Reduction of Grid Effects Due to Local Sub-Gridding in Simulations Using a Composite Grid

Abstract

Our previous paper [1] described and analyzed the performance and some of the advantages obtained when a reservoir simulator is upgraded with: the possibility of flow between any given grid blocks, especially from one to several blocks;a multiple time step calculation. These improvements were appreciated by modelling engineers. The upgraded simulator was used for the simulation of actual reservoirs with complex geometry (such as faulted reservoirs), and for its possibility of having a composite discretization grid (i.e. a grid with several nested levels of local sub-gridding). This practical need justified further extensive study of the flow behaviour near a sub-gridding boundary in a composite grid. Composite grids possess some features which do not exist in regular classical grids. In a regular grid system, when two blocks are neighbours in a certain direction, their grid centers are always drawn up in that direction. In a composite grid, this is not always the case for neighbouring blocks belonging to different levels of local sub-gridding. As a consequence, in a composite grid, approximation of material exchange between two blocks by the classical two point finite difference scheme may be less accurate than in a regular grid. Tests have shown that a significant grid effect may be induced by computing flows between neighbouring blocks belonging to different sub-gridding levels with this scheme. To eliminate these effects, a more sophisticated numerical scheme, involving more than two points, must be used for flow computation. This paper reports an analysis of this phenomenon. It also gives a description of some schemes especially designed for this purpose. These schemes may be viewed as being the classical two point scheme with one of the two points being a grid point but not a grid block center. Reservoir data are then approximated at this new grid point by interpolating data values at the grid block centers. More or less sophisticated interpolations are considered. It also describes some numerical examples performed to compare these schemes when they are used for the simulation of multiphase flows with a composite grid. One of the schemes is selected for its accuracy, its reliability, and its simplicity. On numerical examples, it is shown to nearly reproduce, with a composite grid, results obtained with a fine regular grid having a much larger number of blocks. Moreover, this scheme is implemented on a black oil simulator using a Sequential Solution Method and designed for the simulation of three dimensional three phase flows in large reservoirs with composite grids and several thousands of blocks.

Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis

“…Wi,j+2~ 2Ki,j+2\yi,j+2 3->i-l,;+l 3>i-l ,j+4>-Taking into account formulas (3.6) and (3.10), we can rewrite these expressions in the following form: (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) </«=-i*öfiVu+i.…”

“…Local grid refinement for such types of problems has been used widely in the petroleum engineering literature, e.g., [7,13,14,20]. The importance of very careful treatment of the difference stars near the composite grid interfaces has been demonstrated experimentally in [14].…”

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

“…The discretization represented by Eq. 26 results in a local leading truncation error on the order of O(1/h), when applied toh adaptive nonuniform grids [32,72]. The analysis of [29] indicated that in grids with large aspect ratios the truncation error depends on the grid interface and aspect ratios, the permeabilities, and the pressure gradient acting tangentially on the grid interface (between two unequal blocks).…”

On some variational acceleration techniques and related methods for local refinement

“…Von Rosenberg used a finite difference refinement scheme based upon Taylor-series expansions for the constant coefficient problem [1]. Quandalle and Besset [2] discussed local refinement with local time stepping for reservoir applications.…”

“…In a later paper by Gjesdal [13] the same connection between ACMs and cell-centered multigrid methods was shown. Careful treatment of the finite difference stars near composite grid interfaces is important [14,15,2]. For illustration purposes, the discretization is given for both for a regular uniform grid and a non-uniform composite grid arising from the use of a locally refined grid.…”