly used in reservoir simulation while gridding the near wellbore area is only applied to investigate single well problems, such as coning behavior.

Summary A technique for the sequential generation of perpendicular-bisectional (PEBI) grids adapted to flow information is presented and applied. The procedure includes a fine-scale flow solution, the generation of an initial streamline-isopotential grid, grid optimization, and upscaling. The grid optimization is accomplished through application of a hybrid procedure with gradient and Laplacian smoothing steps, while the upscaling is based on a global-local procedure that makes use of the global solution used in the grid-determination step. The overall procedure is successfully applied to a complex channelized reservoir model involving changing well conditions. The gridding and upscaling procedures presented here may also be suitable for use with other types of structured or unstructured grid systems. Introduction Modern geological and geostatistical tools provide highly detailed descriptions of the spatial variation of reservoir properties, resulting in fine-grid models consisting of 107 to 108 gridblocks. As a consequence of this high level of detail, these models cannot be used directly in numerical reservoir simulators, but need to be coarsened significantly. Coarsening requires the averaging of rock parameters from the fine scale to the coarse scale. This process is referred to as upscaling. For simulation of flow in porous media, the upscaling of permeability is of particular interest. A large body of literature exists on this topic; for a comprehensive review of existing techniques, see Durlofsky (2005). To preserve as much of the geological information of the fine grid as possible, the grid coarsening should not be performed uniformly, but with more refinement in areas that are expected to have large impact on the flow, including structural features, such as faults. Although grid-generation techniques based on purely static, nonflow-based considerations have been shown to produce reasonable results(Garcia et al. 1992), the application of flow-based grids is often preferable. Flow-based grids require the solution of some type of fine-scale problem. They are then constructed by exploiting the information obtained from streamlines (and possibly isopotentials) either directly or indirectly. Depending on the type of grid used, points will be defined as cell vertices or nodes, resulting in either a corner-point geometry or point-distributed grid. Several gridding techniques for reservoir simulation have been introduced along these lines, as we now discuss.

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This paper presents the general and practical applicability of the "windowing technique" 1 to model wells in full-field reservoir simulation. Windows modeling the near-wellbore area and the wellblock itself, constructed by the Perpendicular-Bisectional (PEBI) or k-orthogonal PEBI (k-PEBI) method, will be introduced for all wells of a reservoir or for selected "problem wells."The windowing technique, introduced by Deimbacher and Heinemann, 1 allows a time-dependent replacement of grids for a defined area during a simulation run. A window can represent any kind of well with a gridded wellbore and an appropriate grid pattern around the well. Such an approach makes the generally used Peaceman well model superfluous. Because the gridded wellbore and the gridblocks around it are small (some cubic feet), the computational stability requires small timesteps and a greater number of Newton-Raphson iterations. It is obvious that this is not feasible if the solution for the full-scale model and the well windows must be performed simultaneously. Therefore, in a first step, the fully implicit solution for the full-scale model will be calculated, but the inner blocks of the windows are solved for the pressure only, without updating the saturations and mole fractions. This solution provides the boundary influx for the windows. In a second step, the windows are calculated for the same overall timestep with up to 1,000 small local steps.This paper presents the general and practical applicability of this method. Windows, constructed by the PEBI (k-PEBI) method, can be introduced automatically for all the wells (vertical, horizontal, and slanted) or for certain "problem wells."For testing purposes, real case field models were used. It will be shown that the quality of the results obtained for the model calculated with integrated radial grids around the wells and small overall timesteps are equal to those obtained for the same model using the windowing and local timestepping techniques. Further, it will be shown that solving the window model with large timestep lengths for the first solution step and small lengths for the second solution step results in equal or smaller CPU times in comparison to the conventional model.