Automatic Grid Generation for Modeling Reservoir Heterogeneities

Abstract

Summary An automatic grid-generation and -adjustment method is proposed to producegrids that may be used for finite-difference simulators. The produce grids thatmay be used for finite-difference simulators. The resulting blocks are ashomogeneous as possible with respect to specified parameters (e.g., porosityand permeability). Creating a grid of parameters (e.g., porosity andpermeability). Creating a grid of homogeneous blocks is desirable in flowsimulation studies because it allows easier evaluation of the block average(effective) properties. Introduction For reservoir simulation based on the finite-difference method, the domainis divided into gridblocks and effective rock properties (such as porosity andpermeability) are specified for each block. The boundaries of the externalgridblocks must be as close as possible to reservoir boundaries to approximatereservoir volume correctly. Gaps and overlapping of blocks are not allowed, andadjacent blocks must completely share their common interfaces. Blocks should besmaller where the pressure or saturation variations are larger (e.g., close towells). The volume difference between adjacent blocks should not be too large(e.g., should not exceed a ratio of two), and as much as possible, each blockshould contain only one lithofacies. Blocks should be as close to rectangularboxes as possible. If one wants to account for all these constraints, gridpossible. If one wants to account for all these constraints, grid constructioncan become very difficult. The availability of geostatistical tools, particularly stochastic simulation tools, allows very high-resolutiondescription of the spatial distribution of reservoir parameters. Lithofaciesdistribution, for example, can be described by up to several million pixels, aresolution much finer than that of the typical grid used for flow simulation. Acoarse grid of, for example, 1,000 blocks associated with a finergeo-statistically generated grid of 1 million points or pixels would haveblocks comprising hundreds to thousands of pixels informed with reservoirproperties. Thus, a need exists to coarsen that information, and the averagingprocess would be facilitated if the coarse gridblocks were defined with somecriterion of homogeneity. Two types of approaches are generally considered forthis problem of averaging. The first one uses pseudofunctions (essentially formultiphase flow in large blocks) to provide the blocks with" pseudopermeability" values. Flow simulation with these pseudovalues isequivalent to the "real" flow computed with fine grids generatedlocally in selected parts of the reservoir. The second approach is theupscaling of flow properties by statistical methods. Both approaches improvethe simulations by providing average property values that are better than thoseobtained from arithmetic (for permeabilities in parallel) and harmonic (forpermeabilities in series) averages. The main feature of the proposedpermeabilities in series) averages. The main feature of the proposed automaticgrid-generation method is the production of gridblocks that are as homogeneousas possible in terms of internal block variance of the data- point (pixel)properties. The method also can be used to refine the grid point (pixel)properties. The method also can be used to refine the grid locally (aroundwells, for example) and can control the volume difference between adjacentblocks. In addition, the method can account for any type of faults by buildingblocks with faces abutted against the fault plane; thus, faults intersectingblocks are avoided. The resulting grid may not be rectangular or orthogonal;this is the price paid for having a flexible grid adapted to local spatialheterogeneities. The proposed method is general, however, and can be applied toany type of mesh element (tetrahedrons, hexahedrons, prisms, etc.). Whenelements with greater number of faces (e.g., octahedrons, decahedrons, andmixed meshes) are used, orthogonality constraints can be introduced. Theproposed gridding method consists of a three-step process. The first stepproduces an initial grid mat covers the reservoir or the portion of thereservoir to be simulated. The second and most important step adjusts the nodesof the initial grid to minimize internal block heterogeneities. Relevant rockproperty values must be provided at each node of the fine-scale grid throughgeostatistical simulations, for example. The third step identifies and reformsall poorly shaped blocks (e.g., blocks with nonplanar faces). The key idea inthe method involves interpreting the grid as a 3D graph consisting of nodes andlinks between nodes that are called edges. A certain elasticity or rigidity isprovided to the graph through its edges to make it a "physicalstructure." The coefficients of elasticity of the edges are made functionsof the internal heterogeneity and volume of their adjacent blocks. With aniterative process, the blocks are adjusted by moving all the graph nodessimultaneously. The discussion in this paper is limited to the generation andadjustment of finite-difference grids-that is, the production of quadrilateralelements in 2D and hexahedral elements in 3D. Grid Description. When working with regular grids, regardless of the coordinate system used(for example, Cartesian or cylindrical), we can define the gridblocks from agrid origin and block sizes along the three directions of the grid. Integerindices i, j, and k, are then used as block coordinates. For nonregular grids, the coordinates of the eight apexes of a block and their interconnections mustbe explicitly provided for a complete description of the block faces. Moregenerally, a block can be defined by the combination of two types ofinformation.Geometric information about its location, which could be thecoordinates of its eight apexes.Topologic information about its shape (ablock is topologically equivalent to a cube). This information could consist ofthe description of the 12 links between apexes, or the equations giving the sixblock faces as function of the apexes' coordinates (this is necessary when thefaces are nonplanar). The previous split between geometric and topologicinformation is often used in computer-aided modeling. Any two blocks of a gridshare the same topologic information, specifying, for example, how their nodesare connected, but differ in their geometric information (see Fig. 1). Anyblock can be displaced or deformed just by changing its geometric information, while its topology remains unchanged. Grids also may be described in terms ofgeometric and topologic information by the use of graphs. A graph consists ofnodes whose coordinates represent the geometric information, while the linksbetween nodes define the topologic information. The idea of the grid-adjustmentprocess is to keep the topologic information unchanged while modifying theprocess is to keep the topologic information unchanged while modifying thegeometric information by moving some of the grid nodes (see Fig 2).