Flow Simulation of Geologic Models
Authors: King Michael J. , Mansfield Mark 
Published: December 2007

Abstract
The increased use of fine scale three dimensional geologic models has offered both new opportunities and new challenges for the reservoir engineer. These models offer new opportunities to assess the impact of reservoir heterogeneity on performance prediction. Through the use of multiple models, we are in better position to evaluate the uncertainty of these predictions. Performance prediction either relies on direct simulation (e.g., streamline techniques) or on simulation of upscaled models. Unfortunately, the fundamental development work for each has been performed in relatively simple flow geometries, whilst by their nature, geologic models tend to include more irregular geometries: layers that are removed by scour or by internal pinch-out, isopach thicknesses that vary significantly, and well trajectories that intersect many fine cells at arbitrary orientations to the local stratigraphy. We report on our experience, and the new theoretical advances required to resolve these difficulties. The most important advances have had to do with upscaling and transmissibility, although we also re-examine the management of pinched-out cells, and the calculation of well PI. A new upscaling formulation is introduced which emphasizes three dimensional permeability; it is especially well suited to upscaling from irregularly shaped regions. Re- sampling from the geologic grid to a computational grid has forced us to a new, more fundamental, derivation of transmissibility. Unlike the standard construction, it is guaranteed to never give a negative transmissibility. We also suggest a new treatment of pinched-out cells, which regularizes the vertical non-nearest neighbor connections. Finally, we revisit Peaceman's well PI equations. and show their generalization to inclined wells, full tensor permeability, and computational cells with arbitrary numbers of faces.

Introduction
The last three years have seen an explosive growth in the ability of the petroleum industry to develop flow simulation models based upon detailed three dimensional geologic descriptions. Until recently, such modeling activities have required access to research codes, and experts to utilize them, for example. With a wide range of vendor tools currently available (StrataModel, IRAP/RMS, Storm, RC**2, …) it is now possible to build such models within asset teams without the direct involvement of experienced technologists. At the same time, these tools provide new challenges to the technologist, as the simple (I,J,K) "shoebox" topologies which underlie many of the theoretical algorithms are either not present or have been substantially modified within the geologic framework. We find that it is in the transition from the geologic static model to flow simulation that our theoretical foundations, and the vendor products, are most in need of reevaluation. For this reason we will report upon our experience in performing flow simulation at the scale of the fine geologic model, and the enhanced theoretical understanding which has developed as a result.
Four fundamental technical issues arise:the construction of the three dimensional geologic grid, which defines the data structure for all subsequent calculations,the definition of physical transport properties on this geologic grid, in particular, permeability and net-to-gross,their representation within a finite difference scheme in terms of transmissibility and well PI (well connection factor), andthe methodology for upscaling the flow properties.
Aspects of these issues have been reported in the literature but with surprisingly little emphasis on the vertically complex geologic structures which have tested our theoretical understanding.
New theoretical results are presented for each of these topics, with the most important ones having to do with upscaling and with transmissibility. Nonetheless, we reserve the discussion of these keys points until late in the paper, as they rely upon the other elements. The single most important advance involves the introduction of three dimensional permeability. which is based on a homogenized form of Darcy's equation. It extends the usual one dimensional upscaling computation, and is especially well suited to upscaling from irregularly shaped regions. P. 331^