Efficient Implementation Of Local Grid Refinement Ina Generalized Compositonal Model
Authors: Shiralkar G.S. , Peng C.P. , Steffensen R.J. 
Published: March 2010

Abstract
An efficient implementation of local grid refinement (LGR) in a generalized compositional model js presented, Highlights include:implementation for both block-centred and point-distributed grids,a highly efficient linear solver, with a novel multi-colour ordering, for both five-point and nine-point computational molecules, andefficient data structure which is essential for optimal vector and scalar performance, to maximize the benefits of LGR.
For block-centred grids, refinement is performed by subdividing coarse gridblocks. For point-distributed grids, however, refinement is performed by subdividing along coarse grid lines. This gives rise to several additional complexities at the interfaces between different grids. The actual implementation is discussed.
The linear solver differs from other methods (e.g., BEPS) in that it operates on the composite grid system. For a single grid system (no LGR), the solver uses redlblack ordering for five point molecules and four colour ordering for nine-point molecules. For a composite grid system, a novel repeated multi-colour ordering is employed. This solver- has good convergence properties due to better coupling between.subgrids, as compared to domain decomposition methods. In fact, convergence rates were found to be in line with those for single grids.
The result is a linear solver which avoids the multiple linear solutions that domain decomposition methods entail. Further, the same solver can be used efficiently for both single grid and composite grid systems, eliminating the need to maintain extra solvers.
Finally, simulations showing application to water and miscible floods are presented. They demonstrate the significant savings in CPU expense achieved with this model, while obtaining results of good accuracy.

Introduction
In recent years, local grid refinement techniques have become increasingly popular(l-8). They have been driven largely by the need to better characterize reservoir processes, which occur on many different scales. Modelling of near well behaviour (e.g., liquid dropout in gas condensate, coning), pilot studies, and geostatistically derived reservoir characterizations are some examples that would benefit greatly from local grid refinement. As an example, pilot studies have hitherto employed grids that are globally refined. This can be computationally expensive, and some times compromises must be made with regard to the boundary conditions (modelling of the interaction with the surrounding reservoir). Local grid refinement can generate accurate simulation results at significantly reduced computational expense.
Local grid refinement, however, brings with it its own special problems, such as the calculation of intergrid fluid transfer and the solution of a pressure matrix with non-standard connections due to the multiple grids used. In fact, most of the literature on local grid refinement is devoted to the solution of these problems(4–7).
This paper deals with the implementation of the local grid refinement feature for Cartesian grids in Amoco's Generalized Compositional Model (GCOMP)(9). We discuss technical questions such as generation of the composite grid pressure coefficients and their solution with a novel multiple grid, multi-colour z-line successive line over relaxation (SLOR) linear solver, and we present results of some simulation studies using the local grid refinement feature.