Analytical Upscaling of Small-Scale Permeability Using a Full Tensor

Abstract

Abstract Computational costs limit large scale reservoir simulations to coarse grid systems. Determination of an effective permeability for a simulation grid block requires a proper scale-up of small scale permeability heterogeneities within that grid block. Conventional scale-up techniques are limited to a diagonal tensor representation of effective permeability. Therefore, such techniques cannot handle general permeability anisotropy (full tensor) exemplified by cross-bedded permeability structures that may be present on a smaller scale. An analytical method is developed to calculate an effective permeability tensor for a grid block by accounting for small scale heterogeneities within the grid block. The method honors both the location and the orientation of the small scale heterogeneities. Effective permeability tensors calculated using the analytical method and a numerical method show excellent agreement. Miscible displacement simulations show that the effective permeability tensor method outperforms conventional scale-up techniques in predicting flood front locations in cases of general permeability anisotropy. Introduction Simulation of fluid flow in a porous medium on a field scale require large scale numerical simulations. Although more powerful computers and simulation techniques are continuously being developed, the size of the grid blocks used in field scale fluid flow simulations is too large to explicitly account for the effect of small scale heterogeneities. Such heterogeneities may be comprised of interwell laminations and cross-bedding structures as well as sand/shale sequences. For the large scale simulator grid blocks, the effect of small scale heterogeneities can only be accounted for by calculating an effective permeability. An effective permeability preserves the ratio of the fluid flux and the potential drop across a heterogeneous block and an equivalent homogeneous block. Several methods are presented in the literature for calculating effective permeability. These methods can, in general, be divided into numerical and analytical methods. Numerical (simulation) methods may be used to handle complex heterogeneous systems, whereas analytical method usually are restricted by simplifying assumptions. Analytical methods have the advantage of being less expensive than numerical methods in terms of computational cost. In this study, an analytical effective permeability method is developed. This method combines the advantages of numerical and analytical methods. The proposed method is general in that it allows for full tensor representation of effective permeability. Both the location and orientation of permeability heterogeneities are considered. The method, in essence, captures the effect of the pressure distribution within the grid block. A directional search procedure in local areas (four quadrants of the grid block) identifies the principal axes of permeability and their orientation with respect to the simulation coordinate axes. Coordinate rotation yields effective permeability tensors. These tensors are combined into an effective permeability tensor for the entire grid block based on the coupling of cross-flow-averaged and no-cross-flow-averaged effective permeability tensors from the four quadrants. Darcy's law is manipulated throughout this process. P. 679^