Development and Testing of a Static/Dynamic Local Grid-Refinement Technique
Authors: Biterge Mustafa B. , Ertekin Turgay 
Published: December 2007

Summary
Simple, efficient, and stable static and dynamic local grid-refinementprocedures for multi-dimensional, multiphase reservoir problems are developedand tested problems are developed and tested by isothermal reservoirsimulation.

Introduction
There has been great interest in reducing excessive computer usage innumerical modeling without sacrificing the degree of accuracy. To achieve this, various mesh-refinement methods have been developed that use denser gridpointsonly in localized regions where truncation errors may be fatal.
The physical domain of interest is subdivided by equally or unequally spacedgrid-points to form a computational domain. A grid model is the geometricrepresentation of the resulting computational domain. A coarse- (or base) gridmodel represents the preliminary subdivision of the physical preliminarysubdivision of the physical domain with a grid spacing of (Fig. 1a). The basegrid model can be refined by reducing A. A fine-grid model is formed byreducing A in the entire domain, as shown in Fig. 1b. For a conventionallyrefined grid model, the grid spacing is reduced only within a sub-region of thecoarse model. However, the fine mesh lines are extended to the boundaries ofthe physical domain (Fig. 1c). A locally refined grid model is a special caseof a conventional] refined model. In this case, the fine mesh lines are notextended to the exterior boundary of the reservoir (fig. 1d).
For dynamically refined grid models, the number of gridpoints istime-dependent, and the locally refined region varies spatially. Composite grid(also known as hybrid grid) models combine cylindrical (or elliptical) andrectangular grid systems. They have been used extensively in studies ofnumerical heat transfer and in fluid-mechanics problems. Multigrid models usecoarse- and problems. Multigrid models use coarse- and fine-grid networkssequentially in the entire computational domain. Static grid-refinement studiesfocused on developing techniques for grid interactions at the periphery of thecoarse and fine grids.
Lam and Simpson developed a local mesh-refinement technique for numericalsolution of advection-diffusion transport equations. Graham and Smart describeda reservoir simulator that uses a fine-grid model nested in a coarse-grid modelfor areal simulation representing a reservoir in communication with a largepressure-supporting aquifer. In 1982, Rosenberg developed a mesh-refinementstrategy similar to Lam and Simpson's technique. Quandalle and Besset studiedthe efficiency of the grid-refinement technique described by Rosenberg bysimulating displacement of oil by water in a confined five-spot pattern. In1983, Heinemann et al. developed a procedure for dynamic and local gridprocedure for dynamic and local grid refinement in conjunction with amultiple-application reservoir simulator. Forsyth and Sammon generalized Heinemann el al.'s grid-refinement technique for modeling faults andpinchouts.
Heinemann et al. described a dynamic grid-refinement method as an addendumto their static grid-refinement scheme. Coarse- and refined-grid interactionsat the periphery of the two grid systems are similar to those in Heinemann etal.'s static grid-refinement technique. Additionally, specially developed datamanagement for dynamic subdivision is needed. They used preset limiting valuesfor saturatoin, composition, or temperature changes when subdividing the -basegrid. Han et al. modified and improved Heinemann et al.'s dynamicgrid-refinement technique.
Pedrosa and Aziz developed a procedure to improve wellblock calculations byprocedure to improve wellblock calculations by combining an orthogonalcurvilinear grid network with a rectangular grid network in the vicinity of awell point. They compared the performance of their hybrid grid against acoarse-grid model using waterflooding and water-coning problems.
Brand's multigrid technique has been used extensively for a wide range ofproblems. The multigrid method uses two problems. The multigrid method uses twogrid patterns, one fine and one coarse, that sequentially envelope the entireregion. Basically, in Brandt's technique, a preliminary solution is obtained inthe coarse-grid preliminary solution is obtained in the coarse-grid domain withthe aid of a predetermined iteration criterion. Then, information istransferred from the coarse grid to the fine grid by interpolation of thissolution. The iteration is continued with the fine-grid model. After theiteration is satisfied, the improved solution and residual of the fine grid aretransferred back into the base grid.
JPT
P. 487