Computing Absolute Transmissibility in the Presence of Fine-Scale Heterogeneity

Abstract

SPE Members Abstract This paper presents an algorithm to compute transmissibility when there is permeability heterogeneity and anisotropy at the subgrid scale. The need for a tensor representation of macroscopic transmis-sibility in order to scale properly the effects of microscale permeability variations is demonstrated. Methods for computing tensor transmissibility and the use oftensor transmissibility in reservoir simulation are described. The proposed method is applicable even when the variance of permeability is large and the principal directions of the transmissibility tensor are not aligned with the coordinate axes. Examples demonstrate that the general tensor scaling procedure can give accurate, efficient production estimates on a coarse grid. Introduction All petroleum reservoirs contain variations in permeability and porosity at length scales that are smaller than the smallest grid blocks used in reservoir simulations. Thus any simulation of displacements at the reservoir scale must utilize some sort of averaging of the local varia-tions, so that properties such as permeability and porosity may be assigned to the grid blocks. In most simulations, however, only the most rudimentary averaging techniques are applied. Typically, the reservoir is divided into a small number of layers that are assigned uniform properties, even though log and core data show significant variation within these layers. To obtain effective permeabilities for the layers, the core or log data might be averaged arithmetically, geometrically or harmonically. Similar techniques are used for areal variations, though data characterizing areal variations are usually sparse. Because computation costs climb rapidly as the number of grid blocks increases, there is considerable incentive to use the coarsest grid possible. Representation of the effects of small scale permeability variations is made more difficult by the variety of variations that are possible. Most of the attempts to represent such effects have focussed on specific types of variations. For example, the methods of Haldorsen and Lake and Begg and King apply only to horizontal (or parallel), impermeable shale lenses. The single phase simulations of Warren and Price dealt with spatially random (uncorrelated) permeability distributions. They used a log-normal probability density function for permeability, and found that for distributions with moderate variability, effective permeability for a grid block (macropermeability) could be estimated as the geometric mean of the permeabilities of all the small scale blocks (microblocks) within the larger block. Weber used repetitive symmetry elements and Darcy flow analysis to estimate the effective permeability and anisotropy ratio for festoon cross bed sets and sandstone with low-permeability intercalations. Martin and Cooper extended Darcy flow analysis to stochastically generated sand/shale sequences, and considered two-phase flow. They found that the harmonic mean is a poor predictor of vertical permeability. Kortekaas performed very fine scale two-phase simulations on repetitive arrays intended to mimic cross-bedded sandstone geometry. He generated pseudofunction relations for relative permeability and capillary pressure, and noted that the pseudofunctions were anisotropic with respect to the sedimentary structure. Equally important, Kortekaas found that neglecting the cross-bedding results in production predictions which are invariably optimistic. Lasseter et al. performed multiphase simulations to assess the effects of different scales of variability on displacement behavior, using micro-permeameter measurements, random fractals and deterministic per meability variations. They found that the results depended strongly on the boundary conditions used in the microscale simulation. Also, they observed that repeated multiphase simulations are expensive and their interpretation is laborious. None of the previous investigations considered the tensor aspects of permeability anisotropy, although macrolevel permeability anisotropy is likely to be present when large grid blocks are used to simulate displacements with smaller scale heterogeneities present. It is clear, for example, that anisotropy can be induced by heterogeneities such as lamination, burrows or cross-bedding. The presence of such heterogeneities distorts the local streamlines, and that distortion is reflected in the local pressure field. When such a flow is modeled using a macrolevel representation, however, the local pressure field is represented by a single grid-block pressure. While the finite difference of pressure between grid two blocks is necessarily directed from block center to block center, the distortion in the flow directions can only be reproduced if the flux between blocks is not aligned with the macrolevel pressure gradient. P. 209^